Appendix A Tensor Algebra, Linear Algebra, Matrix Algebra

Appendix ATensor Algebra, Linear Algebra, MatrixAlgebra, Multilinear Algebra

The key word “algebra” is derived from the name and the workof the nineth-century scientist Mohammed ibn Musa al-Khowarizmiwho was born in what is now Uzbekistan and worked in Bagdad atthe court of Harun al-Rashid’s son. “al-jabr” appears in the title ofhis book Kitab al-jabr wail muqabala where he discusses symbolicmethods for the solution of equations, K. Vogel: Mohammed IbnMusa Alchwarizmi’s Algorismus; Das forested Lehrbuch zum Rechnenmit indischen Ziffern, 1461, Otto Zeller Verlagsbuchhandlung, Aalen1963). Accordingly what is an algebra and how is it tied to the notionof a vector space, a tensor space, respectively? By an algebra wemean a set S of elements and a finite set M of operations. Eachoperation .opera/k is a single-valued function assigning to every finiteordered sequence .x1; : : : ; xn/ of n D n.k/ elements of S a value.opera/k .x1; : : : ; xk/ D xl in S. In particular for .opera/k.x1; x2/ theoperation is called binary, for .opera/k.x1; x2; x3/ ternary, in generalfor .opera/k.x1; : : : ; xn/ n-ary. For a given set of operation symbols.opera/1, .opera/2; : : :, .opera/k we define a word. In linear algebrathe set M has basically two elements, namely two internal relations.opera/1 worded “addition” (including inverse addition: subtraction)and .opera/2 worded “multiplication” (including inverse multiplication:division). Here the elements of the set S are vectors over the field R

of real numbers as long as we refer to linear algebra. In contrast, inmultilinear algebra the elements of the set S are tensors over the fieldof real numbers R. Only later modules as generalizations of vectors oflinear algebra are introduced in which the “scalars” are allowed to befrom an arbitrary ring rather than the field R of real numbers.

Let us assume that you as a potential reader are in some way familiar with theelementary notion of a threedimensional vector space Xwith elements called vectorsx 2 R3, namely the intuitive space “ we locally live in”. Such an elementary vectorspace X is equipped with a metric to be referred to as threedimensional Euclidean.

E. Grafarend and J. Awange, Applications of Linear and Nonlinear Models,Springer Geophysics, DOI 10.1007/978-3-642-22241-2,

571

© Springer-Verlag Berlin Heidelberg 2012

572 A Tensor Algebra, Linear Algebra, Matrix Algebra, Multilinear Algebra

As a real threedimensional vector space we are going to give it a linear andmultilinear algebraic structure. In the context of structure mathematics based upon

(a) Order structure(b) Topological structure(c) Algebraic structure

an algebra is constituted if at least two relations are established, namely oneinternal and one external. We start with multilinear algebra, in particular with themultilinearity of the tensor product before we go back to linear algebra, in particularto Clifford algebra.

A-1 Multilinear Functions and the Tensor Space Tpq

Let X be a finite dimensional linear space, e.g. a vector space over the field R of realnumbers, in addition denote by X

� its dual space such that n D dimX D dimX�.

Complex, quaternion and octonian numbersC, H and O as well as rings will only beintroduced later in the context. For p; q 2 Z

C being an element of positive integernumbers we introduce

Tpq .X;X

�/

as the p-contravariant, q-covariant tensor space or space of multilinear functions

f W X� � : : : � X� � X � : : : �X �! R

p dimX��q dimX :

If we assume x1; : : : ;xp 2 X� and x1; : : : ;xq 2 X, then

�

�

�

�

x1 ˝ : : :˝ xp ˝ x1 ˝ : : :˝ xq 2 Tpq .X

�;X/

holds. Multilinearity is understood as linearity in each variable. “˝” identifiesthe tensor product, the Cartesian product of elements .x1; : : : ;xp;x1; : : : ;xq/.

�

�

�

Example A.1: Bilinearity of the tensor product x1 ˝ x2For every x1;x2 2 X, x;y 2 X and r 2 R bilinearity implies

.x C y/˝ x2 D x ˝ x2 C y ˝ x2 .internal left-linearity/x1 ˝ .x C y/ D x1 ˝ x C x1 ˝ y .internal right-linearity/rx ˝ x2 D r.x ˝ x2/ .external left-linearity/x1 ˝ ry D r.x1 ˝ y/ .external right-linearity/: |

A-1 Multilinear Functions and the Tensor Space Tpq 573

The generalization of bilinearity of x1 ˝ x2 2 T02 to multilinearity of

x1 ˝ : : :˝ xp ˝ x1 ˝ : : :˝ xq 2 Tpq

is obvious.

Definition A.1. (multilinearity of tensor space Tpq ):

For every x1; : : : ;xp 2 X� and x1; : : : ;xq 2 X as well as u; v 2 X

�, x;y 2 X

and r 2 R multilinearity implies

.uC v/˝ x2 ˝ : : :˝ xp ˝ x1 ˝ : : :˝ xqD u˝ x2 ˝ : : :˝ xp ˝ x1 ˝ : : :˝ xq C v˝ x2 ˝ : : :˝ xp ˝ x1˝ : : :˝ xq .internal left-linearity/

x1 ˝ : : :˝ xp ˝ .x C y/˝ x2 ˝ : : :xqD x1 ˝ : : :˝ xp ˝ x ˝ x2 ˝ : : :˝ xq C x1 ˝ : : :˝ xp ˝ y ˝ x2˝ : : :˝ xq .internal right-linearity/

ru˝ x2 ˝ : : :˝ xp ˝ x1 ˝ : : :˝ xqD r.u˝ x2 ˝ : : :˝ xp ˝ x1 ˝ : : :˝ xq/ .external left-linearity/

x1 ˝ : : :˝ xp ˝ rx ˝ x2 ˝ : : :˝ xqD r.x1 ˝ : : :˝ xp ˝ x ˝ x2 ˝ : : :˝ xq/ .external right-linearity/:

A possible way to visualize the different multilinear functions which spanfT00;T10;T01;T20;T11;T02; : : : ;Tpq g is to construct a hierarchical diagram or a specialtree as follows.

{0}∈ 00S

x1 ∈ 10 x1 ∈ 1

0

x1⊗ x2∈ 02 x1 ⊗ x1 ∈ 11 x1 ⊗ x2 ∈ 2

0

x1 ⊗ x2 ⊗ x3∈ 30 ; x1 ⊗ x2 ⊗ x1 ∈ 21 ; x1 ⊗ x2 ⊗ x1∈ 12; x1 ⊗ x2 ⊗ x3 ∈ 03


When we learnt first about the tensor product symbolised by “˝” as well as itsmultilinearity we were left with the problem of developing an intuitive under-standing of x1 ˝ x2 , x1 ˝ x1 and higher order tensor products. Perhaps it ishelpful to represent ”the involved vectors” in a contravariant or in a covariantbasis. For instances, x1 D e1x1 C e2x2 C e3x3 or x1 D e1x1 C e2x2 C e3x3 isa left representation of a three-dimensional vector in a 3-left basis fe1; e2; e3gl ofcontravariant type or in a 3-left basis fe1; e2; e3gl of covariant type. Think in terms ofx1 or x1 as a three-dimensional position vector with right coordinates fx1; x2; x3g orfx1; x2; x3g , respectively. Since the intuitive algebras of vectors is commutative wemay also represent the three-dimensional vector in a in a 3-right basis fe1; e2; e3grof contravariant type or in a 3-right basis fe1; e2; e3gr of covariant type such thatx1 D x1e1 C x2e2 C x3e3 or x1 D x1e1 C x2e2 C x3e3 is a right representationof a three-dimensional position vector which coincides with its left representationthanks to commutativity. Further on, the tensor product x ˝ y enjoys the left andright representations

�e1x1 C e2x2 C e3x3

�˝ �e1y1 C e2y2 C e3y3� D

3X

iD1

3X

jD1ei ˝ ej xi yj

and

�x1e1 C x2e2 C x3e3

�˝ �y1e1 C y2e2 C y3e3� D

3X

iD1

3X

jD1xiyj ei ˝ ej

which coincides again since we assumed a commutative algebra of vectors. Theproduct of coordinates .xiyj /; i; j 2 f1; 2; 3g is often called the dyadic product.Please do not miss the alternative covariant representation of the tensor productx˝ y which we introduced so far in the contravariant basis, namely

�e1x1 C e2x2 C e3x3

�˝ �e1y1 C e2y2 C e3y3� D

3X

iD1

3X

jD1ei ˝ ej xi yj

of left type and

�x1e1 C x2e2 C x3e3

�˝ �y1e1 C y2e2 C y3e3� D

3X

iD1

3X

jD1xi yj ei ˝ ej

of right type. In a similar way we produce

x˝ y˝ z D3;3;3X

i;j;kD1ei ˝ ej ˝ ekxiyj zk D

3;3;3X

i;j;kD1xi yj zkei ˝ ej ˝ ek

of contravariant type and


x˝ y˝ z D3;3;3X

i;j;kD1ei ˝ ej ˝ ekxiyj zk D

3;3;3X

i;j;kD1xi yj zkei ˝ ej ˝ ek

of covariant type. “Mixed covariant-contravariant” representations of the tensorproduct x1 ˝ y1 are

x1 ˝ y1 D3X

iD1

3X

jD1ei ˝ ejxiyj D

3X

iD1

3X

jD1xiyj ei ˝ ej

or

x1 ˝ y1 D3X

iD1

3X

jD1ei ˝ ejxiyj D

3X

iD1

3X

jD1xiyj ei ˝ ej :

In addition, we have to explain the notion x1 2 X�; x1 2 X: While the vector x1 is

an element of the vector space X , x1 is an element of its dual space. What is a dualspace? Indeed the dual space X

� is the space of linear functions over the elementsof X. For instance, if the vector space X is equipped with inner product, namelyhei je j i D gij ; i; j 2 f1; 2; 3g, with respect to the base vectors fe1; e2; e3g whichspan the vector space X, then

ej D3X

iD1ei gij

transforms the covariant base vectors fe1; e2; e3g into the contravariant base vectorsfe1; e2; e3g D span X

� by means of Œgij � D G�1, the inverse of the matrix Œgij � DG 2 R

3�3 . Similarly the coordinatesgij of the metric tensor g are used for “raising”or “lowering” the indices of the coordinates xi ; xj , respectively, for instance

xi D3X

jD1gij xj ; xi D

3X

jD1gij x

j :

In a finite dimensional vector space, the power of a linear space X and its dualX? does not show up. In contrast, in an infinite dimensional vector space X the

dual space X? is the space of linear functionals which play an important role

in functional analysis. While through the tensor product “˝” which operated onvectors, e.g. x˝ y, we constructed the p-contravariant, q-covariant tensor space orspace of multilinear functions Tpq .X;X�/ , e.g. T20;T

02;T

11 , we shall generalise the

representation of the elements of Tpq by means of

f DnDdim X

�X

i1;:::;ipD1ei1 ˝ : : :˝ eipfi1:::ip 2 T

p0


f DnDdim X

�X

i1;:::;iqD1ei1 ˝ : : :˝ eiqf

i1:::iq 2 T0q

f DnDdim X

�X

i1;:::;ipD1

nDdim XX

j1;:::;jqD1ei1 ˝ : : :˝ eip ˝ ej1 ˝ : : :˝ ejqf

i1;:::;ipj1;:::;jq

2 Tpq

for instance

f D3X

i;jD1ei ˝ ej fij D

3X

i;jD1fij ei ˝ ej 2 T

20

f D3X

i;jD1ei ˝ ej f ij D

3X

i;jD1f ij ei ˝ ej 2 T

02

f D3X

i;jD1ei ˝ ej f i

j D3X

i;jD1f ij ei ˝ ej 2 T

11

We have to emphasize that the tensor coordinates fi1;:::;ip ; fi1;:::;iq , f

i1;:::;ipj1;:::;jp

are nolonger of dyadic or product type. For instance, for

�

�

�

�(2,0)-tensor : bilinear functions:

f DnX

i;jD1ei ˝ ej fij D

nX

i;jD1fij ei ˝ ej 2 T

20

fij ¤ fifj�

�

�

�(2,1)-tensor : trilinear functions:

f DnX

i;j;kD1ei ˝ ej ˝ ekf k

ij DnX

i;jD1f kij ei ˝ ej ˝ ek 2 T

21

f kij ¤ fifj f k

�

�

�

�

(3,1)-tensor : ternary functions:

f DnX

i;j;k;lD1ei ˝ ej ˝ ek ˝ el f l

ijk DnX

i;jD1f lijkei ˝ ej ˝ ek ˝ el 2 T

31

f lijk ¤ fifj fkf l


holds. Table A.1 is a list of .p; q/-tensors as they appear in various sciences.Of special importance is the decomposition of multilinear functions as elementsof the space T

pq into their symmetric, antisymetric and residual constituents we are

going to outline.

Table A.1 Various examples of tensor spaces Tpq (p& q W rank of

tensor) (2,0) tensor, tensor space T20

Metric tensor DifferentialGauss curvature tensor geometryRicci curvature tensor

Gravity gradient tensor GravitationFaraday tensor, Maxwell tensor ElectromagneticTensor of dielectric constantTensor of permeability

Strain tensor, stress tensor Continuum mechanics

Energy momentum tensor MechanicsElectromagnetismGeneral relativity

Second order multipole Gravitostaticstensor Magnetostatics

Electrostatics

Variance-covariance Mathematicalmatrix statistics

Table A.2 Various examples of tensor spaces Tpq (p C q W rank of

tensor) (2,1) tensor, tensor space T21

Cartaan torsion tensor DifferentialDeometry

Third order multipole Gravitostaticstensor Magnetostatics

Electrostatics

skewness tensor MathematicalThird momentum tensor Statisticsof probability distribution

Tensor of piezoelectric Coupling of stressconstant and electrostatic field


Table A.3 Various examples of tensor spaces Tpq (p C q W rank of

tensor) (3,1) tensor, (2,2) tensor, tensor space T31, T

22

Riemann curvature tensor DifferentialGeometry

Fourth order multipole Gravitostaticstensor Magnetostatics

Electrostatics

Hooke tensor Stress-strain relationConstitutive equationContinuum mechanicsElasticity, viscosticity

Curtosis tensor MathematicalFourth moment tensor of Statisticsa probability distribution

A-2 Decomposition of Multilinear Functions intoSymmetric Multilinear Functions, AntisymmetricMulti-linear Functions and Residual MultilinearFunctions TT pq D S

pq ˚ Apq ˚ R

pq

Tpq as the space of multilinear functions follows the decomposition T

pq D S

pq ˚

Apq ˚R

pq into the subspace Spq of symmetric multilinear functions, the subspace Ap

q

of antisymmetric multilinear functions and the subspace Rpq of residual multilinear

functions:

Box A.2i. (Antisymmetry of the symbols fi1:::ip ):

fij D �fj ifijk D �fikj ; fjki D �fj ik; fkij D �fkj i

fijkl D �fijlk; fjkli D �fjkil ; fklij D �fklj i ; fkijk D �flikj

Box A.2ii. (Symmetry of the symbols fi1:::ip ):

fij D fj ifijk D fikj ; fjki D fj ik; fkij D fkj i

fijkl D fijlk; fjkli D fjkil ; fklij D fklj i ; fkijk D flikj

A-2 Decomposition of Multilinear Functions into Symmetric Multilinear Functions 579

Box A.2iii. (The interior product of bases, span Sp, n D dim X D dimX

� D3):

S1 W 11Šei

S2 W 12Š.ei ˝ ej C ej ˝ ei /

DW ei _ ej ; ei _ ej D Cej _ e i

S3 W 13Š.ei ˝ ej ˝ ek C ei ˝ ek ˝ ej C ej ˝ ek ˝ ei

C ej ˝ ei ˝ ek C ek ˝ ei ˝ ej C ek ˝ ej ˝ ei /DW ei _ ej _ ek

ei _ ej _ ek D ei _ ek _ ej D ek _ e i _ ej D ek _ ej _ e i D ej _ ek _ ei

D ej _ ei _ ek

Box A.2iv. (The exterior product of bases, span Ap, nD dim XD dimX�D 3):

A1 W 11Šei

A2 W 12Š.ei ˝ ej � ej ˝ ei /

DW ei ^ ej ; e i ^ ej D �ej ^ e i

A3 W 13Š.ei ˝ ej ˝ ek � ei ˝ ek ˝ ej C ej ˝ ek ˝ ei

� ej ˝ ei ˝ ek C ek ˝ ei ˝ ej � ek ˝ ej ˝ ei /DW ei ^ ej ^ ek

ei ^ ej ^ ek D �e i ^ ek ^ ej D Cek ^ ei ^ ej D �ek ^ ej ^ e i

D ej ^ ek ^ ei D �ej ^ e i ^ ek


Box A.2v. (Sp, symmetric multilinear functions):

T10 � S

1 3 f D8<

:

nD dimX�

X

iD1ei fi

9=

;

T20 � S

2 3 f D8<

:1

2Š

nD dimX�

X

i;jD1ei _ ej fij

9=

;

D8<

:

nD dimX�

X

i�jei _ ej f.ij /jf.ij / WD 1

2Š.fij C fj i /

9=

;

T30 � S

3 3 f D8<

:1

3Š

nD dimX�

X

i;j;kD1e i _ ej _ ekfijk

9=

;D8<

:

nX

i<j<k

ei _ ej _ ekf.ijk/jf.ijk/

WD 1

3Š.fijk C fikj C fjki C fj ik C fkij C fkj i /

9=

;

Tp0 � S

p 3 f D8<

:1

pŠ

nD dimX�

X

i1;i2;:::;ipD1ei1 _ ee2 _ : : : _ eip fi1i2:::ip

9=

;

D8<

:

nD dimX�

X

i1�i2�:::�ipei1 _ ei2 _ : : : _ eipf.i1i2:::ip/jf.i1i2:::ip/

WD 1

pŠ.fi1:::ip�1ip C fi1:::ip ip�1 C : : :C fip:::i1i2 C fip:::i2i1 /

9=

;:

�

�

Corollary: dimSpD

�nCp� 1

p

�, in particular if nDp, then dimS

pD�2p� 1p

�.

Box A.2vi. (Ap, antisymmetric multilinear functions):

T10 � A1 3 f D

8<

:

nD dimX�

X

iD1ei fi

9=

;


T20 � A2 3 f D

8<

:1

2Š

nD dimX�

X

i;jD1ei ^ ej fij

9=

;

D8<

:

nD dimX�

X

i<j

ei ^ ej fŒij �jfŒij � WD 1

2Š.fij C fj i /

9=

;

T30 � A3 3 f D

8<

:1

3Š

nD dimX�

X

i;j;kD1ei ^ ej ^ ekfijk

9=

;

D8<

:

nX

i<j<k

ei ^ ej ^ ekfŒijk�jfŒijk�

WD 1

3Š.fijk � fikj C fjki � fj ik C fkij � fkj i /

9=

;

Tp0 � Ap 3 f D

8<

:1

pŠ

nD dimX�

X

i1;i2;:::;ipD1ei1 ^ ee2 ^ : : : ^ eip fi1i2:::ip

9=

;

D8<

:

nD dimX�

X

i1<i2<:::<ip

ei1 ^ ei2 ^ : : : ^ eipfŒi1i2:::ip�jfŒi1i2:::ip �

WD 1

pŠ.fi1:::ip�1ip � fi1:::ip ip�1 C : : :C fip:::i1i2 � fip:::i2i1 /

9=

;:

�

�

Corollary: dim Ap D nŠ=.pŠ.n�p/Š/D�n

p

�in particular if nDp, then dim ApD1.

Box A.2vii. (Aq, antisymmetric multilinear functions, exterior productProposition):

(a) For every x1; : : : ;xi�1;xiC1; : : : ;xq 2 X as well as x;y 2 X and r; s 2 R

multilinearity implies

x1 ^ : : : ^ xi�1 ^ .rx C sy/ ^ xiC1 ^ : : : ^ xqD r.x1 ^ : : : ^ xi�1 ^ x ^ xiC1 ^ : : : ^ xqC s.x1 ^ : : : ^ xi�1 ^ y ^ xiC1 ^ : : : ^ xq:


(b) For every permutation � of f1; 2; : : : ; qg we have

x�1 ^ x�2 ^ : : : ^ x�q D sign .�/x1 ^ x2 ^ : : : ^ xq:(c) Let A 2 Aq.X/, B 2 As.X/; then

B ^ A D .�1/qsA ^B:(d) For every q, 0 � q � n, the tensor space Aq of antisymmetric multilinear

functions has dimension

dim Aq D�n

q

�D nŠ=.qŠ.n � q/Š/:

As detailed examples we like to decompose fT10;T01;T20;T02g in R2 and R

3,respectively, into symmetric and antisymmetric constituents.

Example A.2. (Tpq D Spq ˚ Ap

q ˚ Rpq , decomposition of multilinear functions

into symmetric and antisymmetric constituents):

As a first example of the decomposition of multilinear functions (tensor space)into symmetric and antisymmetric constituents we consider a linear space X (vectorspace) of dimension dimX D n D 2. Its dual space X�, dimX

� D dimX D n D 2,is spanned by orthonormal contravariant base vectors fe1; e2g. Choose q D 0,p D 1 and p D 2.

X D span fe1; e2g versus X� D span fe1; e2g

T10 D A1 D S

1 3 f D(

2X

iD1ei fi

)

D e1f1 C e2f2 2 X

T20 D A2 ˚ S

2

T20 3 f D

8<

:

2X

i;jD1ei ˝ ej fij

9=

;

D e1 ˝ e1f11 C e1 ˝ e2f12 C e2 ˝ e1f21 C e2 ˝ e2f22D e1 ˝ e1f11 C e2 ˝ e2f22 C 1

2.e1 ˝ e2 � e2 ˝ e1/f12

C 1

2.e1 ˝ e2 C e2 ˝ e1/f12 � 1

2.e1 ˝ e2 � e2 ˝ e1/f21

C 1

2.e1 ˝ e2 C e2 ˝ e1/f21

D e1 _ e1f11C e2 _ e2f22C e1 ^ e2.f12� f21/=2Ce1 _ e2.f12Cf21/=2dimS

2 D 3; dim A2 D 1:


As a second example of the decomposition of multilinear functions (tensor space)into symmetric and antisymmetric constituents we consider a linear space X (vectorspace) of dimension dimX D n D 3, spanned by orthonormal contravariant basevectors fe1; e2; e3g. Choose p D 0, q D 1 and 2.

spanX D fe1; e2; e3g

T01 D A1 D S1 3 f D

(3X

iD1ei f

i

)

D e1f 1 C e2f 2 C e3f 3 2 X

T02 D A2 ˚ S2

T02 3 f D

8<

:

3X

i;jD1ei ˝ ej f ij

9=

;

D(

3X

iD1ei ˝ e1f i1 C

3X

iD1ei ˝ e2f i2 C

3X

iD1ei ˝ e3f i3

)

D e1 ˝ e1f 11 C e2 ˝ e1f 21 C e3 ˝ e1f 31 C e1 ˝ e2f 12 C e2 ˝ e2f 22

C e3 ˝ e2f 32 C e1 ˝ e3f 13 C e2 ˝ e3f 23 C e3 ˝ e3f 33

C 1

2.e1 ˝ e2 � e2 ˝ e1/f 12 C 1

2.e1 ˝ e2 C e2 ˝ e1/f 12

� 12.e1 ˝ e2 � e2 ˝ e1/f 21 C 1

2.e1 ˝ e2 C e2 ˝ e1/f 12

C 1

2.e2 ˝ e3 � e3 ˝ e2/f 23 C 1

2.e2 ˝ e3 C e3 ˝ e2/f 23

� 12.e2 ˝ e3 � e3 ˝ e2/f 32 C 1

2.e2 ˝ e3 C e3 ˝ e2/f 32

C 1

2.e3 ˝ e1 � e1 ˝ e3/f 31 C 1

2.e3 ˝ e1 C e1 ˝ e3/f 31

� 12.e3 ˝ e1 � e1 ˝ e3/f 13 C 1

2.e3 ˝ e1 C e1 ˝ e3/f 13 |

Since the subspaces Spq , Ap

q and Rpq are independent, Spq ˚ Ap

q ˚ Rpq denotes

the direct sum of subspace Spq , Ap

q and Rpq . Unfortunately T

pq as the space of

multilinear functions cannot be completely decomposed in the space of symmetricand antisymmetric multilinear functions: For instance, the dimension identitiesapply dimTp D np , dimSp D .nCp�1

p /, dim Ap D .np/ with respect of a vectorspace X of dimension dimX D n, such that dimRp D dimTp �dim Sp �dim Ap Dnp � .nCpC1

p / � .np/ < np , in general. There is one exception, namely the (2,0) or(1,1) or (0,2) tensor space where the dimension of the subspace R

20 , R1

1 or R02 of

residual multilinear functions is zero. An example is dimR2 D n2 � .nC1

2 / � .n2/ Dn2 � .nC 1/n=2� n.n � 1/2 D 0.


A-3 Matrix Algebra, Array Algebra, Matrix Norm andInner Product

Symmetry and antisymmetry of the symbols fi1:::ip can be visualized by thetrees of Box A.2i and A.2ii. With respect to the symbols of the interiorproduct “_” and the exterior product “^” (“wedge product”) we are able toredefine symmetric and antisymmetric functions according to Box A.2iii–vi.Note the isomorphism of tensor algebra T

pq and array algebra, namely of

(a) Œfi � 2 Rn (onedimensional array, “column vector”,

dim Œfi � D n � 1)(b) Œfij � 2 R

n�n (twodimensional array, column-row array, “matrix”,dim Œfij � D n � n)

(c) Œfijk � 2 Rn�n�n (threedimensional array, “indexed matrix”,

dim Œfijk � D n � n � n)

etc. For the base space x 2 ˝ � R3 to be threedimensional Euclidean we had

answered the question how to measure the length of a vector (“norm”) and the anglebetween two vectors (“inner product”). The same question will finally been raisedfor tensors tpq 2 T

pq . The answer is constructively based on the vectorization of the

arrays Œfij �, Œfijk �, : : : Œfi1:::ip � by taking advantage of the symmetry-antisymmetrystructure of the arrays and later on applying the Euclidean norm and the Euclideaninner product to the vectorized array.

For a 2-contravariant, 0-covariant tensor we shall outline the procedure.

(a) Firstly let F D Œfij � be the quadratic matrix of dimension dimF D n � n,an element of T20. Accordingly vecF is the vector

vecF D

2

666664

fi1fi2:::

fin�1

fin

3

777775; dim vecF D n2 � 1

which is generated by stacking the elements of the matrix F columnwise in avector. The Euclidean norm and the Euclidean inner product of vecF , vecG ,respectively is

�

�

�

�

kvecF k2 WD .vecF /T .vecF / D trF TF ;

hvecF jvecG i WD .vecF /T vecG D trF TG :

(b) Secondly let F D Œfij � D Œfj i � be the symmetric matrix of dimensiondimF Dn�n, an element of S2. Accordingly vechF (read “vector half”) is

A-3 Matrix Algebra, Array Algebra, Matrix Norm and Inner Product 585

the n.nC 1/=2 � 1 vector which is generated by stacking the elements on andunder the main diagonal of the matrix F columnwise in a vector:

F D Œfij � D Œfj i � D F T H) vechF WD

2

666666666664

f11:

fn1

f22:

f2n

:

fnn

3

777777777775

; dim vechF D n.nC 1/=2

vechF D H vecF ; dimH D n.nC 1/=2 � n2:

The Euclidean norm and the Euclidean inner product of vechF , vechG ,respectively is

F D Œfij � D Œfj i � D F T

G D Œgij � D Œgj i � D GT

#

H)

�

�

�

�

kvechF k2 WD .vechF /T .vechF /

hvechF jvechG i WD .vechF /T .vechG /:

(c) Thirdly let F D Œfij � D �Œfj i � be the antisymmetric matrix of dimensiondimF D n � n, an element of A2. Accordingly veckF (read “vector skew”)is the n.n� 1/=2� 1 vector which is generated by stacking the elements underthe main diagonal of the matrix F columnwise in a vector:

F D Œfj i �D �Œfj i �D �F T H) veckF WD

2

666666666664

f21:

fn1

f32

:

fn2

:

fn�1n

3

777777777775

; dim veckF Dn.n� 1/=2

veckF D K vecF ; dimK D n.n � 1/=2 � n2:

The Euclidean norm and the Euclidean inner product of veckF , veckG ,respectively


F D Œfij � D �Œfj i � D �F T

G D Œgij � D �Œgj i � D �GT

#

H)

�

�

�

�

kveckF k2 WD .veckF /T .veckF /

hveckF jveckG i WD .veckF /T .veckG /:

You will find the forms (a) vec F and (b) VechF in the standard book of(Harville, 1997, Chap. 16, 2001, Chap. 16) including many examples. Here wewill present Example A.3 in computing the norm as well as the inner product ofa 2-contravariant, 0-covariant tensor for the operator vec, vech and veck. The veckoperator has been introduced by (Grafarend and Schaffrin, 1993, pp. 418–419).

Example A.3. (Norm and inner product of a 2-contravariant, 0-covarianttensor):

.a/ A WD2

4a d g

b e h

c f k

3

5 ; dim A D 3 � 3 H) vec A D Œa; b; c; d; e; f; g; h; k�T ;

dim vec A D 9 � 1k vec Ak2 D .vec A/T .vec A/ D tr ATA D a2 C : : :C k2

.b/ A WD2

4a b c

b d e

c e f

3

5D AT ; dim A D 3 � 3 H) vech A D Œa; b; c; d; e; f �T ;

dim vech A D 6 � 1vech A D H vec A

8 H WD

2

66666664

1 0 0 0 0 0 0 0 0

0 1=2 0 1=2 0 0 0 0 0

0 0 1=2 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1=2 0 1=2 0

0 0 0 0 0 0 0 0 1

3

77777775

; dimH D 6 � 9

kvech Ak2 D .vech A/T .vech A/ D a2 C b2 C c2 C d2 C e2 C f 2

(Henderson and Searle (1978, p. 68–69))

.c/ A WD

2

664

0 �a �b �ca 0 �d �eb d 0 �fc e f 0

3

775D �AT ; dim A D 3 � 3 H)

veck A D Œa; b; c; d; e; f �T ; dim veck A D 6 � 1veck A D K vec A

A-4 The Hodge Star Operator, Self Duality 587

8 K WD 1

2

2

66666664

0 1 0 0 �1 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 �1 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 �1 0 0 0

0 0 0 0 0 0 1 0 0 �1 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0 �1 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0 �1 0

3

77777775

;

dimK D 6 � 16kveck Ak2 D a2 C b2 C c2 C d2 C e2 C f 2 |

A-4 The Hodge Star Operator, Self Duality

In Chap. 3, we took advantage of the Hodge star operator, for instance fortransforming an overdetermined system of linear equations (inconsistent system)into a system of condition equations. There we referred to multilinear operations“join” and“meet”.The most important operator of the algebra of antisymmetric multilinear functionsis the “Hodge star operator” which we shall present finally. In addition, we shallbring to you the surprising special feature of skew algebra called “selfduality”.The algebra Ap

q of antisymmetric multilinear functions has been based on theexterior product “^” (“wedge product”). There has been created a duality operatorcalled the Hodge star operator � which is a linear map of Ap �! An�p wheren D dimX D dimX

� denotes the dimension of the base space X � R3. The

basic idea of such a map of antisymmetric multilinear functions f 2 Ap intoantisymmetric linear functions �f 2 An�p originates according to Box A.2vii fromthe following situation: The multilinear base of Ap is spanned by

f1; ei1 ; ei1 ^ ei2 ; : : : ei1 ^ e i2 ^ : : : ^ eip g

once we focus on p D 0; 1; 2; : : : n, respectively. Obviously for any dimensionnumber n and p-contravariant, q-covariant index of the skew tensor space Ap

q thereis an associated cobasis, namely

n D 1 ; p D 0; 1 W

8<

:

basis W f1; ei1gassociatedcobasis W fei1 ; 1g

n D 2 ; p D 0; 1; 2 W

8<

:

basis W f1; ei1 ; ei1 ^ ei2gassociatedcobasis W fei1 ^ ei2 ; ei2 ; 1g


n D 3 ; p D 0; 1; 2; 3 W

8<

:

basis W f1; ei1 ; ei2 ^ ei3 ; e i1 ^ ei2 ^ ei3gassociatedcobasis W fei1 ^ e i2 ^ ei3 ; ei2 ^ ei3 ; ei3 ; 1g:

in general, for arbitrary n 2 N, p D 0; 1; : : : ; n � 1; n�

�

�

�

basis Wf1; ei1 ; : : : ; e i1 ^ : : : ^ eingassociated cobasis Wfei1 ^ ei2 ^ : : : ^ ein�1 ^ ein ; ei2 ^ : : : e in�1 ^ e in ; : : : ; e in�1 ^ ein ; ein ; 1;

as long as we concentrate on p-contravariant Ap only. A similar set-up of basis-associated cobasis for q-covariant Aq and mixed Ap

q can be made. The linear mapAp �! An�p, the Hodge star operator

�

�

�

�

�.ei1 ^ : : : ^ eip / WD 1

.n � p/Š�i1:::ipipC1:::in

eipC1 ^ : : : ^ ein

maps by means of the permutation symbol

�i1 ::: ipipC1 ::: in

WD

2

6666664

C 1 for an even permutationof f1; 2; : : : ; n � 1; ng

� 1 for an odd permutationof f1; 2; : : : ; n � 1; ng

0 otherwise

– sometimes called Eddington’s epsilons – on orthonormal (“unimodular”) base ofAp onto an orthonormal (“unimodular”) base of An�p.

For antisymmetric multilinear functions also called antisymmetric tensor-valuedfunctions represented in an orthonormal (“unimodular”) base the Hodge staroperator is the following linear map

Tp0 � Ap 3 f D

8<

:1

pŠ

nDdimX�

X

i1;:::;ipD1

ei1 ^ : : : ^ eip fi1:::ip

9=

;;

�Tp0 �An�p 3 � f WD

8<

:1

.n � p/ŠnDdimX�

X

ipC1;:::;in

nDdimXX

i1;:::;ip

1

pŠ�i1:::ipipC1:::in

eipC1 ^ : : : ^ einfi1:::ip

9=

;:

As soon as the base space x 2 ˝ � R3 is not covered by Cartesian coordinates,

rather by curvilinear coordinates, its coordinates base


fb1;b2;b3g D fdy1; dy2; dy3g versus fb1;b2;b3g D�@

@y1;@

@y2;@

@y3

�

of contravariant versus covariant type is neither orthogonal nor normalized. It is forthis reason that finally we present �f , the Hodge star operator of an antisymmetricmultilinear function f , also called the dual of f , in a general coordinate base.

Definition A.2. (Hodge star operator, the dual of an antisymmetric multilin-ear function):

If an antisymmetric .p; 0/ multilinear function is an element of the skew algebraAp with respect, to a general base fbi1 ^ : : : ^ bipg is given

f D8<

:1

pŠ

nDdimX�

X

i1;:::;ipD1bi1 ^ : : : ^ bip fi1:::ip

9=

;

then the Hodge star operator, the dual of f , can be uniquely represented by

(a) �f D8<

:1

.n � p/ŠnDdimX�

X

ipC1;:::;in

nDdimX�

X

i1;:::ip

nDdimX�

X

j1;:::;jp

1

pŠbipC1 ^ : : : ^ bip

� pg�i1:::ip ipC1:::ingi1j1 : : : gipjpfj1:::jp

)

(b) �f D8<

:1

.n�p/ŠnDdimX�

X

ipC1;:::;in

nDdimX�

X

i1;:::;ip

1

pŠbipC1 ^ : : : ^ binpg�i1:::ipipC1:::inf

i1:::ip

9=

;

(c) .�f /k1:::kn�p D(nDdimX�P

i1;:::ip

pg�i1:::ipk1:::kn�pf

i1:::ip

)

as an element of the skew algebra An�p in the general associated cobase fbpC1^: : :^bingwith respect to the base space x 2 X � R

n of dimension n D dimX DdimX

� and Œgkl � D G�1 D adjG=detG ,pg Dpjgkl j.

If we extend the algebra Ap of antisymmetric multilinear functions by �1 D e1 ^: : : en 2 An and �e1 ^ : : : ^ en D 1 2 A0 D R, respectively, let us collect someproperties of �f , the dual of f .

�

�

�

�

Proposition A.1. (Hodge star operator, the dual of an antisymmetric multi-linear function):

Let the linearly ordered base fe1; : : : ; eng be orthonormal (“unimodular”).Then the Hodge star operator of an antisymmetric multilinear function f , thedual of f , with respect to fe1; : : : eng satisfies the following:


(a) � maps antisymmetric p-contravariant tensor-valued functions to antisymmet-ric .n � p/-contravariant tensor-valued functions:� W Ap �! An�p

(b)

( �1 D e1 ^ : : : ^ en DW e for every 1 2 A0; e 2 A

�e D 1 for every e 2 An; 1 2 Ap

(c) � � f D .�1/p.n�p/f for every f 2 Ap

(d) f ^ �f D kf k2e1 ^ : : : ^ en with respect to the norm

kf k2 WD 1

pŠ

nDdimXX

i1;:::;ipD1fi1:::ip f

i1:::ip :

Example A.4. (Hodge star operator n D dimX D dimX� D 3, spanX

� Dfe1; e2; e3g, Ap �! An�p):

n D 3 ; p D 0 W � 1 D e1 ^ e2 ^ e3

n D 3 ; p D 1 W � ei1 D 1

2�i1i2i3ei2 ^ ei3

2

66666664

�e1 D 1

2.e2 ^ e3 � e3 ^ e2/ D e2 ^ e3

�e2 D 1

2.e3 ^ e1 � e1 ^ e3/ D e3 ^ e1

�e3 D 1

2.e1 ^ e2 � e2 ^ e1/ D e1 ^ e2

n D 3 ; p D 2 W � ei1 ^ ei2 D �i1i2i3e i3

2

64

�e1 ^ e2 D e3�e2 ^ e3 D e1�e3 ^ e1 D e2

n D 3 ; p D 3 W � ei1 ^ ei2 ^ e i3 D 1 |

Example A.5. (Hodge star operator of an antisymmetric tensor-valued func-tion, n D dimX D dimX

� D 3, Ap �! An�p):

Throughout we apply the summation convention over repeated indices.

n D 3 ; p D 0 W(

f “0-differential form”

�f D fdx1 ^ dx2 ^ dx3 “3-differential form’’


n D 3 ; p D 1 W

8ˆˆˆ<

ˆˆ:

f D dxi1fi1 “1-differential form”

�f D 1

2�i1i2i3dxi2 ^ dxi3fi1

D f1dx2 ^ dx3 C f2dx3 ^ dx1C f3dx1 ^ dx2 “2-differential form”

n D 3 ; p D 2 W

8ˆˆ<

ˆˆ:

f D 1

2dxi1 ^ dxi2fi1i2 “2-differential form”

�f D 1

2�i1i2

i3dxi3fi1i2

D f23dx1 C f31dx2 C f12dx3 “1-differential form”

n D 3 ; p D 3 W

8ˆ<

ˆ:

f D 1

6dxi1 ^ dxi2 ^ dxi3fi1i2i3 “3-differential form”

�f D 1

6�i1i2i3fi1i2i3 D f123 “0-differential form” |

Example A.6. (Hodge star operator, n D dimX D dimX� D 3, “�” product

(cross product)):

By means of the Hodge star operator we are able to interpret the “�” product(“cross product”) in threedimensional vector space. If the vectorsx;y 2X, dimX D3, presented in the orthonormal (“unimodular”) base fe1; e2; e3g the followingequivalence between �x ^ y and x � y holds:

x D e ixi ; y D ejyj .summation convention

x 2 X ; y 2 X i; j 2 f1; 2; 3; g/

#

H)

x ^ y D ei ^ ejxiyj

D e1 ^ e2.x1y2 � x2y1/C e2 ^ e3.x2y3 � x3y2/C e3 ^ e1.x3y1 � x1y3/

H) �.x ^ y/ D �.e1 ^ e2/.x1y2 � x2y1/C �.e2 ^ e3/.x2y3 � x3y2/C �.e3 ^ e1/.x3y1 � x1y3/ D �kij ekxiyj

�.x ^ y/ D e3.x1y2 � x2y1/C e1.x2y3 � x3y2/C e2.x3y1 � x1y3/D e1.x2y3 � x3y2/C e2.x3y1 � x1y3/C e3.x1y2 � x2y1/

x � y D ei � ejxixjei � ej WD � k

ij ek

#

H)�

�

�

�

x � y D �.x ^ y/ |


Example A.7. (Hodge star operator, n D dimX D dimX� D 4;X 2 R

4,Minkwinski space, self-duality (Atiyah et al., 1978)):

By means of the Examples A-5–A-7 we like to make you familiar with (a) the Hodgestar operator of an antisymmetric tensor-valued function overR3, (b) its equivalenceto the “x” product (“cross product”) and (c) selfduality in a fourdimensional space.Such a selfduality plays a key role in differential geometry and physics as beingemphasized by Atiyah et al. (1978).

�

�

�

Historical Aside

Thus we have constructed an anticommutative algebra by implement-ing the “exterior product” “^”, also called “wedge product”, initiated byH. Grassmann in “Ausdehnungslehre” (second version published in 1882).See also his collected works, H. Grassmann (1911). In addition the work byG. Peano (Calcolo geometrico secondo, l’Ausdehnungslehre di Grassmann,Fratelli Bocca Editori, Torino 1888) should be mentioned here. The historicaldevelopment may be documented by the work of Forder (1960). A modernversion of the “wedge product” is given by Berman (1961). In particular wemention the contribution by Barnabei et al. (1985) where by avoiding the notionof the dual X� of a linear space X and based upon operations like union,intersection, and complement – i.e. known in Boolean algebra – have developeda double algebra with exterior products of type one (“wedge product”, “thejoin”) and of type two (“the meet”), namely “to restore H. Grassmanns originalideas to full geometrical power”. The star operator “�” has been introduced byW. V. D. Hodge, being implemented into algebra in the work Hodge (1941) andHodge and Pedoe (1968), pp. 232–309). Here the star operation has been called“dual Grassmann coordinates”; in addition “intersections and joins” have beenintroduced.

A-5 Linear Algebra

Multilinear algebra is built on linear algebra we are going into now. At first we givea careful definition of linear algebra which secondly we deepen by the diagrams“Ass”, “Uni” and “Comm”. The subalgebra “ring with identity” which is of centralimportance for solving polynomial equations by means of Groebner bases, theBuchberger algorithm and the multipolynomial resultant method is our third subject.Section 4 introduces the motion of division algebra and the non-associative algebra.Fifthly, we confront you with Lie algebra (“God is a lie Group”); in particular withWitt algebra. Section 6 compares Lie algebra and Killing analysis. Here we addsome notes on the difficulties of a composition algebra in Sect. 7. Finally in Sect. 8

A-5 Linear Algebra 593

matrix algebra is presented again, but this time as a division algebra. As examplesof a division algebra as well as composition algebra we introduce complex algebra(Clifford algebra C l.0; 1/ ) in Sect. 9 and quaternion algebra in Sect. 10 (Cliffordalgebra C l.0; 2/ ) which is followed by an interesting letter of W. R. Hamilton(16 October 1943) to his son reproduced in Sect. 11. Octonian algebra (Cliffordalgebra with respect to H � H) in Sect. 12 is an example for a “non associative”algebra as well as a composition algebra. Of course, we have reserved “Sect. 13”for the fundamental Hurwitz theorem of composition algebra and the fundamentalFrobenius theorem of division algebra.

A-51 Definition of a Linear Algebra

Up to now we have succeeded to introduce the base space X of vectors x 2 X D R3

equipped with a metric and specialized to be threedimensional Euclidean. We haveextended the base space to a tensor space, namely from vector-valued functions totensor-valuedfunctions on fRn; gij g. Now we proceed to give linear and multilinearfunctions an algebraic structure, namely by the definition of two binary operations,(two internal relations) .opera/1 D ˛, .opera/2 D � and one binary operation(external relation) .opera/3 D ˇ.

Definition A.51. (linear algebra over the field of real numbers, linearity ofvector space X):

Let R be the field of real numbers. A linear algebra over R or R-algebraconsists of a set X of objects, two internal relations (either “additive” or“multiplicative”) and one external relation

.opera/1 DW ˛ W X �X �! X

.opera/2 DW ˇ W R �X �! X or X �R �! X

.opera/3 DW � W X �X �! X:

1 With respect to the internal relation ˛ (“join”) X as a linear space is a vectorspace over R, an Abelian group written “additively” or “multiplicatively” :

x;y ; z 2 X

additively written multiplicatively writtenAbelian group Abelian group

˛.x;y/ DW x C y ˛.x;y/ DW x ı y.G1C/ .x C y/C z D x C .y C z/ .G1ı/ .x ı y/ ı z D x ı .y ı z/

.additive associativity/ .multiplicative associativity/


.G2C/ x C 0 D x .G2ı/ x ı 1 D x.additive identity; .multiplicative identity;neutral element/ neutral element/

.G3C/ x C .�x/ D 0 .G3ı/ x ı x�1 D 1.additive inverse/ .multiplicative inverse/

.G4C/ x C y D y C x .G4ı/ x ı y D y ı x.additive commutativity; .multiplicative commutativity;Abelian axiom/ Abelian axiom/:

The triplet of axioms f.G1C/; .G2C/; .G3C/g or f.G1ı/; .G2ı/; .G3ı/gconstitutes the set of group axioms.

2 With respect to the external relation ˇ the following compatibility conditionsare satisfied:

x;y 2 X;r; s 2 R

ˇ.r;x/ DWr � x.D1C/ r � .x C y/ D .x C y/ � r .D1ı/ r � .x ı y/ D .x ı y/ � r

D r � x C r � y D .r � x/ ı yD x � r C y � r D x ı .y ı r/.1st additive distributivity/ .1st multiplicative

distributivity/

.D2C/ .r C s/ � x D x � .r C s/ .D2ı/ .r ı s/ � x D x � .r ı s/D r � x C s � x D r � .s � x/D x � r C x � s D .x � r/ � s.2nd additive distributivity/ .2nd multiplicative

distributivity/

.D3/ 1 � x D x � 1 D x.left and right identity/

3 With respect to the internal relation � (“meet”) the following compatibilityconditions are satisfied:

x;y ; z 2 X; r 2 R

�.x;y/ DW x � y.G1�/ .x � y/ � z D x � .y � z/

.associativity w.r.t internal multiplication/


.D1 � C/ x � .y C z/ D x � y C x � z.x C y/ � z D x � zC y � z

.left and right additive distributivityw.r.t internal multiplication/

.D1 � ı/ x � .y ı z/ D .x � y/ ı z.x ı y/ � z D x ı .y � z/

.left and right multiplicative distributivityw.r.t internal multiplication/

.D2 � �/ r � .x � y/ D .r � x/ � y.x � y/ � r D x � .yr/.left and right distributivity of internaland external multiplication/

Please, in the following pay attention to the 3 axiom sets of the linear algebra.We specify now the three axiomatic sets called “associativity” (ass), “unity” (Uni)and “commutativity” (comm).

A-52 The Diagrams “Ass”, “Uni” and “Comm”

Conventionally a linear algebra is minimally constituted by the triplet (X; ˛; ˇ/where X as a linear space is a vector space equipped with the linear maps˛ W X � X �! X and ˇ W R � X �! X satisfying the axioms (Ass) and (Uni)according to the following diagrams:

(Uni):(Ass):The diagram

The square

Commutes.

a × idb × id id × b

α

αid × a

Commutes.

(Comm):

The triangle

α α

× ×

×

× ×× ×


Axiom (Ass) expresses the requirement that the multiplication ˛ is associativewhereas Axiom (Uni) means that the element ˇ.1/ of X is a left as well as a rightunit for ˛. The algebra .X; ˛; ˇ/ is commutative if in addition it satisfies the axiom(Comm): commutes where �X;X is the flip switching the factors: �X;X.x ı y/ Dy ı x. |Indeed we have expressed a set of axioms both explicitly as well as in a diagram-matic approach which minimally constitute a linear algebra .X; ˛; ˇ/. In addition,beside the first internal relation ˛ called “join” we have experienced a secondinternal relation � called “meet” which had to be made compatible with the otherrelations ˛ and ˇ, respectively. Actually the diagram for the axiom (Dis) is left asan exercise.Obviously we have experienced the words “addition” and “multiplication” forvarious binary operations. Note that in the linear algebra isomorphic to the vectorspace as its geometric counterpart we have not specified the inner multiplication�.x;y/ 2 X. In a threedimensional vector space of Euclidean type

�.x;y/ DW �.x ^ y/ D x � y

namely the star � of the exterior product x ^ y or the “cross product” x � y, forx 2 R

3, y 2 R3 is an example. Sometimes

�.x;y/ DW Œx;y �

is written by rectangular brackets.

�

�

�

Historical Aside

Following a proposal of L. Kronecker (“Uber die algebraisch auflosbarenGleichungen (Abhandlung, 1929) the axiom of commutativity .G4C/ or .G4ı/is called after N. H. Abel (Memoire sur un classe particuliere d’equationsresolable algebraique, Crelle’s J. reine angewandte Mathematik 4 (1828)131–156 Oeuvres vol. 1, pp. 478–514, vol. 2, pp. 217–243, 329–331 editedby S. Lie and L. Sylow, Christiana 1881) who dealt with a particular class ofequations of all degrees which are solvable by radicals, e.g. the cyclotomicequation xn�1 D 0. N. H. Abel has proved the following general theorem: If theroots of an equation are such that all roots can be expressed as rational functionsof one of them, say x, and if any two of the roots, say r1x and r2x where r1 andr2 are rational functions are connected in such a way that r2r1x D r1r2x, thenthe equation can be solved by radicals. Refer r2r1x D r1r2x to .G1/.


A-53 Ringed Spaces: The Subalgebra “Ring with Identity”

In .G2ı/ the neutral element 1 as well as in .G3ı/ the inverse element has beenmultiplied from the right. Similarly left multiplication .G2ı/ by the neutral element1 as well as .G3ı/ by the inverse element are defined. Indeed it can be shown thatthere exist exactly one neutral element which is both left-neutral and right-neutralas well as exactly one inverse element which is both left-inverse and right-inverse.A subalgebra is called a “ring with identity” if the following seven conditions hold:

A ring with identity .G3�/ is a division ring if every nonzero element of thering has a multiplicative inverse. A commutative ring is a ring with commutativemultiplication .G4�/. Modules are generalizations of the vector spaces of linearalgebra in which the “scalars” are allowed to be from an arbitrary ring, rather than afield of real numbers. They will be discussed as soon as we introduce superalgebras.Now we take reference to�

�

�

�

Lemma A.53. (anticommutativity): x ı x D 0 for all x 2 X” x ı y D�y ı x for all x;y 2 X. “ı” is used in the notation “^” accordingly.

Proof.

“ H) ”x ı y C y ı x D x ı x C x ı y C y ı x C y ı yD x ı .x C y/C y ı .x C y/ D .x C y/ ı .x C y/ D 0

“(H ”x D y H) x ı x D �x ı x H) x ı x D 0:

|Later on we refer to the following algebras.


A-54 Definition of a Division Algebra and Non-AssociativeAlgebra

Definition A.23. (division algebra):

A R-algebra is called division algebra over R, if all non-null elements ofX

0 WDX n f0g form additionally a group with respect to inner multiplication� DW x ı y , namely

x;y; z 2 Xnf0g.G1ı/ .x ı y/ ı z D x ı .y ı z/

(associativity of inner multiplication)

.G2ı/ x ı 1 D x(identity of inner multiplication)

.G3ı/ x ı x�1 D 1(inverse of inner multiplication)

Definition A.24. (non-associative algebra):

A weaking of a R-algebra is the non-associative algebra over R, if the axioms1 , 2 and 3 of linear algebra according to Definition A.21 hold with theexception of .G1ı/, that is the associativity of inner multiplication is cancelled.

A-55 Lie Algebra, Witt Algebra

“perhaps god is a lie group”

Many physicists believe that all modern physics is based on the operation called“Lie algebra”. Indeed more than 1000 textbooks and papers are written on thismost important subject. Indeed many physicists believe in “perhaps god is a liegroup”. Here, we can give a very short notice of “Lie algebra”. We skip a note ofcomparison “Lie algebra” and its brother “Killing analysis”.

Definition A.25. (Lie algebra):

A non-associative algebra is called Lie algebra over R, if the following opera-tions with respect to inner multiplication � WD x ı y hold:


.L1/ x ı x D 0

.L2/ .x ı y/ ı zC .y ı z/ ı x C .z ı x/ ı y D 0:.Jacobi identity/

The examples of the Lie algebra are numerous. As a special Lie algebra we presentthe Witt algebra which is applied to Laurent polynomials.

Example A.8. (Witt algebra on the ring of Laurent polynomials (Chen,1995)):

The Witt algebra W is the complex Lie algebra of polynomial fields on the unitcircle S

1. An element of W is a linear combination of the elements of the formein� @

@�, where � is a real parameter, and the Lie bracket of W is given by

�eim�

@

@�; ein�

@

@�

D i.n�m/ei.mCn/ @

@�:

A-56 Definition of a Composition Algebra

Various algebras are generated by adding an additional structure to the minimal setof axioms of a linear algebra blocked by (a), (b) and (c). Later on we use such anadditional structure for matrix algebra.

Definition A.26. (composition algebra):

A non-associative algebra with 1 as identity of inner multiplication is calledcomposition algebra over R, if there exists a regular quadratic formQ WX �!R

which is compatible with the corresponding operations that is the followingoperations hold:

.K1/ Q W X �! R is a regular quadratic form;

x; y; z 2 R; r 2 R

Q.r � x/ D r2 �Q.x/ .quadratic form/

Q.xCyC z/DQ.xC y/CQ.xC z/CQ.yC z/�Q.x/�Q.y/�Q.z/Q.r � x C y/� r �Q.x C y/ D .r � 1/ � Œr �Q.x/�Q.y/�Q.x/ D 0” x D 0 .regularity/

.K2/ Q.x ^ y/ D Q.x/ �Q.y/ .multiplicativity/

Q.1/ D 1


The quadratic form introduced by Definition A.55 leads to the topological notion ofscalar products, norm and metric we already used:

Lemma A.56. (scalar product, norm, metric):

In a composition algebra with a positive-definite quadratic form a scalar product(“inner product”) is defined by the bilinear form h�j�i W X � X

� �! R with

hxjyi WD 1

2ŒQ.x C y/ �Q.x/�Q.y/�I

a norm is defined by k � k W X �! R with

kxk WD CŒQ.x/�1=2

and metric is defined by the bilinear form

.x;y/ WD CŒQ.x � y/�1=2:

Thus to the algebraic structure a topological structure is added, if in addition

.K3/ Q.x/ 0

for all x 2 X holds.

Proof.�

�

�

�

(i) scalar product

h�j�i W X �X �! R is a scalar product since

.1/ hxjyi D 1

2ŒQ.x C y/ �Q.x/�Q.y/�

D 1

2ŒQ.y C x/ �Q.y/ �Q.x/� D hyjxi .symmetry/

.2/ hx C yjzi D 1

2ŒQ.x C y C z/�Q.x C y/ �Q.z/�

D 1

2ŒQ.x C z/ �Q.x/�Q.z/�

C 1

2ŒQ.y C z/ �Q.y/ �Q.z/�

D hxjzi C hyjzi .additivity/


.3/ hrxjyi D 1

2ŒQ.rx C y/ �Q.rx/�Q.y/�

D 1

2rŒQ.x C y/ �Q.x/�Q.y/�

D r � hxjyi .homogenity/

.4/ hxjxi D 1

2ŒQ.x C x/�Q.x/�Q.x/�

D 1

2ŒQ.2x/� 2 �Q.x/� D Q.x/ 0 .positivity/

�

�

�

�(ii) norm

k � k W X �! R is a norm since

.N1/ kxk D CŒQ.x/�1=2 0(positivity) and

kxk D 0 ” x D 0

.N2/ krxk D CŒQ.rx/�1=2D CŒr2 �Q.x/�1=2Djr j � kxk(homogenity)

.N3/ kx C yk D CŒQ.xCy/�1=2 D CŒQ.x/CQ.y/C 2hxjyi�1=2

D Chkxk2 C kyk2 C 2kxk � kyk

i1=2 � kxk C kyk:.Cauchy-Schwarz’ inequality/“triangle inequality”/

�

�

�

�(iii) metric

W X �X �! R is a metric since

.M1/ .x;y/ D CŒQ.x � y/�1=2 D kx � yk 0 .posi tivi ty/

and

.x;y/ D 0 ” x � y D 0 ” x D y.M2/ .x;y/ D kx � yk D j � 1j � ky � xk D .y ;x/ .symmetry/

.M3/ .x;y/ D kx � yk D k.x � z/C .z � y/k� kx � zk C kz � xk D .x; z/C .z;y/

(triangle inequality) |


We treat in the next section a special case of a division algebra, namely matrixalgebra.

A-6 Matrix Algebra Revisited, Generalized Inverses

It is known that every nonsingular matrix A has a unique inverse, usually denotedby A�1 such that AA�1 D A�1A D I, where I is the unit matrix. A matrix has ainverse only if it is square, and even then only if it is nonsingular, or alternativelyif its columns or rows are linearly independent. Over the many years past it wasfelt the need of some kind of partial inverse of a matrix that is singular or evenrectangularsingular matrix. By the term generalized inverse of a given matrix A weshall mean a matrix X associated in some way with the matrix A that (a) exists fora class of matrices larger than the class of nonsingular matrices, (b) has some of theproperties of the usual inverse, and (c) reduces to the usual inverse when A is nonsingular. Some others have used the term “pseudoinvese” rather than general inversewithout specifying the chosen type of general inverse.For a given matrix A, the matrix equation AXA D A alone characterizes thosegeneralized inverses X that are of use in analyzing the solutions of the linear systemAx D b. For other purposes, other relationships play an essential role. Thus if weare concerned with least squares properties, the equation AXA D A is not enoughand must be supplemented by other relations. There results a more restricted class ofgeneralized inverses. Here we are limited to generalized inverses finite matrices, butextension to infinite-dimensional space and to differential and integral operationsare very well known.E.H. Moore is attributed to have written one of the first papers on the topicof generalized inverses called by him the “general reciprocal” of any finitematrix,square or rectangular. His first publication on the subject was an abstracttalk given at a meeting of the American Mathematical Society which appeared in1920. Details of his talk were published only in 1935 (Moore, 1920), Moore (1935);(Moore and Nashed, 1974) after Moore’s death! Little note was taken of Moore’sdiscovery for 30 years after his first publication, during which time generalizedinverses were taken for matrices by C. L. Siegel and for operators by Tseng (1936),Tseng (1949a)–Tseng (1949c), Tseng (1956), Murray and von Neumann (1936) andmany others. A surprising revival of interest in general inverses appeared in 1950swhen least squares properties were analyzed. The important properties were realizedby Bjerhammar (1951a), Bjerhammar (1951b), (1968), one swedish colleagueteaching LESS and geodetic applications of the subject. In 1955 R. Penrose extendedA. Bjerhammar’s results on linear systems and showed that E.H. Moore inversesatisfied four equations of type (a) AXA D A, (b)XAX D X, (c) AX

0 D AX, and(d) XA

0 D XA nowadays called the axioms of the Moore-Penrose inverse, oftendenoted by AC.There are excellent textbooks on matrix algebra and generalized inverses ofmatrices including many examples upto 500! We like to mention the excellent text

A-6 Matrix Algebra Revisited, Generalized Inverses 603

of Ben-Israel and Greville (1974), Bjerhammar (1973), Gere and Weaver (1965),Graybill (1963), Mardia et al. (1979), Nashed (1974), Rao and Mitra (1971), Raoand Rao (1998), Scarle (1982), Styan (1983). Here we will be unable to competewith the dense information given by these special texts.

We assume that the reader is familiar with the standard notion of a matrixas a rectangular array of numbers offered by any undergraduate text of appliedmathematics. Familiarity of various notion, for instance the dimension of a matrixof type n � m, multiplication and addition of two matrices of type (a) “Cayley”or simply the matrix product C WDA:B, (b)“Kronecker-Zehfuss” denoted by C WDB˝ A, (c) “Katri-Rao” denoted by C WD Bˇ A, and (d)“Hadamard” attributed byK WD G �H.

How are the various matrix product defined? Who defined the variousalgebras?

Matrix algebra is the special division algebra over the field of real numbers. Thereis a natural generalization in terms of complex numbers for instance.

The three axioms

Let A D Œaij � 2 Rn�m be a rectangular matrix

1 A;B;C 2 Rnm; first set of axiom ˛.A;B/ DW ACB

.G1C/ .ACB/C C D AC .B C C /

.G2C/ AC 0 D A

.G3C/ A �A D 0

.G4C/ ACB D B C A

2 A;B 2 Rnm ; second set of axiom r; s 2 R; ˇ.r;A/ DW r � A

.D1C/ r � .ACB/ D r � AC r �B

.D2C/ .r C s/ � A D r � AC sA

.D3/ 1 � A D A

3 “multiplication of matrices”

(a) “Cayley-product” (just “the matrix product”)

A D Œaij � 2 Rn�l ; dim A D n � l

B D Œbij � 2 Rl�m ; dimB D l �m

#

H) C WD A �B D Œcij � 2 Rnm ;

dimC D n �m cij WDlP

kD1aikbkl :


The product was introduced by Cayley (1857); see also his CollectedWorks, vol. 2, 475–496. A historical perspective is given in Feldmann(1962).

(b) “Kronecker-Zehfuß-product”

A D Œaij � 2 Rn�m ; dim A D n �m

B D Œbij � 2 Rk�l dimB D k � l

#

H) C WD B ˝ A D Œcij � 2 Rkn�lm;

dimC D kn � lm; B ˝ A WD ŒbijA�:

The product was early referenced to L. Kronecker by Mac Duffee (1946).The other reference is Zehfuss (1858). See also Henderson et al. (1983)and Horn and Johnson (1990). Reference is also made to Steeb (1991) andGraham (1981).

(c) “Khatri-Rao-product” (two rectangular matrices of identical columnnumber)

A D Œa1; : : : ; am� 2 Rn�m ; dim A D n �m

B D Œb1; : : : ;bm� 2 Rk�m ; dimB D k �m

#

H) C WD B ˇ A WD Œb1 ˝ a1; : : : ;bm ˝ am� 2 Rkn�m

dimC D kn �m:

“two rectangular matrices of identical column numbers are multiplied.”The product was introduced by Khatri and Rao (1968).

(d) “Hadamard product” (two rectangular matrices of the same dimension,elementwise product)

G D Œgij � 2 Rn�m ; dimG D n �m

H D Œhij � 2 Rm�m ; dimH D n �m

#

H) K WD G �H D Œkij � 2 Rn�m ;

dimK D n �m; kij WD gij hij .no summation/:

“two rectangular matrices of the same dimension define the elementwiseproduct.”The product was introduced by Hadamard (1899). See also Moutard(1894), as well as Hadamard (1949) and Schur (1911) and Horn andJohnson (1991) and Styan (1973).


Example:

(a)

A D�1 2 3

4 5 6

2 Z

2�3; B D2

42 3

4 5

6 7

3

5 2 Z3�2 H)

(b)

H) A �B D�28 34

64 79

2 Z

2�2 .“integer numbers”/:

A D�1 2 3

4 5 6

2 Z

2�3 ; B D2

41 2

3 4

5 6

3

5 2 Z3�2 H) B ˝A D ŒbijA�

D2

4

2

41 2

3 4

5 6

3

5˝ A

3

5 D

2

6666666664

1 2 3 2 4 6

4 5 6 8 10 12

3 6 8 4 8 12

12 15 18 16 20 24

5 10 15 6 12 18

20 25 30 24 30 36

3

7777777775

2 Z6�6

(c)

A D�1 2 3

4 5 6

2 Z

2�3; B D2

41 2 3

4 5 6

7 8 9

3

5 2 Z3�3

H) B ˇ A D2

4

2

41

4

7

3

5˝2

41

4

3

5 ;

2

42

5

8

3

5˝2

42

5

3

5 ;

2

43

6

9

3

5˝2

43

6

3

5

3

5

D

2

66666664

1 4 9

4 10 18

4 10 18

16 25 36

7 16 27

23 40 54

3

77777775

2 Z6�3


(d)

G D�1 2 3

4 5 6

2 Z

2�3 ; H D�2 3 4

5 6 7

2 Z

2�3

H) G �H D Œgij hij � D�2 6 12

20 30 42

2 Z

2�3:

A-61 Special Matrices: Helmert, Hankel, and Vandemonte

All the quoted textbooks review effectively special matrices of various types:symmetric, antisymmetric, diagonal, unity, zero, triangular, idempotent, normal,orthogonal, orthonormal, positive-definite, positive-semidefinite, permutation, andcommutation matrices.Here, we review only orthonormal matrices of various representations. The Helmertrepresentation of orthonomal matrix is given in various forms. In addition, we giveexamples for an orthogonal matrix, for instance the Hankel matrix of power sumsand the Vandermonde matrix.

Definition (orthogonal matrix) :

The matrix A is called orthogonal if AA0 and A0A are diagonal matrices. (The rowsand columns of A are orthogonal.)

Definition (orthonormal matrix): The matrix A is called orthonormal ifAA0 D A0A D I. (The rows and columns of A are orthonormal.)

Facts (representation of a 22 orthonormal matrix) X 2 SO.2/: A 2 � 2 orthonormalmatrix X 2 SO.2/ is an element of the special orthogonal group SO(2) defined by

SO.2/ WD fX 2 R2�2 jX0X D I2 and det X D C1g

8<

:X D

�x1 x2

x3 x4

2 R

2�2ˇˇˇˇˇ

x21 C x22 D 1x1x3 C x2x4x23 C x24 D 1

D 0; x1x4 � x2x3 D C19=

;

(a)

X D�

cos� sin �� sin � cos�

2 R

2�2; � 2 Œ0; 2�

is a trigonometric representation of X 2 SO.2/.


(b)

X D"

xp1 � x2

�p1 � x2 x

#

2 R2�2; x 2 ŒC1;�1�

is an algebraic representation of X 2 SO.2/

x211C x212D 1; x11x21Cx12x22D �x

p1�x2C x

p1� x2D 0; x221Cx222D 1

�:

(c)

X D"

1�x21Cx2 C 2x

1Cx2� 2x1Cx2

1�x21Cx2

#

2 R2�2; x 2 R

is called a stereographic projection of X (stereographic projection of SO(2) S1

onto L1).

(d)

X D .I2 C S/.I2 � S/�1; S D�0 x

�x 0;

where S D �S0 is a skew matrix (antisymmetric matrix), is called a Cayley-Lipschitzrepresentation of X 2 SO.2/. (e) X 2 SO.2/ is a commutative group (“Abel”)(Example: X1 2 SO.2/, X2 2 SO.2/, then X1X2 D X2X1) (SO(n) for n D 2 is theonly commutative group, SO(nj n ¤ 2) is not “Abel”).

Facts (representation of an n � n orthonormal matrix) X 2 SO.n/:

An n � n orthonormal matrix X 2 SO.n/ is an element of the special orthogonalgroup SO(n) defined by

SO.n/ WD fX 2 Rn�njX0X D In and det X D C1g:

As a differentiable manifold SO(n) inherits a Riemann structure from the ambientspace n2 with a Euclidean metric (vecX0 2 R

n2 ; dim vecX0 D n2). Any atlas ofthe special orthogonal group SO(n) has at least four distinct charts and there is onewith exactly four charts. (“minimal atlas”: Lusternik-Schnirelmann category) (a)

X D .In C S/.In � S/�1;

where

S D �S0


is a skew matrix (antisymmetric matrix), is called a Cayley-Lipschitz representationof X 2 SO.n/. (nŠ=2.n� 2/Š is the number of independent parameters/coordinatesof X) (b) If each of the matrices R1; : : : ;Rk is an n � n orthonormal matrix, thentheir product

R1R2 � � �Rk�1Rk 2 SO.n/

is an n � n orthonormal matrix.

Facts (orthonormal matrix: Helmert representation):

Let a0 D Œa1; : : : ; an� represent any row vector such that ai ¤ 0 .i 2 f1; : : : ; ng/ isany row vector whose elements are all nonzero. Suppose that we require an n � northonormal matrix, one row which is proportional to a0. In what follows one suchmatrix R is derived. Let Œr0

1; : : : ; r0n� represent the rows of R and take the first row

r01 to be the row of R that is proportional to a0. Take the second row r0

2 to beproportional to the n-dimensional row vector

Œa1;�a21=a2; 0; 0; : : : ; 0�; .H2/

the third row r03 proportional to

�a1; a2;�.a21 C a22/=a3; 0; 0; : : : ; 0

.H3/

and more generally the first through nth rows r01; : : : ; r0

n proportional to

"

a1; a2; : : : ; ak�1;�k�1X

iD1a2i =ak; 0; 0; : : : ; 0

#

.Hn � 1/

for k 2 f2; : : : ; ng;

respectively confirm to yourself that the n-1 vectors (Hn�1) are orthogonal to eachother and to the vector a0. In order to obtain explicit expressions for r0

1; : : : ; r0n it

remains to normalize a0 and the vectors (Hn�1). The Euclidean norm of the kth ofthe vectors (Hn�1) is

8<

:

k�1X

iD1a2i C

k�1X

iD1a2i

!2.a2k

9=

;

1=2

D(

k�1X

iD1a2i

! kX

iD1a2i

!.

a2k

) 1=2

:

Accordingly for the orthonormal vectors r01; : : : ; r0

n we finally find

(1st row) r01 D

"nX

iD1a2i

#�1=2.a1; : : : ; an/


(kth row) r0k D

2

66664

a2k k�1PiD1

a2i

! kP

iD1a2i

!

3

77775

�1=2

�

a1; a2; : : : ; ak�1;�k�1X

iD1

a2ia k; 0; 0; : : : ; 0

!

.nth row/ r0n D

2

6664

a2n�n�1PiD1

a2i

� �nP

iD1a2i

�:

3

7775

�1=2"

a1; a2; : : : ; an�1;�n�1X

iD1

a2ian

#

:

The recipe is complicated: When a0 D Œ1; 1; : : : ; 1; 1�, the Helmert factors in the1st row, : : : ; kth row,. . . , nth row simplify to

r01 D n�1=2Œ1; 1; : : : ; 1; 1� 2n

r0k D Œk.k � 1/��1=2Œ1; 1; : : : ; 1; 1� k; 0; 0; : : : ; 0; 0� 2n

r0n D Œn.n � 1/��1=2Œ1; 1; : : : ; 1; 1 � n� 2 R

n:

The orthonormal matrix2

66666666664

r01

r02

� � �r0k�1r0k

� � �r0n�1r0n

3

77777777775

2 SO.n/

is known as the Helmert matrix of order n. (Alternatively the transposes of such amatrix are called the Helmert matrix.)

Example (Helmert matrix of order 3):

2

664

1=p3 1=

p3 1=

p3

1=p2 �1=p2 0

1=p6 1=

p6 �2=p6

3

775 2 SO.3/:

Check that the rows are orthogonal and normalized.


Example (Helmert matrix of order 4):

2

66664

1=2 1=2 1=2 1=2

1=p2 �1=p2 0 0

1=p6 1=

p6 �2=p6 0

1=p12 1=

p12 1=

p12 �3=p12

3

777752 SO.4/:

Check that the rows are orthogonal and normalized.

Example (Helmert matrix of order n):

2

66666666664

1=pn 1=

pn 1=

pn 1=

pn � � � 1=

pn 1=

pn

1=p2 �1=p2 0 0 � � � 0 0

1=p6 1=

p6 �2=p6 0 � � � 0 0

� � � � � �1p

.n�1/.n�2/1p

.n�1/.n�2/1p

.n�1/.n�2/ � � � � � �1�.n�1/p.n�1/.n�2/ 0

1pn.n�1/

1pn.n�1/

1pn.n�1/ � � � � � � 1p

n.n�1/1�npn.n�1/

3

77777777775

2 SO.n/:

Check that the rows are orthogonal and normalized. An example is the nth row

1

n.n � 1/ C � � � C1

n.n � 1/ C.1 � n/2n.n � 1/ D

n� 1n.n � 1/ C

1 � 2nC n2n.n � 1/

D n2 � nn.n � 1/ D

n.n � 1/n.n � 1/ D 1;

where .n � 1/ terms 1=Œn.n � 1/� have to be summed.

Definition (orthogonal matrix):

A rectangular matrix A D Œaij � 2 Rn�m is called “a Hankel matrix” if the nCm�1

distinct elements of A,

2

666664

a11a21� � �an�11an1 an2 � � � anm

3

777775

only appear in the first column and last row.


Example: Hankel matrix of power sums

Let A 2 Rn�m be a nm rectangular matrix (n 6 m) whose entries are power sums.

A WD

2

6666666664

nP

iD1˛ixi

nP

iD1˛i x

2i � � �

nP

iD1˛ix

mi

nP

iD1˛ix

2i

nP

iD1˛i x

3i � � �

nP

iD1˛ix

mC1i

::::::

::::::

nP

iD1˛ix

ni

nP

iD1˛ix

nC1i � � �

nP

iD1˛ix

nCm�1i

3

7777777775

A is a Hankel matrix.

Definition (Vandermonde matrix):

Vandermonde matrix: V 2 Rn�n

V W D

2

66664

1 1 � � � 1

x1 x2 � � � xn:::

::::::

:::

xn�11 xn�1

2 � � � xn�1n

3

77775;

det V DY

i;ji>j

n

.xi � xj /:

Example: Vandermonde matrix V 2 R3�3

V WD

2

64

1 1 1

x1 x2 x3

x21 x22 x

23

3

75 ; det V D .x2 � x1/.x3 � x2/.x3 � x1/:

Example: Submatrix of a Hankel matrix of power sums Consider the submatrixP D Œa1; a2; : : : ; an� of the Hankel matrix A 2 R

n�m .n 6 m/ whose entries arepower sums. The determinant of the power sums matrix P is

det P D

nY

iD1˛i

! nY

iD1xi

!

.det V/2;

where det V is the Vandermonde determinant. Example: Submatrix P 2 R3�3 of a

3 � 4 Hankel matrix of power sums (n D 3, m D 4)


A D

2

6664

˛1x1 C ˛2x2 C ˛3x3 ˛1x21 C ˛2x

22 C ˛3x

23 ˛1x

31 C ˛2x

32 C ˛3x

33 ˛1x

41 C ˛2x

42 C ˛3x

43

˛1x21 C ˛2x

22 C ˛3x

23 ˛1x

31 C ˛2x

32 C ˛3x

33 ˛1x

41 C ˛2x

42 C ˛3x

43 ˛1x

51 C ˛2x

52 C ˛3x

53

˛1x31 C ˛2x

32 C ˛3x

33 ˛1x

41 C ˛2x

42 C ˛3x

43 ˛1x

51 C ˛2x

52 C ˛3x

53 ˛1x

61 C ˛2x

62 C ˛3x

63

3

7775

P D Œa1; a2; a3�

2

6664

˛1x1 C ˛2x2 C ˛3x3 ˛1x21 C ˛2x

22 C ˛3x

23 ˛1x

31 C ˛2x

32 C ˛3x

33

˛1x21 C ˛2x

22 C ˛3x

23 ˛1x

31 C ˛2x

32 C ˛3x

33 ˛1x

41 C ˛2x

42 C ˛3x

43

˛1x31 C ˛2x

32 C ˛3x

33 ˛1x

41 C ˛2x

42 C ˛3x

43 ˛1x

51 C ˛2x

52 C ˛3x

53

3

7775:

A-62 Scalar Measures of Matrices

All reference textbooks review various scalar measures of matrices like

• Linear independence• Column and row rank• Rank identities

to which we refer. Permanently, we referred to a special technique called IPM(“Inverse Partitioned Matrix”) of a symmetric matrix we will review now.

Facts: (Inverse Partitional Matrix/IPM/of a symmetric matrix):

Let the symmetric matrix A be partitioned as

A WD"

A11 A12

A012 A22

#

; A011 D A11; A0

22 D A22:

Then its Cayley inverse A�1 is symmetric and can be partitioned as well as

A�1 D2

4A11 A12

A0

12 A22

3

5

�1

D2

4ŒI C A�1

11 A12.A22 � A0

12A�111 A12/

�1A0

12�A�111 �A�1

11 A12.A22 � A0

12A�111 A12/

�1

�.A22 � A0

12A�111 A12/

�1A0

12A�111 .A22 � A0

12A�111 A12/

�1

3

5 ;

if A�111 exists;

A�1 D2

4A11 A12

A0

12 A22

3

5

�1

D2

4.A11 � A0

12A�122 A12/

�1 �.A11 � A0

12A�122 A12/

�1A12A�122

�A�122 A0

12.A11 � A0

12A�122 A12/

�1 ŒI C A�122 A0

12.A11 � A12A�122 A0

12/�1A12�A�1

22

3

5 ;


if A�122 exists:

S11 WD A22 � A012A�1

11 A12 and S22 WD A11 � A012A�1

22 A12

are the minors determined by properly chosen rows and columns of the matrix Acalled “Schur complements” such that

A�1 D"

A11 A12

A012 A22

#�1D".IC A�1

11 A12S�111 A0

12/A�111 �A�1

11 A12S�111

�S�111 A0

12A�111 S�1

11

#

if A�111 exists;

A�1 D"

A11 A12

A012 A22

#�1D"

S�122 �S�1

22 A12A�122

�A�122 A0

12S�122 ŒIC A�1

22 A012S�1

22 A12�A�122

#

if A�122 exists;

are representations of the Cayley inverse partitioned matrix A�1 in terms of “Schurcomplements”.The formulae S11 and S22 were first used by J. Schur (1917). The term “Schurcomplements” was introduced by Haynsworth (1968). Albert (1969) replaced theCayley inverse A�1 by the Moore-Penrose inverse AC. For a survey we recommendCottle (1974), Ouellette (1981), Carlson (1986) and Styan (1985).

Proof.

For the proof of the “inverse partitioned matrix” A�1 (Cayley inverse) of thepartitioned matrix A of full rank we apply Gauss elimination (without pivoting).

AA�1 D A�1A D I

A D�

A11 A12

A012 A22

; A0

11 D A11; A022 D A22

"A11 2 R

m�m; A12 2 Rm�l

A012 2 R

l�m; A22 2 Rl�l

A�1 D�

B11 B12B012 B22

; B0

11 D B11; B022 D B22

"B11 2 R

m�m; B12 2 Rm�l

B012 2 R

l�m; B22 2 Rl�l

AA�1 D A�1A D I,

2

6664

A11B11 C A12B012 D B11A11 C B12A0

12 D Im .1/

A11B12 C A12B22 D B11A12 C B12A22 D 0 .2/

A012B11 C A22B0

12 D B012A11 C B22A0

12 D 0 .3/

A012B12 C A22B22 D B0

12A12 C B22A22 D Il : .4/


Case (a) W A�111 exists

“forward step”

A11B11 CA12B012 D Im .first left equation:multiply by �A0

12A�111 /

A012B11 C A22B0

12 D 0 .secondrightequation)

#

, �A012B11 �A0

12A�111 A12B0

12 D A012A�1

11

A012B11 CA22B0

12 D 0

#

,"

A11B11 C A12B012 D Im

.A22 � A012A�1

11 A12/B012 D �A0

12A�111

) B012 D �.A22 �A0

12A�111 A12/

�1A012A�1

11

B012 D �S�1

11 A012A�1

11

or"

Im 0

�A012A�1

11 Il

#"A11 A12

A012 A22

#

D"

A11 A12

0 A22 � A012A�1

11 A12

#

:

Note the “Schur complement” S11 WD A22 �A012A�1

11 A12.

“backward step”

A11B11 C A12B012 D Im

B012 D �.A22 � A0

12A�111 A12/

�1A012A�1

11

#

) B11 D A�111 .Im � A12B0

12/ D .Im � B12A012/A�1

11

B11 D ŒIm C A�111 A12.A22 �A0

12A�111 A12/

�1A012�A�1

11

B11 D A�111 C A�1

11 A12S�111 A0

12A�111

A11B12 C A12B22 D 0 .second left equation/

) B12 D �A�111 A12B22 D �A�1

11 A12.A22 � A012A�1

11 A12/�1

, B22 D .A22 �A012A�1

11 A12/�1

B22 D S�111 :

Case (b) W A�122 exists

“forward step”


A11B12 C A12B22 D 0 .third right equation)

A012B12 C A22B22 D Il .fourth left equation: multiply by � A12A�1

22 /

#

, A11B12 C A12B22 D 0

�A12A�122 A0

12B12 �A12B22 D �A12A�122

#

,"

A012B12 C A22B22 D Il

.A11 � A12A�122 A0

12/B12 D �A12A�122

) B12 D �.A11 �A12A�122 A0

12/�1A0

12A�122

B12 D �S�122 A12A�1

22 or"

Im �A12A�122

0 Il

#"A11 A12

A012 A22

#

D"

A11 �A12A�122 A0

12 0

A012 A22

#

:

Note the “Schur complement”

S22 WD A11 � A12A�122 A0

12:

“backward step”

A012B12 C A22B22 D Il

B12 D �.A11 � A12A�122 A0

12/�1A12A�1

22

#

) B22 D A�122 .Il � A0

12B012/ D .Il � B0

12A12/A�122

B22 D ŒIl C A�122 A0

12.A11 � A12A�122 A0

12/�1A12�A�1

22

B22 D A�122 C A�1

22 A012S�1

22 A12A�122

A012B11 C A22B0

12 D 0 .third left equation/

) B012 D �A�1

22 A012B11 D �A�1

22 A012.A11 � A12A�1

22 A012/

�1

,B11 D .A11 � A12A�1

22 A012/

�1

B11 D S�122 :

|

The representations B11;B12;B21DB012;B22 in terms of A11;A12;A21 D A0

12;A22

have been derived by Banachiewicz (1937). Generalizations are referred to Ando(1979), Brunaldi and Schneider (1963), Burns et al. (1974), Carlson (1980), Meyer(1973) and Mitra (1982), Li and Mathias (2000). We leave the proof of the followingfact as an exercise.


Fact (Inverse Partitioned Matrix /IPM/ of a quadratic matrix):

Let the quadratic matrix A be partitioned as

A WD"

A11 A12

A21 A22

#

:

Then its Cayley inverse A�1 can be partitioned as well as

A�1 D"

A11 A12

A21 A22

#�1D"

A�111 C A�1

11 A12S�111 A21A�1

11 �A�111 A12S�1

11

�S�111 A21A�1

11 S�111

#

;

if A�111 exists

A�1 D"

A11 A12

A21 A22

#�1D"

S�122 �S�1

22 A12A�122

�A�122 A21S�1

22 A�122 CA�1

22 A21S�122 A12A�1

22

#

;

if A�122 exists

and the “Schur complements” are defined by

S11 WD A22 �A21A�111 A12 and S22 WD A11 � A12A�1

22 A21:

Facts: Cayley Inverse, DG matrix identity, SMW matrix identity, sum of twomatrices

.DG id/ BD.AC CBD/�1 D .B�1 C DA�1C/�1DA�1(Duncan-Guttman matrix identity):

.SMW id/ .AC CBD/�1 D A�1 � A�1C.B�1 C DA�1C/�1DA�1(Sherman - Morrison - Woodbury matrix identity):

Duncan (1944) calls (DG) the Sherman-Morrison-Woodbury matrix identity. If thematrix A is singular consult Henderson and Searle (1981a), Henderson and Searle(1981b), Ouellette (1981), Hager (1989), Stewart (1977) and Riedel (1992). (SMW)has been noted by Duncan (1944) and Guttman (1946): The result is directly derivedfrom the identity

.AC CBD/.AC CBD/�1 D I

) A.ACCBD/�1 CCBD.AC CBD/�1 D I

.AC CBD/�1 D A�1 � A�1CBD.AC CBD/�1


A�1 D .AC CBD/�1 C A�1CBD.AC CBD/�1

DA�1 D D.AC CBD/�1 C DA�1CBD.AC CBD/�1

DA�1 D .ICDA�1CB/D.AC CBD/�1

DA�1 D .B�1 C DA�1C/BD.AC CBD/�1

.B�1 C DA�1C/�1DA�1 D BD.AC CBD/�1:

Up to now we presented IPM of a symmetric matrix. Alternatively, we review rankfactorization of a symmetric and positive semidefinite/definite matrix.

Facts (rank factorization):

(a) If the n � n matrix is symmetric and positive semidefinite, then its rankfactorization is

A D"

G1

G2

#�

G01 G0

2

;

where G1 is a lower triangular matrix of the order O.G1/ D rk A � rk A withrk G2 D rk A, whereas G2 has the format O.G2/ D .n � rk A/ � rk A: In thiscase we speak of a Choleski decomposition.

(b) In case that the matrix A is positive definite, the matrix block G2 is not neededanymore: G1 is uniquely determined. There holds

A�1 D .G�11 /

0G�11 :

Various notions, for instance submatrices adjoint, traces and determinants, as scalarmeasures of a matrix are given by D. A. Harville (2001, examples, Sects. 2,5 and 13)to which we refer. We already introduced vector-valued matrix forms by means ofthe operations vec, vech and veck. Please pay attention to the vec-forms

(a) vecA � B � C0 D .A˝ C/vecB(b) .A/0vecB D tr.A � B0/(c) vec xy0 D x˝ y

(d) vec(A � B/ D .B0 ˝ In/vecA D .B0 ˝ A/vecIm

D .Iq ˝ A/vecB; for all A 2 Rn�m; B 2 R

m�q

given by Pukelsheim (1977), pp. 324, 326 or Harville (2001), examples,Sect. 16, 22 pp.


A-63 Three Basic Types of Generalized Inverses

In Chaps. 1, 3 and 5 we obtained the complete class of solution of equation “AxD y00and showed that every solution can be expressed in the form x D Gy where G isthe properly chosen generalized inverse. Here we inquire to find solutions for suchgeneralized inverses which are of the following type

.i/ rk A D rk .AA0/ D n W “minimum norm”

Consistent system of observation equations: Underdetermine.

Theorem (“minimum norm”: Rao and Mitra (1971), pp. 44–47)

Let G be a generalized inverse of the matrix A such that Gy is a minimum normsolution of the equation Ax D y; rkA D rk .AA0/ D n. Then it is necessary andsufficient that

(a) kxk2 D kGyk2 D Min

Ax D yor

(b) AGA D A and .GA/0 D GA. or(c) A�

mr D A0.AA0/� (reflexive generalized inverse which is minimum norm).

If we use the weighted generalized inverse of the typenMin

Ax D ykxk2 D x0Gx

o

we get the Gx-MINOS solution of Chap. 1, case (a), case (b) and case (c).

.ii/ rk A D rk .A0A/ D m W “generalized least-squares solution for Ax D y”

Inconsistent system of linear observation equations: overdetermine.

Theorem (“Least-squares”: Rao and Mitra (1971), pp. 48–50)

Let G be a matrix (not necessarily a generalized inverse) such that Gy is a least-squares solution of Ax D y such that rkA D rk .A0A/ D m. Then it is necessaryand sufficient that

(a) ky �Axk2 D min .inconsistent/or

(b) AGA D A and .GA/0 D GA:or

(c) Alr D .A0A/�A (reflexive generalized inverse which is least-squares).

If we use the weighted generalized inverse of the typenMin

xky � Axk2Gy

D

.y � Ax/0Gy.y � Ax/o

we enjoy receiving the Gy-LESS solution of Chap. 3,

case (a), case (b) and case (c).

A-7 Complex Algebra, Quaternion Algebra, Octonian Algebra, Clifford Algebra 619

.iii/ rkA < min fm; ng W “generalized inverse for a minimum norm least-squares solution”

Inconsistent system of linear observation equations with a datum defect:overdetermine-underdetermined. ut

Here we start with a definition

Definition (“MINOLESS”: Rao and Mitra (1971), p. 51)

G is said to be a minimum norm, least-squares generalized inverse of the matrix A,if G is A�

l and for any y 2 Y

kGykm � kxkm for all x 2 fx j ky � Axkm � ky � Azkm for all z 2 X

where k:kn and k:km are norms in Xm and Y

n respectively. Finally we will refer tothe basic results generating “MINOLESS”.

Theorem (“MINOLESS”: Rao and Mitra (1971), pp. 50–54,64)

Let G be a matrix such that Gy is a minimum norm, least-squares solution of theinconsistent equation Ax D yC i. Then it is necessary and sufficient that x D ACynamely, (a) AGA D A, (b) .AG/0 D AG, (c) GAG/ D G, (d) .GA/0 D GA(Penrose 1955). or

AC D A0AŒ.A0A/2��A0 D .IC A0A/�A0ŒA0A.IC A0A/�A��A� or

AC D k lim�!1kX

kD1A0.IC AA0/�k D k lim�!1

kX

kD1.IC A0A/�kA0 or

AC D � lim�! 0CA0.�IC AA0/�1 D � lim�! 0C.I�C A0A/�1A0

If we use the weighted norms kxk2Px and ky � Axk2Py , we receive the Gx-minimumnorm, Gy- least squares solution of Chap. 5, case (a), case (b) and case (c).

A-7 Complex Algebra, Quaternion Algebra, OctonianAlgebra, Clifford Algebra, Hurwitz Theorem

The greatest advantage of Clifford algebra is the following fact: We are able to writedown the field equations namely “rot” and “div” equation of gravitational field andelectromagnetic field in one equation. The related literature is very rich and wecould recommend the book by Meetz and Engl (1980). A more detailed introductioninto Clifford algebra and its fascinating chess board as well as Clifford analysis islisted in Grafarend (2004) with more than 300 references. From some point of view,the variona algebra presented here are rather out of fashion. Indeed


• Complex algebra, Clifford algebra Cl.0; 1/• Quaternion algebra, Clifford algebra Cl.0; 2/

A-71 Complex Algebra as a Division Algebra as wellas a Composition Algebra, Clifford algebra Cl.0; 1/

Here we present to you “complex algebra” and refer it to Clifford algebra C l.0; 1/.The “complex algebra” C due to C. F. Gauß is a division algebra over R as well asa composition algebra.

x 2 C

x D e0x0 C e1x1 subject to fx0; x1g 2 R2

span C D f1; e1g

1 x;y; z 2 R2

˛.x;y/ DW x C y :

The axioms .G1C/, .G2C/, .G3C/, .G4C/ of an Abelian group apply.

2 x;y 2 R2; r; s 2 R

ˇ.r;x/ DW r � x:

The axioms .D1C/, .D2C/, .D3/ of additive distributivity apply.

3 One way to explicitly describe a multiplicative group with finitely manyelements is to give a table listing the multiplications just representing the map� W C � C �! C.

�

�

�

�multiplication diagram, Cayley diagram

1 e1

1 1 e1

e1 e1 �1

Note that in the multiplication table each entry of a group appears exactly oncein each row and column. The multiplication has to be read from left to rightthat is, the entry at the intersection of the row headed by e1 and the columnheaded by e1 is the product e1 � e1. Such a table is called a Cayley diagramof the multiplicative group.


Here note in addition the associativity of the internal multiplication given inthe table. Such a “complex algebra” C is not a Lie algebra since neither x �x D 0 .L1/ nor .x �y/� zCC.y � z/�xC .z�x/�y D 0 (Jacobi identity).L2/ hold. Just by means of the multiplication table compute

x � x D 1f.x0/2 � .x1/2g C 2e1x0x1 6D 0:

4 Begin with the choice

x�1 D 1

.x0/2 C .x1/2 .1x0 � e1x1/

in order to end up with

x � x�1 D .1x0 C e1x1/ � 1x0 � e1x1.x0/2 C .x1/2 D 1

Accordingly .G1�/, .G2�/, .G3�/ of a division algebra apply.


Q.x/ D Q.1x0 C e1x1/ WD .x0/2 C .x1/2

in order to prove (K1), (K2) and (K3). We only focus on (K2i):

Q.x � y/ D Q.x/ �Q.y/ .multiplicativity/

Q.x � y/ D Qf1.x0y0 � x1y1/C e1.x1y0 C x0y1/gD .x0/.y0/2 C .x1/2.y1/2 C .x1/2.y0/2 C .x0/2.y1/2

Q.x/ �Q.y/ D f.x0/2 C .x1/2g � f.y0/2 C .y1/2gD .x0/2.y0/2 C .x1/2.y1/2 C .x1/2.y0/2 C .x0/2.y1/2�

�

�

�

Q.x � y/ D Q.x/ �Q.y/ q:e:d:

How can be one dream about such a complex algebra C? Gauss (1887) had beenmotivated in his number theory to introduce complex numbers with i WD p�1 asthe “imaginary unit”. Identify 1x0 with the “real part” and e1x1 D ix1 with the“imaginary part” of x and we are left with the standard theory of complex numbers.x�1 is based upon the complex conjugate 1x0�e1x1 of x being divided by the normof x. There is a remarkable isomorphism between complex numbers and complexalgebra.The proper algebraic interpretation of complex numbers is in terms of Cliffordalgebra C l.0; 1/. Observe g.e1; e1/ D �1 which interprets the binary operationof the base vector which spans the vector part of a complex number. Now translatethe multiplication table into the language of the Clifford product, namely


�

�

!

1�^ 1 D 1; 1

�^ e1 D e1e1

�^ 1 D e1; e1�^ e1 D �1

in order to convince yourself that the Clifford algebra C l.0; 1/ is algebraicallyisomorphic to the space of complex numbers. |How can we relate complex numbers to Clifford algebra C l1? Observe g.e1; e1/ D�1 which interprets the binary operation of the base vector which spans the vectorpart of a complex number. While the scalar part of a complex number is an elementof A0, its vector part can be considered to be an element ofA1. The direct sum

A0 ˚A1

of spaces A0, A1 is algebraically isomorphic to the space of complex numbers,being an element of the Clifford algebra C l1. |

A-72 Quaternion Algebra as a Division Algebra as wellas a Composition Algebra, Clifford algebra Cl.0; 2/

Next we review “quaternion algebra” quoting W. R. Hamilton and refer it to Cliffordalgebra C l.0; 2/. The ”quaternion algebra” H due to Hamilton (1843) is a divisionalgebra over R as well as a composition algebra:

x 2 H

x D 1x0 C e1x1 C e2x2 C e3x3 subject to fx0; x1; x2; x3g 2 R4

span H D f1; e1; e2; e3g

1 x;y ; z 2 R4

˛.x;y/ DW x C y :

The axioms .G1C/, .G2C/, .G3C/, .G4C/ of an Abelian additive groupapply.

2 x;y 2 R4; r; s 2 R

ˇ.r;y/ DW r � x:



3 One way to explicitly describe a multiplicative group with finitely manyelements is to give a table listing the multiplications just representing the map� W H �H �! H.

�

�

�

�

multiplication table, Cayley diagram

1 e1 e2 e3

1 1 e1 e2 e3

e1 e1 �1 e3 �e2e2 e2 �e3 �1 e1

e3 e3 e2 �e1 �1

:

Note that in the multiplication table each antry of a group appears exactly oncein each row and column. The multiplication has to be read from left to rightthat is, the entry at the intersection of the row headed by e1 and the columnheaded by e2 is the product e1 � e2. Such a table is called a Cayley diagram ofthe multiplicative group.Here note in addition the associativity of the internal multiplication given bythe table, e.g. e1 � .e2 � e3/ D e1 � e1 D �1 D e3 � e3 D .e1 � e2/ � e3 or

x � y D 1

xıyı �3X

kD1xkyk

!

CX

i;j;k

ek.xıyk C xkyı C �kij xiyj /

such that .x � y/ � z D x � .y � z/.Such a “Hamilton algebra” H is not a Lie algebra since neither x � x D 0

(L1) nor .x � y/ � zC .y � z/ � x C .z � x/ � y D 0 (L2) (Jacobi identity)hold. Just by means of the multiplication table compute

x � x D 1f.xı/2 � .x1/2 � .x2/2 � .x3/2gC 2e1xıx1 C 2e2xıx2 C 2e3xıx3 6D 0


x�1 D 1

.xı/2 C .x1/2 C .x2/2 C .x3/2 .1xı � e1x1 � x2x2 � e2x3/

in order to to end up with

x � x�1D Œ1xıC e1x1C e2x2C e3x3� � 1xı � e1x1 � e2x2 � e3x3.xı/2C .x1/2C .x2/2C .x3/2 D 1:

Accordingly .G1�/, .G2�/, .G3�/ of a division algebra apply.



Q.x/ D Q.1C e1x1 C e2x2 C e3x3/ WD .x0/2 C .x1/2 C .x2/2 C .x3/2

in order to prove .K1/. .K2/ and .K3/.

The laborious proofs are left as an exercise.How can one dream about such a “quaternion algebra” H? Hamilton (16 October1843) invented quaternion numbers as outlined in a letter (1865) to his sonA. H. Hamilton for the following reason:

“If I may be allowed to speak of myself in connexion with the subject,I might do so in a way which would bring you in, by referring to anantequaternionic time, when you were a mere child, but had caught fromme the conception of a Vector, as represented by a Triplet; and indeedI happen to be able to put the finger of memory upon the year andmonth – October, 1843 – when having recently returned from visits to Corkand Parsonstown, connected with a Meeting of the British Association, thedesire to discover the laws of the multiplication referred to regained withme a certain strength and carnestness, which had for years been dormant,but was then on the point of being gratified, and was occasionally talkedof with you. Every morning in the early part of the above cited month, onmy coming down to breakfast, your (then) little brother William Edwin, andyourself, used to ask me, “Well, Papa, can you multiply triplets”? WheretoI was always obliged to reply, with a sad shake of the head: “No, I can onlyadd and subtract them.”

But on the 16th day of the same month – which happened to be a Monday,and a Council day of the Royal Irish Academy – I was walking in to attendand preside, and your mother was walking with me, along the Royal Canal,to which she had perhaps driven; and although she talked with me nowand then, yet an under-current of thought was going on in my mind, whichgave at last a result, whereof it is not too much to say that I felt at oncethe importance. An electric circuit seemed to close; and a spark flashedforth. The herald (as I foresaw, immediately) of many long years to comeof definitely directed thought and work, by myself if spared, and at allevents on the part of others, if should even be allowed to live long enoughdistinctly to communicate the discovery. Nor could I resist the impulse –unphilosophical as it may have been – to cut with a knife on a stoneof Brougham Bridge, as we passed it, the fundamental formula with thesymbols, i; j; k; namely

i 2 D j 2 D k2 D ijk D �1


which contains the Solution of the Problem, but of course, as an inscription,has long since moldered away. A more durable notice remains, however, on theCouncil Books of the Academy for that day (October 16th, 1843), which recordsthe fact, that I then asked for and obtained base to read a Paper on Quaternion, atthe First General Meeting of the Session: which reading took place accordingly,on Monday the 13th of the November following.”

Obviously the vector part e1x1Ce2x2Ce3x3 D ix1Cj x2Ckx3 of a quaternionnumber replaces the imaginary part of a complex number, the scalar part 1x0 thereal part. The quaternian conjugate

1xı �3X

kD1ekx

k DW x�

substitutes the complex conjugate of ta complex number, leading to the quaternioninverse

x�1 D x�

Q.x/:

The proper algebraic interpretation of quaternion numbers is in terms of Cliffordalgebra C l.0; 2/. If n D dimX D 2 is the dimension of the linear space X whichwe base the Clifford algebra on, its basis elements are

f1; e1; e2; e1 �^ e2gsubject to

e1�^ e2 C e2 �^ e1 D 0;

e1�^ e1 D g.e1; e1/1 D �1 ; e2 �^ e2 D g.e2; e2/1 D �1;

.e2�^ e2/2 D .e1 �^ e2/ �^ .e1 �^ e2/ D �e2 �^ .e1 �^ e1/ �^ e1 D Ce2 �^ e2 D �1

.e2�^ e2/ �^ e1 D �e2 �^ .e1 �^ e1/ D e2

.e2�^ e2/ �^ e2 D e1 �^ .e2 �^ e2/ D �e1

and

e3 WD e1 �^ e2by classical notation. Obviously

x D 1xı C e1x1 C e2x2 C e1 �^ e2x3 2 C l.0; 2/

is an element of Clifford algebra C l.0; 2/.There is an algebraic isomorphism between the quaternion algebra H of vectors andthe quaternion algebra of matrices, namely either M.R4�4/ of 4 � 4 real matricesor M.C2�2/ of 2 � 2 complex matrices.


Firstly we define the 4 � 4 real matrix basis E and decompose it into thefour constituents ˙ 0, ˙ 1, ˙ 2, ˙ 3 of 4 � 4 real Pauli matrices which form amultiplicative group of the multiplication table of Hamilton type

E WD

2

664

1 e1 e2 e3

�e1 1 �e3 e2

�e2 e3 1 �e1�e3 �e2 e1 1

3

775 D 1˙ 0 C e1˙ 1 C e2˙ 2 C e3˙ 3 2 R

4�4

˙ 0 D

2

664

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

3

775 ; ˙ 1 WD

2

664

0 1 0 0

�1 0 0 0

0 0 0 �10 0 1 0

3

775 ;

˙ 2 D

2

664

0 0 1 0

0 0 0 1

�1 0 0 0

0 �1 0 0

3

775 ; ˙ 3 WD

2

664

0 0 0 1

0 0 �1 00 1 0 0

�1 0 0 0

3

775 ;

�

�

�

�


˙ 0 ˙ 1 ˙ 2 ˙ 3

˙ 0 ˙ 0 ˙ 1 ˙ 2 ˙ 3

˙ 1 ˙ 1 �˙ 0 ˙ 3 �˙ 2

˙ 2 ˙ 2 �˙ 3 �˙ 0 ˙ 1

˙ 3 ˙ 3 ˙ 2 �˙ 1 �˙ 0

or

˙ 0˙ 0 D ˙ 0;˙ 0˙ i D ˙ i˙ 0 D ˙ i for all i 2 f1; 2; 3g˙ i˙ j D �ıij˙ 0 C �ijk˙ k for all i; j; k 2 f1; 2; 3g

LetA, B, C 2 M (R4�4, Hamilton) constituted by means of

2

664

a1 a2 a3 a4�a2 a1 �a4 a3�a3 a4 a1 �a2�a4 �a3 a2 a1

3

775 DW A

2

664

b1 b2 b3 b4�b2 b1 �b4 b3

�b3 b4 b1 �b2�b4 �b3 b2 b1

3

775 DW B


such that the Cayley-product

AB D

2

664

c1 c2 c3 c4

�c2 c1 �c4 c3�c3 c4 c1 �c2�c4 �c3 c2 c1

3

775 DW C 2M .R4�4; Hamilton/

c1 D a1b1 � a2b2 � a3b3 � a4b4c2 D a1b2 C a2b1 C a3b4 � a4b3c3 D a1b3 � a2b4 C a3b1 C a4b2c4 D a1b4 C a2b3 � a3b2 C a4b1

fulfilling the axioms .G1ı/, .G2ı/, .G3ı/ of a non-Abelian multiplicative group,namely

.G1ı/ .AB/C D A.BC/ .associativity/

.G2ı/ AI D A .identity/

.G3ı/ AA�1 D I .inverse/;

but .G4ı/ does not apply, in particular

detA D a21 C a22 C a23 C a24; detB D b21 C b22 C b23 C b24det .AB/ D .detA/.detB/:

Secondly we define the 2 � 2 complex matrix basis E and decompose it into thefour constituents ˙0, ˙1, ˙2, ˙3 of 2 � 2 complex Pauli matrices which form amultiplicative group of the multiplication table of Hamilton type

E WD"1C ie1 e2 C ie3�e2 C ie3 1 � ie1

#

D 1˙ 0 C e1˙ 1 C e2˙ 2 C e3˙3 2 C2�2

˙ 0 WD�1 0

0 1

; ˙ 1 WD

�i 0

0 �i;

˙ 2 WD�0 1

1 0

; ˙ 3 WD

�0 i

i 0

�

�

�

�



˙0 ˙1 ˙2 ˙3

˙0 ˙0 ˙1 ˙2 ˙3

˙1 ˙1 �˙0 ˙3 �˙2

˙2 ˙2 �˙3 �˙0 ˙1

˙3 ˙3 ˙2 �˙1 �˙0

:

Note thatE 2 C2�2 is “Hermitean”. LetA,B,C 2M .C2�2, Hamilton) constituted

by means of

"a1 C ia2 a3 C ia4�a3 C ia4 a1 � ia2

#

DW A"b1 C ib2 b3 C ib4�b3 C ib4 b1 � ib2

#

DW B

such that the Cayley-product

AB D"c1 C ic2 c3 C ic4�c3 C ic4 c1 � ic2

#

DW C 2M .C2; Hamilton/

det .AB/ D .detA/.detB/ D �a21 C a22 C a23 C a24� �b21 C b22 C b23 C b24

�

D c21 C c22 C c23 C c24c1 C ic2 D .a1 C ia2/.b1 C ib2/C .a3 C ia4/.�b3 C ib4/

D a1b1 � a2b2 � a3b3 � a4b4 C i.a1b2 C a2b1 C a3b4 � a4b3/c3 C ic4 D .a1 C ia2/.b3 C ib4/C .a3 C ia4/.b1 � ib2/

D a1b3 � a2b4 C a3b1 C a4b2 C i.a1b4 C a2b3 � a3b2 C a4b1/

fulfilling the axioms .G1ı/, G2ı/, .G3ı/ of a non-Abelian multiplicative group.The spinor

s WD"1C ie1e2 C ie3

#

D"s1

s2

#

as a vector of length zero relates to the 2 � 2 complex matrix basis by

E D"

s1 s2

�s�2 s

�1

#

: |


A-73 Octanian Algebra as a Non-Associative Algebraas well as a Composition Algebra, Clifford algebrawith Respect to H � H

The octanian algebra O also called “the algebra of octaves” due to Graves (1843)and Cayley (1845) is a composition algebra over R as well as a non-associativealgebra:

x 2 O

x D 1xıC e1x1C e2x2C e3x3C e4x4C e5x5C e6x6C e7x7 subject to

fxı; x1; : : : ; x6; x7g 2 R8

spanO D f1; e1; e2; e3; e4; e5; e6; e7g

1 x;y; z 2 R8

˛.x;y/ DW x C y


2 x;y 2 R8; r; s 2 R

ˇ.r;x/ DW r � x

The axioms .D1C/, .D2C/, .D3/ of additive distributivity apply.3 One way to explicitly describe a multiplicative group with finitely many

elements is to give a table listing the multiplications just representing the map� W O �O �! O.

�

�

�

�


1 e1 e2 e3 e4 e5 e6 e7

1 1 e1 e2 e3 e4 e5 e6 e7

e1 e1 �1 e3 �e2 e5 �e4 �e7 e6

e2 e2 �e3 �1 e1 �e6 e7 �e4 �e5e3 e3 e2 �e1 �1 e7 �e6 e5 �e4e4 e4 �e5 �e6 �e7 �1 �e1 �e2 �e3e5 e5 e4 �e7 e6 �e1 �1 �e3 e2

e6 e6 e7 e4 �e5 �e2 e3 �1 �e1e7 e7 �e6 e5 e4 �e3 �e2 e1 �1


Note that in the multiplication table each entry of a group appears exactly oncein each row and column. The multiplication has to be read from left to right thatis, the entry at the intersection of the row headed by e5 is the product e3 � e5.Such a table is called a Cayley diagram of the multiplicative group.Note the non-associativity of the internal multiplication given by the table, e.g.e2�.e3�e4/ 6D .e2�e3/�e4, namely by means of e3�e4 D e7, e2�.e3�e4/ De2 � e7 D �e5 versus e2 � e3 D e1, .e2 � e3/ � e4 D e1 � e4 D Ce5. Suchan “octonian algebra” O is not a Lie algebra since neither x � x D 0 (L1) nor.x � y/zC .y � z/ � x C .z � x/ � y D 0 (L2) (Jacobi identity) hold. Just bymeans of the multiplication table compute

.e2 � e3/ � e4 C .e3 � e4/ � e2 C .e4 � e2/ � e3 D e5 6D 0:

4 does not apply.


Q.x/ D Q.1xı C e1x1 C � � � C e6x6 C e7x7/D .xı/2 C .x1/2 C � � � C .x6/2 C .x7/2

in order to prove (K1), (K2), (K3). The laborious proofs are left as an exercise.

The proper algebraic interpretation of octonian numbers is in terms of Cliffordalgebra, namely with respect to the eight dimensional set H � H DW H2 whereH is the usual skew field of Hamilton’s quaternions, algebraically isomorphic toC l.0; 2/. Indeed it would have been temptating to base “octonian algebra” O onC l.0; 3/, dimC l.0; 3/ D 23 D 8, but its generic elements

f1; e1; e2; e3; e1 �^ e2; e2 �^ e3; e3 �^ e1; e1 �^ e2 �^ e3g

are not representing the octonian multiplication table e.g.

e1�^ e2 �^ e3/2 D g.e1 �^ e2 �^ e3; e1 �^ e2 �^ e3/

D .e1 �^ e2 �^ e3/ �^ .e1 �^ e2 �^ e3/

D �e1 �^ e2 �^ e1 �^ e3 �^ e2 �^ e3 D e1 �^ e2 �^ e1 �^ e2 �^ .e3 �^ e3/

D �e1 �^ e2 �^ e1 �^ e2 D �e1 �^ .e2 �^ e2/ �^ e1 D �.e2 �^ e1/ D C1

In contrast, let us introduce the pair

x WD .a;b/ 2 fX j a 2 H;b 2 Hg


1 x 2 H2; x

0 2 H2; x

00 2 H2

˛.x;x0 DW x C x0

/

x C x0 D .aC a0

;bC b0

/:


2 x:x0 2 H

2; r; r0 2 R

ˇ.r;x/ DW r � xr � x D .r � a; r � b/:


3 x 2 H �H D H2; x

0 2 H �H D H2

�.x;x0

/ DW x � x0

x � x0 DW .aa0 � b0

b;b0

aC ba0

/

a, b denote the conjugate of a 2 H, b 2 H, respectively. If .a;b/, .a0

;b0

/ arerepresented by0

@1˛ı C3X

iD1ei ˛

i ; 1ˇı C3X

jD1ej ˇ

j

1

A ;

0

@1˛0ı C

3X

i 0D1ei 0˛

0i ; 1ˇ0ı C

3X

jD1ej 0ˇ

0j

1

A

respectively, where

spanH D f1; e1; e2; e1 �^ e2 D e3g or

spanH D f10

; e10 ; e20 ; e10

�^ e20 D e30g D f10

; e5; e6; e7g

the “octonian product” x � x0

results in�

�

�

x � x0 D�1

�˛ı˛0ı � ˇıˇ0ı �

3P

kD1.˛k˛‘k � ˇkˇ0k/

C3P

i;j;kD1ekŒ˛

ıˇk C ˛kˇı C �kij .˛i˛0j � ˇjˇ0i /�;

� 1�˛ıˇ0ı � ˛0ıˇı �

3P

kD1.˛kˇ‘k � ˛0kˇk/

C3P

i;j;kD1ekŒ˛

ıˇ0k C ˛kˇ0ı C �kij .˛j ˇ0i � ˛0j ˇi /�

!

:


the axioms .D1 � C/, .D2 � �/ of distributivity apply. The pair .1; 0/ is theneutral element.

4 Begin with the choice of

.1st/ the transpose x WD a;�b/ of x 2 H2

.2nd/ x � x D Q.x/ or x � x D .aaC bb; 0/

.3rd/ x�1 D x

Q.x/if x 6D 0

in order to end up with

x � x�1 D 1:

A historical perspective of octonian numbers is given by Van Der Waerden(1950). Reference is made to Graves (1848) and A. Caley: Collected Mathe-matical Papers, vol. 1, p. 127 and vol. 11, pp. 368–371. |

The exceptional role of the examples on complex, quaternion and octonian algebraillustrating Clifford algebra C l.0; 1/, C l.0; 2/ as well as Clifford algebra withrespect to H

2 is established by the following theorems:�

�

�

Theorem (“Hurwitz’ theorem of composition algebras”):

A complete list of composition algebras over R consists of(a) the real numbers R,

(b) the complex numbers C,

(c) the quaternions H,

(d) the octonians O .

�

�

�

Theorem (“Frobenius’ theorem of division algebras”):

The only finite-dimensional division algebra over R are(˛) the real numbers R,

(ˇ) the complex numbers C,

(� ) the quaternions H .

Historical Aside

For details consult the historical texts Hurwitz (1898), Frobenius (1878) as well asHazlett (1917). A more recent reference is Jacobson (1974), pp. 425 and 430.


A-74 Clifford Algebra

We already took advantage of the notion of Clifford algebra when we referredto complex algebra, quaternion algebra, and octonian algebra. Here we finallyconfront you with the definition of “orthogonal clifford algebra C l.p; q/”. But onour way to Clifford algebra we have to generalize at first the notion of a basis, inparticular its bilinear form.�

�

�

Theorem (bilinear form):

Suppose that the bracket h�j�i or g.�; �/ W X�X� �! R is a bilinear form ona finite dimensional linear space X, e. g. a vector space, over the field R ofreal numbers, in addition X

� its dual space such that n D dimX� D dimX.

There exists a basis fe1; : : : ; eng such that

(a) hei j ej i D 0 or g.ei ; ej / D 0 for i 6D j

(b)

2

664

hei1 j ei1i D C1 or g.ei1 ; e i1 / D C1 for 1 � i1 < p ;hei2 j ei2i D �1 or g.e i2 ; ei2 / D �1 for p C 1 � i2 � p C q D r ;hei3 j ei3i D 0 or g.e i3 ; ei3 / D 0 for r C 1 � i3 � n

holds.

The numbers r and p are determined exclusively by the bilinear form. r is called therank, r � p D q is called the index and the ordered pair .p; q/ the signature. Thetheorem assures that any two spaces of the same dimension with bilinear forms ofthe same signature are isometrically isomorphic. A scalar product (“inner product”)in this context is a nondegenerate bilinear form, i.e., a form with rank equal tothe dimension of X. When dealing with low dimensional spaces as we do, we willoften indicate the signature with a series of plus and minus signs and zeroes whereappropriate. For example, the signature of R41 may be written .C C C�/ insteadof .3; 1/. If the bilinear form is nondegenerate, a basis with the properties listed incorresponding Theorem is called an orthonormal basis (“unimodular”) for X withrespect to the bilinear form.

Definition (orthogonal Clifford algebra C l.p; q/):

The orthogonal Clifford algebraC l.p; q/ is the algebra of polynomials generatedby the direct sum of the space of multilinear functions

˚nmD0�^m

.X�/

on a linear space X, respectively its dual X� over the field of real numbers Rnp ofdimension


dim˚nmD0�^m

.X�/ D 2n

and signature .p; q/, namely�

�

�

�

1f0 CnDdimX

�

P

i1D1ei1fi1 C

nDdimX�

P

i1;i2D1ei1

�^ ei2fi1i2

CnDdimX

�

P

i1;i2;i3D1ei1

�^ ei2 �^ ei3fi1i2i3 C � � � CnDdimX

�

P

i1;:::;inD1ei1

�^ � � � �^ einfi1��in

subject to the Clifford product, also called the Clifford dualizer,

"

#

$

%

.i/ ei�^ ej D �ej �^ ei for i 6D j

.ii/

2

664

ei1�^ ei1 D g.ei1 ; ei1/ D C1 for 1 � i1 � p ;

ei2�^ ei2 D g.ei2 ; ei2/ D �1 for p C 1 � i2 � p C q D r ;

ei3�^ ei3 D g.ei3 ; ei3/ D 0 for r C 1 � i3 � n ;

or

e1�^ ej C ej �^ ei

D 2g.ei ; ej /ıij 1 subject to

2

664

g.e i1 ; ei1 / D C1 for 1 � i1 � pg.e i2 ; ei2 / D �1 for pC 1� i2�pC qD rg.e i3 ; ei3 / D 0 for r C 1 � i3 � n

1 being the neutral element. If ek�^ ek D 0 or g.ek; ek/ D 0 holds uniformly

the orthogonal Clifford algebra C l.p; q/ reduces to the polynomial algebra ofantisym-metric multilinear functions

˚nmD0Am D ˚nmD0�m.X�/ D �.X�/

dim ˚nmD0 Am D dim ˚nmD0 �m.X�/ D dim .X�/ D 2n

represented by

"

#

$

%

1f0 C 1

1Š

nDdimX�

X

i1D1ei1fi1 C

1

2Š

nDdimX�

X

i1;i2D1ei1 ^ ei2fi1i2

C 13Š

nDdimX�

X

i1;i2;i3D1ei1 ^ ei2 ^ ei3fi1i2i3 C � � � C

1

nŠ

nDdimX�

X

i1;:::;inD1ei1 ^ � � � ^ e infi1��in :


�

�

�

Example: Clifford product: x � y, x � y 2 X; signX D .3; 0/Let x � y 2 X be a real three-dimensional vector space X of signature (3,0).then x � y (read: “x Clifford y”) with respect to a set of the bases fe1; e2; e3gaccounts for

x�^ y D

3X

iD1

3X

jD1ei

�^ ej xiyj D3X

iD1

3X

jD1xiyj ei

�^ ej

x�^ y D .e1x1 C e2x2 C e3x3/

�^.e1y1 C e2y2 C e3y3/

x�^ y D 1.x1y1 C x2y2 C x3y3/C e1 ^ e2.x1y2 � x2y1/

C e2 ^ e3.x2y3 � x3y2/C e3 ^ e1.x3y1 � x1y3/

Indeed x�^ y as a Clifford number is decomposed into a scalar part and an

antisymmetric tensor part with respect to the bilinear basis fe1 ^ e2; e2 ^ e3;e3 ^ e1g.

Tensor algebra or the algebra of multilinear functions, namely

1f0 CnDdimX

�

X

i1D1ei1fi1 C

nDdimX�

X

i1;i2D1ei ˝ ei2fi1i2

CnDdimX

�

X

i1;i2;i3D1ei1 ˝ ei2 ˝ ei3fi1i2i3 C � � � C

nDdimX�

X

i1;:::;inD1ei1 ˝ � � � ˝ einfi1��in

2 ˚nmD0 ˝ .X�/ D ˝.X/ D T C ˚ T �

subject to

"T C D ˚hD0 ˝2h .X/ .“even”/

T � D ˚kD0 ˝2kC1 .X/ .“odd”/

is the sum of two spaces, T C and T �, respectively, in particular

C lC 3 1f0C 1

2Š

nX

i1;i2D1ei1

�^ ei2fi1i2 C1

4Š

nX

i1;i2;i3;i4

ei1�^ ei2 �^ e i3 �^ ei4fi1i2i3i4 C � � � ;

C l� 3 1

1Š

nX

i1D1ei1fi1 C

1

3Š

nX

i1;i2;i3D1ei1

�^ ei2 �^ ei3fi1i2i3 C � � � :


Obviously C lC as well as C l� are subalgebras of C l . Let the Clifford numbers zbe divided into zC 2 C lC and z� 2 C l�, then the properties

�

�

�

�zC �^ zC 2 C lC; z� �^ z� 2 C lC; zC �^ z� 2 C l�

prove that C l.p; q/ is graded over the cydric group Z2 D f0; 1g. |

Appendix BSampling Distributions and Their Use:Confidence Intervals and Confidence Regions

We begin with sampling distributions by two sections on the transformation ofrandom variables. A first confidence interval for Gauss-Laplace normal distributionobservations �1�2 known is constructed, including its famous Three Sigma Rule.The second confidence interval of Gauss-Laplace i.i.d. observations is derived forthe sample mean b� BLUE of �1 when the variance is known. We take advantageof the forward-backward Helmert transformation leading to Helmert chi-square 2 probability distribution (Helmert, 1875). In contrast, the sampling distributionsfrom Gauss-Laplace i.i.d. normal distributions within the third confidence intervalis analyzed for the mean, but an unknown variance. Our basis is student’st-distribution on the random variable

pn.b� � �/=b� where the sample mean b� –

is BLUE of � whereas the sample confidence for the “true” mean �, variance�2 unknown which is based on student’s probability distribution. In detail, wederive the related Uncertainty Principle generated by the Magic Triangle of (a)length of the confidence interval, (b) coefficient of negative confidence called theuncertainty number and (c) the number of observations. For more details we referto Lemma B.12 of W.S. Gosset. The sampling from the Gauss-Laplace normaldistribution, namely the fourth confidence interval for the variance is reviewed: Weintroduce the missing confidence interval for the “true” variance �2: Table B.18as a flow chart summarizes the various steps in computing a confidence intervalfor the variance. This time the related Uncertainty Principle is built on (a) thecoefficient of complementary confidence ˛, also called uncertainty number (b) thenumber of observations, n, and (c) the length ı�2.n; ˛/ of the confidence intervalthe “true” variance �2. The most detailed sampling theory from multidimensionalGauss-Laplace normal distribution, namely for the confidence region for the fixedparameters in the linear Gauss–Markov, is stated by an extensive Example B.19.We need the Principle Component Analysis PCA, also called Singular ValueDecomposition (SVD) or Eigenvalue Analysis (EIGEN). Theorem B.17 summarizesthe marginal distributions special linear Gauss–Markov model with datum defect.In our appendix we concentrate on multidimensional variance analysis, namelysampling form the multivariate Gauss-Laplace normal distribution. In short


637


638 B Sampling Distributions and Their Use: Confidence Intervals and Confidence Regions

introduction we review the distribution of sample mean and variance-covariance.We define the k-dimensional noncentral WHISHART distribution by (B377) and thecharacteristic function of a vector-valued random variate of type Gauss-Laplacedistribution. Our highlight is the distribution of xN and SN of type WHISHART.Another highlight is the distribution related to the correlation coefficients originatedby R.A. Fisher (1915), programmed by F.N. David (1954).

B-1 A First Vehicle: Transformation of Random Variables

If the probability density function (pdf ) of a random vector yD Œy1; : : : ; yn�0 isknown, but we want to derive the probability density function (pdf ) of a randomvector x D Œx1; : : : ; xn�

0 (pdf ) which is generated by an injective mapping x =g(y)or xi D gi Œy1; : : : ; yn� for all i 2 f1; : : : ; ng we need the results of Lemma B.1.

Lemma B.1. (transformation of pdf):

Let the random vector y WD Œy1; : : : ; yn�0 be transformed into the random vector

x D Œx1; : : : ; xn�0 by an injective mapping xDg.y/ or xi Dgi Œy1; : : : ; yn� for all

i 2 f1; : : : ; ng which is of continuity class C1 (first derivatives are continuous).

Let the Jacobi matrix Jx W D .@gi =@yi / be regular (det Jx ¤ 0), then the inversetransformation yDg � 1.x/ or yi Dg�1

i Œx1; : : : ; xn� is unique. Let fx.x1; : : : ; xn/be the unknown pdf, but fy.y1; : : : ; yn/ the given pdf, then

fx.x1; : : : ; xn/ D f .g�11 .x1; : : : ; xn/; : : : ; g

�11 .x1; : : : ; xn//

ˇdet Jy

ˇ

with respect to the Jacobi matrix

Jy WD�@yi

@xj

D�@g�1

i

@xj

for all i; j 2 f1; : : : ; ng holds. Before we sketch the proof we shall present twoexamples in order to make you more familiar with the notation.

Example B.1. (“counter example”):

The vector-valued random variable .y1; y2/ is transformed into the vector-valuedrandom variable .x1; x2/ by means of

x1 D y1 C y2; x2 D y21 C y22

Jx W D�@x

@y0

D�@x1=@y1 @x1=@y2

@x2=@y1 @x2=@y2

D�1 1

2y1 2y2

B-1 A First Vehicle: Transformation of Random Variables 639

x21 D y21 C 2y1y2 C y22 ; x2 C 2y1y2 D y21 C 2y1y2 C y22x21 D x2 C 2y1y2; y2 D .x21 � x2/=.2y1/

x1 D y1 C y2 D y1 C1

2

x21 � x2y1

; x1y1 D y21 C1

2.x21 � x2/

y21 � x1y1 C1

2.x21 � x2/ D 0

y1 D �1

2x1 ˙

sx214� 12.x21 � x2/

y2 D1

2x1

sx214� 12.x21 � x2/:

At first we have computed the Jacobi matrix Jx , secondly we aimed at an inversionof the direct transformation .y1; y2/ 7! .x1; x2/. As the detailed inversion stepproves, namely the solution of a quadratic equation, the mapping xDg.y/ is notinjective.

Example B.2.

Suppose .x1; x2/ is a random variable having pdf

fx.x1; x2/ D�

exp .�x1 � x2/; x1 0; x2 00 ; otherwise:

We require to find the pdf of the random variable

.x1 C x2; x2=x1/:

The transformation

y1 D x1 C x2; y2 D x2

x1

has the inverse

x1 D y1

1C y2 ; x2 Dy1y2

1C y2 :

The transformation provides a one-to-one mapping between points in the firstquadrant of the .x1; x2/ – plane P2x and in the first quadrant of the .y1; y2/ – planeP2y . The absolute value of the Jacobian of the transformation for all points in the

first quadrant is


ˇˇˇˇ@.x1; x2/

@.y1; y2/

ˇˇˇˇ D

ˇˇˇˇ

@x1@y1

@x1@y2

@x2@y1

@x2@y2

ˇˇˇˇDˇˇˇˇ.1C y2/�1 �y1.1C y2/�2y2.1C y2/�1 y1.1C y2/�2Œ.1C y2/ � y2�

ˇˇˇˇ

D y1.1C y2/�3 C y1y2.1C y2/�3 D y1

.1C y2/2 :

Hence we have found for the pdf of .y1; y2/

fy.y1; y2/ D"

exp.�y1/ y1.1Cy2/2 ; y1 > 0; y2 > 0

0 ; otherwise:

Incidentally it should be noted that y1 and y2 are independent random variables,namely

fy.y1; y2/ D f1.y1/f .y2/ D y1 exp.�y1/.1C y2/�2: |

Proof. The probability that the random variables y1; : : : ; yn take on values in theregion˝y is given by

Z� � �

˝y

Zfy.y1; : : : ; yn/dy1 � � �dyn:

If the random variables of this integral are transformed by the functionxi Dgi .y1; : : : ; yn/ for all i 2 f1; : : : ; ng which map the region ˝y onto theregions˝x, we receive

Z� � �

˝y

Zfy.y1; : : : ; yn/dy1 � � �dyn

DZ� � �

˝x

Zfy.g

�11 .x1; : : : ; xn/; : : : ; g

�1n .x1; : : : ; xn//

ˇˇdet Jy

ˇˇ dx1 � � �dxn

from the standard theory of transformation of hypervolume elements, namely

dy1 � � �dyn D j det Jy jdx1 � � �dxnor

�.dy1 ^ � � � ^ dyn/ D j det Jy j � .dx1 ^ � � � ^ dxn/:Here we have taken advantage of the oriented hypervolume element dy1 ^ � � � ^ dyn(Grassmann product, skew product, wedge product) and the Hodge star operator *applied to the n – differential form dy1 ^ � � � ^ dyn 2 �n (the exterior algebra�n).

B-1 A First Vehicle: Transformation of Random Variables 641

The star * : �p ! �n�p in Rn maps a p – differential form onto a .n � p/ –

differential form, in general. Here p D n, n � p D 0 applies. Finally we define

fx.x1; : : : ; xn/ WD f .g�11 .x1; : : : ; xn/; : : : ; g

�1n .x1; : : : ; xn//j det Jyj

as a function which is certainly non-negative and integrated over˝x to one. |In applying the transformation theorems of pdf we meet quite often the problem thatthe function xi D gi .y1; : : : ; yn/ for all i 2 f1; : : : ; ng is given but not the inversefunction yi Dg�1

i .x1; : : : ; xn/ for all i 2 f1; : : : ; ng. Then the following results arehelpful.

Corollary B.1. (Jacobian):

If the inverse Jacobian j det Jx j D j det.@gi =@yj /j is given, we are able to compute

j det Jyj D j det@g�1

i .x1; : : : ; xn/

@xjj D j det Jj�1 D j det

@gi .y1; : : : ; yn/

@yjj�1:

Example B.3. (Jacobian):

Let us continue Example B.2. The inverse map

y D�g�11 .y1; y2/

g�12 .y1; y2/

D�x1 C x2

x2= x1

) @y

@x0 D�

1 1

�x2= x21 1= x1

jy D jJyj Dˇˇˇ @y@x0

ˇˇˇ D 1

x1C x2

x21D x1 C x2

x21

jx D jJxj D j�1y D jJyj�1 D

ˇˇˇˇ@x

@y0

ˇˇˇˇ D

x21x1 C x2 D

x21y1

allows us to compute the Jacobian Jx from Jy . The direct map

x D�g1.y1; y2/

g2.y1; y2/

D�x1

x2

D�y1=.1C y2/y1y2=.1C y2/

leads us to the final version of the Jacobian.

jx D jJxj D y1

.1C y2/2 :

For the special case that the Jacobi matrix is given in a partitioned form, the resultsof Corollary B.3 are useful.


Corollary B.2. (Jacobian):

If the Jacobi matrix Jx is given in the partitioned form

jJxj D�@gi

@yj

�D�

UV

;

then

det Jx Dqj det JxJ0

x j Dp

det.UU0/ detŒVV0 � .VU0/.UU0/�1UV0�

if det.UU0/ ¤ 0

det Jx Dqj det JxJ0

x j Dp

det.VV0/ detŒUU0 � UV0.VV0/�1VU0�;

if det.VV0/ ¤ 0j det Jyj D j det Jxj�1:

Proof. The Proof is based upon the determinantal relations of a partitioned matrixof type

det

�A UV D

D det A det.D � VA�1U/ if det A ¤ 0

det

�A UV D

D det D det.A � UD�1U/ if det D ¤ 0

det

�A UV D

D D det A �V.adjA/U;

which have been introduced by Frobenius (1908) and Schur (1917).

B-2 A Second Vehicle: Transformation of Random Variables

Previously we analyzed the transformation of the pdf under an injective map ofrandom variables y 7! g.y/Dx. Here we study the transformation of polarcoordinates Œ�1; �2; : : : ; �n�1; r� 2 Y as parameters of an Euclidian observationspace to Cartesian coordinates Œy1; : : : ; yn� 2 Y. In addition we introduce thehypervolume element of a sphere S

n�1 � Y; dim YDn. First, we give threeexamples. Second, we summarize the general results in Lemma B.4.

B-2 A Second Vehicle: Transformation of Random Variables 643

Example B.4. (polar coordinates: “2d”):

Table B.1 collects characteristic elements of the transformation of polar coordinates.�1; r/ of type “longitude, radius” to Cartesian coordinates .y1; y2/; their domainand range, the planar elements dy1; dy2 as well as the circle S1 embedded into E

2 WDfR2; ıklg, equipped with the canonical metric I2D Œıkl � and its total measure ofarc !1.

Table B.1

Cartesian and polar coordinates of a two-dimensional observation space, totalmeasure of the arc of the circle

.�1; r/ 2 Œ0; 2� � �0;1Œ; .y1; y2/ 2 R2

dy1dy2 D rdrd�1S1 WD fy 2 R

2jy21 C y22 D 1g

!1 D2Z

0

d�1 D 2:


Table B.2 is a collectors’ item for characteristic elements of the transformationof polar coordinates .�1; �2; r/ of type “longitude, latitude, radius” to Cartesiancoordinates .y1; y2; y3/; their domain and range, the volume element dy1; dy2; dy3as well as of the sphere S

2 embedded into E3 WD fR3; ıklg equipped with the

canonical metric I3 D Œıkl � and its total measure of surface !2.

Table B.2

Cartesian and polar coordinates of a three-dimensional observation space, totalmeasure of the surface of the circle

y1 D r cos�2 cos�1; y2 D r cos�2 sin �1; y3 D r sin�2

.�1; �2; r/ 2 Œ0; 2� � � � 2;

2Œ� �0; rŒ; .y1; y2/ 2 R

2

.y1; y2; y3/;2 R3

dy1dy2dy3 D r2dr cos�2d�1d�2

S2 WD fy 2 R

�jy21 C y22 C y23 D 1g

!2 D2Z

0

d�1

C=2Z

�=2d�2 cos�2 D 4:



Table B.3 is a collection of characteristic elements of the transformation of polarcoordinates .�1; �2; �3; r/ to Cartesian coordinates .y1; y2; y3; y4/; their domain andrange, the hypervolume element dy1; dy2; dy3; dy4 as well as of the 3 - sphere S

3

embedded into E4 WD fR4; ıklg equipped with the canonical metric I4D Œıkl � and

its total measure of hypersurface.

Table B.3

Cartesian and polar coordinates of a four-dimensional observation space totalmeasure of the hypersurface of the 3-sphere

y1 D r cos �3 cos�2 cos �1; y2 D r cos �3 cos �2 sin�1;

y3 D r cos �3 sin�2; y4 D r sin�3

.�1; �2; �3; r/ 2 Œ0; 2� � ��

2;

2Œ� ��

2;

2Œ� �0; 2Œ

dy1dy2dy3dy4 D r3 cos2 �3 cos �2drd�3d�2d�1

Jy WD @.y1; y2; y3; y4/

@.�1; �2; �3; r/

D

2

6664

�r cos �3 cos �2 sin �1 �r cos�3 sin�2 cos �1 �r sin �3 cos �2 cos �1 cos �3 cos �2 cos�1Cr cos �3 cos �2 cos �1 �r cos�3 sin�2 sin�1 �r sin�3 cos �2 sin�1 cos �3 cos �2 sin�1

0 Cr cos �3 cos �2 �r sin�3 sin�2 cos �3 cos �20 0 r cos�3 sin�3

3

7775

j det Jy j D r3 cos2 �3 cos�2

S3 WD fy 2 R

4jy21 C y22 C y23 C y24 D 1g!3 D 22:

Lemma B.2. (polar coordinates, hypervolume element, hypersurface element):

Let

2

66666666666664

y1y2

y3y4� � �yn�3yn�2yn�1yn

3

77777777777775

D r

2

66666666666664

cos�n�1 � cos�n�2 � cos�n�3 � � � � � cos�2 � cos �1cos�n�1 � cos�n�2 � cos�n�3 � � � � � cos�2 � sin �1cos�n�1 � cos�n�2 � cos�n�3 � � � � � sin �2cos�n�1 � cos�n�2 � � � � � cos�3� � �cos�n�1 � cos�n�2 � sin �n�3cos�n�1 � cos�n�2cos�n�1 � sin �n�2sin �n�1

3

77777777777775


be a transformation of polar coordinates .�1; �2; : : :,�n�2; �n�1; r/ to Cartesiancoordinates .y1; y2; : : :,yn�1; yn/, their domain and range given by

.�1; �2; : : : ; �n�2; �n�1; r/ 2 Œ0; 2� � � � 2;C

2Œ � � � � � � �

2;

C2Œ � � �

2;C

2Œ � �0;1Œ ;

then the local hypervolume element

dy1 � � �dynD rn�1dr cosn�2 �n�1 cosn�3 �n�2 : : : cos2 �3 cos�2d�n�1d�n�2 : : : d�3d�2d�1;

as well as the global hypersurface element

!n�1 D 2 � .n�1/=2

� .n�12/WD

C=2Z

�=2cos�n�2

n�1d�n�1 � � �C=2Z

�=2cos�2d�2

2Z

0

d�1;

where �.X/ is the gamma function. Before we care for the proof, let us defineEuler’s gamma function.

Definition B.1. (Euler’s gamma function):

� .x/ D1Z

0

e�t tx�1dt .x > 0/

is Euler’s gamma function which enjoys the recurrence relation

� .x C 1/ D x� .x/

subject to

� .1/ D 1 or �

�1

2

�D p

� .2/ D 1 �

�3

2

�D 1

2�

�1

2

�D 1

2

p

� � � � � �� .nC 1/ D nŠ �

�p

q

�D p � q

q�

�p � qq

�

if n is integer, n 2 ZC if

p

qis a rational number, p=q 2 Q

C.


Example B.7. (Euler’s gamma function):

(a) � .1/ D 1 .i/ �

�1

2

�D p

(b) � .2/ D 1 .ii/ �

�3

2

�D 1

2�

�1

2

�D 1

2

p

(c) � .3/ D 1 � 2 D 2 .iii/ �

�5

2

�D 3

2�

�3

2

�D 3

4

p

(d) � .4/ D 1 � 2 � 3 D 6 .iv/ �

�7

2

�D 5

2�

�5

2

�D 15

8

p:

Proof. Our proof of Lemma B.4 will be based upon computing the image of thetangent space TyS

n�1 � En of the hypersphere S

n�1 � En. Let us embed the

hypersphere Sn�1 parameterized by .�1; �2; : : : ; �n�2; �n�1/ in E

n parameterizedby .y1; : : : ; yn/, namely y 2 E

n,

y D e1r cos�n�1 cos�n�2 � � � cos�2 cos�1

C e2r cos�n�1 cos�n�2 � � � cos�2 sin �1 C � � �C en�1r cos�n�1 sin�n�2

C enr sin �n�1:

Note that �1 is a parameter of type longitude, 0 � �1 � 2 , while �2; : : : ; �n�1are parameters of type latitude, �=2 < �2 < C=2; : : : ;�=2 < �n�1 < C=2(open intervals). The images of the tangent vectors which span the local tangentspace are given in the orthonormal n- leg fe1; e2; : : : ; en�1; enj0g by

g1 WD D�1y D � e1r cos�n�1 cos�n�2 � � � cos�2 sin �1

C e2r cos�n�1 cos�n�2 � � � cos�2 cos�1

g2 WD D�2y D � e1r cos�n�1 cos�n�2 � � � sin �2 cos�1

� e2r cos�n�1 cos�n�2 � � � sin �2 sin �1

C e3r cos�n�1 cos�n�2 � � � cos�2

: : :

gn�1 WD D�n�1y D � e1r sin �n�1 cos�n�2 � � � cos�2 cos�1 � � � �� en�1r sin �n�1 sin �n�2

C enr cos�n�1


gn WD D�ny D e1r cos�n�1 cos�n�2 � � � cos�2 cos�1

C e2r cos�n�1 cos�n�2 � � � sin �2 sin �1 C � � �C en�1r cos�n�1 cos�n�2

C enr sin �n�1 D y=r:

fg1; : : : ; gn�1g span the image of the tangent space in En. gn is the hypersphere

normal vector, kgnk D 1. From the inner products hgi jgj i D gij , i; j 2 f1; : : : ; ng,we derive the Gauss matrix of the metric G WD Œgij �.

hg1jg1i D r2 cos2 �n�1 cos2 �n�2 � � � cos2 �3 cos2 �2

hg2jg2i D r2 cos2 �n�1 cos2 �n�2 � � � cos2 �3

� � �hgn�1jgn�1i D r2;hgnjgni D 1:

The off-diagonal elements of the Gauss matrix of the metric are zero. Accordingly

pdet Gn D

pdet Gn�1 D rn�1.cos�n�1/n�2.cos�n�2/n�3 � � � .cos�3/2 cos�2:

The square rootp

det Gn,p

det Gn�1 elegantly represents the Jacobian determinant

Jy WD @.y1; y2; : : : ; yn/

@.�1; �2; : : : ; �n�1; r/Dp

det Gn:

Accordingly we have found the local hypervolume elementp

det Gn dr d�n�1d�n�2 � � �d�3 d�2 d�1. For the global hypersurface element !n�1, we integrate

2Z

0

d�1 D 2

C=2Z

�=2cos�2d�2 D Œsin �2�

C=2�=2 D 2

C=2Z

�=2cos2 �3d�3 D 1

2Œcos�3 sin�3 � �3�C=2�=2 D �=2


C=2Z

�=2cos3 �4d�4 D 1

3Œcos2 �4 sin �4 � 2 sin�4�

C=2�=2 D �

4

3

: : :

C=2Z

�=2.cos�n�1/n�2d�n�1 D 1

n � 2Œ.cos�n�1/n�3�C=2�=2

C 1

n � 3

C=2Z

�=2.cos�n�1/n�4d�n�1

recursively. As soon as we substitute the gamma function, we arrive at !n�1.

B-3 A First Confidence Interval of Gauss–Laplace NormallyDistributed Observations �, � 2 Known, the ThreeSigma Rule

The first confidence interval of Gauss-Laplace normally distributed observationsconstrained to .�; �2/ known, will be computed as an introductory example. Anapplication is the Three Sigma Rule.In the empirical sciences, estimates of certain quantities derived from observationsare often given in the form of the estimate plus or minus a certain amount. Forinstance, the distance between a benchmark on the Earth’s surface and a satelliteorbiting the Earth may be estimated to be

.20; 101; 104:132˙ 0:023/m

with the idea that the first factor is very unlikely to be outside the range

20; 101; 104:155m to 20; 101; 104:109m :

A cost accountant for a publishing company in trying to allow for all factors whichenter into the cost of producing a certain book,

actual production costs, proportion of plant overhead, proportion of executivesalaries,

may estimate the cost to be 21˙ 1; 1 Euro per volume with the implication that thecorrect cost very probably lies between 19.9 and 22.1 Euro per volume. The Bureauof Labor Statistics may estimate the number of unemployed in a certain area to be2:4˙ 0:3 million at a given time though intuitively it should be between 2.1 and 2.7million. What we are saying is that in practice we are quite accustomed to seeingestimates in the form of intervals. In order to give precision to these ideas we shall

B-3 A First Confidence Interval of Gauss–Laplace Normally Distributed Observations 649

consider a particular example. Suppose that a random sample x 2 fR; pdf g is takenfrom a Gauss-Laplace normal distribution with known mean � and known variance�2. We ask the key question.

What is the probability � of the random interval .� � c�; � C c�/ to coverthe mean � as a quantile c of the standard deviation �?

To put this question into a mathematical form we write the probabilistic two-sidedinterval identity.

P.x1 < X < x2/ D P.� � c� < X < �C c�/ D �;x2D�Cc�Z

x1D��c�

1

�p2

exp

�� 1

2�2.x � �/2

�dx D �

with a left boundary l Dx1 and a right boundary r Dx2. The length of the intervalis x2 � x1 D r � l . The center of the interval is .x1 C x2/=2 or �. Here we havetaken advantage of the Gauss-Laplace pdf in generating the cumulative probability

P.x1 < X < x2/ D F.x2/ � F.x1/

F.x2/ � F.x1/ D F.�C c�/ � F.�C c�/:

Typical values for the confidence coefficient � are � D 0:95 (� D 95% or 1�� D 5%negative confidence), � D 0:99 (� D 1% or 1 � � D 1% negative confidence) or� D 0:999 (� D 999%O or 1 � � D 1%O negative confidence).Consult Fig. B.1 for a geometric interpretation. The confidence coefficient � is ameasure of the probability mass between x1 D � � c� and x2 D � C c� . For agiven confidence coefficient �

x2Z

x1

f .xj�; �2/dx D �

establishes an integral equation. To make this point of view to be better under-stoodlet us transform the integral equations to its standard form.

x 7! z D 1�.x � �/, x D �C �z

x2D�Cc�Z

x1D��c�

1

�p2

exp

�� 1

2�2.x � �/2

�dx D

CcZ

�c

1p2

exp

��12

z2�d z D �

x2Z

x1

f .xj�; �2/dx DCcZ

�cf .zj0; 1/d z D �:


μ–3σ μ–2σ μ–σ μ μ+σ μ+2σ μ+3σx

f(x)

Fig. B.1 Probability mass in a two-sided confidence interval x1 < X < x2 or � � c� < X <

�C c� , three cases: (a) c D 1, (b) c D 2 and (c) c D 3

The special Helmert transformation maps x to z, now being standard Gauss-Laplace normal:��1 is the dilatation factor, also called scale variation, but � thetranslation parameter.The Gauss-Laplace pdf is symmetric, namely f .�x/ D f .Cx/ or f .�z/ Df .Cz/. Accordingly we can write the integral identity

x2Z

x1

f .xj�; �2/dx D 2x2Z

0

f .xj�; �2/dx D �

,CcZ

�cf .zj0; 1/d z D 2

cZ

0

f .zj0; 1/d z D �:

The classification of integral equations tells us that

�.z/ D 2zZ

0

f .z�/d z�

is a linear Volterra integral equation the first kind.


In case of a Gauss-Laplace standard normal pdf, such an integral equation is solvedby a table. In a forward computation

F.z/ WDzZ

�1f .z�j0; 1/d z� or ˚.z/ WD

zZ

�1

1

2exp

��12

z�2�d z�

are tabulated in a regular grid. For a given value F.z1/ or F.z2/, z1 or z2are determined by interpolation. C.F. Gauss did not use such a procedure. Hetook advantage of the Gauss inequality which has been reviewed in this contextby Pukelsheim (1994). There also the Vysochanskii-Petunin inequality has beendiscussed. We follow here a two-step procedure. First, we divide the domainz 2 Œ0; 1� into two intervals z 2 Œ0; 1� and z 2 Œ1;1�. In the first interval f .z/ isisotonic, differentiable and convex, f 00.z/ D f .z/.z2 � 1/ < 0; while in the secondinterval isotonic, differentiable and concave, f 00.z/ D f .z/.z2�1/ > 0. z D 1 is thepoint of inflection. Second, we setup Taylor series of f .z/ in the interval z 2 Œ0; 1�at the point z D 0, while in the interval z 2 Œ1;1� at the point z D 1 and z 2 Œ2;1�at the point z D 2.Three examples of such a forward solution of the characteristic linear Volterraintegral equation of the first kind will follow. They establish:

The One Sigma Rule

The Two Sigma Rule The Three Sigma Rule

Box B.1. (Operational calculus applied to the Gauss-Laplace probabilitydistribution):

“generating differential equations”

f 00.z/C 2f 0.z/C f .z/ D 0

subject to

C1Z

�1f .z/d z D 1


“recursive differentiation”

f .z/ D 1p2

exp

��12

z2�

f 0.z/ D �zf .z/ DW g.z/f 00.z/ D g0.z/ D �f .z/ � zg.z/ D .z2 � 1/f .z/f 000.z/ D 2zf .z/C .z2 � 1/g.z/ D .�z3 C 3z/f .z/

f .4/.z/ D .�3z2 C 3/f .z/C .�z3 C 3z/g.z/ D .z4 � 6z2 C 3/f .z/f .5/.z/ D .4z3 � 12z/f .z/C .z4 � 6z2 C 3/g.z/ D .�z5 C 10z3 � 15z/f .z/

f .6/.z/ D .�5z4 C 30z2 � 15/f .z/C .�z5 C 10z3 � 15z/g.z/

D .z6 � 15z4 C 45z2 � 15/f .z/f .7/.z/ D .6z5 � 60z3 C 90z/f .z/C .z6 � 15z4 C 45z2 � 15/g.z/

D .�z7 C 21z5 � 105z3 C 105z/f .z/

f .8/.z/ D .�7z6 C 105z4 � 315z2 C 105/f .z/C.�z7 C 21z5 � 105z3 C 105z/f .z/

D .z8 � 28z6 C 210z4 � 420z2 C 105/f .z/f .9/.z/ D .8z7 � 168z5 C 840z3 � 840z/f .z/

C.z8 � 28z6 C 210z4 � 420z2 C 105/g.z/D .�z9 C 36z7 � 378z5 C 1260z3 � 945z/f .z/

f .10/.z/ D .�9z8 C 252z6 � 1890z4 C 3780z2 � 945/f .z/CC.�z9 C 36z7 � 378z5 C 1260z3 � 945/g.z/

D .z10 � 45z8 C 630z6 � 3150z4 C 4725z2 � 945/f .z/

“upper triangle representation of the matrix transforming f .z/! f n.z/”

2

6666666666666666664

f .z/f 0.z/f 00.z/f 000.z/f .4/.z/f .5/.z/f .6/.z/f .7/.z/f .8/.z/f .9/.z/f .10/.z/

3

7777777777777777775

D f .z/

2

6666666666666666664

1 0 0 0 0 0 0 0 0 0 0

0 �1 0 0 0 0 0 0 0 0 0

�1 0 1 0 0 0 0 0 0 0 0

0 3 0 �1 0 0 0 0 0 0 0

3 0 �6 0 1 0 0 0 0 0 0

0 �15 0 10 0 �1 0 0 0 0 0

�15 0 45 0 �15 0 1 0 0 0 0

0 105 0 �105 0 21 0 �1 0 0 0

105 0 �420 0 210 0 �28 0 1 0 0

0 �945 0 1260 0 �378 0 36 0 �1 0�945 0 �4725 0 �3150 0 630 0 �45 0 1

3

7777777777777777775

2

6666666666666666664

1

zz2

z3

z4

z5

z6

z7

z8

z9

z10

3

7777777777777777775

:


B-31 The Forward Computation of a First Confidence Intervalof Gauss–Laplace Normally Distributed Observations:�; � 2 Known

We can avoid solving the linear Volterra integral equation of the first kind if wepush forward the integration for a fixed value z.

Example B.8. (Series expansion of the Gauss-Laplace integral, firstinterval):

Let us solve the integral

�.z D 1/ WD 21Z

0

f .z�/d z�

in the first interval 0 � z � 1 by Taylor expansion with respect to the successivedifferentiation of f .z/ outlined in Box B.1 and the specific derivatives f n.0/ givenin Table B.4. Based on those auxiliary results, Box B.2 presents us the detailedinterpretation. First, we expand exp.�z2=2/ up to order O.14/. The specific Taylorseries are uniformly convergent. Accordingly, in order to compute the integral,second we integrate termwise up to order O.15/. For the specific value zD 1, wehave computed the coefficient of confidence �.1/ D 0:683. The result

P.� � � < X < �C �/ D 0:683

is known as the One Sigma Rule.

68.3% of the sample are in the interval �� 1�; �C 1� Œ, 0.317% outside. Ifwe make three experiments, one experiment is outside the one interval.

Box B.2. A specific integral

“expansion of the exponential function”

expx D 71C x

1ŠC x2

2ŠC x3

3ŠC � � � C xn

nŠCO.n/ 8 jxj <1

x WD �1

2z2

exp

��1

2z2�D 1� 1

2z2C 1

8z4 � 1

48z6C 1

384z8� 1

3840z10C 1

46080z12CO.14/


“the specific integral”

zZ

0

f .z�/d z� D 1p2

zZ

0

exp�z�2

�d z� D 1p

2

�z � 1

6z3 C 1

40z5 � 1

336z7

C 1

3456z9 � 1

42240z11 C 1

599040z13 CO.15/

�

“the specific values z D 1”

�.1/ D 21Z

0

f .z/d z

D 2p2

�1 � 1

6C 1

40� 1

336C 1

3456� 1

42240C 1

599040CO.15/

�

D 2p2.1 � 0:166;667C 0:025;000/

�0:002;976C 0:000;289�0; 000;024C 0:000;002/

D 2p20:855;624 D 0:682;689

D 0:683

“coefficient of confidence”

0:683 D 1 � 317:311 � 10�3 D 1 � 13:

Table B.4 (Special values of derivatives- Gauss-Laplace probability distribution):

z D 0

1p2D 0:398; 942

f .0/ D 1p2; f 0.0/ D 0;

f 00.0/ D � 1p2;1

2Šf 00.0/ D �0:199; 471


f 000.0/ D 0; f .4/.0/ D C 3p2;1

4Šf .4/.0/ D C0:049; 868

f .5/.0/ D 0; f .6/.0/ D � 15p2;1

6Šf .6/.0/ D �0:008; 311:

Example B.9. (Series expansion of the Gauss-Laplace integral, 2nd inter-val):

Let us solve the integrals

�.z D 2/ WD 2

2Z

0

f .z�/d z� D 21Z

0

f .z�/d z�C22Z

1

f .z�/d z�

�.z D 2/ D �.z D 1/C 22Z

1

f .z�/d z�;

namely in the second interval 1 � z � 2. First, we setup Taylor series of f .z/“around the point z D 1”. The derivatives of f .z/ “at the point z=1” are collectedup to order 10 in Table B.5. Second, we integrate the Taylor series termwise andreceive the specific integral of Box B.3. Note that termwise integrated is permittedsince the Taylor series are uniformly convergent. The detailed computation up toorder O(12) has led us to the coefficient of confidence �.2/ D 0:954. The result

P.� � 2� < X < �C 2�/ D 0:954

is known as the Two Sigma Rule.

95.4% of the sample interval �� 2�; �C 2� Œ, 0.046% outside. If we make22 experiments, one experiment is outside the 2� interval.

Box B.3. (A specific integral):


f .z/ WD 1p2

exp

��12

z2�

f .z/ D f .1/C 1

1Šf 0.1/.z� 1/C 1

2Šf 00.1/.z � 1/2 C � � � C 1

10Šf 10.1/.z� 1/10

C O.11/


“the specific integral”

2Z

1

f .z�/d z� D f .1/.z � 1/C 1

2

1

1Šf 0.1/.z� 1/2 C 1

3

1

2Šf 00.1/.z� 1/3

C 1

4

1

3Šf 000.1/.z� 1/4 C 1

5

1

4Šf .4/.1/.z� 1/5

C 1

6

1

5Šf .5/.1/.z � 1/6 C � � � C 1

11

1

10Šf .10/.1/.z� 1/11 C O.12/

case z D 2

�.2/ D �.1/C 22Z

1

f .z/d z D�.1/C 2.0:241; 971

� 0:120; 985C 0:020; 122� 0:004; 024� 0:002; 016C 0:000; 768C 0:000; 120� 0:000; 088� 0:000; 002� 0:000; 050C O.12//

D 0:682; 690C 0:271; 632D 0:954


0:954 D 1 � 45:678 � 10�3 D 1 � 1

22:

Table B.5 (Special values of derivatives – Gauss-Laplace probability distri-bution):

z D 1

1p2D 0:398; 942; exp

��12

�D 0:606; 531

f .1/ D 1p2

exp

��12

�D 0:241; 971

f 0.1/ D �f .1/ D � 1p2

exp

��12

�D �0:241; 971; f 00.1/ D 0

f 000.1/ D 2f .1/ D 2p2

exp

��12

�D 0:482; 942


1

3Šf 000.1/ D C0:080; 490

f .4/.1/ D �2f .1/ D �0:482; 9421

4Šf .4/.1/ D �0:020; 122

f .5/.1/ D �6f .1/ D � 6p2

exp

��12

�D �1:451; 826

1

5Šf .5/.1/ D �0:012; 098

f .6/.1/ D 16f .1/ D 16p2

exp

��12

�D 3:871; 536

1

6Šf .6/.1/ D 0:005; 377

f .7/.1/ D 20f .1/ D 20p2

exp

��12

�D 4:829; 420

1

7Šf .7/.1/ D 0:000; 958

f .8/.1/ D �132f .1/ D �31:940; 1721

8Šf .8/.1/ D �0:000; 792

f .9/.1/ D �28f .1/ D �6:775; 1881

9Šf .9/.1/ D �0:000; 019

f .10/.1/ D �8234f .1/ D �1992:3891

10Šf .10/.1/ D �0:000; 549:

Example B.10. (Series expansion of the Gauss-Laplace integral, third inter-val):

Let us solve the integrals

�.z D 3/ WD 2

3Z

0

f .z/d z D 21Z

0

f .z/d zC 22Z

1

f .z/d zC 23Z

2

f .z/d z

�.z D 3/ D �.z D 1/C �.z D 2/C 23Z

2

f .z/d z;


namely in the third interval 2 � z � 3. First, we setup Taylor series of f(z) “aroundthe point z D 2”. The derivatives of f .z/ “at the point z D 2” are collected up toorder 10 in Table B.6. Second, we integrate the Taylor series termwise and receivethe specific integral of Box B.4. Note that termwise integration is permitted sincethe Taylor series are uniformly convergent. The detailed computation up to orderO(12) leads us to the coefficient of confidence �.3/ D 0:997. The result

P.� � 3� < X < �C 3�/ D 0:997is known as the Three Sigma Rule.

99.7% of the sample are in the interval �� 3�; �C 3�Œ, 0.003% outside.If we make 377 experiments, one experiment is outside the 3� interval.

Table B.6 (Special values of derivatives Gauss-Laplace probability distribu-tion):

z D 21p2D 0:398; 942; exp .�2/ D 0:135; 335

f .2/ D 1p2

exp .�2/ D 0:053; 991

f 0.2/ D �2f .2/ D �0:107; 982

f 00.2/ D 3f .2/; 1

2Šf 00.2/ D C0:080; 986

f 000.2/ D �2f .2/; 1

3Šf 000.2/ D �0:017; 997

f .4/.2/ D �5f .2/; 1

4Šf .4/.2/ D �0:011; 248

f .5/.2/ D 18f .2/; 1

5Šf .5/.2/ D C0:008; 099

f .6/.2/ D �11f .2/; 1

6Šf .6/.2/ D �0:000; 825

f .7/.2/ D �86f .2/; 1

7Šf .7/.2/ D �0:000; 921

f .8/.2/ D C249f .2/; 1

8Šf .8/.2/ D C0:000; 333

f .9/.2/ D 190f .2/; 1

9Šf .9/.2/ D C0:000; 028

f .10/.2/ D �2621f .2/; 1

10Šf .10/.2/ D �0:000; 039:


Box B.4 (A specific integral)


f .z/ WD 1p2

exp

��12

z2�

f .z/ D f .2/C 1

1Šf 0.2/.z� 2/C 1

3Šf 00.2/.z� 2/2 C � � � C 1

10Šf .10/.2/.z� 2/10

CO.11/“the specific integral”

zZ

2

f .z�/d z� D f .2/.z� 2/C 1

2

1

1Šf 0.2/.z� 2/2 C 1

3

1

2Šf 00.2/.z � 2/3

C 1

4

1

3Šf 000.2/.z � 2/4 C 1

5

1

4Šf .4/.2/.z� 2/5

C 1

6

1

5Šf .5/.2/.z� 2/6 C � � � C 1

11

1

10Šf .10/.2/.z� 2/11 CO.12/

case z D 3

�.3/ D �.1/C �.2/C 23Z

2

f .z/d z

D 0:682; 690C 0:271; 672C 2.0:053; 991� 0:053; 991C 0:026; 995� 0:004; 499� 0:002; 250C 0:001; 350� 0:000; 118C 0:000; 037C 0:000; 003� 0:000; 004CO.12// D 0:682; 690C 0:271; 632C 0:043; 028D 0:997


0:997 D 1 � 2:65 � 10�3 D 1 � 1

377:

B-32 The Backward Computation of a First ConfidenceInterval of Gauss–Laplace Normally DistributedObservations: �; � 2 Known

Finally we solve the Volterra integral equation of the first kind by the techniqueof series inversion, also called series reversion. Let us recognize that the interval


integration of a Taylor series expanded Gauss-Laplace normal density distributionled us to a univariate homogeneous polynomial of arbitrary order. Such a univariatehomogeneous polynomial y D a1x C a2x2 C � � � C anxn (“input”) can be reversedas a univariate homogeneous polynomial x D b1y C b2y2 C � � � C bnyn (“output”)as outlined in Table B.7. Consult Abramowitz and Stegun (1965), p. 16 for a review,but Grafarend (1996) for a derivation based upon Computer Algebra.

Table B.7 (Series inversion (Grafarend, 1996)):

“input: univariate homogeneous polynomial”

y D a1x C a2x2 C a3x3 C a4x4 C a5x5 C a6x6 C a7x7 CO.x8/

“output: reverse univariate homogeneous polynomial”

x D b1y C b2y2 C b3y3 C b4y4 C b5y5 C b6y6 C b7y7 CO.y8/

“coefficient relation”

(a) a1b1 D 1(b) a31b2 D �a2(c) a51b3 D 2a22 � a1a3(d) a71b4 D 5a1a2a3 � a21a4 � 5a32(e) a91b5 D 6a21a2a4 C 3a21a23 C 14a42 � a31a5 � 21a1a22a3(f) a111 b6 D 7a31a2a5C7a31a2a4C84a1a32a3�a41a6�28a1a2a23�42a52�28a21a22a4(g) a131 b7 D 6a41a2a6 C 8a41a2a5 C 4a41a24 C 120a21a32a4 C 180a21a22a23 C 132a62� a51a7 � 36a31a22a5 � 72a31a2a3a4 � 12a31a33 � 330a1a42a3.

Example B.11. (Solving the Volterra integral equation of the first kind):

Let us define the coefficient of confidence � D 0:90 or 90%. We want to know thequantile c0:90 which determines the probability identity

P.� � c� < X < �C c�/ D 0:90:

If you follow the detailed computation of Table B.5, namely the input as well as theoutput data up to order O(5), you find the quantile

c0:90 D 1:64;

as well as the confidence interval

P.� � 1:64� < X < �C 1:64�/ D 0:90:


90% of the sample are in the interval �� 1:64�; �C 1:64�Œ, 10% outside. If wemake 10 experiments one experiment is outside the 1.62� interval.

Table B.8 (Series inversion quantile c0:90):

(i) input

�.z/ D 21Z

0

f .z�/d z� C 2zZ

0

f .z�/d z� D 0:682; 689C 2Œf .1/.z� 1/

C 1

2

1

1Šf 0.1/.z� 1/2 C � � � C 1

n

1

.n � 1/Šf.n�1/.1/.z� 1/n CO.nC 1/�

1

2Œ�.z/ � 0:682; 689�D f .1/.z� 1/C 1

2

1

1Šf 0.1/.z� 1/2 C � � �

C 1

n

1

.n � 1/Šf.n�1/.1/.z� 1/n CO.nC 1/�

y D a1x C a2x2 C � � � C anxn CO.nC 1/x WD z � 1y WD .0:900; 000� 0:682; 689/=2D 0:108; 656a1 WD f .1/ D 0:241; 971

a2 WD 1

2

1

1Šf 0.1/ D �0:241; 971

a3 WD 1

3

1

2Šf 00.1/ D 0

a4 WD 1

4

1

3Šf 000.1/ D 0:020; 125

a5 WD 1

5

1

4Šf .4/.1/ D �0:004; 024� � �

an WD 1

n

1

.n� 1/Šf.n�1/.1/:

(ii) output

b1 D 1

a1D 4:132; 726

b2 D � 1a31a2 D 8:539; 715


b3 D 1

a51.2a22 � a1a3/ D 35:292; 308

b4 D 1

a71.5a1a2a3 � a21a4 � 5a32/ D 158:070

b5 D 1

a91.6a21a2a4 C 3a21a23 C 14a42 � a31a5 � 21a1a22a3/ D 475:452; 152

b1y D 0:449; 045; b2y2 D 0:100; 821;b3y

3 D 0:045; 273; b4y4 D 0:022; 032;b5y

5 D 0:007; 201x D b1y C b2y2 C b3y3 C b4y4 C b5y5 CO.6/ D 0:624; 372z D xC 1 D 1:624; 372 D c0:90:

At this end we would like to give some sample references on computing the “inverseerror function”

y D F.x/ WDxZ

0

1p2

exp

��12

z2�d z DW erfx versus

x D F�1.y/ D inverfy;

namely Carlitz (1963) and Strecok (1968).

B-4 Sampling from the Gauss–Laplace Normal Distribution:A Second Confidence Interval for the Mean, VarianceKnown

The second confidence interval of Gauss-Laplace i.i.d. observations will be con-structed for the mean b� BLUUE of �, when the variance �2 is known. “n” is thesize of the sample, namely to agree to the number of observations. Before we presentthe general sampling distribution we shall work through two examples. ExampleB.12 has been chosen for a sample size n D 2, while Example B.12 for nD 3observations. Afterwards the general result is obvious and sufficiently motivated.

Example B.12. (Gauss-Laplace i.i.d. observations, observation space Y,dim Y)

In order to derive the marginal distributions of b� BLUUE of � and b�2 BIQUUEof �2 for a two dimensional Euclidean observation space we have to introduce

B-4 Sampling from the Gauss–Laplace Normal Distribution: 663

ˆx: = σ2 / σ2

ˆf2(x) = f2(σ2 / σ2 )

Fig. B.2 Special Helmert pdf of 2p for one degree of freedom, p D 1

various images. First, we define the probability function of two Gauss-Laplace i.i.d.observations and consequently implement the basic .b�;b�2/ decomposition into thepdf. Second, we separate the quadratic form .y � 1�/0.y � 1�/ into the quadraticform .y � 1b�/0.y � 1b�/, the vehicle to introduce b�2 BIQUUE of �2, and into thequadratic form .b� � �/2 the vehicle to bring in b� BLUUE of �. Third, we aim attransforming the quadratic form .y � 1b�/0.y � 1b�/D y0 My; rk MD 1, into thespecial form 1=2.y1 � y2/2 DW x. Fourth, we generate the marginal distributionsf1.b�

ˇˇ�; �2=2/ of the mean b� BLUUE of � and f2.�2/ of the sample varianceb�2

BIQUUE of �2. The basic results of the example are collected in Corollary B.6. Thespecial Helmert pdf 2 with one degree of freedom is plotted in Fig. B.2, but thespecial Gauss-Laplace normal pdf of variance �2=2 in Fig. B.3.

The first action item

Let us assume an experiment of two Gauss-Laplace i.i.d. observations. Their pdf isgiven by

f .y1; y2/ D f .y1/f . y2/;

f .y1; y2/ D 1

2�2exp

�� 1

2�2Œ.y1 � �/2 C .y2 � �/2�

�;

f .y1; y2/ D 1

2�2exp

�� 1

2�2.y � 1�/0.y � 1�/

�:


–5 –4 –3 –2 –1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

f1(z | 0, 1)

ˆ s2

2z = (m − m)/

Fig. B.3 Special Gauss-Laplace normal pdf of .b�� /=�2=2

The second action item

The coordinates of the observation vector have been denoted by Œy1; y2�0 D

y2Y; dim YD 2. The quadratic form .y � 1�/0.y � 1�/ allows the fundamentaldecomposition

y � 1� D .y � 1b�/C 1.b� � �/.y � 1�/0.y � 1�/ D .y � 1b�/0.y � 1b�/C 101.b� � �/2

.y � 1�/0.y � 1�/ D b�2 C 2.b�� /2:

Here, b� is BLUUE of � and b�2 BIQUUE of �2. The detailed computation provesour statement.

Œ.y1 �b�/C .b�� /�2 C Œ.y2 �b�/C .b� � �/�2

D .y1 �b�/2 C .y2 �b�/2 C 2.b�� /2 C 2.b�� /Œ.y1 �b�/C .y2 �b�/�D .y1 �b�/2 C .y2 �b�/2 C 2.b�� /2 C 2b�.y1 �b�/C 2b�.y2 �b�/� 2.y1 �b�/� � 2.y2 �b�/�


b� D 1

2.y1 C y2/

b�2 D .y1 �b�/2 C .y2 �b�/2:

As soon as we substituteb� andb�2 we arrive at

.y1 � �/2 C .y2 � �/2 D Œ.y1 �b�/C .b� � �/�2 C Œ.y2 �b�/C .b� � �/�2

D b�2 C 2.b�� /2;

since the residual terms vanish:

2b�.y1 �b�/C 2b�.y2 �b�/� 2.y1 �b�/�� 2.y2 �b�/�D 2b�y1 � 2b�2 C 2b�y2 � 2b�2 � 2�y1 C 2�b� � 2�y2 C 2�b�D 2b�.y1 C y2/� 4b�2 � 2�.y1 C y2/C 4�b�D 4b�2 � 4b�2 � 4�b�C 4�b� D 0:

The third action item

The cumulative pdf

dF D f .y1; y2/dy1dy2 D f1.b�/f2.x/db�dx D f1.b�/f2�b�2

�2

�db� d

b�2

�2

has to be decomposed into the first pdf f1.b� j� ; �2=n/ representing the pdf ofthe sample mean b� and the second pdf f2.x/ of the new variable x WD .y1 �y2/

2=.2�2/ D b�2=�2 representing the sample varianceb�2, normalized by �2.

How can the second decomposition f1f2 be understood?

Let us replace the quadratic form .y � 1�/0.y � 1�/ D b�2 C 2.b� � �/2 in thecumulative pdf

dF D f .y1; y2/dy1dy2 D 1

2�2exp

�� 1

2�2.y � 1�/0.y � 1�/

�dy1dy2

D 1

2�2exp

�� 1

2�2Œb�2 C 2.b�� /2�

�dy1dy2

dF D f .y1; y2/dy1dy2

D 1p2� 1

�p2

exp

��12

.b�� /2�2=2

�� 1p

2� 1p

2�exp

��12

b�2

�2

�dy1dy2:


The quadratic formb�2, conventionally given in terms of the residual vector y � 1b�will be rewritten in terms of the coordinates Œy1; y2�0 D y of the observation vector.

b�2 D .y1 �b�/2 C .y2 �b�/2 D�y1 � 1

2.y1 C y2/

2C�y2 � 1

2.y1 C y2/

2

b�2 D 1

4.y1 � y2/2 C 1

4.y2 � y1/2 D 1

2.y1 � y2/2:

The fourth action item

The new variable x WD 12�2.y1 � y2/2 will be introduced in the cumulative pdf

dF D f .y1; y2/dy1dy2. The new surface element db�dx will be related to the oldsurface element dy1dy2.

db�dx Dˇˇˇˇ det

"Dy1b� Dy2b�

Dy1x Dy2x

#ˇˇˇˇ dy1dy2 D jJjdy1dy2

Dy1b� WD@b�@y1D 1

2; Dy2b� WD

@b�@y2D �1

2

Dy1x WD@x

@y1D y1 � y2

�2; Dy2x WD

@x

@y2D �y1 � y2

�2:

The absolute value of the Jacobi determinant amounts to

jJj Dˇˇˇˇ det

"12

12

y1�y2�2� y1�y2

�2

# ˇˇˇˇ D y1 � y2

�2; jJj�1 D �2

y1 � y2 :

In consequence, we have derived

dy1dy2 D �2

y1 � y2 db�dx D�p2pxdb�dx

based upon

x D 1

2�2.y1 � y2/2)

p2x D 1

�.y1 � y2/:

In collecting all detailed partial results we can formulate a corollary.


Corollary B.3. (marginal probability distributions ofb�; �2 given, and b�2):

The cumulative pdf of a set of two observations is represented by

dF D f .y1; y2/dy1dy2Df1.b� j�; �2=2/f2.x/db�dx subject to

f1.b� j�; ; �2=2/ WD 1p2� 1

�p2

exp

��12

.b�� /2�2=2

�

f2.x/ D f2

�b�2

�2

�WD 1p

2

1px

exp

��12x

�subject to

C1Z

�1f1.b�/ db� D 1 and

C1Z

0

f2.x/ dx D 1:

f1.b�/ is the pdf of the sample mean b� D .y1 C y2/=2 and f2.x/ the pdf of thesample variance b�2 D .y1 � y2/2=2 D �2x. f1.b�/ is a Gauss-Laplace pdf withmean� and variance �2=n, while f2.x/ is a Helmert 2 with one degree of freedom.

Example B.13. (Gauss-Laplace i.i.d. observations, observation space Y,dim YD 3):

In order to derive the marginal distribution of b� BLUUE of � and b�2 BIQUUE of�2 for a three-dimensional Euclidean observation space, we have to act in variousscenes. First, we introduce the probability function of three Gauss-Laplace i.i.d.observations and consequently implement the .b�; b�2/ decomposition in the pdf.Second, we force the quadratic form .y � 1b�/0.y � 1b�/ to be decomposed into b�2

and .b��/2, actually a way to introduce the sample varianceb�2 BIQUUE ofb�2 andthe sample meanb� BLUUE of �. Third, we succeed to transform the quadratic form.y � 1b�/0.y � 1b�/ D yMy; rkM D 2 into the canonical form z21 C z22 by means ofŒz1; z2�

0 D HŒy1; y2; y3�0. Fourth, we produce the right inverse H�k D H0 in order to

invert H, namely Œy1; y2; y3�0 D H0Œz1; z2�0. Fifth, in order to transform the originalquadratic form ��2.y�1b�/0.y�1b�/ into the canonical form z21Cz22Cz23 we reviewthe general Helmert transformation z D ��1H.y��/ and its inverse y�� D �H0zsubject to H 2 SO.3/ and identify its parameters translation, rotation and dilatation(scale). Sixth, we summarize the marginal probability distributions f1.b� j� ; �2=3/of the sample mean b� BLUUE of � and f2.2b�2=�2/ of the sample variance b�2

BIQUUE of �2. The special Helmert pdf 2 with two degrees of freedom is plottedin Fig. B.4 while the special Gauss-Laplace normal pdf of variance �2=3 in Fig. B.5.The basic results of the example are collected in Corollary B.7.


f2(2s2 | s2)ˆ

x := 2s2 | s2

Fig. B.4 Special Helmert pdf of 2p for two degrees of freedom, p D 2

3ˆ )s2

f1(m | m,

ˆ(m − m) / s2

3

Fig. B.5 Special Gauss-Laplace normal pdf of .b�� /= �2

3



Let us assume an experiment of three Gauss-Laplace i.i.d. observations. Their pdfis given by

f .y1; y2; y3/ D f .y1/f . y2/f . y3/;

f .y1; y2; y3/ D .2/�3=2��3 exp

�� 1

2�2Œ.y1 � �/2C .y2��/2C .y3 ��/2�

�;

f .y1; y2; y3/ D .2/�3=2��3 exp

�� 1

2�2.y � 1�/0.y � 1�/

�:

The coordinates of the observation vector have been denoted by Œy1; y2; y3�0D y2Y,dimY D 2.


The coordinates of the observation vector have been denoted byD y2Y, dimYD 2.The quadratic form .y1 � 1�/0.y2 � 1�/ allows the fundamental decomposition

y � 1� D .y � 1b�/C 1.b� � �/;.y � 1�/0.y � 1�/ D .y � 1b�/0.y � 1b�/C 101.b� � �/2

.y � 1�/0.y � 1�/ D 2b�2 C 3.b�� /2:

Here, b� is BLUUE of � and b�2 BIQUUE of �2. The detailed computation provesour statement.

Œ.y1 �b�/C .b� � �/�2 C Œ.y2 �b�/C .b�� /�2 C Œ.y3 �b�/C .b� � �/�2

D .y1 �b�/2 C .y2 �b�/2 C .y3 �b�/2 C 3.b�� /2 C 2b�.y1 �b�/C 2b�.y2 �b�/C 2b�.y3 �b�/� 2.y1 �b�/�� 2.y2 �b�/� � 2.y3 �b�/�

b�BLUUE of� W b� D 1

3.y1 C y2 C y3/

b�2 BIQUUE of �2 W b�2 D 1

2Œ.y1 �b�/2 C .y2 �b�/2 C .y3 �b�/2�:

As soon as we substituteb� andb�2 we arrive at

.y1 � �/2 C .y2 � �/2 C .y3 � �/2

D Œ.y1 �b�/2C .b��/�2C Œ.y2 �b�/2C .b��/2�2C Œ.y3 �b�/2C .b��/2�2

D 2b�2 C 3.b�� /2:



Let us begin with transforming the cumulative probability function

dF D f .y1; y2; y3/dy1dy2dy3

D .2/�3=2��3 exp

�� 1

2�2.y � 1�/0.y � 1�/

�dy1dy2dy3

D .2/�3=2��3 exp

�� 1

2�2Œ2b�2 C 3.b� � �/2�

�dy1dy2dy3

into its canonical form. In detail, we transform the quadratic form b�2 BIQUUE of�2 canonically.

y1 �b� D y1 � 13y1 � 1

3.y2 C y3/ D 2

3y1 � 1

3.y2 C y3/ D 2

3y1 � 1

3y2 � 1

3y3

y2 �b� D y2 � 13y2 � 1

3.y1 C y3/ D �1

3y1 C 2

3y2 � 1

3y3

y3 �b� D y3 � 13y3 � 1

3.y1 C y2/ D �1

3y1 � 1

3y2 C 2

3y3

as well as

.y1 �b�/2 D 4

9y21 C

1

9y22 C

1

9y23 �

4

9y1y2 C 2

9y2y3 � 4

9y3y1

.y2 �b�/2 D 1

9y21 C

4

9y22 C

1

9y23 �

4

9y1y2 � 4

9y2y3 C 2

9y3y1

.y3 �b�/2 D 1

9y21 C

1

9y22 C

4

9y23 C

2

9y1y2 � 4

9y2y3 � 4

9y3y1

and

.y1 �b�/2 C .y2 �b�/2 C .y3 �b�/2 D 2

3.y21 C y22 C y23 � y1y2 � y2y3 � y3y1/:

We shall prove

.y � 1b�/0.y � 1b�/ D y0My D z21 C z22; rkM D 2; M 2 R3�3

that the symmetric matrix M,

M D2

42 �1 �1�1 2 �1�1 �1 2

3

5


has rank 2 or rank deficiency 1. Just compute jMj D 0 as a determinant identity aswell as the subdeterminant

ˇˇˇ 2 �1�1 2

ˇˇˇ D 3 ¤ 0

.y � 1b�/0.y � 1b�/ D y0My D 1

3Œy1; y2; y3�

2

42 �1 �1�1 2 �1�1 �1 2

3

5

2

4y1

y2y3

3

5 :

How to transform a degenerate quadratic form to a canonical form?

Helmert (1875) had the bright idea to implement what we call nowadays the forwardHelmert transformation

z1 D 1p1 � 2.y1 � y2/

z2 D 1p2 � 3.y1 C y2 � 2y3/ or

z21 D1

2.y21 � 2y1y2 C y22/

z22 D1

6.y21 C y22 C 4y23 C 2y1y2 � 4y2y3 � 4y3y1/

z21 C z22 D2

3.y21 C y22 C y23 � y1y2 � y2y3 � y3y1/ :

Indeed we found

.y � 1b�/0.y � 1b�/ D .y1 �b�/2 C .y2 �b�/2 C .y3 �b�/2 D z21 C z22:

In algebraic terms, a representation of the rectangular Helmert matrix is

z D�

z1z2

D"

1p1�2 � 1p

1�2 0

1p2�3

1p2�3 � 2p

2�3

#2

4y1y2

y3

3

5

z D H23y; H23 WD"

1p1�2 � 1p

1�2 0

1p2�3

1p2�3 � 2p

2�3

#

:


The rectangular Helmert matrix is right orthogonal,

H23H023 D

"1p1�2 � 1p

1�2 0

1p2�3

1p2�3 � 2p

2�3

#2

664

1p1�2

1p2�3

� 1p1�2

1p2�3

0 � 2p2�3

3

775 D

�1 0

0 1

D I2:


By means of the forward Helmert transformation we could prove that z21 C z22represents the quadratic form .y�1b�/0.y�1b�/. Unfortunately, the forward Helmerttransformation only allows indirectly by a canonical transformation. What would beneeded is the inverse Helmert transformation z 7! y, also called backward Helmerttransformation. The rectangular Helmert matrix has the disadvantage to have noCayley inverse. Fortunately, its right inverse

H�R WD H0

23.H23H023/

�1 D H023 2 R

3�2

solves our problem. The inverse Helmert transformation

y D H�Rz D H0

23z

brings

.y � 1b�/0.y � 1b�/ D 2

3.y21 C y22 C y23 � y1y2 � y2y3 � y3y1/

via

y D H0

23z or

2

664

y1 D 1p1�2 z1 C 1p

2�3 z2

y2 D � 1p1�2 z1 C 1p

2�3 z2

y3 D � 2p2�3 z2;

y21 C y22 C y23 D1

2z21 C

1

6z22 C

1p3

z1z2 C 1

2z22 C

1

6z22 �

1p3

z1z2 C 2

3z22;

y1y2 C y2y3 C y3y1 D �12

z21 C1

6z22 C

1p3

z1z2 � 13

z22 �1p3

z1z2 � 12

z22;

2

3.y21 C y22 C y23 � y1y2 � y2y3 � y3y1/ D

2

3

�3

2z21 C

3

2z22

�D z21 C z22;

into the canonical form.


The fifth action item

Let us go back to the partitioned pdf in order to inject the canonical representationof the deficient quadratic form y0My; M 2 R

3�3, rkM D 2. Here we meet first theproblem to transform

dF D .2/�3=2��3 exp

�� 1

2�2Œ2b�2 C 3.b� � �/2�

�dy1dy2dy3

by an extended vector Œz1; z2; z3�0 DW z into the canonical form dF D .2/�3=2exp

�� 12.z21 C z22 C z23/

�d z1d z2d z3, which is generated by the general forward

Helmert transformation

z D ��1H.y � 1�/

2

4z1z2z3

3

5 D 1

�

2

664

1p1�2

1p1�2 0

1p2�3

1p2�3 � 2p

2�31p3

1p3

1p3

3

775

2

4y1 � �y2 � �y3 � �

3

5

or its backward Helmert transformation, also called the general inverse Helmerttransformation

2

4y1 � �y2 � �y3 � �

3

5 D �

2

664

1p1�2

1p2�3

1p3

1p1�2

1p2�3

1p3

0 � 2p2�3

1p3

3

775

2

4z1z2z3

3

5 :

y � 1� D �H0z

thanks to the orthonormality of the quadratic Helmert matrix H3, namely H3H03 D

H03H3 D I3 or H�1

3 D H03.

Secondly, notice the transformation of the volume element

dy1dy2dy3 D d.y1 � �/d.y2 � �/d.y3 � �/ D �3d z1d z2d z3;

which is due to the Jacobi determinant

J D �3 ˇˇH0ˇˇ D �3 jHj D �3:

Let us prove that

z3 WDp3��1.b� � �/ D b� � �

�p3


brings the first marginal density

f1 D�b�j�; �

2

3

�WD 1p

2� 1

�p3

� exp

�� 1

2�23.b�� /2

�

into the canonical form

f1.z3 j0; 1/ D 1p2

exp

��12

z23

�:

Let us computeb� � � as well as 3.b�� /2 which concludes the proof.

z3 D ��1�1p3;1p3;1p3

2

4y1 � �y2 � �y3 � �

3

5

z3 D ��1 y1 C y2 C y3 � 3�p3

y1 C y2 C y33

D b� ) y1 C y2 C y3 D 3b�

3

775

) z3 D 3��1b� � �p3; z23 D

1

�23.b�� /2:

Indeed the extended Helmert matrix H3 is ingenious to decompose

1

�2.y � 1b�/0.y � 1b�/ D z21 C z22 C z23

into a canonical quadratic form relating z21 C z22 to b�2 and z23 to .b� � �/2. At thispoint, we have to interpret the general Helmert transformation z D ��1H.y � �/:Structure elements of the Helmert transformation

scale or dilatation ��1

rotation H

translation �

��1 2RC produces a dilatation or a scale change, H2SO.3/ WD fH2R3�3 jH0H DI3 and jHj D C1g a rotation (3 parameters) and 1� 2 R

3 a translation. Please,prove for yourself that the quadratic Helmert matrix is orthonormal, that is HH0 DH0H D I3 and jHj D C1.


The sixth action item

Finally we are left with the problem to split the cumulative pdf into one part f1.b�/which is a marginal distribution of the arithmetic meanb� BLUUE of � and anotherpart f2.b�/which is a marginal distribution of the standard deviationb� ,b�2 BIQUUEof �2, Helmert’s 22 with two degrees of freedom.First let us introduce polar coordinates .�1; r/ which represent the Cartesian coor-dinates z1D r cos�1, z2D r sin �1. The index 1 is needed for later generalization tohigher dimension. As a longitude, the domain of �1 is �1 2 Œ0; 2� or 0 � �1 � 2 .The new random variable z21 C z22 D kzk2 DW x or radius r relates to Helmert’s

2 D z21 C z22 D2b�2

�2D 1

�2.y � 1b�/0.y � 1b�/ :

Secondly, the marginal distribution of the arithmetic meanb�, b� BLUUE of �,

f1

�b� j� ; �p

3

�db� D f1.z3 j0; 1 /d z3 � N

��;

�p3

�

is a Gauss-Laplace normal distribution with mean � and variance �2=3 gener-ated by

dF1 D f1�b� j� ; �p

3

�db� D f1.z3 j0; 1/d z3

D .2/�1=2 exp

��12

z23

�d z3

C1Z

�1

C1Z

�1.2/�1 exp

��12.z21 C z22/

�d z1d z2 or

f1

�b� j� ; �p

3

�db� D .2/�1=2

p3

�exp

�� 3

2�2.b�� /2

�db�:

Third, the marginal distribution of the sample variance 2b�2=�2 D z21 C z22 DW x,Helmert’s 2 distribution for p D n � 1 D 2 degrees of freedom,

f2.2b�2=�2/ D f2.x/ D 1

2p=2� .p

2/xp2 �1 exp

�x2

�

is generated by

dF2 DC1Z

�1.2/�1=2 exp

��12

z23

�d z3

2Z

0

d�1.2/�1 12

exp

��12x

�dx


dF2 D .2/�1!2 12

exp

��12x

�dx subject to !2 D

2Z

0

d�1 D 2

dF2 D 1

2exp

��12x

�dx

and

x WD z21 C z22 D r2) dx D 2rdr; dr D dx

2r

d z1d z2 D rdrd�1 D1

2dxd�1

is the transformation of the surface element d z1d z2. In collecting all detailed resultslet us formulate a corollary.

Corollary B.7. (marginal probability distributions ofb�, �2 given, and ofb�2):

The cumulative pdf of a set of three observations is represented by

dF D f .y1; y2; y3/dy1dy2dy3 D f1�b�j�; �

2

3

�f2.x/db�dx

subject to

f1

�b�j�; �

2

3

�WD 1p

2� 1

�=p3� exp

��12

.b� � �/2�2=3

�

f2.x/ D 1

2exp

��12x

�

subject to

x WD z21 C z22 D 2b�2=�2; dx D 2

�2db�2

and

1Z

�1f1.b�/ db� D 1 versus

1Z

0

f2.x/ dx D 1:

f1.b�/ is the pdf of the sample mean b�D .y1 C y2 C y3/=3 and f2.x/ the pdfof the sample variance b�2D .y0 � 1b�/0.y � 1b�/=2 normalized by �2. f1.b�/ is a


Gauss-Laplace pdf with meanb� and variance �2=3, while f2.x/ a Helmert 2 withtwo degrees of freedom.In summary, an experiment with three Gauss-Laplace i.i.d. observations is charac-terized by two marginal probability densities, one for the meanb� BLUUE of � andanother one forb�2 BIQUUE of �2:

Marginal probability densities n D 3, Gauss-Laplace i.i.d. observations

b� W f1�b� j� ; �p

3

�db� b�2 W f2.2b�2=�2/db�2

D .2/�1=2 1

�=p3

exp

��12

.b� � �/2�2=3

�db� D b�2

�2exp

��b�

2

�2

�db�2

or or

f1.z3 j0; 1/d z3 f2.x/dx

D .2/�1=2 exp

��12

z23

�d z3 D 1

2exp

��12x

�dx

z3 WD b� � �3�

x WD 2b�2

�2

B-41 Sampling Distributions of the Sample Mean b�, � 2

Known, and of the Sample Variance b� 2

The two examples have prepared us for the general sampling distribution of thesample mean b�, �2 known, and of the sample varianceb�2 for Gauss-Laplace i.i.d.observations, namely samples of size n. By means of Lemma B.8 on the rectangularHelmert transformation and Lemma B.9 on the quadratic Helmert transformationwe prepare for Theorem B.10 which summaries both the pdfs for b� BLUUE of �,�2 known, for the standard deviationb� and forb�2 BIQUUE of �2. Corollary B.11focusses on the pdf of

�

� D qb� where�

� is an unbiased estimation of the standarddeviation � , namely Ef��g D � .

Lemma B.3. (rectangular Helmert transformation):

The rectangular Helmert matrix Hn�1;n 2 Rn�.n�1/ transforms the degenerate

quadratic form

.n � 1/b�2 WD .y � 1b�/0.y� 1b�/ D y0My; rkMDn�1

subject to

b� D 1

n10y



.n � 1/b�2 D z0n�1zn�1 D z21 C � � � C z2n�1:

The special Helmert transformation yn 7! Hn�1;nyn D zn�1 is represented by

Hn�1;n WD2

6666664

1p1�2

1p1�2 0 0 � � � 0 0

1p2�3

1p2�3 � 2p

2�3 0 � � � 0 0

� � � � � � � � � � � � � � � � � � � � �1p

.n�1/.n�2/1p

.n�1/.n�2/1p

.n�1/.n�2/1p

.n�1/.n�2/ � � � � n�1p.n�1/.n�2/ 0

1pn.n�1/

1pn.n�1/

1pn.n�1/

1pn.n�1/ � � � 1p

n.n�1/ � npn.n�1/

3

7777775

:

The inverse Helmert transformation z 7! y D H�Rz D H0z or yn D H0

n�1�nzn isbased on its right inverse which thanks to the right orthogonality Hn�1;nH0

n;n�1 DIn�1 coincides with its transpose,

H�R D H0.HH0/�1 D H0 2 R

n�.n�1/:

Lemma B.4. (quadratic Helmert transformation):

The quadratic Helmert matrix H 2 Rn�n, also called extended Helmert matrix or

augmented Helmert matrix, transforms the quadratic form

1

�2.y � 1�/0.y � 1�/ D 1

�2Œ.y � 1b�/C 1.b�� /�0Œ.y � 1b�/C 1.b�� /�

subject to

b� D 1

n10y

by means of

z D ��1H.y � 1�/ or y D �H0.z � 1�/


1

�2.y � 1�/0.y � 1�/ D z21 C � � � C z2n�1 C z2n D

n�1X

jD1z2j C z2n

z2n D n.b� � �/2:


Such a Helmert transformation y 7! z D ��1H.y � 1�/ is represented bydisplaylines

H WD2

6666666666666664

1p1�2

1p1�2 0 0 � � � 0 0

1p2�3

1p2�3 � 2p

2�3 0 � � � 0 0

1p3�4

1p3�4

1p3�4 � 3p

34� � � 0 0

� � � � � �1p

.n�1/.n�2/1p

.n�1/.n�2/1p

.n�1/.n�2/1p

.n�1/.n�2/ � � � � n�1p.n�1/.n�2/ 0

1pn.n�1/

1pn.n�1/

1pn.n�1/

1pn.n�1/ � � � 1p

n.n�1/ � npn.n�1/

1pn

1pn

1pn

1pn

� � � 1pn

1pn

3

7777777777777775

Since the quadratic Helmert matrix is orthonormal, the inverse Helmert transfor-mation is generated by

z 7! y � 1� D �H0z:

The proofs for Lemmas B.8 and B.9 are based on generalizations of the special casesfor n D 2, Example B.11, and for n D 3, Example B.12. Any proof will be omittedhere.The highlight of this paragraph is the following theorem.

Theorem B.10. (marginal probability distribution of .b�; �2/ and b�2):

The cumulative pdf of a set of n observations is represented by

dF D f .y1; : : : ; yn/dy1 � � �dynD f1.b�/f4.b�/db�db� D f1.b�/f2.b�2/db�db�2

as the product of the marginal pdf f1.b�/ of the sample mean b�D n�11yand the marginal pdf f2.b�/ of the sample standard deviation b� Dp.y � 1b�/0.y � 1b�/=.n� 1/, also called r.m.s. (root mean square error), or

the marginal pdf f2.b�2/ of the sample variance b�2. Those marginal pdfs arerepresented by

dF1 D f1.b�/db�dF4 D f4.b�/db� and dF2 D f2.b�2/db�2;


(a) sample mean b�

f1.b�/ D f1�b�j�; �

2

n

�WD 1

�pn

p2

exp

��12

.b� � �/2�2=n

z WDpn

�.b� � �/ W f1.z/d z D 1p

2exp

��12

z2�d z

(b) sample r.m.s.b�

p WD n � 1dF4 D f4.b�/db�

f4.b�/ D 2pp

�p2p=2� .p

2/b�p�1 exp

��p2

b�2

�2

�

x WD pn � 1b��Dpp

�b�

f4.x/dx D 2

2p=2� .p

2/xp�1 exp

��12x2�dx

dF4 D f4.x/dx

(c) sample varianceb�2

p WD n � 1

f2.b�2/ D 1

�p2p=2� .p

2/pp=2b�p�2 exp

��12pb�2

�2

�;

x WD .n � 1/b�2

�2D p

�2b�2 W

f2.x/dx D 1

2p=2� .p

2/xp2 �1 exp

��12x

�dx:

f1.b�j�; �2n / as the marginal pdf of the sample mean BLUUE of � is a Gauss-Laplace pdf with mean � and variance �2=n. f2.x/; x WDppb�=� , is the standardpdf of the normalized root-mean-square error with p degrees of freedom. Incontrast, f2.x/; x W Dpb�2=�2 is a Helmert Chi Square 2p pdf with p D n � 1degrees of freedom.Before we present a sketch of a proof of Theorem B.10 which will be run withfive action items and a special reference to the first and second vehicle, we give


some historical comments. Kullback (1934) refers the marginal pdf f1.b�/ of the“arithmetic mean” b� to Poisson (1827), Haussdorff (1901) and Irwins (1937). Hehas also solved the problem to find the marginal pdf of the “geometric mean”.The marginal pdf f2.b�2/ of the sample variance b�2 has been originally derivedby Helmert (1875), Helmert (1876a), Helmert (1876b). A historical discussion ofHelmert’s distribution is offered by David (1957), Kruskal (1946), Lancaster (1965),Lancaster (1966), Pearson (1931) and Sheynin (1995).The marginal pdf f4.b�/ has not found any interest in practice so far. The reason maybe found in the effect that b� is not an unbiased estimate of the standard deviation� , namely Efb�gD� . According to Czuber (1891), p. 162, Roseen (1948), p. 37,Schmetterer (1956), p. 203, Stoom (1967), p. 199, 218, Fisz (1971), p. 240 andRichter and Mammitzsch (1973), p. 42 have documented that

�

� Drp

2

�p2

�

�

�p C 12

�b� D qb�

is an unbiased estimation�

� of the standard deviation � , namely Efb�g D � . p Dn � 1 again denotes the number of degrees of freedom. Schaffrin (1979), p. 240 hasproven that

b�p WDp.y � 1b�/0.y � 1�/

r

p � 12

D b�p2pp

2p � 1

r

p � 12

is an asymptotic (“from above”) unbiased estimation b�p of the standard deviation� . Let us implement

�

� BLUUE of � into the marginal pdf f .b�/:

Corollary B.8. (marginal probability distributions of�

� for Gauss-Laplace i.i.d.observations, Ef��g D �):

The marginal pdf of�

� , an unbiased estimation of the standard deviation, isrepresented by

dF4 D f4.�

�/d�

�

f4.�

�/ D 2pp

�p2p=2

� p

�p C 12

�

p2

� p2� pC1

p2

��

�p�1 exp

0

BBB@�

0

BB@

�

�p C 12

�

�p2

�

1

CCA

2

�

�2

1

CCCA


Fig. B.6 Marginal pdf for the sample standard deviation�

� (r.m.s.)

and

dF4 D f4.x/dx

x WDp2

�

�

�p C 12

�

�p2

� b�

f4.x/ D 2

2p=2�p2

�xp�1 exp

��12x2�

subject to

Efxg D p2�

�p C 12

�

�p2

� :

Figure B.6 illustrates the marginal pdf for “degrees of freedom” p 2 f1; 2; 3; 4; 5g.


Proof.


The pdf of n Gauss-Laplace i.i.d. observations is given by

f .y1; : : : ; yn/ D f .y1/ � � �f .yn/ DnY

iD1f .yi /

f .y1; : : : ; yn/ D .2/�n=2��n exp

�� 1

2�2.y � 1�/0.y � 1�/

�:

The coordinates of the observation space Y have been denoted by Œy1; : : : ; yn�0 D y.Note dim Y D n.


The quadratic form .y� 1�/0.y � 1�/ > 0 allows the fundamental decomposition

y � 1� D .y� 1b�/C 1.b�� /;.y � 1�/0.y � 1�/ D .y� 1b�/0.y � 1b�/C 101.b� � �/2

101 D n

.y � 1�/0.y � 1�/ D .n � 1/b�2 C n.b� � �/2:

Here,b� is BLUUE of � andb�2 BIQUUE of �2. The decomposition of the quadratic.y�1�/0.y�1�/ into the sample varianceb�2 and the square .b��/2 of the shiftedsample meanb�� has already been proved for n D 2 and n=3. The general resultis obvious.


Let us transform the cumulative probability into its canonical forms.

dF D f .y1; : : : ; yn/dy1 � � �dynD .2/�n=2��n exp

�� 1

2�2.y � 1�/0.y� 1�/

�dy1 � � �dyn

D .2/�n=2��n exp

�� 1

2�2

�.n � 1/b�2 C n.b� � �/2

�dy1 � � �dyn

z D ��1H.y� 1�/ or y � 1� D �H0z1

�2.y � 1�/0.y � 1�/ D z21 C z22 C � � � C z2n�1 C z2n:

Here, we have substituted the divert Helmert transformation (quadratic Helmertmatrix H) and its inverse. Again ��1 is the scale factor, also called dilatation,


H an orthonormal matrix, also called rotation matrix, and 1� 2 Rn the translation,

also called shift.

dF D f .y1; : : : ; yn/dy1 � � �dynD 1p

2exp

��12

z2n1

.2/.n�1/=2

�exp

��12.z21 C � � � C z2n/

�dz1dz2 � � � dzn�1dzn

based upon

dy1dy2 � � �dyn�1dyn D �nd z1d z2 � � �d zn�1d zn

J D �njH0j D �njHj D �n:

J again denotes the absolute value of the Jacobian determinant introduced by thefirst vehicle.


First, we identify the marginal distribution of the sample meanb�.

dF1 D f1�b�j�; �

2

n

�db�:

According to the specific structure of the quadratic Helmert matrix zn is gener-ated by

zn D ��1�1pn; : : : ;

1pn

2

64

y1 � �:::

yn � �

3

75 D ��1 y1 C � � � C yn � n�p

n;

upon substituting

b� D 1

n10y D y1 C � � � C yn

n) y1 C � � � C yn D nb�)

zn D ��1 n.b� � �/pn

) z2n D n.b� � �/2�2

; d zn D 1

�

pn db�:

Let us implement d zn in the marginal distribution.

dF1 D 1p2

exp

��12

z2n

�d zn�


�C1Z

�1� � �

C1Z

�1.2/�.n�1/=2 exp

��12.z21 C � � � C z2n�1/

d z1 � � �d zn�1;

C1Z

�1� � �

C1Z

�1.2/�.n�1/=2 exp

��12.z21 C � � � C z2n�1/

d z1 � � �d zn�1 D 1

dF1 D 1p2

exp

��12

z2n

�d zn

dF1 D 1p2

1�pn

exp

�12

.b� � �/2�2

n

!

db� D f1�b�j�; �

2

n

�db�:


Second, we identify the marginal distribution of the sample varianceb�2. We departthe ansatz

dF2 D f2.b�/db� D f2.b�2/db�2

in order to determine the marginal distribution f2.b�/ of the sample root-mean-square errors b� and the marginal distribution f2.b�2/ of the sample variance b�2.A first version of the marginal probability distribution dF2 is

dF2 D 1p2

1Z

�1exp

��12

z2n

�d zn�

� 1

.2/.n�1/=2 exp

��12.z21 C � � � C z2n�1/

d z1 � � �d zn�1:

Transform the Cartesian coordinates .z1 C � � � C zn�1/ 2 Rn�1 to spherical

coordinates .˚1; ˚2; : : : ; ˚n�2; rn�1/. From the operational point of view,p D n�1,the number of “degrees of freedom”, is an optional choice. Let us substitute theglobal hypersurface element !n�1 or !p into dF2, namely

1p2

1Z

�1exp

��12

z2n

�d zn D 1

dF2 D 1

2.n�1/=2 rn�2 exp

��12r2�dr�


�C=2Z

�=2cosn�3�n�2d�n�2

C=2Z

�=2cosn�4�n�3d�n�3 � � �

C=2Z

�=2cos�2d�2

2Z

0

d�1

dF2 D 1

.2/p=2rp�1 exp

��12r2�dr�

�C=2Z

�=2cosp�2�p�1

C=2Z

�=2cosp�3�p�2d�p�2 � � �

C=2Z

�=2cos2�3d�3

C=2Z

�=2cos�2d�2

2Z

0

d�1

!n�1 D !p D 2

�

�n � 12

�.n�1/=2 D 2

�p2

�p=2

D=2Z

�=2cosp�2�p�1d�p�1

=2Z

�=2cosp�3�p�2d�p�2 � � �

=2Z

�=2cos2�3d�3

=2Z

�=2cos�2d�2

2Z

0

d'1

dF2 D !p

.2/p=2rp�1 exp

��12r2�dr

dF2 D 2

2p=2��p

2

� rp�1 exp

��12r2�dr :

The marginal distribution of the r.m.s. f2.b�/ is generated as soon as we substitute

the radius r Dq

z21 C � � � C z2n�1 Dpn � 1 b�1=� . Alternatively the marginal

distribution of the sample variance f2.b�2/ is produced when we substitute the radiussquare r2 D z21 C � � � C z2n�1 D .n � 1/b�2=�2.Project A

r D pn � 1 b�=� D ppb��) dr D

pp

�db�

dF2 D f2.b�/db�

f2.b�/ D 2pp

�p2p=2b�p�1 exp

��12

p

�2b�2�:


Indeed, f2.b�/ establishes the marginal distribution of the root-mean-square errorb�with p D n � 1 degrees of freedom.

Project B

x WD r2 D .n � 1/b�2

�2D pb�

2

�2DW 2p )

2

64dx D 2rdrdr D dx

2rD 1

2

dxpx

rp�1dr D 1

2xp2 �1dx

dF2 D f2.x/dx

f2.x/ WD 1

2p=2� .p

2/xp2 �1 exp

��12x

�:

Finally, we have derived Helmert’s Chi Square 2p distribution f2.x/dxDdF2 bysubstituting r2, rp�1 and dr in factor of x WD r2 and dx D 2rdr .

Project CReplace the radical coordinate squared r2 D .n � 1/b�2=�2 D pb�2=�2 by rescalingon the basis p=�2

x D r2 D z21 C � � � C z2n�1 D z21 C � � � C z2p D .n� 1/b�2

�2D p

�2b�2

dx D p

�2db�

within Helmert’s 2p with p D n � 1 degrees of freedom

dF2 D f2.b�2/db�2

f2.b�2/ D 1

�p2p=2� .p

2/pp=2b�p�2 exp

��12

p

�2b�2�:

Recall that f2.b�2/ establishes the marginal distribution of the sample varianceb�2

with pDn � 1 degrees of freedom. Both, the marginal pdf f2.b�/ of the samplestandard deviation b� , also called root-mean-square error, and the marginal pdff2.b�2/ of the sample varianceb�2 document the dependence on the variance �2 andits power �p .

“Here is our journey’s end.”(W. Shakespeare: Hamlet) |


B-42 The Confidence Interval for the Sample Mean,Variance Known

An application of Theorem B.10 is Lemma B.12 where we construct the confidenceinterval for the sample mean b�, BLUUE of �, variance �2 known, on the basis ofits sampling distribution. Example B.14 is an example of a random sample of sizen D 4.

Lemma B.12. (confidence interval for the sample mean, variance known):

The sampling distribution of the sample meanb� D n�1 10y, BLUUE of �, is Gauss-Laplace normal,

�

� � N .b�j�; �2=n/, if the observations yi ; i 2 f1; : : : ; ng, areGauss-Laplace i.i.d. The “true” mean � is an element of the two-sided confidenceinterval.

� 2�b� � �pnc1�˛=2;b�C �p

nc1�˛=2Œ

with confidence

P

�b� � �p

nc1�˛=2 < � < b�C �p

nc1�˛=2

�D 1 � ˛

of level 1 � ˛. For three values of the coefficient of confidence � D 1 � ˛, TableB.9 is a list of associated quantiles c1�˛=2.

Example B.14. (confidence interval for the sample mean b�, �2 known):

Suppose that a random sample

y WD

2

664

y1y2y3y4

3

775 D

2

664

1:2

3:4

0:6

5:6

3

775 ; b� D 2:7; �2 D 9; r:m:s: W 3

of four observation is known from a Gauss-Lapace normal distribution withunknown mean � and a known standard deviation � D 3. b� BLUUE of � is thearithmetic mean. We intend to determine upper and lower limits which are rathercertain to contain the unknowns parameter� between them. Previously, for samplesof size 4 we have known that the random variable

z D b� � ��pn

D b� � �32


is normally distributed with mean zero and unit variance. b� is the sample mean2.7 and 3/2 is �=

pn. The probability � D 1 � ˛ that z will be between any two

arbitrarily chosen numbers c1 D �c and c2 D c is

P f�c1 < z < Cc2g Dc2Z

c1

f .z/d z D � D 1 � ˛;

P f�c < 2 < Ccg DCcZ

�cf .z/d z D � D 1 � ˛;

cZ

�1f .z/d z D 1 � ˛

2;

�cZ

�1f .z/d z D ˛

2:

� is the coefficient of confidence, ˛ the coefficient of negative confidence, also calledcomplementary coefficient of confidence. The four representations of the probability� D 1 � ˛ to include z in the confidence interval �c < z < Cc have led to thelinear Voltera integral equation of the first kind

cZ

�1f .z/d z D 1� ˛

2D 1C �

2:

Three values of the coefficient of confidence � or its compliment ˛ are popular andlisted in Table B.9.

Table B.9 (Values of the coefficient of confidence):

� 0:950 0:990 0:999

˛ 0:050 0:010 0:001˛2

0:025 0:005 0:000; 5

1 � ˛2D 1C�

20:975 0:995 0:999; 5

In solving the linear Voltera integral equation of the first kind

zZ

�1f .z�/d z� D 1 � 1

2˛.z/ D 1

2Œ1C �.z/�:

Table B.10 collects the quantiles c1�˛=2 given the coefficients of confidence on theircomplements which we listed in Table B.10.


Table B.10 (Quantiles for the confidence interval of the sample mean,variance unknown):

1 � ˛2D 1C�

2� ˛ c1�˛=2

0:975 0:95 0:05 1:960

0:995 0:99 0:01 2:576

0:999; 5 0:999 0:001 3:291

Given the quantiles c1�˛=2, we are going to construct the confidence interval forthe sample mean b�, the variance �2 to be known. For this purpose, we convert theforward transformationsb�! z D pn.b� � �/=� to �.

b�� pn

z D �

b�1 WD b� � �pnc1�˛=2 < � < b�C �p

nc1�˛=2 DW b�2:

The interval b�1 < � < b�2 for the fixed value z D c1�˛=2 contains the “true” mean� with probability � D 1 � ˛.

P

�b� � �p

nc1� ˛

2< � < b�C �p

nc1� ˛

2

�

DcZ

�cf .z/d z D

b�2Z

b�1

f

�b�j�; �

2

n

�db� D � D 1 � ˛

since

b�2Z

�1f .b�/db� D

b�C �p

nc1�˛=2Z

�1f .b�/db� D

c1�˛=2Z

�1f .b�/db� D 1 � ˛

2:

An animation of the coefficient of confidence and the probability functions of aconfidence interval is offered by Figs. B.7 and B.8.

b�1 D b� � c1�˛=2 �pn; b�2 D b�C c1�˛=2 �p

n

Let us specify all the integrals to our example

c1�˛=2Z

�1f .z/d z D 1 � ˛=2


ˆf(m | m, σ2) ˆf(m | m, σ2)

a / 2 a / 2g = 1-a

m1ˆm1ˆ

μ m2ˆ

Fig. B.7 Two-sidedconfidence interval� 2�b�1;b�2Œ; f .b�j�; �2=n/pdf :

m1 = m − c1−a /2mˆ ˆ

n

σ m1 = m + c1−a /2ˆ ˆn

σ

Fig. B.8 Two-sided confidence interval, quantile c1�˛=2

1:960Z

�1f .z/d z D 0:975;

2:576Z

�1f .z/d z D 0:995;

3:291Z

�1f .z/d z D 0:999; 5:

Those data lead to a triplet of confidence intervals.

case .a/ D 0:95; D 0:05; c1�˛=2 D 1:960

P

�2:7 � 3

21:96 < � < 2:7C 3

21:96

�D 0:95

P f�0:24 < � < C5:64g D 0:95case .b/ D 0:99; D 0:01; c1�˛=2 D 2:576

P

�2:7 � 3

22:576 < � < 2:7C 3

22:576

�D 0:99

P f�1:164 < � < 6:564g D 0:99case .c/ D 0:999; D 0:001; c1�˛=2 D 3:291

P

�2:7 � 3

23:291 < � < 2:7C 3

23:291

�D 0:999

P f�2:236 < � < 7:636g D 0:999 :

With probability 95% the “true” mean � is an element of the interval ]�0.24,C5.64[. In contrast, with probability 99% the “true” mean � is an element of thelarger interval ]�1.164, C6.564[. Finally, with probability 99.9% the “true” mean� is an element of the largest interval [�2.236,C7.636].


B-5 Sampling from the Gauss–Laplace Normal Distribution:A Third Confidence Interval for the Mean, VarianceUnknown

In order to derive the sampling distributions for the sample mean, variance un-known, of Gauss-Laplace i.i.d. observation B51 introduces two examples (two andthree observations, respectively, for generating Student’s t distribution. Lemma B.13reviews Student’s t-distribution of the random variable

pn0.b� � �/=b� where the

sample mean b� is BLUUE of �, whereas the sample variance b�2 is BIQUUE of�2. B52 by means of Lemma B.13 introduces the confidence interval for the “true”mean � variance �2 unknown, which is based on Student’s probability distribution.For easy computation, Table B.12 is its flow chart. B53 discusses The UncertainlyPrinciple generated by The Magic Triangle of (a) length of confidence interval,(b) coefficient of negative confidence, also called the uncertainty number, and (c) thenumber of observations. Various figures and examples pave the way for the routineanalyst’s use of the confidence interval for the mean, variance unknown.

B-51 Student’s Sampling Distribution of the Random Variable.b� � �/=b�

Two examples for nD 2 or nD 3Gauss-Laplace i.i.d. observations keep us to deriveStudent’s t-distribution for the random variable

pn.b� � �/=b� where b� is BLUUE

of �, whereas p D n�1 is BIQUUE of 2. Lemma B.12 and its proof is the highlightof this paragraph in generating the sampling probability distribution of Student’s t .

Example B.15 (Student’s t-distribution for two Gauss-Laplace i.i.d. observa-tions):

First, assume an experiment of two Gauss-Laplace i.i.d. observations called y1 andy2: We want to prove that .y1 C y2/=2 and .y1 � y2/2=2 or the sample meanb� andthe sample varianceb�2 are stochastically independent. y1 and y2 are elements of thejoint pdf.

f .y1; y2/ D f .y1/f .y2/

f .y1; y2/ D 1

2�2exp

�� 1

2�2Œ.y1 � �/2 C .y2 � �/2�

�

f .y1; y2/ D 1

2�2exp

�� 1

2�2.y � 1�/0.y � 1�/

�:

The quadratic form .y � 1�/0.y � 1�/ is decomposed into the sample varianceb�2,BIQUUE of �2, and the deviate of the sample mean b�, BLUUE of �, from � bymeans of the fundamental separation

B-5 Sampling from the Gauss–Laplace Normal Distribution: A Third Confidence 693

y � 1� D y � 1b�C 1.b� � �/.y � 1�/0.y � 1�/ D .y � 1b�/0.y � 1b�/C 101.b� � �/2

.y � 1�/0.y � 1�/ D b�2 C 2.b�� /2

f .y1; y2/ D 1

2�2exp

�� 1

2�2b�2�

exp

�� 2

2�2.b� � �/2

�:

The joint pdf f .y1; y2/ is transformed into a special form if we replace

b� D .y1 C y2/=2;

b�2 D .y1 �b�/2 C .y2 �b�/2 D 1

2.y1 � y2/2;

namely

f .y1; y2/ D 1

2�2exp

�� 1

2�21

2.y1 � y2/2

�exp

�� 1�2.y1 C y22

� �/2�:

Obviously the product decomposition of the joint pdf documented that (y1+y2)/2and (y1-y2)2/2 orb� andb�2 are independent random variables.Second, we intend to derive the pdf of Student’s random variable t WD p2.b� ��/=b� , the deviate of the sample meanb� form the “true” mean �, normalized by thesample standard deviationsb� . Let us introduce the direct Helmert transformation

z1 D 1

�

y1 � y2p2D b��; z2 D

p2

�

�y1 C y22

� ��D 2

�

b� � �p2;

or

�z1z2

D 1

�

"1p2� 1p

21p2

1p2

#�y1 � �y2 � �

;

as well as the inverse

�y1 � �y2 � �

D �

"1p2

1p2

� 1p2

1p2

#�z1z2

;

which brings the jointpdf dF Df .y1; y2/dy1dy2 D f .z1; z2/d z1d z2 into thecanonical form.


dF D 1p2

exp

��12

z21

�1p2

exp

��12

z22

�d z1d z2

1

�2dy1dy2 D

ˇˇˇˇ

1p2

1p2

� 1p2

1p2

ˇˇˇˇd z1d z2 D d z1d z2:

The Helmert random variable x WD z21 or z1 D px replaces the random variable z1such that

d z1d z2 D 1

2

1pxdxd z2

dF D 1p2

1

2px

exp

��12x

�1p2

exp

��12

z22

�dxd z2

is the joint pdf of x and z2. Finally, we introduce Student’s random variable

t W D p2b�� b�

z2 Dp2

�.b�� /) b�� D �p

2z2

z1 Dpx D b�

�) b� D �z1 D �

px

t D z2px, z2 D

px t ) z22 D x t2 :

Let us transform dF D f .x; z2/dxd z2 to dF Df .t; x/dtdx, namely from the jointpdf of the Helmert random variable x and the Gauss-Laplace normal variate z2 tothe joint pdf of the Student random variable t and the Helmert random variable x.

d z2dx DˇˇˇˇDt z2 Dxz2Dtx Dxx

ˇˇˇˇ dt dx

d z2dx Dˇˇˇˇˇ

px

p22x�3=2t

0 1

ˇˇˇˇˇdt dx

d z2dx Dpx dt dx

dF D f .t; x/dtdx

f .t; x/ D 1

2

1

2exp

��12.1C t2/x

:


The marginal distribution of Student’s random variable t , namely dF3 D f3.t/dt ,is generated by

f3.t/ WD 1

2

1

2

1Z

0

exp

��12.1C t2/x

dx

subject to the standard integral

1Z

0

exp.�ˇx/dx D�� 1ˇ

exp .�ˇx/1

0

D 1

ˇ

ˇ WD 1

2.1C t2/; 1

ˇD 2

1C t2

such that

f3.t/ D 1

2

1

1C t2 ;

dF3 D 1

2

1

1C t2 dt

and characterized by a pdf f3.t/ which is reciprocal to .1C t2/.

Example B.16. (Student’s t-distribution for three Gauss-Laplace i.i.d. obser-vations):

First, assume an experiment of three Gauss-Laplace i.i.d. observations called y1; y2and y3: We want to derive the joint pdf f .y1; y2; y3/ in terms of the sample meanb�, BLUUE of �, and the sample varianceb�2, BIQUUE of �2.

f .y1; y2; y3/ D f .y1/f .y2/f .y3/

f .y1; y2; y3/ D 1

.2/3=2�3exp

�� 1

2�2Œ.y1 � �/2 C .y2 � �/2 C .y3 � �/2�

�

f .y1; y2; y3/ D 1

.2/3=2�3exp

�� 1

2�2.y � 1�/0.y � 1�/

�:

The quadratic form .y � 1�/0.y � 1�/ is decomposed into the sample varianceb�2

and the deviate sample meanb� from the “true” mean� by means of the fundamentalseparation


y � 1� D y � 1b�C 1.b� � �/.y � 1�/0.y � 1�/ D .y � 1b�/0.y � 1b�/C 101.b� � �/2

.y � 1�/0.y � 1�/ D 2b�2 C 3.b� � �/2

dF D f .y1; y2; y3/dy1dy2dy3D 1

.2/3=2�3exp

�� 1

2�22b�2

�exp

�� 3

2�2.b� � �/2

�dy1dy2dy3:

Second, we intend to derive the pdf of Student’s random variable t WD p3.b��/=b� ,the deviate of the sample meanb� from the “true” mean �, normalized by the samplestandard deviationb� . Let us introduce the direct Helmert transformation

z1 D 1

�p2.y1 � �C y2 � �/ D 1

�p2.y1 C y2 � 2�/

z2 D 1

�p2 � 3.y1 � �C y2 � � � 2y3 C 2�/ D

1

�p2 � 3.y1 C y2 � 2y3/

z3 D 1

�p3.y1 � �C y2 � �C y3 � �/ D 1

�p3.y1 C y2 C y3 � 3�/

or

2

4z1z2z3

3

5 D 1

�

2

664

1p1�2

1p1�2 0

1p2�3

1p2�3 � 2p

2�31p3

1p3

1p3

3

775

2

4y1 � �y2 � �y3 � �

3

5

as well as its inverse

2

4y1 � �y2 � �y3 � �

3

5 D �

2

664

1p1�2

1p2�3

1p3

1p1�2

1p2�3

1p3

0 � 2p2�3

1p3

3

775

2

4z1z2z3

3

5

in general

z D ��1H.y � �/ versus .y � 1�/ D �H0z;

which help us to bring the joint pdf dF.y1; y2; y3/dy1dy2dy3 Df .z1; z2; z3/d z1d z2d z3 into the canonical form.


dF D 1

.2/3=2exp

��12.z21 C z22/

�exp

��12

z23

�d z1d z2d z3

1

�3dy1dy2dy3 D

ˇˇˇˇˇˇ

1p1�2

1p2�3

1p3

1p1�2

1p2�3

1p3

0 � 2p2�3

1p3

ˇˇˇˇˇˇ

d z1d z2d z3 D d z1d z2d z3:

The Helmert random variable x WD z21C z22 orpx D

qz21 C z22 replaces the random

variableq

z21 C z22 as soon as we introduce polar coordinates z1D r cos�1, z2D rsin �1, z21 C z22 D r2 DW x and compute the marginal pdf

dF.z3; x/ D 1p2

exp

��12

z23

�d z3

1

2

1

2exp

��12x

�dx

2Z

0

d�1;

by means of

x WD z21 C z22 D r2) dx D 2rdr; dr D dx

2r;

d z1d z2 D rdrd�1 D 1

2dxd�1;

dF.z3; x/ D 1

2exp

��12x

�1p2

exp

��12

z23

�dxd z3;

the joint pdf of x and z. Finally, we inject Student’s random variable

t WD p3b� � �b�

;

decomposed into

z3 Dp3

�.b� � �/) b� � � D �p

3z3

qz21 C z22 D

px D p2b�

�) b� D �p

2

qz21 C z22 D

�p2

px

t D z3px

p2, z3 D 1p

2

px t ) z23 D

1

2x t2:


Let us transform dF Df .x; z3/dxd z3 to dF Df .t; x/dtdx. Alternatively we maysay that we transform the joint pdf of the Helmert random variable x and the Gauss-Laplace normal variate z3 to the joint pdf of the Student random variable t and theHelmert random variable x.

d z3dx DˇˇˇˇDt z3 Dxz3Dtx Dxx

ˇˇˇˇdtdx

d z3dx Dˇˇˇˇˇ

pxp2

1

2p2x�3=2t

0 1

ˇˇˇˇˇdtdx

d z3dx Dpxp2dtdx

dF D f .t; x/dtdx

f .t; x/ D 1

2p2

1p2

px exp

��12.1C t2

2/x

:

The marginal distribution of Student’s random variable t , namely dF3 D f3.t/dt ,is generated by

f3.t/ WD 1

2p2

1p2

1Z

0

px exp

��12

�1C t2

2

�x

dx

subject to the standard integral

1Z

0

x˛ exp.�ˇx/dx D � .˛ C 1/ˇ˛C1 ; ˛ D 1

2; ˇ D 1

2

�1C t2

2

�

such that1Z

0

px exp

��12

�1C t2

2

�x

dx D 23=2 � . 3

2/

.1C t 2

2/3=2

�

�3

2

�D 1

2

p

f3.t/ D �

�3

2

�1p2

1

.1C t2

2 /3=2;

dF3 Dp2

4.1C t 2

2/3=2

dt:

Again Student’s t-distribution is reciprocal t.1C t2=2/3=2.


Lemma B.13. Student’s t-distribution for the derivate of the mean .b��/=b�(Gosset, 1908)):

Let the random vector of observations y D Œy1; : : : ; yn�0 be Gauss-Laplace i.i.d.

Student’s random variable

t WD b� � �b�pn;

where the sample mean b� is BLUUE of � and the sample varianceb�2 is BIQUUEof �2; associated to the pdf

f .t/ D � .pC12/

� .p

2/

1pp

1

.1C t 2

p/.pC1/=2 :

p D n � 1 is the “degree of freedom” of Student’s distribution fp.t/. Gosset(1908) published the t-distribution of the ratio

pn.b��/=b� under the pseudonym

“Student”: The probable error of a mean.

Proof.

The joint probability distribution of the random variable zn WD pn.b� � �/=� andthe Helmert random variable x WD z21 C � � � C z2n�1 D .n� 1/b�2=�2 represented by

dF D f1.zn/f2.x/d zndx

due to the effect that zn and z21C� � �C z2n�1 orb�� and .n�1/b�2 are stochasticallyindependent. Let us take reference to the specific pdfs with n and n�1 D p degreesof freedom

dF1 D f1.zn/d zn and dF2 D f2.x/dx;

f1.zn/ D 1p2

exp

��12

z2n

�and f2.x/ D 1

� .p

2/

�1

2

�p=2x.p�2/=2 exp

��12x

�;

or

dF1 D f1.b�/db� and dF2 D f2.b�2/db�2;

f1.b�/ Dpn

�p2

exp

��n2

.b��/2�2

�and

f2.b�2/ D 1

�p2p=2� .p=2/pp=2b�p�2

exp

��12

p

�2b�2�

we derived earlier. Here, let us introduce Student’s random variable


t WD znpx

pn � 1 D b� � �

b�pn or zn D

pxp

n � 1t Dpxppt:

By means of the Jacobi matrix J and its absolute Jacobi determinant jJj�dt

dx

D�Dzn t Dxt

Dznx Dxx

�d zndx

D J

�d zndx

J D2

4

pppx� 12x�3=2zn

pp

0 1

3

5 ; jJj Dpppx; jJj�1 D

pxpp

we transform the surface element d zndx to the surface element jJj�1dtdx, namely

d zndx Dpxppdtdx:

Lemma B.14. (Gamma Function):

Our final action item is to calculate the marginal probability distributionsdF3D f3.t/dt of Student’s random variable t , namely

dF3 D f3.t/dt

f3.t/ WD 1p2p

1

� .p=2/

�1

2

�p=2 1Z

0

x.p�1/=2 exp

��12

�1C t2

p

�x

�dx:

Consult (Grobner and Hofreiter, 1973, p. 55), for the standard integral

1Z

0

x˛ exp.�ˇx/dx D ˛Š

ˇ˛C1 D� .˛ C 1/ˇ˛C1 and

1Z

0

x.p�1/=2 exp

��12

�1C t2

p

�x

dx D 2.pC1/=2 � .

pC12/

.1C t 2

p/.pC1/=2 ;

where p D n � 1 is the rank of the quadratic Helmert matrix Hn. Notice a resultof the gamma function � .pC1

2/ D .

p�12/Š. In summary, substituting the standard

integral


dF3 D f3.t/dt

f3.t/ D� .

pC12/

� .p

2/

1pp

1

.1C t 2

p/.pC1/=2

resulted in Student’s t distribution, namely the pdf of Student’s random variable.b� � �/=b� .

B-52 The Confidence Interval for the Mean, VarianceUnknown

Lemma B.12 is the basis for the construction of the confidence interval of the “true”mean, variance unknown, which we summarize in Lemma B.13, Example B.17contains all details for computing such a confidence interval, namely Table B.11,a collection of the most popular values of the coefficient of confidence, as well asTables B.12, B.13 and B.14, listing the quantiles for the confidence interval of theStudent random variable with p D n� 1 degrees of freedom. Figures B.9 and B.10illustrate the probability of two-sided confidence interval for the mean, varianceunknown, and the limits of the confidence interval. Table B.15 as a flow chart pavesthe way for the “fast computation” of the confidence interval for the “true” mean,variance unknown.

Lemma B.15. (confidence interval for the sample mean, varianceunknown):

The random variable t D pn.b� � �/=b� characterized by the ratio of the deviateof the sample mean b� D n�110y, BLUUE of �, from the “true” mean � and thestandard deviation b� , b�2 D .y � 1b�/0.y � 1b�/=.n � 1/ BIQUUE of the “true”variance �2, has the Student t-distribution with p D n � 1 “degrees of freedom”.The “true” mean � is an element of the two-sided confidence interval

� 2�b� � b�pnc1�˛=2; b�C b�p

nc1�˛=2Œ

with confidence

P

�b� � b�p

nc1�˛=2 < � < b�C b�p

nc1�˛=2

�D 1� ˛

of level 1�˛. For three values of the coefficient of confidence � D 1�˛, Table B.12is a list of associated quantiles c1�˛=2.


Example B.17. (confidence interval for the example mean b�, �2 unknown):

Suppose that a random sample2

664

y1

y2y3y4

3

775 D

2

664

1:2

3:4

0:6

5:6

3

775 ; b� D 2:7; b�2 D 5:2; b� D 2:3

of four observations is characterized by the sample mean b� D 2:7 and the samplevarianceb�2 D 5:2. 2.2:7 � �/=2:3 D pn.b� � �/=b� D t has Student’s pdf withp D n�1 D 3 degrees of freedom. The probability � D 1�˛ that t will be betweenany two arbitrarily chosen numbers c1 D �c and c2 D Cc is

Pf�c1 < t < c2g Dc2Z

c1

f .t/dt D � D 1 � ˛

Pf�c < t < cg DCcZ

�cf .t/dt D � D 1 � ˛

Pf�c < t < cg DcZ

�1f .t/dt �

�cZ

�1f .t/dt D � D 1 � ˛

cZ

�1f .t/dt D 1 � ˛

2;

�cZ

�1f .t/dt D˛

2

� is the coefficient of confidence, ˛ the coefficient of negative confidence, also calledcomplementary coefficient of confidence. The four representations of the probability� D 1�˛ to include t in the confidence interval�c < t < Cc have led to the linearVolterra integral equation of the first kind

cZ

�1f .t/dt D 1 � ˛

2D 1

2.1C �/:

Three values of the coefficient of confidence � or its complement ˛ are popular andlisted in Table B.10.

Table B.11 (Values of the coefficient of confidence):

� 0:950 0:990 0:999

˛ 0:050 0:010 0:001˛2

0:025 0005 0:000; 5

1 � ˛2D 1C�

20:975 0:995 0:999; 5


In solving the linear Volterra integral equation of the first kind

tZ

�1f .t�/dt� D 1 � 1

2˛.t/ D 1

2Œ1C �.t/�;

which depends on the degrees of freedom p D n � 1, Tables B.12–B.14 collectthe quantiles c1�˛=2=

pn given the coefficients of confidence or their complements

which we listed in Table B.11.

Table B.12 (Quantiles c1�˛=2=pn for the confidence interval of the Student

random variable with p D n � 1 degrees of freedom):

1 � ˛=2 D .1C �/=2 D 0:975; � D 0:95; ˛ D 0:05p n 1p

nc1�˛=2 p n 1p

nc1�˛=2

1 2 8:99 14 15 0:554

2 3 2:48 19 20 0:468

3 4 1:59 24 25 0:413

4 5 1:24 29 30 0:373

5 6 1:05 39 40 0:320

6 7 0:925 49 50 0:284

7 8 0:836 99 100 0:198

8 9 0:769 199 200 0:139

9 10 0:715 499 500 0:088



1 � ˛=2 D .1C �/=2 D 0:995; � D 0:990; ˛ D 0:010p n 1p

nc1�˛=2 p n 1p

nc1�˛=2

1 2 45:01 14 15 0:769

2 3 5:73 19 20 0:640

3 4 2:92 24 25 0:559

4 5 2:06 29 30 0:503

5 6 1:65 39 40 0:428

6 7 1:40 49 50 0:379

7 8 1:24 99 100 0:263

8 9 1:12 199 200 0:184

9 10 1:03 499 500 0:116




1 � ˛=2 D .1C �/=2 D 0:999:5; � D 0:999; ˛ D 0:001p n c1�˛=2 c1�˛=2=

pn p n c1�˛=2 c1�˛=2=

pn

1 2 636:619 450:158 14 15 4:140 1:069

2 3 31:598 18:243 19 20 3:883 0:868

3 4 12:941 6:470 24 25 3:725 0:745

4 5 8:610 3:851 29 30 3:659 0:668

5 6 6:859 2:800 30 41 3:646 0:655

6 7 5:959 2:252 40 41 3:551 0:555

7 8 5:405 1:911 60 61 3:460 0:443

8 9 5:041 1:680 120 121 3:373 0:307

9 10 4:781 1:512 1 1 3:291 0

Since Student’s pdf depends on p D n � 1, we have tabulated c1�˛=2=pn for

the confidence interval we are going to construct. The Student’s random variablet Dpn.b� � �/=b� is solved for �, evidently our motivation to introduce theconfidence intervalb�1 < � < b�2

t D pnb�� b�) b�� D b�p

nt ) � D b� � b�p

nt

b�1 WD b� � b�pnc1�˛=2 < � < b�C b�p

nc1�˛=2 DW b�2:

The interval b�1 < � < b�2 for the fixed value t D c1�˛=2 contains the “true” mean� with probability � D 1 � ˛.

P

�b�� b�p

nc1�˛=2 < � < b�C b�p

nc1�˛=2

�

DCcZ

�cf .t/dt D � D 1 � ˛

because of

cZ

�1f .t/dt D

c1�˛=2Z

�1f .t/dt D 1 � ˛

2:


t

a / 2 a / 2g = 1–a

f(t)

−c1−a / 2 +c1−a / 2

Fig. B.9 Two-sidedconfidence interval� 2�b�1;b�2Œ, Student’s pdffor p D 3 degrees of freedom.n D 4/b�1 Db��b�c1�˛=2=

pn,

b�2 Db�Cb�c1�˛=2=pn

m1ˆ m2m ˆ

ˆ ˆm – sc1−a /2 / n ˆ ˆm + sc1−a /2 / n

Fig. B.10 Two-sided confidence interval for the “true” mean �, quantile c1�˛=2

Figures B.9 and B.10 illustrate the coefficient of confidence and the probabilityfunction of a confidence interval.Let us specify all the integrals to our example.

c1�˛=2Z

�1f .t/dt D 1 � ˛=2

c0:975Z

�1f .t/dt D 0:975;

c0:995Z

�1f .t/dt D 0:995;

c0:999;5Z

�1f .t/dt D0:999; 5:

These data substituted into Tables B.12–B.14 lead to the triplet of confidenceintervals for the “true” mean.

case .a/ W � D 0:95; ˛ D 0:05; 1 � ˛=2 D 0:975p D 3; n D 4; c1�˛=2=

pn D 1:59

Pf2:7 � 2:3 � 1:59 < � < 2:7C 2:3 � 1:59g D 0:95

Pf�0:957 < � < 6:357g D 0:95:

case .b/ W � D 0:99; ˛ D 0:01; 1 � ˛=2 D 0:995p D 3; n D 4; c1�˛=2=

pn D 2:92


Pf2:7 � 2:3 � 2:92 < � < 2:7C 2:3 � 2:92g D 0:99

Pf�4:016 < � < 9:416g D 0:99:

case .c/ W � D 0:999; ˛ D 0:001; 1� ˛=2 D 0:999; 5

p D 3; n D 4; c1�˛=2=pn D 6:470Pf2:7 � 2:3 � 6:470 < � < 2:7C 2:3 � 6:470g D 0:999

Pf�12:181 < � < 17:581g D 0:999:

The results may be summarized as follows:

With probability 95% the “true” mean � is an element of the interval � �0:957, C6:357Œ. In contrast, with probability 99% the “true” mean � is anelement of the larger interval � � 4:016, C9:416Œ. Finally, with probability99.9is an element of the largest interval ��12:181,C17:581Œ. If we comparethe confidence intervals for the mean �, �2 known, versus �2 unknown, werealize much larger intervals for the second model. Such a result is not toomuch a surprise, since the model “�2 unknown” is much weaker than themodel “�2 known”.

Table B.15 (Flow chart: Confidence interval for the mean �, �2 unknown):

Choose a coefficient of confidence g according toTable B11.

Solve the linear Volterra integral equation of the first kindF(c1−a / 2) = 1−a / 2 = (1 + g ) / 2 by Table B12, Table B13 or TableB14 for a Student pdf with p = n−1 degrees of freedom: readc1−α / 2 / n.

Compute the sample mean m and the sample standard deviation s.

ˆ ˆ ˆ ˆ ˆ

ˆˆ

Compute sc1−a / 2 / n and m−sc1−a / 2 / n, m + sc1−a / 2 / n.

g


B-53 The Uncertainty Principle

Figure B.11 is the graph of the function

��.nI˛/ WD 2b�c1�˛=2=pn;

where �� is the length of the confidence interval of the “true” mean �; �2

unknown. The independent variable of the function is the number of observations n.The function ��.nI˛/ is plotted for fixed values of the coefficient of comple-mentary confidence ˛, namely ˛ D 10%; 5%; 1%: For reasons given later thecoefficient of complementary confidence ˛ is called uncertainty number. The graphfunction��.nI˛/ illustrates two important facts.

Fact #1: For a constant uncertainty number ˛, the length of the confidence interval�� is smaller the larger the number of observations n is chosen.

Fact #2: For a constant number of observations n, the smaller the number ofuncertainly ˛ is chosen, the larger is the confidence interval��.

Evidently, the diversive influences of (a) the length of the confidence interval��; (b) the uncertainly number ˛ and (c) the number of observations n, whichwe collected in The Magic Triangle of Figure B.12, constitute The UncertainlyPrinciple, in formula

��.n� 1/ W k�˛;

Fig. B.11 Length of the confidence interval for the mean against the number of observations


length of the confidence interval

uncertainty number a number of observations n

Fig. B.12 The Magic Triangle, constituents: (a) uncertainty number ˛ (b), number of observationsn, (c) length of the confidence interval ��

where k�˛ is called the quantum number for the mean � which depends on theuncertainty number ˛. Table B.15.2 is a list of those quantum numbers. Let usinterpret the uncertainty relation��.n� 1/ k�˛ . The product��.n� 1/ definesgeometrically a hyperbola which we approximated out of the graph of Fig. B.12.Given the uncertainty number ˛, the product ��.n � 1/ has a smallest number,here denoted by k�˛ . For instance, choose ˛ D 1% such that k�˛=.n� 1/ � �� or16:4=.n�1/ � ��. For n taken values 2, 11, 101, we get the inequalities 8:2 � ��,1:64 � ��, 0:164 � ��.

Table B.15.2 (Coefficient of complementary confidence ˛, uncertaintynumber ˛, versus quantum number of the mean k�˛ (Grafarend, 1970)):

˛ k�˛10% 6:6

5% 9:6

1% 16:4

! 0 !1

B-6 Sampling from the Gauss–Laplace Normal Distribution:A Fourth Confidence Interval for the Variance

Theorem B.10 already supplied us with the sampling distribution of the samplevariance, namely with the probability density function f .x/ of Helmert’s randomvariable x D .n � 1/b�2=�2 of Gauss-Laplace i.i.d. observations n of sample vari-anceb�2, BIQUUE of �2. B61 introduces accordingly the so far missing confidenceinterval for the “true” variance �2. Lemma B.15 contains the details, followedby Example B.16. Tables B.17–B.19 contain the properly chosen coefficients ofcomplementary confidence and their quantiles

c1.pI˛=2/; c2.pI 1 � ˛=2/;

B-6 Sampling from the Gauss–Laplace Normal Distribution: A Fourth 709

dependent on the “degrees of freedom” p D n � 1. Table B.20 as a flow chartsummarizes various steps in computing a confidence interval for the variance �2.B62 reviews The Uncertainty Principle which is built on (a) the coefficient ofcomplementary confidence ˛, also called uncertainty number, (b) the number ofobservations n and (c) the length��2.nI˛/ of the confidence interval for the “true”variance �2.

B-61 The Confidence Interval for the Variance

Lemma B.15 summaries the construction of a two-sided confidence interval forthe “true” variance based upon Helmert’s Chi Square distribution of the randomvariable .n�1/b�2=�2 where b�2 is BIQUUE of �2. Example B.18 introduces arandom sample of size n D 100 with an empirical variance b�2 D 20:6. Basedupon coefficients of confidence and complementary confidence given in Table B.14,related confidence intervals are computed. The associated quantiles for Helmert’sChi Square distribution are tabulated in Table B.15 .� D 0:95/, Table B.18.� D 0:99/ and Table B.19 .� D 0:998/: Finally, Table B.20 as a flow chartpares the way for the “fast computation” of the confidence interval for the “true”variance �2.

Lemma B.16. (confidence interval for the variance):

The random variable x D .n � 1/b�2=�2, also called Helmert’s 2, characterizedby the ration of the sample variance b�2 BIQUUE of �2, and the “true” variance�2, has the 2' pdf of p D n � 1 degrees of freedom, if the random observationsyi ; i 2 f1; : : : ng are Gauss-Laplace i.i.d. The “true” variance �2 is an element ofthe two-sided confidence interval

�2 2 � .n � 1/b�2c2.pI 1 � ˛=2/;

.n � 1/b�2c1.pI˛=2/Œ

with confidence

P

�.n � 1/b�2

c2.pI 1 � ˛=2/ < �2 <

.n � 1/b�2c1.pI˛=2/

�D 1 � ˛ D �

of level 1 � ˛. Tables D.8, D.18 and D.16 list the quantiles c1.pI 1 � ˛=2/and c2.pI˛=2/ associated to three values of Table B.19 of the coefficient ofcomplementary confidence 1 � ˛.In order to make yourself more familiar with Helmert’s Chi Square distribution werecommend to solve the problems of Exercise B.1.


Exercise B.1. (Helmert’s Chi Square 2' distribution):

Helmert’s random variable x WD .n � 1/b�2=�2 D pb�2=�2 has the non-symmetric 2' pdf. Prove that the first four central moments are,

(a) 1 D 0, Efxg D � D p(b) 2 D �2x D 2p(c) 3 D .2p/3=2.8=p/1=4 (coefficient of skewness 23 =

22 D

p8=p/

(d) 4 D 6�4.1C 2p/ (coefficient of curtosis 4=22 � 3 D 3C 12=p/.Guide

.a/Efxg D p

�2Efb�2g

Efb�2g D �2.unbiasedness/

3

5) Efxg D p

.b/

Dfb�2g D 2

n � 1�4 D 2

p�4

Dfxg D p2

�4Dfb�2g

3

775) Dfxg D 2p:

Example B.18. (confidence interval for the sample variance b�2):

Suppose that a random sample .y1; : : : yn/ 2 Y of size n D 100 has led to anempirical varianceb�2 D 20:6. xD .n�1/b�2=�2 or 99 � 20:6=�2 D 2039:4=�2 hasHelmert’s pdf with p D n� 1 D 99 degrees of freedom. The probability � D 1�˛that x will be between c1.pI˛=2/ and c2.pI 1 � ˛=2/ is

Pfc1.pI˛=2/<x < c2.pI 1�˛=2/g Dc2Z

c1

f .x/dx D � D 1 � ˛

Pfc1.pI˛=2/<x < c2.pI 1�˛=2/g Dc2Z

0

f .x/dx �c1Z

0

f .x/dx D � D 1 � ˛

c2Z

0

f .x/dx D F.c2/ D 1 � ˛=2 D .1C �/=2;

c1Z

0

f .x/dx D F.c1/ D ˛=2 D .1 � �/=2

Pfc1.pI˛=2/<x < c2.pI 1�˛=2/g D F.c2/� F.c1/D .1C �/=2� .1� �/=2D�

c1.pI˛=2/ < x < c2.pI 1 � ˛=2/, c1.pI˛=2/ < .n � 1/b�2

�2< c2.pI 1 � ˛=2/


or

1

c2<

�2

.n � 1/b�2 <1

c1, .n � 1/b�2

c2< �2 <

.n � 1/b�2c1

P

�.n � 1/b�2

c2.pI 1 � ˛=2/ < �2 <

.n � 1/b�2c1.pI˛=2/

�D 1 � ˛ D �:

Since Helmert’s pdf f .xIp/ is now-symmetric there arises the question how todistribute the confidence � or the complementary confidence ˛ D � � 1 on theconfidence interval limits c1 and c2, respectively. If we setup F.c1/ D ˛=2, we definea cumulative probability half of the complementary confidence. If F.c2/ � F.c1/ DPfc1.pI˛=2/ < x < c2.pI 1 � ˛=2/g D 1 � ˛ D � is the cumulative probabilitycontained in the interval c1 < x < c2, we derive F.c2/ D 1 � ˛=2. Accordinglyc1.pI˛=2/ < x < c2.pI 1�˛=2/ is the confidence interval based upon the quantilec1 with cumulative probability ˛=2 and the quantile c2 with cumulative probability1 � ˛=2. The four representations of the cumulative probability of the confidenceinter-val c1 < x < c2 establish two linear Volterra integral equations of the firstkind

c1Z

0

f .x/dx D ˛=2 and

c2Z

0

f .x/dx D 1 � ˛=2;

dependent on the degree of freedom p D n � 1 of Helmert’s pdf f .x; p/.As soon as we have established the confidence interval c1.pI˛=2/ < x < c2.pI 1�˛=2/ for Helmert’s random variable x D .n�1/b�2=�2 D pb�2=�2, we are left withthe problem of how to generate a confidence interval for the “true” variance �2, thesample varianceb�2 given. If we take the reciprocal interval c�1

2 < x�1 < c�11 for

Helmert’s inverse random variable 1=x D �2=Œ.n � 1/b�2�, we are able to multiplyboth sides by .n � 1/b�2. In summary, a confidence interval which corresponds toc1 < x < c2 is .n� 1/b�2=c2 < �2 < .n � 1/b�2=c1.Three values of the coefficient of confidence � or of complementary confidence˛ D 1 � � which are most popular we list in Table B.16.

Table B.16 (Values of the coefficient of confidence � ,c1.pI˛=2/ and c2.pI 1�˛=2/):

˛ 0:950 0:990 0:998

˛=2 0:050 0:010 0:002

˛=2 0:025 0:005 0:001

1 � ˛=2 0:975 0:995 0:999


In solving the linear Volterra integral equations of the first kind

c1Z

0

f .x/dx D F.c1/ D ˛=2 and

c2Z

0

f .x/dx DF.c2/ D 1 � ˛=2;

which depend on the degrees of freedom p D n � 1, Tables B.17–B.19 collect thequantiles c1.pI˛=2/ and c2.pI 1 � ˛=2/ for given values p and ˛.

Table B.17 (Quantiles c1.pI˛=2/, c2.pI 1 � ˛=2/ for the confidence intervalof the Helmert random variable with p D n � 1 degrees of freedom):

˛=2 D 0:025; 1 � ˛=2 D 0:975; ˛ D 0:05; � D 0:95

p n c1.pI˛=2/ c2.pI 1 � ˛=2/ p n c1.pI˛=2/ c2.pI 1 � ˛=2/1 2 0:000 5:02 14 15 5:63 26:1

2 3 0:506 7:38 19 20 8:91 32:9

3 4 0:216 9:35 24 25 12:4 39:4

4 5 0:484 11:1 29 30 16:0 45:7

5 6 0:831 12:8 39 40 23:7 58:1

6 7 1:24 14:4 49 50 31:6 70:2

7 8 1:69 16:0 99 100 73:4 128

8 9 2:18 17:5

9 10 2:70 19:0


˛=2 D 0:005; 1 � ˛=2 D 0:995; ˛ D 0:01; � D 0:99


2 3 0:010 9 10 1:73 23:6

3 4 0:072

4 5 0:207 14 15 4:07 31:3

19 20 6:84 38:6

5 6 0:412 24 25 9:89 45:6

6 7 0:676 29 30 13:1 52:3

7 8 0:989

39 40 20 65:5

49 50 27:2 78:2

99 100 66:5 139


0.5

f(x)

0 c1(p;a / 2) c2 (p;1−a / 2)

Fig. B.13 Two-sided confidence interval �2 2�p�2=c2; p�2=c1Œ Helmert’s pdf, f(x)


˛=2 D 0:001; 1 � ˛=2 D 0:999; ˛ D 0:002; � D 0:998


2 3 0:00 13:82 14 15 3:04 36:12

3 4 0:02 16:27 19 20 5:41 43:82

4 5 0:09 18:47 24 25 8:1 51:2

5 6 0:21 20:52 29 30 11:0 58:3

6 7 0:38 22:46 50 51 24:7 86:7

7 8 0:60 24:32 70 71 39:0 112:3

8 9 0:86 26:13 99 100 60:3 147:4

100 101 61:9 149:4

Those data collected in Tables B.17–B.19 lead to the triplet of confidence intervalsfor the “true” variance

case .a/ W � D 0:95; ˛ D 0:05; ˛=2 D 0:025; 1 � ˛=2 D 0:975p D 99; n D 100; c1.pI˛=2/ D 73:4; c2.pI 1 � ˛=2/ D 128

P

�99 � 20:6

128< �2 <

99 � 20:673:4

�D 0:95

Pf15:9 < �2 < 27:8g D 0:95:


Fig. B.14 Two-sided confidence interval for the “true” variance �2, quantiles c1.pI˛=2/ andc2.pI 1� ˛=2/

case .b/ W � D 0:99; ˛ D 0:01; ˛=2 D 0:005; 1 � ˛=2 D 0:995p D 99; n D 100; c1.pI˛=2/ D 66:5; c2.pI 1 � ˛=2/ D 139

P

�99 � 20:6

139< �2 <

99 � 20:666:5

�D 0:99

Pf14:7 < �2 < 30:7g D 0:99:

case .c/ W � D 0:998; ˛ D 0:002; ˛=2 D 0:001; 1 � ˛=2 D 0:999p D 99; n D 100; c1.pI˛=2/ D 60:3; c2.pI 1 � ˛=2/ D 147:3

P

�99 � 20:6

147:3< �2 <

99 � 20:660:3

�D 0:996

Pf13:8 < �2 < 33:8g D 0:998:

The results can be summarized as follows. With probability 95%, the“true” variance �2 is an element of the interval [15.9, 27.8]. In contrast,with probability 99%, the “true” variance is an element of the largerinterval [14.7, 30.7]. Finally, with probability 99.8% the “true” variance is anelement of the largest interval [13.8, 33.8]. If we compare the confidenceintervals for the variance �2, we realize much larger intervals for smallercomplementary confidence namely 5%, 1% and 0.2%. Such a result issubject of The Uncertainty Principle.


Table B.20 (Flow chart – confidence interval for the variance �2):

Compute the sample variance s2and the quantiles

.

Choose a coefficient of confidence g according to Table B16.

Solve thelinear Volterra integral equation of the first kindF(c1(p;a /2)) – a / 2, F(c2(p;1−a / 2))−1−a / 2

by Table B17, Table B18 or Table B19 for a Helmert pdf withp = n−1 degrees of freedom.

s2 ∈ ] [(n−1)s2 (n−1)s2

c2(p;a / 2) c1(p;a / 2)

B-62 The Uncertainty Principle

Figure B.15 is the graph of the function

��2.nI˛/ WD .n � 1/b�2�

1

c1.n � 1I˛=2/ �1

c2.n � 1I 1 � ˛=2/�;

where ��2 is the length of the confidence interval of the “true” variance �2.The independent variable of the functions is the number of observations n. Thefunction ��2.nI˛/ is plotted for fixed values of the coefficient of complementaryconfidence ˛, namely ˛ D 5%, 1%, 0.2%. For reasons given later on the coefficientof complementary confidence ˛ is called uncertainty number. The graph of thefunction��2.nI˛/ illustrates two important facts.

• Fact #1: For a contrast uncertainty number ˛; the length of the confidence interval��2 is smaller, the larger number of observations n is chosen.

• Fact #2: For a contrast number of observations n, the smaller the number ofuncertainty ˛ is chosen, the larger is the confidence interval��2.

Evidently, the divisive influences of (a) the length of the confidence interval ��2,(b) the uncertainty number ˛ and (c) the number of observations n, which wecollect in The Magic Triangle of Fig. B.16, constitute The Uncertainty Principle,formulated by the inequality

��2.n � 1/ k�2˛;


Fig. B.15 Length of the confidence interval for the variance against the number of observations

length of the confidence interval

uncertainty number a number of observations n

Fig. B.16 The magic triangle, constituents: (a) uncertainty number ˛ (b), number of observationsn, (c) length of the confidence interval ��2

where k�2˛ is called quantum number for the variance �2. The quantum numberdepends on the uncertainty number ˛. Let us interpret the uncertainty relation��2.n � 1/ k�2˛. The product ��2.n � 1/ defines geometrically a hyperbolawhich we approximated to the graph of Fig. B.15. Given the uncertainty number˛, the product ��2.n � 1/ has a smallest number denoted by k�2˛. For instance,choose ˛ D 1% such that k�2˛=.n � 1/ � ��2 or 42=.n � 1/ � ��2. For thenumber of observations n, for instance n D 2; 11; 101, we find the inequalities42 � ��2; 4:2 � ��2; 0:42 � ��2.Table B.21 (Coefficient of complementary confidence ˛, uncertaintynumber ˛, versus quantum number of the mean k�2˛ (Grafarend, 1970):

˛ k�2˛10% 19:5

5% 25:9

1% 42:0

! 0 !1

At the end, pay attention to the quantum numbers k�2˛ listed in Table B.21.

B-7 Sampling from the Multidimensional Gauss–Laplace Normal Distribution 717

B-7 Sampling from the Multidimensional Gauss–LaplaceNormal Distribution: The Confidence Region for theFixed Parameters in the Linear Gauss–Markov Model

Example B.19. (special linear Gauss–Markov model, marginal probabilitydistributions):

For a simple linear Gauss–Markov model Efyg D A�, A 2 Rn�m, rkA D m,

namely n D 3, m D 2, Dfyg D I�2 of Gauss-Laplace i.i.d. observations, we aregoing to compute

• The sample pdf ofb� BLUUE of �• The sample pdf ofb�2 BIQUUE of �2.

We follow the action within seven items. First, we identify the pdf of Gauss-Laplacei.i.d. observations y 2 R

n. Second, we review the estimations of b� BLUUE of �as well as b�2 BIQUUE of �2. Third, we decompose the Euclidean norm of theobservation space ky �Efygk2 into the Euclidean norms ky �Ab�k2 and kb� � �k2.Fourth, we present the eigenspace A0A analysis and the eigenspace synthesis of theassociated matrices M WD In � A.A0A/�1A0 and N WD A0A within the Eulideannorms ky � A�k2 D y0My and kb� � �k2A0A D .b� � �/0N.b� � �/, respectively.The eigenspace representation leads us to canonical random variables .z1; : : : ; zn�m/relating to the norm ky � A�k2 D y0My and .zn�mC1; : : : ; zn/ relating to the normkb��k2N which are standard Gauss-Laplace normal. Fifth, we derive the cumulativeprobability of Helmert’s random variable x WD .n � rkA/b�2=�2 D .n �m/b�2=�2and of the unknown parameter vectorb� BLUUE of � or its canonical counterpartb�BLUUE of �, multivariate Gauss-Laplace normal. Action six generates the marginalpdf ofb� orb�, both BLUUE of � or �, respectively. Finally, action seven leads us toHelmert’s Chi Square distribution 2p with p D n� rkA D n�m (here: n�m D 1)“degrees of freedom”.


Let us assume an experiment of three Gauss-Laplace i.i.d. observationsŒy1; y2; y3�

0 D y which constitute the coordinates of the observation space Y,dim The observationsyi , i 2 f1; 2; 3g are related to a parameter space � withcoordinates Œ�1; �2� D � in the sense to generate a straight line yfkg D �1 C �2k.The fixed effects �j ; j 2 f1; 2g, define geometrically a straight line, statistically aspecial linear Gauss–Markov model of one variance component, namely


the first moment

Efyg D A� � E8<

:

2

4y1y2y3

3

5

9=

;D2

41 k11 k21 k3

3

5��1�2

D2

41 0

1 1

1 2

3

5��1�2

; rkA D 2:

The central second moment

Dfyg D In�2 � Dfyg D D8<

:

2

4y1y2

y3

3

5

9=

;D2

41 0 0

0 1 0

0 0 1

3

5 �2; �2 < 0:

k represents the abscissa as a fixed random, y the ordinate as the observation,naturally a random effect. Samples of the straight line are taken at k1 D 0,k2 D 1, k3 D 2, this calling for y.k1/ D y.0/ D y1, y.k2/ D y.1/ D y2,y.k3/ D y.2/ D y3, respectively. Efyg is a consistent equation. Alternatively, wemay sayEfyg 2 R.A/. The matrix A 2 R

3�2 is rank deficient by p D n�rkA D 1,also called “degree of freedom”. The dispersion matrix Dfyg, the central momentof second order, is represented as a linear model, too, namely by one-variancecomponent �2. The joint probability function of the three Gauss-Laplace i.i.d.observations

dF D f .y1; y2; y3/dy1dy2dy3 D f .y1/f .y2/f .y3/dy1dy2dy3

f .y1; y2; y3/ D .2/�3=2��3 exp

�� 1

2�2.y �Efyg/0.y �Efyg/

�

will be transformed by means of the special linear Gauss–Markov model with one-variance component.


For such a transformation, we needb� BLUUE of � andb�2 BIQUUE �2.

b� BLUUE of � W

2

666666664

b� D .A0A/�1A0y;

A0A D�3 3

3 5

; .A0A/�1 D 1

6

�5 �3�3 3

;

b� D 1

6

�5 2 �1�3 0 3

2

4y1y2

y3

3

5 ;

b�2 BIQUUE of �2 W

2

6666664

b�2 D 1

n � rkA.y � A�/0.y � A�/

D 1

n � rkAy0.In �A.A0A/�1A0/y;

b�2 D 1

6.y21 � 4y1y2 C 2y1y3 C 4y22 � 4y2y3 C y23/:



The quadratic form .y� Efyg/0.y� Efyg/ allows the fundamental decomposition

y � Efyg D y � A� D y �A� C A.b� � �/.y � Efyg/0.y � Efyg/ D .y � A�/0.y� A�/

D .y � A�/0.y �A�/C .b� � �/0A0A.b� � �/

.y �Efyg/0.y �Efyg/ D .n � rkA/b� 2 C .b� � �/0A0A.b� � �/

.y �Efyg/0.y �Efyg/ D b� 2 C .b� � �/0�3 3

3 5

.b� � �/:


In order to bring the quadratic form .y � Efyg/0.y � Efyg/ D .n � rkA/b�2CC.b� � �/0A0A.b� � �/ into a canonical form, we introduce the generalised forwardand backward Helmert transformation

HH0 D In

z D ��1H.y� Efyg/ D ��1H.y � A�/ and

y �Efyg D �H0z

.y �Efyg/0.y �Efyg/ D �2z0HH0z D �2z0z

1

�2.y � Efyg/0.y � Efyg/ D z0z D z21 C z22 C z23:

How to relate the sample variance b�2 and the sample quadratic form .b� ��/0A0A.b� � �/ to the canonical quadratic form z0z?

Previously, for the example of direct observations in the special linear Gauss–Markov model Efyg D 1�, Dfyg D In�2 we succeeded to relate z21 C � � � C z2n�1to b�2 and z2n to .b� � �/2. Here the sample variance b�2, BIQUUE �2, as well asthe quadratic form of the deviate of the sample parameter vectorb� from the “true”parameter vector � have been represented by

b�2 D 1

n � rkA.y� A�/0.y �A�/ D 1

n � rkAy0ŒIn � A.A0A/�1A0�y

rkŒIn � A.A0A/�1A0� D n � rkA D n �m D 1 versus.b� � �/0A0A.b� � �/; rk.A0A/ D rkA D m:


The eigenspace of the matrices M and N, namely

M WD In � A.A0A/�1A and N WD A0A

D 1

6

2

41 �2 1

�2 4 �21 �2 1

3

5 D�3 3

3 5

;

will be analyzed.

The eigenspace analysis of the The eigenspace analysis of the

matrix M j 2 f1; : : : ; ng matrices N;N�1 i 2 f1; : : : ; mgV0MV U0NU D Diag.�1; : : : ; �m/ D �N

D"

V0

1

V0

2

#

M ŒV1;V2� U0N�1U D Diag.�1; : : : ; �m/ D �N�1

D Diag.�1; : : : ; �n�m; 0; : : : ; 0/ D �M

�1 D 1

�1; : : : ; �m D 1

�m

�1 D 1

�1; : : : ; �m D 1

�m

rkM D n�m rkN D rkN�1 D m

Orthonormality of the eigencolumns Orthonormality of the eigencolumns

V0

1V1 D In�m; V0

2V2 D Im

V0

1V2 D 0 2 R.n�m/�m

2

4 V0

1

V0

2

3

5h

V1 V2

iD2

4 In�m 0

0 Im

3

5

U0U D Im

v01v1 D 1

v01v2 D 0: : :

v0n�1vn D 0v0nvn D 1

u01u1 D 1

u01u2 D 0: : :

u0m�1um D 0u0mum D 1

V 2 SO.n/ W eigencolumns W.M� �j In/vj D 0

eigenvaluesjM � �j Inj D 0

U 2 SO.m/ W eigencolumns W.N � �j In/uj D 0eigenvaluesjN � �i Imj D 0

in particular


eigenspace analysis of the matrix M;

rkM D n�m; M 2 R3�3; A 2 R

3�2; rkM D 1

�M D Diag.1; 0; 0/

V D ŒV1; V2� D

2

664

0:4082 0:7024 0:5830

�0:8165 0:5667 �0:11090:4082 0:4307 �0:8049

3

775

V1 2 R3�1 V2 2 R

3�2

eigenspace analysis of the matrix N;

rkN D m; N 2 R2�2; rkN D 2

�N D Diag.0:8377; 7:1623/

U D"0:8112 0:5847

�0:5847 0:8112

#

U 2 R2�2

to be completed by

eigenspace synthesis of the matrix M

M D V�MV0

M D ŒV1;V2��M

"V0

1

V0

2

#

D ŒV1;V2� Diag.�1; : : : ; �n�m; 0; : : : ; 0/

"V0

1

V0

2

#

M D V1 Diag.�1; : : : ; �n�m/V0

1

eigenspace synthesis of the matrix N;N � 1

N D U�NU0; N�1 D U�N�1U0

N D U Diag.�1; : : : ; �m/U0

N�1 D U Diag.�1; : : : ; �m/U0

in particular

M D2

64

0:4082

�0:81650:4082

3

75�1

h0:4082 �0:8165 0:4082

iN D

"0:8112 0:5847

�0:5847 0:8112

#"�1 0

0 �2

#

"0:8112 �0:58470:5847 0:8112

#

�1 D 1 versus �1 D 0:8377; �2 D 7:1623

N�1 D�0:8112 0:5847

�0:5847 0:8112 �

�1 0

0 �2

�0:8112 �0:58470:5847 0:8112

�1 D 1 versus �1 D ��11 D 1:1937; �2 D ��1

2 D 0:1396:

The non-vanishing eigenvalues of the matrix M have been denoted by.�1; : : : ; �n�m/, m eigenvalues are zero such that eigen, 0; : : : ; 0/. The eigenvaluesof the regular matrix N span eigen .N/ D .�1; : : : ; �m/. Since the dispersion matrixDfb�g D .A0A/�1�2 D N�1�2 is generated by the inverse of the matrix A0A D N,we have computed, in addition, the eigenvalues of the matrix N�1 by means ofeigen.N�1/ D .�1; : : : ; �m/. The eigenvalues of N and N�1, respectively, arerelated by

�1 D ��11 ; : : : ; �m D ��1

m or �1 D ��11 ; : : : ; �m D ��1

m :


In the example, the matrix M had only one non-vanishing eigenvalue �1 D 1. Incontrast, the regular matrix N was characterized by two eigenvalues �1 D 0:8377

and �2 D 7:1623, its inverse matrix N�1 by �1 D 1:1937 and �2 D 0:1396.The two quadratic forms, namely

y0My D y0V�MV0y versus .b� � �/0N.b� � �/ D .b� � �/0U�MU0.b� � �/;

build up the original quadratic form

1

�2.y �Efyg/0.y �Efyg/ D 1

�2y0MyC 1

�2.b� � �/0N.b� � �/

D 1

�2y0V�MV0yC 1

�2.b� � �/0U�NU0.b� � �/

in terms of the canonical random variables

V0y D y� , y D Vy� and U0.b� � �/ Db��

such that

1


�2.y�/0�My� C 1

�2.b� � �/�N.b� � �/

1


�2

n�mX

jD1.y�j /2�j C 1

�2

mX

iD1.b�i � �i /2�i

1

�2.y �Efyg/0.y �Efyg/ D z21 C � � � C z2n�m C z2

n�mC1C � � � C z2n:

The quadratic form z0z splits up into two terms, namely

z21 C � � � C z2n�m

D 1

�2

n�mX

jD1.y�j /2�j

and

z2n�mC1 C � � � C z2n

D 1

�2

mX

iD1.b�i � �i /2�j

here

z21 D1

�2y21� D

b�2

�2and

z22 C z23 D1

�2Œ.b�1 � �1/2�1 C .b�2 � �2/2�2�; or

z21 D1

6�2.y21 � 4y1y2 C 2y1y3 C 4y22 � 4y2y3 C y23/ and


z22 C z23 D1

�2Œ0:8377.b�1 � �1/2 C 7:1623.b�2 � �2/2� D 1

�2.b� � �/0

�3 3

3 5

.b� � �/:


We are left with the problem to transform the cumulative probability dF Df .y1; y2;y3/dy1dy2dy3 into the canonical form dF Df .z1; z2; z3/d z1d z2d z3. Here we takeadvantage of Corollary B.3. First, we introduce Helmert’s random variable x WD z21and the random variablesb�1 and b�2 of the unknown parameter vector b� of fixedeffects .b� � �/0A0A.b� � �/ D z22 C z23 D kzk2A0A if we denote z WD Œz2; z3�0.

d z1d z2 Dp

det A0A db�1db�2 Dp6 db�1db�2;

according to Corollary B.3 is special representation of the surface element by meansof the matrix of the metric A0A. In summary, the volume element

d z1d z2d z3 D dxpx

pdet A0A db�1db�2

leads to the first canonical representation of the cumulative probability

dF D 1p2

exp

��12x

�dxpx

jA0Aj1=22�2

exp

�� 1

2�2.b� � �/0A0A.b� � �/

db�1db�2:

The left pdf establishes Helmert’s pdf of x D z21 D b�2=�2, dx D ��2db�2. Incontrast, the right pdf characterizes the bivariate Gauss-Laplace pdf of kb� � �k2.Unfortunately, the bivariate Gauss-Laplace A0A normal pdf is not given in thecanonical form. Therefore, second we do correlate kb� � �k2A0A by means ofeigenspace synthesis.

A0A D U�A0AU0 D U Diag.�1; �2/U0 D U Diag

�1

�1;1

�2

�U0

kb� � �k2A0A WD .b� � �/0A0A.b� � �/ D .b� � �/0U Diag

�1

�1;1

�2

�U0.b� � �/

jA0Aj1=2 D p�1�2 D 1p�1�2

b� � � WD U0.b� � �/,b� � � D U.b� � �/

kb� � �k2A0A D .b� � �/0 Diag

�1

�1;1

�2

�.b� � �/ D kb� � �k2D


kb� � �k2A0A D .b� � �/0�3 3

3 5

.b� � �/

D .b� � �/02

64

1

1:19370

01

0:1396

3

75 .b�� / D kb� � �k2D:

By means of the canonical variablesb� D U0b� we derive the cumulative probability

dF D 1

�p2

1px

exp

��12x

�dx

1

2�2p�1�2

� exp

�� 1

2�2

�.b�1 � �1/2

�1C .b�2 � �2/2

�2

�db�1db�2 or

dF D f .x/f .b�1/f .b�2/dxdb�1db�2:Third, we prepare ourselves for the cumulative probability dF D f .z1; z2; z3/d z1d z2d z3. We depart from the representation of the volume element

d z1d z2d z3 D dx

2px

pdet A0Adb�1db�2

subject to

x D z21 D1

�2b�2 D 1

�2y0My

b� D .A0A/�1A0y:

The Helmert random variable is a quadratic form of the coordinates .y1; y2; y3/of the observation vector y. In contrast, b� BLUUE of � is a linear form of thecoordinates .y1; y2; y3/ of observation vector y. The transformation of the volumeelement

dxdb�1db�2 D jJxjdy1dy2dy3is based upon the Jacobi matrix Jx.y1; y2; y3/

Jx D

2

64

D1x D2x D3x

D1b�1 D2

b�1 D3b�1

D1b�2 D2

b�2 D3b�2

3

75 D

"a0

.A0A/�1A0

#1 � 32 � 3


a D2

4D1x

D2x

D3x

3

5 D 2

�2My D 1

3�2

2

4y1 � 2y2 C y3�2y1 C 4y2 � 2y3y1 � 2y2 C y3

3

5

x D 1

�2y0My D 1

6�2.y21 � 4y1y2 C 2y1y3 C 4y22 � 4y2y3 C y23 /

.A0A/�1A0 D 1

6

�5 2 �1�3 0 3

Jx D 1

6�2

2

42y1 � 4y2 C 2y3 �4y1 C 8y2 � 4y3 2y1 � 4y2 C 2y3

5 2 �1�3 0 3

3

5

det Jx Dp

det.A0A/�1 detŒa0a � a0A.A0A/�1A0a�

det Jx Dp

detŒa0a � a0A.A0A/�1A0a�p

det.A0A/

a0a D 4

�4y0M0My D 4

�4y0My

a0A.A0A/�1A0a D 4

�4y0M0A.A0A/�1A0My

D 4

�4y0ŒI3 �A.A0A/�1A0�A.A0A/�1A0ŒI3 � A.A0A/�1A0�y D 0

det Jx D 2p

y0My

�2p

det.A0A/

j det Jyj D j det Jx j�1 D �2

2

pdet.A0A/p

y0My:

The various representations of the Jacobian will finally lead us to the special formof the volume element

1

2dx db�1 db�2 D 1

�2

py0My

jA0Aj1=2 dy1dy2dy3

and the cumulative probability


dF D 1

�3.2/3=2exp

��12.z21 C z22 C z23/

d z1d z2d z3

D 1

�p2

exp

��12x

�dxpx

jA0Aj1=22�2

exp

�� 1

2�2.b� � �/0A0A.b� � �/

db�1db�2

D 1

�3.2/3=2exp

��12.y � Efyg/0.y � Efyg/

dy1dy2dy3:


The first target is to generate the marginal pdf of the unknown parameter vectorb�,BLUUE of �.

dF1 D1Z

0

1p2

exp

��12x

�dxpx

jA0Aj1=22�2

exp

�� 1

2�2.b� � �/0A0A.b� � �/

�db�1db�2

dF1 D1Z

0

1p2

exp

��12x

�dxpx

1

2�2p�1�2

exp

� 1

2�2

2X

iD1

.b�i � �i /2�i

!

db�1db�2

Let us substitute the standard integral

1Z

0

1p2

exp

��12x

�dxpxD 1;

in order to have derived the marginal probability

dF1 D f1.b�j�; .A0A/�1�2/db�1db�2

f1.b�/ WD jA0Aj1=22�2

exp

�� 1

2�2.b� � �/0A0A.b� � �/

�

dF1 D f1.b�j�;�N�1�2/db�1db�2

f1.b�/ D 1

2�21p�1�2

exp

�12

2X

iD1

.b�i � �i /2�i

!

:

The seventh action item

The second target is to generate the marginal pdf of Helmert’s random variablex WD b�2=�2, b�2 BIQUUE �2, namely Helmert’s Chi Square pdf 2p with p Dn � rkA (here p D 1) “degree of freedom”.


dF2 D 1p2

exp

��12x

�dxpx

C1Z

�1

C1Z

�1

jA0Aj1=22�2

exp

�� 1

2�2kb� � �k2A0A

�db�1db�2:

Let us substitute the integral

C1Z

�1

C1Z

�1

jA0Aj1=22�2

exp

�� 1

2�2kb� � �k2A0A

�db�1db�2 D 1

in order to have derived the marginal distribution

dF2 D f2.x/dx; 0 � x � 1

f2.x/ D 1

2p=2� .p=2/xp�22 exp

��12x

�; subject to

p D n � rkA D n �m; here W p D 1; �

�1

2

�D p

f2.x/ D 1p2

1px

exp

��12x

�:

The results of the example will be generalized in Lemma B.

Theorem B.11. (marginal probability distributions, special linear Gauss-Markov model):

Efyg D A�

Dfyg D In�2subject to

"A 2 R

n�m; rkA D m; Efyg 2 R.A/�2 2 R

C

defines a special linear Gauss–Markov model of fixed effects � 2 Rm and �2 2 R

Cbased upon Gauss-Laplace i.i.d. observations y WD Œy1; : : : ; yn�0.

b� D .A0A/�1A0y subject to

"Efb�g D �Dfb�g D .A0A/�1�2

and

b�2 D 1

n � rkA.y� A�/0.y �A�/ subject to

2

4Efb�2g D b�2

Dfb�2g D 2�4

n � rkA


identify b� BLUUE � and b�2 BIQUUE of �2. The cumulative pdf of the multidi-mensional Gauss-Laplace probability distribution of the observation vector y DŒy1; : : : ; yn�

0 2 Y

f .yjEfyg;Dfyg D In�2/dy1 � � �dyn

D 1

.2/n=2�nexp

�� 1

2�2.y �Efyg/0.y �Efyg/

�dy1 � � �dyn

D f1.b�/f2.b�2/db�1 � � �db�mdb�2

can be split into two marginal pdfs f1.b�/ of b�, BLUUE of �, and f2.b�2/ of b�2,BIQUUE of �2.

(i) b� BLUUE of �The marginal pdf ofb�;BLUUE of �, is represented by(1st version)

dF1 D f1.b�/d�1 � � �d�m

f1.b�/ D 1

.2/m=2�m

ˇˇA0A

ˇˇ1=2 exp

�� 1

2�2.b� � �/0A0A.b� � �/

�d�1 � � �d�m or

(2nd version)

dF1 D f1.b�/d�1 � � �d�m

f1.b�/ D 1

.2/m=2.�2/m=2.�1�2 � � ��m�1�m/�1=2 exp

� 1

2�2

mX

iD1

.b�i � �/2�i

!

;

by means of Principal Component Analysis PCA also called Singular ValueDecomposition (SVD) or Eigenvalue Analysis (EIGEN) of .A0A/�1,

� D U0��

b� D U0�b�

subject to

"U0�.A0A/�1U� D � D Diag.�1; �2; : : : ; �m�1; �m/

U0�U� D Im; det U� D C1

f1.b�j�;��2/ D f .b�1/f .b�2/ � � �f .b�m�1/f .b�m/

f .b�i / D 1

�p�ip2

exp

�� 1

2�2.b�i � �i /2

�i

�8i 2 f1; : : : ; mg:

The transformed fixed effects .b�1; : : : ;b�m/, BLUUE of .�1; : : : ; �m/, are mutu-ally independent and Gauss-Laplace normal

b�i � N .�i j�2�i / 8i 2 f1; : : : ; mg:


(third version)

zi WD b�i � �p�2�i

W f1.zi /d zi D 1p2

exp

��12

z2i

�d zi 8i 2 f1; : : : ; mg:

(ii) b�2 BIQUUE �2

The marginal pdf ofb�2, BIQUUE �2, is represented by(first version)

p D n � rkA


f2.b�2/ D 1

�p2p=2� .p=2/pp=2b�p�2 exp

��12pb�2

�2

�:

(second version)

dF2 D f2.x/dx

x WD .n � rkA/b�2

�2D p

�2b�2 D 1

�2.y �A�/0.y �A�/

f2.x/ D 1

2p=2� .p=2/xp2 �1 exp

��12x

�:

f2(x) as the standard pdf of the normalized sample variance is a Helmert ChiSquare 2p pdf with p D n � rkA “degree of freedom”.

Proof.


First, let us decompose the quadratic form ky�Efygk2 into estimates 1Efyg ofEfyg.y � Efyg D y �1Efyg C .1Efyg � Efyg/y � Efyg D y �Ab� C A.b� � �/

and

.y�Efyg/0.y�Efyg/D .y�1Efyg/0.y�1Efyg/C .1Efyg�Efyg/0.1Efyg�Efyg/ky � Efygk2 D ky �1Efygk2 C k1Efyg � Efygk2

.y �Efyg/0.y �Efyg/ D .y �A�/0.y �A�/C .b� � �/0A0A.b� � �/ky � Efygk2 D ky � A�k2 C kb� � �k2A0A :

Here, we took advantage of the orthogonality relation.


.b� � �/0A0.y �A�/0 D .b� � �/0A0.In � A.A0A/�1A0/y

D .b� � �/0.A0 � A0A.A0A/�1A0/y D 0:


Second, we implementb�2 BIQUUE of �2 into the decomposed quadratic form.

ky � A�k2 D .y � A�/0.y� A�/ D y0.In � A.A0A/�1A0/y

D y0My D .n � rkA/b�2

ky �Efygk2 D .n � rkA/b�2 C .b� � �/0A0A.b� � �/

ky �Efygk2 D y0MyC .b� � �/0N.b� � �/:

The matrix of the normal equations N WD A0A; rkN D rkA0A D rkA D m, andthe matrix of the variance component estimation M WD In � A.A0A/�1A0, rkM Dn � rkA D n � m have been introduced since their rank forms the basis of thegeneralized forward and backward Helmert transformation.

HH0 D Inz D ��1H.y� Efyg/ D ��1H.y � A�/ and

y �Efyg D �H0z1

�2.y � Efyg/0.y � Efyg/ D z0H0Hz D z0z

1

�2ky � Efygk2 D kzk2:

The standard canonical variable z 2 Rn has to be associated with norms ky � Ab�k

and kb� � �kA0A.


Third, we take advantage of the eigenspace representation of the matrices .M;N/and their associated norms.

y0My D y0V�MV0y versus .b� � �/0N.b� � �/ D .b� � �/0U�NU0.b� � �/�M D Diag.�1; : : : ; �n�m; 0; : : : ; 0/

2 Rn D R

n�m � Rm

versus�N D Diag.�1; : : : ; �m/:

2 Rm

m eigenvalues of the matrix M are zero, but n�rkA D n�m is the number of its non-vanishing eigenvalues which we denote by .�1; : : : ; �n�m/. In contrast,m D rkA is


the n-m number of eigenvalues of the matrix N, all non-zero. The canonical randomvariable

V0y D y� , y D Vy� and U0.b� � �/ Db��

lead to

1


�2.y�/�My� C 1

�2.b�� /0�N.b� � �/

1


�2

n�mX

jD1.y�j /2�j C 1

�2

mX

iD1.b�i � �i /2�i

1

�2.y �Efyg/0.y �Efyg/ D z21 C � � � C z2n�m C z2n�mC1 C � � � C z2n

subject to

z21 C � � � C z2n�m and z2n�mC1 C � � � C z2n

D 1�2

n�mPjD1

.y�j /2�j D 1

�2

mP

iD1.b�i � �i /2�i

kzk2 D z0z D z21 C � � � C z2n�m C z2n�mC1 C � � � C z2n

D 1

�2y0MyC 1

�2.b� � �/0N.b� � �/

D 1

�2ky �Efygk2 D 1

�2.y � Efyg/0.y � Efyg/:

Obviously, the eigenspace synthesis of the matrices N D A0A and M D In��A.A0A/�1A0 has guided us to the proper structure synthesis of the generalizedHelmert transformation.


Fourth, the norm decomposition unable us to split the cumulative probability

dF D f .y1; : : : ; yn/dy1 � � �dyninto the pdf of the Helmert random variable x WD z21 C � � � C z2n�m D ��2.n �rkA/b�2 D ��2.n � m/b�2 and the pdf of the difference random parameter vectorz2n�mC1 C � � � C z2n D ��2.b� � �/0A0A.b� � �/.

dF D f .z1; : : : ; zn�m; zn�mC1; : : : ; zn/d z1 � � �d zn�md zn�mC1d zn


f .z1; : : : ; zn/ D 1

.2/n=2exp

��12

z0z�

D�1

2

� n�m2

exp

��12.z21 C � � � C z2n�m/

��1

2

�m2

� exp

��12.z2n�mC1 C � � � C z2n/

�:

Part A

x WD r2 D z21 C � � � C z2n�m) dx D 2.z1d z1 C � � � C zn�md zn�m/

z1 D r cos�n�m�1 cos�n�m�2 � � � cos�2 cos�1z2 D r cos�n�m�1 cos�n�m�2 � � � cos�2 sin�1

� � �zn�m�1 D r cos�n�m�1 sin �n�m�2

zn�m D r sin �n�m�1

2

4z1� � �

zn�m

3

5 D 1

�Diag.

p�1; : : : ;

p�n�m/V0

1y

V D ŒV1;V2�; V01V1 D In�m; V0

2V2 D Im; V01V2 D 0

VV0 D In; V 2 Rn�n; V1 2 R

n�.n�m/; V2 2 Rn�m and

2

4zn�mC1� � �zn

3

5 D 1

�Diag.

p�1; : : : ;

p�m/U0.b� � �/

altogether

2

66666664

z1: : :

zn�mzn�mC1: : :

zn

3

77777775

D ��1"

Diag.p�1; : : : ;

p�n�m/V0

1y

Diag.p�1; : : : ;

p�m/U0.b� � �/

#

:

The partitioned vector of the standard random variable z is associated with the normkzn�mk2 and kzmk2, namely


kzn�mk2 C kzmk2

D z21 C � � � C z2n�m C z2n�mC1 C � � � C z2n

D 1

�2y0V1Diag.

p�1; : : : ;

p�n�m/Diag.

p�1; : : : ;

p�n�m/V0

1y

C 1

�2.b� � �/0U Diag.

p�1; : : : ;

p�m/Diag.

p�1; : : : ;

p�m/U0.b� � �/

D 1

�2y0V1Diag.�1; : : : ; �n�m/V0

1yC 1

�2.b� � �/0U Diag.�1; : : : ; �m/U0.b� � �/

D d z1d z2 � � �d zn�m�1d zn�mD rn�m�1dr.cos�n�m�1/n�m�1.cos�n�m�2/n�m�2

� � � cos2 �3 cos�2d�n�m�1d�n�m�2 � � �d�3d�2d�1:

The representation of the local .n�m/ dimensional hypervolume element in termsof polar coordinates .�1; �2; : : : ; �n�m�1; r/ has already been given by Lemma B.4.Here, we only transform the random variable r to Helmert’s random variable x.

x WD r2 W dx D 2rdr; dr D dx

2px; rn�m�1 D x.n�m�1/=2

rn�m�1dr D 1

2x.n�m�1/=2dx:

Part A concludes with the representation of the left pdf in terms of Helmert’s polarcoordinates

dF` D�1

2

� n�m2

exp� 12.z21 C � � � C z2n�m/d z1 � � �d zn�m

D 1

2

�1

2

� n�m2

xn�m�2

2 dx.cos�n�m�1/n�m�1.cos�n�m�2/n�m�2

� � � cos2 �3 cos�2d�n�m�1d�n�m�2 � � �d�3d�2d�1:

Part BPart B focuses on the representation of the right pdf, first in terms of the randomvariablesb�, second in terms of the canonical random variablesb�.

dFr D�1

2

�m2

exp

��12.z2n�mC1 C � � � C z2n/

�d zn�mC1 � � �d zn

z2n�mC1 C � � � C z2n D1

�2.b� � �/0A0A.b� � �/

d zn�mC1 � � �d zn D 1

�m=2jA0Aj1=2db�1 � � �db�m:


The computation of the local m-dimensional hyper volume element d zn�mC1 � � �d znhas followed Corollary B.3 which is based upon the matrix of the metric ��2A0A.The first representation of the right pdf is given by

dFr D�1

2

�m2

exp

��12.z2n�mC1 C � � � C z2n/

�d zn�mC1 � � �d zn

D�1

2

�m2 jA0Aj1=2

�m=2exp

�� 1

2�2.b� � �/0A0A.b� � �/

�db�1 � � �db�m:

Let us introduce the canonical random variables .b�1; : : : ;b�m/ which are generatedby the correlating quadratic form kb� � �k2A0A.

.b� � �/0A0A.b� � �/ D .b� � �/0U Diag

�1

�1; : : : ;

1

�m

�U0.b� � �/:

Here, we took advantage of the eigenspace synthesis of the matrix A0A DW N and.A0A/�1 DW N�1. Such an inverse normal matrix is the representing dispersionmatrix Dfb�g D .A0A/�1�2 D N�1�2.

UU0 D Im � U 2 SO.m/ WD fU 2n�m jUU0 D Im; jUj D C1gN WD A0A D U Diag.�1; : : : ; �m/U0 versus

N�1 WD .A0A/�1 D U Diag.�1; : : : ; �m/U0 subject to

�1 D ��11 ; : : : ; �m D ��1

m or �1 D ��11 ; : : : ; �m D ��1

m

jA0Aj1=2 D p�1 � � ��m D 1p�1 � � ��m

b�� WD U0.b� � �/,b� � � WD U0.b� � �/

kb� � �k2A0A DW .b� � �/0A0A.b� � �/ D .b� � �/0Diag

�1

�1; : : : ;

1

�m

�.b� � �/:

The local m-dimensional hypervolume element db�1 � � �db�m is transformed to thelocal m-dimensional hypervolume element db�1 � � �db�m by

db�1 � � �db�m D jUjdb�1 � � �db�m D db�1 � � �db�m:Accordingly we have derived the second representation of the right pdf f .b�/.

dFr D�1

2

�m2 1

�m=2p�1 � � ��m

� exp

�� 1

2�2.b� � �/0Diag

�1

�1; : : : ; y

1

�m

�.b�� /

�db�1 � � �db�m:


Part CPart C is an attempt to merge the left and right pdf

dF D dF`dFr D 1

2

�1

2

� n�m2

x.n�m�2/=2dxd!n�m�1

��1

2

�m2 jA0Aj1=2

�mexp

��12.b� � �/0A0A.b� � �/

�db�1 � � �db�m or

dF D dF`dFr D 1

2

�1

2

� n�m2

x.n�m�2/=2dxd!n�m�1

��1

2

�m2 1

�mp�1 � � ��m

exp

�� 1

2�2.b�� /0Diag

�1

�1; : : : ;

1

�m

�.b��/

�db�1 � � �db�m:

The local .n � m � 1/-dimensional hypersurface element has been denoted byd!n�m�1 according to Lemma B.4.


Fifth, we are going to compute the marginal pdf ofb� BLUUE of �.

dF1 D f1.b�/db�1 � � �db�mas well as

dF1 D f1.b�/db�1 � � �db�minclude the first marginal pdf f1.b�/ and f1.b�/, respectively.The definition

f1.b�/ WD1Z

0

dx

Zd!n�m�1

1

2

�1

2

� n�m2

x.n�m�2/=2

��1

2

�m2 jA0Aj1=2.�2/m=2

exp

��12.b� � �/0A0A.b� � �/

�

subject to

1Z

0

dx

Zd!n�m�1

1

2

�1

2

� n�m2

x.n�m�2/=2 D 1

leads us to

f1.b�/ D�1

2

�m2 jA0Aj1=2

�mexp

��12.b� � �/0A0A.b� � �/

�:


Unfortunately, such a general multivariate Gauss-Laplace normal distributioncannot be tabulated. An alternative is offered by introducing canonical unknownparametersb� as random variables.The definition

f1.b�/ WD1Z

0

dx

Zd!n�m�1

1

2

�1

2

� n�m2

x.n�m�2/=2

��1

2

�m2 1

�m.�1�2 � � ��m�1�m/�1=2

� exp

�� 1

2�2.b�� /0Diag

�1

�1; : : : ;

1

�m

�.b� � �/

�

subject to

1Z

0

dx

Zd!n�m�1

1

2

�1

2

� n�m2

x.n�m�2/=2 D 1

alternatively leads us to

f1.b�/ D�1

2

�m2 1

�m1p

�1 � � ��mexp

� 1

2�2

mX

iD1

.b�i � �/2�i

!

f1.b�1; : : : ;b�m/ D f1.b�1/ � � �f1.b�m/

f1.b�i / WD 1

�p�ip2

exp

��12

.b�i � �/2�i

�8i 2 f1; : : : ; mg:

Obviously the transformed random variables .b�1; : : : ; yb�m/ BLUUE of .�1; : : : ; �m/are mutually independent and Gauss-Laplace normal.


Sixth, we shall compute the marginal pdf of Helmert’s random variable x D.n � rkA/b�2=�2 D .n �m/b�2=�2,b�2 BIQUUE �2,

dF2 D f2.x/dxincludes the second marginal pdf f2.x/.The definition

f2.x/ WDZd!n�m�1

1

2

�1

2

�n�m2

x.n�m�2/=2 �C1Z

�1db�1 � � �

C1Z

�1db�m

�1

2

�m2 jA0Aj1=2

�m

� exp

�� 1

2�2.b� � �/0A0A.b� � �/

�


subject to

!n�m�1 DZd!n�m�1 D 2 � .n�m�1/=2

� .n�m�12

/;

according to Lemma B.4

C1Z

�1� � �

C1Z

�1db�1 � � �db�m

�1

2

�m2 jA0Aj1=2

�mexp

�� 1

2�2.b� � �/0A0A.b� � �/

�

DC1Z

�1d z1 � � �

C1Z

�1d zm

�1

2

�m2

exp

��12.z21 C � � � C z2m/

�

leads us to

p WD n � rkA D n �mp�

�n�m � 1

2

�D �

�1

2

��

�n �m � 1

2

�D �

n �m2

�D �

p2

�

f2.x/ D 1

2p=2� .p=2/xp2 �1 exp

��12x

�;

namely the standard pdf of the normalised sample variance, known as Helmert’s ChiSquare pdf 2p with p D n � rkA D n �m “degree of freedom”. If you substitutex D .n�rkA/b�2=�2 D .n�m/b�2=�2, dx D .n�rkA/��2db�2 D .n�m/��2db�2we arrive at the pdf of the sample varianceb�2, in particular


f2.b�2/ D 1

�r2p=2� .p=2/pp=2b�p�2 exp

��12pb�2

�2

�:

Here is our proof’s end.

Theorem B.12. (marginal probability distributions, special linear Gauss–Markov model with datum defect):

Efyg D A�

Dfyg D V�2subject to

A 2n�m; r WD rkA < minfn;mgEfyg 2 R.A/V 2n�n; rkV D n


defines a special linear Gauss–Markov model with datum defect of fixed effects� 2Rm and a positive definite variance-covariance matrix Dfyg D ˙ y ofmultivariate Gauss-Laplace distributed observations y WD Œy1; : : : ; yn�0.

b� D ACy subject to

2

64

b� D Ly “linear”

kLA � Imk2 D min “minimum bias”

trDfb�g D tr L˙ yL0 D min “best” and

b�2 D 1

n � rkA.y �Ab�/0˙�1

y .y � Ab�/ subject to

2

64Efb�2g D �2

Dfb�2g D 2�4

n � rkA

identifyb� BLIMBE of � andb�2 BIQUUE of �2.Part AThe cumulative pdf of the multivariate Gauss-Laplace probability distribution ofthe observation vector y D Œy1; : : : ; yn�0 2 Y

f .yjEfyg; Dfyg D ˙ y/dy1 � � �dyn

D 1

.2/n=21

j˙ y j1=2 exp

��12.y � Efyg/0˙�1

y .y� Efyg/�dy1 � � �dyn

is transformed by

˙ yDW0Diag.�1; : : : ; �n/WDW0Diag.p�1; : : : ;

p�n/Diag.

p�1; : : : ;

p�n/W

˙ 1=2y WD Diag.

p�1; : : : ;

p�n/W

˙ y D .˙ 1=2y /0˙ 1=2

y versus

˙�1y DW0Diag

�1

�1; : : : ;

1

�n

�W

DW0Diag

�1p�1; : : : ;

1p�n

�Diag

�1p�1; : : : ;

1p�n

�W

˙�1=2 WD Diag

�1p�1; : : : ;

1p�n

�W

˙�1 D .˙�1=2y /0˙�1=2

y

subject to the orthogonality condition

WW0 D In

ky � Efygk2˙�1yWD .y � Efyg/0˙�1

y .y� Efyg/

B-8 Multidimensional Variance Analysis, Sampling from the Multivariate 739

D .y �Efyg/0W0Diag

�1p�1; : : : ;

1p�n

�Diag

�1p�1; : : : ;

1p�n

�

�W.y �Efyg/0

D .y� � Efy�g/0.y� � Efy�g/ DW ky� �Efy�gk2Insubject to the star or canonical coordinates

y� D Diag

�1p�1; : : : ;

1p�n

�Wy D ˙�1=2

y y

f .yjEfy�g;Dfy�g D In/dy�1 � � �dy�

n

D 1

.2/n=2exp

��12.y� � Efy�g/0.y� �Efy�g/

�dy�

1 � � �dy�n

D 1

.2/n=2exp

��12ky� �Efy�gk2

�dy�

1 � � �dy�n

into the canonical Gauss-Laplace pdf.Part BThe marginal pdf ofb� BLUUE of �, is represented by

(1st version)

dF1 D f1.b�/d�1 � � �d�n:

B-8 Multidimensional Variance Analysis, Sampling fromthe Multivariate Gauss–Laplace Normal Distribution

Let x1; : : : ; xN denote a random sample from as k-variate Gauss-Laplace normaldistribution with � as the mean vector and ˙ as the variance-covariance matrix,such that

.B 370/ xi WD .Xi;1; : : : ;Xi;k/0

for i 2 f1; : : : ; N g. Let X�1 WD .x1; : : : ; xN /. The joint density of the random sam-ple is generated by the product of N multivariate normal densities. Let us denote theparameter space by f�;˙ I� 2 R

k;˙ positive definiteg DW ˝ . Here we describeestimation of the parameters from a multivariate Gauss-Laplace normal distributionand describe properties of these estimates. In addition, we care for inference for asimple, multiple, and partial correlation coefficients based on the normal randomsample. Of course, we have to proof that we can assume a multivariate Gauss-Laplace normal distribution. We add some methods for assessing Gauss-Laplacenormality and describe suitable transformation to normality.


B-81 Distribution of Sample Mean and Variance-Covariance

By the representation of the sample mean of the random sample x1; : : : ; xN by(B371) and the sample variance-covariance matrix by (B372)–(B375), namely thek � k dimensional matrix SN=.N � 1/ D fSjlg for all j; l 2 f1; : : : ; kg subject

to XDNP

iD1Xi;j =N . The sample variance-covariance matrix has k variances Sjj for

j 2 f1; : : : ; kg and k.k � 1/=2 possibly distinct covariances. Sjl for j; l; j; l 2f1; : : : ; kg. The generalized sample variance is the scalar quality jSN=.N � 1/jwhich determines the degree of “peakedness” of the joint density of x1; : : : ; xN andis a normal summary measure of variability in the sample.

�

�

�

�Table B.1 Sample mean, sample variance-covariance matrix

Sample mean

x^

N WDNP

iD1

xi (B371)

Sample variance-covariance matrix

SN WD NP

iD1

.xi � x^

N /.xi � x^

N /0 D NP

iD1

xi x0

i �N xi � x^

N xi � x^

N0 (B372)

SN =.N � a/ DW fSjl g for j; l 2 f1; : : : ; kg (B373)

Sjl WD NP

iD1

.Xi;j � X^

j /.Xi l � Xl /=.N � 1/ (B374)

Subject to

X^

j D NP

iD1

Xi;j N (B375)

Given a univariate random sample X1; : : : ;XN from a N.�; � 2/ population, thatthe sample mean X^ has a normal distribution with mean � and variance � 2=N ,the static .N � 1/S2=� 2 is distributed as a �2 variable and the two distributions areindependent. The corresponding distributional result for the k-variable situations isgiven by our previous result. We first define a multivariate distribution called theWishart distribution which is derived from the multivariate Gauss-Laplace normaldistribution as its sampling distribution of the sample statistics

NX

iD1.Xi �XN /.Xi �XN /

0 .B 376/

The Wishart distribution is a multivariate extension of the chi-square distribution.

Definition B.18. Wishart distribution

A random k-dimensional matrix W is said to follow a k-dimensional noncentralWishart distribution. Wk.˙ ; m; x/ with m degrees of freedom, parameter ˙ ,


and noncentrality parameter � WD �0P�1�0=2 if W can be represented as,

W WDmP

jD1xixj where xj for all j 2 f1; : : : ; mg are i:i:i:Nk.�;

P/ vectors. Provide

m:k the density function of W is

f .WI�;˙ / D jWj.m�k�1/=2 expftr.�˙�1W=2/g2km=2j˙ jm=2�k.m=2/ .B 377/

where �k.m=2/Dk.k�1/=4˘kjD1� . 12 .m C 1 � j // is the multivariate Gamma

functions. If�D 0, we say that W follows a central Wishart distribution Wk.˙ ; m/;since ˙ is p.d., it is clear that � D 0 if and only if � D 0. The additivityproperty of Wishart matrices states that if Wj

ind

� Wk.˙ ; mj /; j D 1; : : : ; J , thenJP

jD1Wj �Wk

�˙ ;

JP

jD1mj

�.

End of Definition B.18: Wishart distribution

Intermezzo

Before we proceed we must introduce results about the Fourier transform ofa random vector, namely called “characteristic function” being generalized inAppendix C to the “characteristic functional” for random functions, namelystochastic processes. We will introduce some lemmas and definitions.

Definition B.19. Characteristic function of a random vector

The characteristic function of a random vector X is Efexp i t 0xg define for everyvector t .

Definition B.20. Decomposition of a complex-valued function

Let the complex-valued function g.x/ be written as

g.x/ D g1.x/C ig2.x/ .B 378/

where g1.x/ and g2.x/ are real-valued. Then the expected value of g.X/ is

Efg.X/g D Efg1.x/g C iEfg2.x/g .B 379/

in particular,

Efexp.i t 0x/g D Efcos.t 0x/g C iEfsin.t 0x/g .B 380/


Lemma B.18. Product rule for independent variables

Let X0 WD .X.1/0X.2/0/. If X.1/ and X.2/ are independent and g.x/ D g.1/.x.1//g.2/.x.2//, then

Efg.X/g D Efg.1/.X.1//gEfg.2/.X.2//g .B 381/

Lemma B.19. Components of independent distribution data

If the component of X are independently distributed, then

Efexp.i t 0XN /g DNY

jD1Efi tjXj g .B 382/

Theorem B.20. Characteristic function of a vector-valued random variant oftype Gauss-Laplace distribution

The characteristic function of X which is Gauss-Laplace distribution according toN.�;˙ / is

Efexp.i t 0X/g D exp

�t 0�C 1

2t 0˙ t

�.B 383/

for any real vector t .

Theorem B.21. Characteristic function �.t/

If the random vector X has the density f .x/ and the characteristic function f .t/,then

f .x/ D 1

.2/

C1Z

�1: : :

C1Z

�1exp.�i t 0x/N .t/d!1 : : : d! .B 384/

and

f N .!/ D EfCi!0Xg DC1Z

�1: : :

C1Z

�1exp.Ci!0x/f .x/dx1 : : : dx .B 385/

illustrating synthesis and analysis

Here ends our intermezzo


f N .!/ D EfCi!0Xg DC1Z

�1: : :

C1Z

�1exp.Ci!0x/f .x/dx1 : : : dx .B 386/

Results: Distribution of xN and SN

Let x1; : : : ; xN denote a random sample from the Gauss-Laplace normal distribu-tion Nk.�; � /.

(a) The distribution of xN is Nk.��=N/(b) For N 2, SN follows a Wishart distribution Wk.˙

01N � 1/

(c) xN and SN are independently distributed.

Proof

Efexp.i t 0xN /g DNY

jD1Efexp.i t 0j xj /=N g D exp.t 0�C 1

2t 0.˙=N/t .B 387/

for N D 2 it follows

S2 D�x1� 1

2.x1 C x2/

�x1� 1

2.x1 C x2/

0C�x2 � 1

2.x1 C x2/

�x2 � 1

2.x1Cx2/

0

) S2 D 1

2.x1 � x2/.x1 � x2/0 .B 388/

Obviously .x1 � x2/p2 is distributed according to Nk.0;˙ / and S2 according to

Wk.˙ ; 1/

S3 D�

x1 � 13.x1 C x2 C x3/

�x1 � 1

3.x1 C x2 C x3/

0

C�

x2 � 13.x1 C x2 C x3/

�x2 � 1

3.x1 C x2 C x3/

0

C�

x3 � 13.x1 C x2 C x3/

�x3 � 1

3.x1 C x2 C x3/

0.B 389/

S3 D S2 C 2

3.x3 � x2 /.x3 � x2 /

0 .B 390/

subject to

x2 D1

3.2x2 C x3/ .B 391/


in general

xm D1

mC 1.mxm C xmC1/ and

SmC1 D Sm C m

mC 1.xmC1 � xm/.xmC1 � xm/0 .B 392/

B-82 Distribution Related to Correlation Coefficients

Since xi for all i 2 f1; : : : ; m C 1g are mutually independent and therefore�xmC1;Sm

�are functions only of xi for i 2 f1; : : : ; mg, it follows that

�xmC1;Sm

�

are independent of xmC1.By the hypothesis of induction we enjoy the independence of xm and Sm

xmC1;Sm and xmC1 .B 393/

are mutually independent. Thus, Sm is independent of�xm; xmC1

�and therefore

independent of xmC1�xm which follows aNk�0; mC1

m˙�mC1m

�xmC1 � xm

�.xmC1�

xm/0 are distributed as Wk .˙; 1/. Note that

cov�xi � xm; xn

� D 0 (B 394)

Which implies thath�

x1 � xN�; : : : ;

�xN � xN

�0i0is uncorrelated and therefore

independent of xN . In consequence, Sn is distributed independently of xN .

Results: Maximum likelihood estimation if � andP

Based on a random sample x1; : : : ; xN from a Nk .�; ˙/ distribution, the maximumlikelihood estimates of � and

Pare

b�ML D1

N

NX

iD1xi D xN ; .B 395/

and

bML D 1

N

NX

iD1

�xi � xN

� xi � X

^N

�0 D N � 1N

SN .B 396/

Proof

The likelihood function L .�; ˙ I x1; : : : ; xN / has the form shown on the rightside. The MLE’s of � and ˙ are denoted by b�ML and b

ML, and are the


values that maximize L .�; ˙ I x1; : : : ; xN /. Since ˙�1 is p.d., the distance�xN ��

�0˙�1 �xN ��

�in the exponent of the likelihood function is positive

unless� D XN . The MLE of � is then xN , since it is the value of � that maximizesthe likelihood function. We next maximize the following function with respect to˙ :

L.b�ML;˙/ Dexp

�� 12tr

�˙�1PN

iD1�xi � xN

� xi � X

^N

�0�

.2/Nk=2 j˙ jN=2 .B 397/

With AD˙;B D PNiD1

�xi � xN

� xi �X

^N

�0, and bDN=2, we can show that

bMLD 1

N

PNiD1

�xi � xN

� xi �X

^N

�0maximizes L.b�ML;˙/. The maximum

likelihood is

Lb�ML; bML

�D exp � Nk

2

.2/Nk=2 jbMLjN=20.B 397/

which completes the proof.

Corollary: Invariance property of MLE

Using the invariance property of MLE’s which states that the MLE of a function

g .�/ of a parameter � is gb�ML

�, we see that

1. The MLE of the function �0˙�1� is b�0MLb�1MLb�ML,

2. The MLE ofp�lj , the (l,j)th element of

Pis given by

pb�lj , the square root of

the MLE of the(l,j)th entry in bP

ML

Although the estimator b�ML is an unbiased estimator of �; bML is a biasedestimator of ˙ . analogous to the univariate case, an unbiased estimator of ˙ isb D 1

N�1SN . The unbiased estimators of� abd˙ denoted byb� and b respectivelyare complete sufficient statistics.Of course, you have realized that we gave no space to derive the Wishart distri-bution. Instead we refer to his works in Wishart (1928), Wishart (1931), Wishart(1955) as well as Wishart and Bartlett (1932). the general theory of multivariatestatistical analysis including the Gauss-Laplace multivariate normal distribution,the estimation of the mean vector and the variance-covariance matrix the distributionof sample correlation coefficients, the generalized t2 statistics and the distributionof the sample covariance matrix can be taken from the excellent review of Anderson(1958) including 413 references. finally we advice the readers to consider the worksof Koch (1988a), Koch (1988b), pp. 160–173

B.8 Distributions related to correlation coefficientsTraditionally we divide simple, multiple and partial correlations. Here we restrictour study to the analysis of the simple correlation based on i id bivariateGauss-Laplace normal samples.


�Xi;j;Xi;l

�for all i 2 f1; : : : ; N g .B 399/

where Xi;j and Xi;l denote the j th and l th components of the k - l dimensionalvector x. SupposeE

�Xi;j

� D�j ;E .Xi;l/ D�l , var�Xi;j

� D � zj ; var .Xi;l/ D � z

l andcorr

�Xi;j;Xi;l

� D jl .The exact sampling distribution of was first derived by Fisher (1955) who showedthat for D 0 the statistic

pN � 2^p1� is distributed as t with (N-2) degrees

of freedom. He introduced the Z-transformation,

Z WD 1

2

�1C ^=1 � ^� .B 400/

and proved that even for relative small samples it is approximately Gauss-Laplacenormally distributed, and that as N increases the variance of the Z-transformationrapidly becomes nearly independent of converging to 1=.N � 3/. In 1938,F. N. David (1938) published tables of the exact sampling distribution of ^ for D 0.0:1/0:9,N DB.1/25; 50; 100; 200; 400; ^D�1.0:01/C1, and investigatedthe accuracy of the approximate distributions suggested by R.H. Fisher. She foundapproximate formulas for the mean/variance of Z: They were “extraordinarilyaccurate even for the low values of N provided jj is not too near unity”. Hotdling(1953) intensively studied the mathematical properties of the distribution functionof ^ and the suggested improvements of the Z-transformation in terms of ahypergeometric function. Beside discussing methods of interpolating the density healso calculated moments of ^. From the vast reference list of the subject we liketo mention the contribution of Soper et al. (1955), Kraemer (1973), Wilcox (1997)and Green (1952) as well as Khan (1994).

Example B: Correlation coefficient, “iid” Gauss-Laplace two dimensional normal

We start from the two dimensional Gauss-Laplace normal observations .xi ; yi /for N-independent identical distributed data .“i id 00/ estimated by BLUUE andBIQUUE. By a special decomposition we shift the sample distribution into twonormal distributions relating to.b�1;b�2/ on the one side and to

�b�21b�

22b�12

�or�b�21b�

2212

�on the other side.

b12 indicates the sample correlation coefficient via the characteristic function basedon Fourier-Stieltjes integration we derive the sampling distribution of the correlationcoefficientb12 .12/ using tables of David (1954).“N independently, identically distributed observations : iid”“Gauss-Laplace normal, two-dimensional observations: .xi ; yi / for all i 2f1; : : : ; N g”


“Probability density”

f .x; y/ D 1

2

1

�1�2p1 � 2 exp� 1

2 .1 � 2/

�".x � �1/�12

C 2 .xi�1/ .y � �2/�1�2

C .y � �2/2�22

#

.B 31/

f .x1; : : : ; xN ; y1; : : : ; yN / DNY

iD1f .xi ; yi / D f .x1; y1/ � � � .xN ; yN /

D

1

2�1�2p1 � 2

!N

exp

� 1

2 .1 � 2/NX

iD1

.xi � �1/2

�21� 2 .xi � �1/ .yi � �2/

�1�2C .y � i �mu2/

2

�22

!!

.B 32/

BLUUE W �1 D1

N

NX

iD1xi D k0x=N .B 33/

BLUUE W �2 D1

N

NX

iD1yi D k0y=N .B 34/

BIQUUE:

�^21 D

1

N � 1NX

iD1

�xi � �1

�2 D �x � k�1�0 �

x � k�1�= .N � 1/ .B 35/

�^22 D

1

N � 1NX

iD1

�yi � �1

�2 D �y � k�1�0 �

y � k�2�= .N � 1/ .B 36/

�12 D1

N � 1NX

iD1.xi ��1 /.yi ��2 / D .x�k�1 /0.y�k�2 /=.N � 1/ .B 37/

“Decomposition”


NX

iD1.xi ��1/2 D

NX

iD1Œ.xi �b�1/.�1��1 /�2 D .N�1/�1 2CN.�1��1 /2 .B39/

NX

iD1.yi ��2/2 D

NX

iD1Œ.yi �b�2/.�2 ��2 /�2 D .N� 1/�2 2CN.�2��2 /2 .B40/

NX

iD1.xi ��2/.yi ��2/D .N� 1/12�1 �2 C N.�1 � �1 /.�2 � �2 / .B41/

f .x1; : : : ; xN ; y1; : : : ; yN / D

1

2�1�2p1 � 2

!N

� exp

�� N

2.1�2/�.�1 �b�1/2

�21� 212.�1 �b�1/.�2 �b�2/

�1�2C .�2 �b�2/2

�22

�

� exp

�� N � 12.1� 212/

�b�21�21� 212b12b�1b�2

�1�2C b�

22

�22

�.B42/

“Product of two normal distributions”

First part, marginal distribution

C1Z

�1

C1Z

�1

0

B@

1

2�1�2

q1 � 212

1

CA exp

� N

2.1� 212/

".�1 �b�1/2

�21

� 212.�1 �b�1/.�2 �b�2/�1�2

C .�2 �b�2/2�22

#

db�1db�2

!

.B43/

“Fourier-Stieltjes analysis”

f �.!/ DZ.exp i!x/f .x/dx , .B44/

, f .x/ D 1

2

C1Z

�1.exp i!x/f .x//f ^.!/d! .B45/

First part of the marginal normal distribution


1

2�1�2p1 � 12

Z C1Z

�1db�1db�2 exp� N

2.1� 12/21 � i t1�21

.�1 �b�1/2

� expC N

2.1� 212/212

1 � i t2�1�2

.�1 �b�1/.�2 �b�2/2

� exp� N

2.1� 12/21 � i t3�22

.�2 �b�2/2 .B46/

“abbreviations”

�1 WD N

2.1� 212/.�1 �b�1/2

�21D �1.b�1/ .B47/

expCit1�1 D exp it1N

2.1� 212/.�1 �b�1/2

�21

�2 WD N

2.1� 212/212

.�1 �b�1/.�2 �b�2/�1�2

D �2.b�1;b�2/ .B48/

expCit2�2 D expCit2N

2.1� 212/212

.�1 �b�1/.�2 �b�2/�1�2

“First part of the marginal normal distribution”

1

2�1�2

q1 � 212

Z C1Z

�1exp� N

2.1� 212/�1 � i!1�21

.�1 �b�1/2

� 212.1C i!2/�1�2

.�1 �b�1/.�2 �b�2/

C 1 � i!3�22

.�2 �b�2/2db�1dcmu2

D .1 � 212/1=2Œ.1 � i!1/.1 � i!3/� 212.1C i!2/2�1=2

.B49/

“Check yourself the integral’

expŒ�˛x2 C ˇxy � �y2�dxdy”

“Second part of the marginal normal distribution”


.1 � 212/N�12

Œ.1� it1/.1� it3/� 212.1C it2/�N�12

exp

�N � 1

2.1�212/�b�21�21� 212b12b�1b�2

�1�2C b�

22

�22

�

.B50/

f .x1; : : : ; nN ; y1; : : : ; yN /dx1 : : : dxN dy1 : : : dyN

) f .x1; y1/ : : : f .xN ; yN /dx1dy1 : : : dxN dyN

f .b�1;b�2/f .b�1;b�2;b�12/db�1db�2db�1db�2db�12 .B51/

f .x1; : : : ; xN ; y1; : : : ; yN / D p1.b�1;b�2/p2.b�1;b�2;b�12/“first”

f N1 .t1; t2; t3/ f1.b�1;b�2/

pN1 .t1; t2; t3/ D.1� 212/

N�12

Œ.1 � i t1/.1 � i t3/� 212.1C i t2/2�.B52/

“second”

p.g1; g2; g3/ D .1 � 212/N�12

.2n/3

Z Z C1Z

�1

exp.�i!1�1 � i!/2�2 � i!3�3/Œ.1 � i t1/.1 � i t3/� 212.1C i t2/2�

� d!1d!2d!3 .B53/

p.b12/ D .1 � 212/N�12

.N � 3/Š .1 �b212/

N�42

dN�2

d.b12/

N�20

B@

arccos.�12b�12/q1 � 212b212

1

CA (B.1)

End of Lemma Example

As a specific example, we consider the confidence interval of the confidencecoefficient 0.95 on the correlation of 0.782 observed by a sample of ten. UsingGraph 4 of David (1954) we find the two limits are 0.34 and 0.98 . This result leadsus to the interval 0:34 < 12 < 0:98 (Anderson 1958, p 72).

Lemma B (probability interval for the correlation coefficient, functional determi-nant: David 1954, pp. xxxv–xxxviii):

ˇˇˇ

@.x1; : : : ; xN ; y1; : : : ; yN /

@.b�1;b�2I�1; : : : ; �N�1; �1; : : : ; �N�1/

ˇˇˇ D N

subject to the Helmert transformation


x1 D b�1 C 1p2 � 1�1 C

1p3 � 2�2 C � � � C

1pN.N � 1/�N�1

x2 D b�1 � 1p2 � 1�1 C

1p3 � 2�2 C � � � C

1pN.N � 1/�N�1

x3 D b�1 � 2p3 � 2�2 C � � � C

1pN.N � 1/�N�1

: : :

xN D b�1 � 1pN.N � 1/�N�1

y1 D b�2 C 1p2 � 1�1 C

1p3 � 2�2 C � � � C

1pN.N � 1/�N�1

y2 D b�2 � 1p2 � 1�1 C

1p3 � 2�2 C � � � C

1pN.N � 1/�N�1

y3 D b�2 � 2p3 � 2�2 C � � � C

1pN.N � 1/�N�1

: : :

yN D b�2 � 1pN.N � 1/�N�1

p.b�1;b�2; �j ; �j / D p.xi ; y; i/ˇˇˇˇ

@.xi ; yi /

@.b�1;b�2; �j ; �j /

ˇˇˇˇ

for all j D 1; 2; : : : ; N � 1I i D 1; 2; : : : ; N

Appendix CStatistical Notions, Random Eventsand Stochastic Processes

Definitions and lemmas relating to statistics, random events as well as stochasticprocesses are given, neglecting their proofs. First, we review statistical momentsof a probability distribution of random vectors and list the Gauss-Laplace normaldistribution by introducing the notion of a quasi-normal distribution. As the end ofAppendix C1 we take reference to two lemmas about the Gauss-Laplace 3� rule,namely the Gauss-Laplace inequality and the Vysochainskii-Potunin inequality.Appendix C2 reviews the spatial linear error propagation as well as the generalnonlinear error propagation based on yDg.x/ introducing the moments of second,third and fourth order. The special role Jacobi matrix as well as the Hesse matrix isclassified. Appendix C3 reviews useful identities likeEfyy0˝j g andEfyy0˝yy0gas well as D Ef.y �Efyg/.y �Efyg/0˝ .y �Efyg/g relating to the matrix ofobliquity and � WD Ef.y � Efyg/.y � Efyg/0 ˝ .y � Efyg/.y � Efyg/0g C : : :relating to the matrix of courtosis.

Appendix C.4 newly introduces scalar-valued stochastic processes of one param-eter enriched by Appendix C.5 reviewing characteristics of a one parameterstochastic processes like non-stationary, stationary and cryodic stochastic pro-cesses. Extensive simple examples are presented in Appendix C.6 including ARIMAprocesses namely. Appendix C.7 introduce the important WIENER process of typeORNSTEIN- UHLENBECK, WIENER process with drift integrated WIENERprocess. The spectral analysis of one parameter is enriched by 14 exampleslike the DIRAC representation, spectral densities, band limited white noise. Anextensive introduction into scalar-, vector-, and tensor-valued stochastic processesin Appendix C.9 refers to the characteristic functional the moment representationsof stochastic processes for scalar-valued and vector-valued qualities, for statis-tical homogeneous and isotropic fields of multi-point systems. The Appendix isspecified by Example C.9.1 on the topic of two-dimensional Euclidean networksunder the postulates of homogeneity and isotropy in the statistical sense, byExample C.9.2 on criterion matrices for absolute coordinates in a space time astro-nomic frame of reference, for instance represented in scalar-valued spherical har-monic, by Example C.9.3 on space gravity spectroscopy illustrating the benefits of


753


754 C Statistical Notions, Random Events and Stochastic Processes

TAYLOR-KARMAN structured criterion matrices, by Example C.9.4 on Nonlocalprediction and Example C.9.5 on Nonlocal Time Series Analysis.

C-1 Moments of a Probability Distribution,the Gauss–Laplace Normal Distribution and theQuasi-Normal Distribution

First, we define the moments of a probability distribution of a vector valued randomfunction and explain the notion of a Gauss-Laplace normal distribution and itsmoments. Especially we define the terminology of a quasi - Gauss-Laplace normaldistribution.

Definition C.1. (statistical moments of a probability distribution):

(1) The expectation or first moment of a continuous random vector ŒXi � for all i D1; : : : ; n of a probability density f .x1; : : : xn/ is defined as the mean vector � WDŒ�1; : : : ; �n� of EfXig D �i

�i WD EfXig DZ C1

�1: : :

Z C1

�1xif .x1; : : : ; xn/dx1 : : : dxn: (C1)

The first moment related to the random vector Œei � WD ŒXi � EfXig� is called firstcentral moment

i WD EfXi ��i g D EfXig � �i D 0: (C2)

(2) The second moment of a continuous random vector ŒXi � for all i D 1; : : : ; n ofa probability density f .x1; : : : ; xn/ is the mean matrix

�ij WD EfXiXj g DZ C1

�1: : :

Z C1

�1xixj f .x1; : : : ; xn/dx1 � � �dxn: (C3)

The second moment related to the random vector Œei � WD ŒXi �EfXi g� is calledvariance - covariance matrix or dispersion matrix Œ�ij �, especially variance ordispersion �2 for i D j or covariance for i ¤ j :

ii D �2i D V fXig D DfXi g WD Ef.Xi ��i /2g (C4)

ij D �ij D C fXi ;Xj g WD Ef.Xi ��i /.Xj � �j /g (C5)

Dfxg D Œ�ij � D ŒC fXi ;Xj g� D Ef.x�Efxg/.x�Efxg/0g: (C6)

x WD ŒX1; : : : ;Xn�0 is a collection of the n� 1 random vector. The random variables

Xi ;Xj for i ¤ j are called with respect to the central moment of second orderuncorrelated if �ij D 0fori ¤ j . (3) The third central moment with respect to therandom vector Œei � WD ŒXi �EfXig� defined by

C-1 Moments of a Probability Distribution, the Gauss–Laplace Normal 755

ijk WD Efeiej ekg D Ef.Xi � �i /.Xj ��j /.Xk � �k/g (C7)

contains for i D j D k the vector of obliquity with the components

� i WD Efe3i g (C8)

Uncorrelation with respect to the central moment up to third order is defined by

Efen1i1 en2i2 en3i3 g D3Y

jD1Efenjij g

2

481 � i1 ¤ i2 ¤ i3 � n

and0 � n1 C n2 C n3 � 3:

(C9)

(4) The fourth central moment relative to the random vector Œei � WD ŒXi �EfXig�

ijk` WD Efeiej eke`g D Ef.Xi � �i /.Xj ��j /.Xk ��k/.X` � �`/g (C10)

leads for i1 D i2 D i3 D i4 to the vector of curtosis with the components

i WD Efe4i g � 3f�2i g2 (C11)

Uncorrelation with respect the central moments up the fourth order is defined by

Efen1i1 en2i2 en3i3 en4i4 g D4Y

jD1Efenjij g

2

481 � i1 ¤ i2 ¤ i3 ¤ i4 � n

und0 � n1 C n2 C n3 C n4 � 4 :

(C12)

(5) The central moments of the nth order relative to the random vector Œei � WDŒXi �EfXig� are defined by

i1:::in WD Efei1 : : : eing D Ef.Xi1 � �i1/ : : : .Xin � �in/g (C13)

A special distribution is the Gauss-Laplace normal distribution of random vectorsin R

n: Note that alternative distributions on manifolds exist in large numbers, forinstance the von Mises distribution on S

2 or the Fisher distribution on S3 of Chap. 7.

Definition C.2. (Gauss-Laplace normal distribution):

An n � 1 random vector x WD ŒX1; : : : ;Xn�0 is a Gauss-Laplace normal distribution

if its probability density f .x1; : : : xi / has the representation

f .x/ D .2/�n=2j˙ j�1=2 expŒ�12.x � Efxg/0˙�1.x � Efxg/� (C14)


Symbolically we can writex � N .�;˙ / (C15)

with the moment of first order or mean vector � WD Efxg and the central momentof second order or variance – covariance matrix˙ WD Ef.x�Efxg/.x�Efxg/0g:The moments of the Gauss-Laplace normal distribution are given next.

Lemma C.3. (moments of the Gauss-Laplace normal distribution):

Let the n � 1 random vector x WD ŒX1; : : : ;Xn�0 follow a Gauss-Laplace normal

distribution. Then all central moments of odd order disappear and the centralmoments of even order are product sums of the central moments of second orderexclusively.

Efei1 : : : eing D 08n D 2mC 1;m D 1; : : : ;1 (C16)

Efei1 : : : eing D fct.�2i1 ; �i1i2 ; : : : ; �2i2; : : : ; �2in /8n D 2m;m D 1; : : : ;1 (C17)

ij D �ij ; i i D �2i ; (C18)

ijk D 0; (C19)

ijk` D ij k` C ikj` C i`jk; (C20)

iijj D iijj C 2ij ij D �2i �2j C 2�2ij ; i i i i D 3.�2i /2 (C21)

ijk`m D 0; (C22)

i1i2:::i2m�2i2m�1i2m D i1i2:::i2m�2i2m�1i2m C i1i2:::i2m�3i2m�1i2m�2i2m C : : :Ci2i3:::i2m�1i1i2m: (C23)

The vector of obliquity WD Œ 1; : : : ; m�0 and the vector of curtosis WDŒ�1; : : : ; �m�

0 vanish.A weaker assumption compared to the Gauss-Laplace normal distribution is theassumption that the central moments up to the order four are of the form (C18)–(C21). Thus we allow for a larger class of distributions which have a similarstructure compared to (C18)–(C21).

Definition C.4. (quasi-Gauss-Laplace normal distribution):

A random vector x is quasi-Gauss-Laplace normally distributed if it has a contin-uous symmetric probability distribution f .x/ which allows a representation of itscentral moments up to the order four of type (C18)–(C21).Of special importance is the computation of error bounds for the Gauss-Laplacenormal distribution, for instance. As an example, we have the case called the 3�

C-2 Error Propagation 757

rule which states that the probability for the random variable for X falling away,from its mean by more than three standard deviations (SDs) is at most 5%,

P fjX � �j 3�g � 4

81< 0:05: (C24)

Another example is the Gauss-Laplace inequality, that bounds the probability forthe deviation from the mode �.

Lemma C.5. (Gauss inequality):

The expected squared deviation from the mode � is

P fjX � �j rg � 4�2

9r2forallr

p4=3 � (C25)

P fjX � �j rg � 1 � .r=p3�/forallr �p4=3 � (C26)

subject to �2 WD Ef.X� �/2g.Alternatively, we take advantage of the Vysochanskii–Potunin inequality.

Lemma C.6. (Vysochanskii–Potunin inequality):

The expected squared deviation from an arbitrary point ˛ 2 is

P fjX � ˛j rg � 42

9r2forallr

p8=3 (C27)

P fjX � ˛j rg � 42

3r2� 13

forallr �p8=3 (C28)

subject to 2 WD Ef.X � ˛/2g:References about the two inequalities are Gauss (1823), Pukelsheim (1994).

C-2 Error Propagation

At the beginning we note some properties of operators “expectation E” and “dis-persion D”. Those derivatives can be taken from Pukelsheim (1993) and Teuniseen(1989b, 1990). Afterwards we review the special and general, in particular nonlinearerror propagation.

Lemma C.7. (expectation operators EfXg ):

E is defined as a linear operator in the space of random variables in Rn, also called

expectation operator. For arbitrary constants ˛; ˇ; ı 2 R there holds the identity


Ef˛Xi C ˇXi C ıg: (C29)

Let A and B be two m � n and m � ` matrices and ı an m � 1 vector of constantsx WD ŒXi ; : : : ;Xn�

0 and y WD ŒYi ; : : : ;Yn�0 two n � 1 and ` � 1 random vectors

such thatEfAXC BXC ıg D AEfxg C BEfyg C ı (C30)

holds. The expectation operator E is not multiplicative that is

EfXiXj g D EfXigEfXj g C C fXi ;Xj g ¤ EfXigEfXj g; (C31)

if Xi and Xj are correlated.

Lemma C.8. (special error propagation):

Let y be an n � 1 dimensional random vector which depends linear of the m � 1dimensional random vector x by means of a constant n � m dimensional matrix Aand of a constant dimensional vector ı of the form y WD Ax C ı. Then hold the“error propagation law”

Dfyg D DfAxC ıg D ADfxgA0: (C32)

The dispersion function D is not addition, in consequence a nonlinear operatorthat is

DfXi C Xj g D DfXig CDfXj g C 2C fXi ;Xj g ¤ DfXi g CDfXj g; (C33)

if Xi and Xj are correlated.The “special error propagation law” holds for a linear transformation x !y D Ax C ı. The “general nonlinear error propagation” will be presented byLemmas C.7 and C.8. The detailed proofs are taken from Chap. 8, Examples.

Corollary C.9. (“nonlinear error propagation”):

Let y D g.x/ be a scalar valued function between one random variable x and onerandom variable y. g(x) is assumed to allow a Taylor expansion around the fixedapproximation point �0 W

g.x/ D g.�0/C g0.�0/.x � �0/C 12g00.�0/.x � �0/2 CO.3/

D �0 C �1.x � �0/C �2g.�0/2 CO.3/ : (C34)

C-2 Error Propagation 759

Then the expectation and dispersion identities hold

Efyg D g.�x/C 1

2g00.�0/�2x CO�.3/ D g.�x/C �2�2x CO�.3/; (C35)

Dfyg D �21 �2x C 4�2x Œ�1�2.�x � �0/C �22 .�x � �0/2� � �22 �4xCEf.x � �x/3gŒ2�1�2 C 4�22 .�x � �0/�CEf.x � �x/4g�22 CO�.3/:

(C36)

For the special case of a fixed approximation point �0 chosen to coincide with meanvalue �x D �0 and x being quasi - Gauss-Laplace normal distributed we arrive atthe identities

Efyg D g.�x/C 1

2g00.�x/�2x CO�.4/; (C37)

Dfyg D Œg0.�/�2�2x C1

2Œg00.�x/�2�4x CO�.3/: (C38)

The representation (C37) and (C38) characterize the nonlinear error propagationwhich is in general dependent of the central moments of the order two and higher,especially of the obliquity and the curtosis.

Lemma C.10. (“nonlinear error propagation”):

Let y D f.x/ be a vector-valued function between the m � 1 random vector x andthe n � 1 random vector y. g.x/ is assumed to allow a Taylor expansion around them � 1 fixed approximation vector �0 W

g.x/ D g.�0/C g0.�0/.x � �0/C 12g00.�0/Œ.x � �0/˝ .x � �0/�CO�.3/

D 0 C J.x � �0/C 12HŒ.x � �0/˝ .x � �0/CO.3/:

(C39)

With the n �m Jacobi matrix J WD ŒJjgi .�0/� and the n � m2 Hesse matrix H WDŒvecH1; : : : ; vecHn�

0;

Hi WD Œ@j @kgi .�0/�.i D 1; : : : ; nI j; k D 1; : : : ; m/ (C40)

there hold the following expectation and dispersion identities (“nonlinear errorpropagation”)

Efyg D �y D g.�x/C1

2Hvec˙ CO�.3/ (C41)


Dfyg D †y D J†J 0 C 1

2J � Œ† ˝ .�x � �0/0 C .�x � �0/0 ˝†�H0

C 12

HŒ† ˝ .�x � �0/C .�x � �0/˝†�J0

C 14

H Œ† ˝ .�x � �0/.�x � �0/0 C .�x � �0/.�x � �0/0 ˝†C.�x � �0/˝† ˝ .�x � �0/0 C .�x � �0/0 ˝† ˝ .�x � �0/�H0

C12

J �Ef.x� �x/.x ��x/0 ˝ .x � �x/0gH0

C12

H �Ef.x� �x/˝ .x � �x/.x ��x/0gJ0

C14

H � ŒEf.x ��x/.x� �x/0 ˝ .x ��x/g � .�x � �0/0

C.�x � �0/ �Ef.x� �x/0 ˝ .x � �x/.x � �x/0gC.I˝ .�x � �0// �Ef.x� �x/.x � �x/0 ˝ .x� �x/0gCEf.x ��x/˝ .x ��x/.x� �x/0g � ..�x � �0/0/˝ I/�H0

C14

H � f.x ��x/.x � �x/0 ˝ .x ��x/.x � �x/0g�vec†.vec†/0�H0 CO�.3/:

(C42)

In the special case that the fixed approximations vector �0 coincides to themean vector �x D �0 and the random vector x is quasi-Gauss-Laplace normallydistributed the following identities hold:

Efyg D �y D g.�x/C 12Hvec† CO�.4/ .C � 43/

Dfyg D †y D J†J 0 � 14Hvec†.vec†/0H0

C14

H �Ef.x� �x/.x � �x/0 ˝ .x � �x/.x � �x/

0g �H0 CO�.3/

D J†J0 C 1

4HŒ† ˝† CEf.x� �x/

0 ˝† ˝ .x ��x/g� �H0 CO�.3/:

(C44)

C-3 Useful Identities

Notable identities about higher order moments are the following.

C-3 Useful Identities 761

Lemma C.11. (identities: higher order moments):

(a) Kronecker-Zehfuss products #1

Efyy0g D Ef.y�Efyg/.y� Efyg/0g C EfygEfyg0 (C45)

Efyy0˝ yg D Ef.y�Efyg/.y� Efyg/0˝ .y �Efyg/gCEfyy0g ˝ Efyg C Efy˝ Efyg0˝ ygCEfyg ˝Efyy0g � 2Efyg ˝ Efyg0˝Efyg (C46)

Efyy0˝ yy0g D G �‰ ˝ Efyg0 �Efyg0˝‰�‰ 0 ˝ Efyg �Efyg ˝‰ 0 CEfyy0g ˝ Efyy0gCEfy0˝ Efyy0g ˝ yg C Efy˝ ygEfy˝ yg0�2EfygEfyg0˝ EfygEfyg0: (C47)

(b) Kronecker products #2

‰ W D Ef.y�Efyg/.y� Efyg/0˝ .y �Efyg/g (C48)

G W D Ef.y�Efyg/.y� Efyg/0˝ .y �Efyg/.y� Efyg/0g�Ef.y�Efyg/.y� Efyg/0˝ Ef.y�Efyg/.y� Efyg/0g�Ef.y�Efyg/0˝ Ef.y� Efyg/.y�Efyg/0g ˝ .y �Efyg/g�Ef.y�Efyg/˝ .y �Efyg/gEf.y� Efyg/0˝ .y � Efyg/0g: (C49)

The n2�nmatrix‰ contains the components of obliquity, the n2�n2 matrixG thecomponents of curtosis relative to the n � 1 central random vector y � Efyg.

(c) Covariances between linear and quadratic forms

C fF1yC d1;F2yC d2g WD Ef.F1yC d1 � EfF1yC d1g/.F2yC d2�EfF2yC d2g/0g D F1†F 2

(C50)

(linear error propagation)

CfFyC d; y0Hyg WD Ef.FyC d � EfFyC d g/.y0Hy� Efy0Hyg/gD FŒEfyy0˝ y0g �EfygEfy0˝ y0g�vecH

D 1

2F‰ 0vec.HCH0/C F†.HCH0/Efyg (C51)


C fy0Gy; y0Hyg WD Ef.y0Gy �Efy0Gyg/.y0Hy� Efy0Hyg/gD .vecG/0ŒEfyy0˝ yy0g � Efy˝ ygEfy0˝ y0g�vecH

D 1

4Œvec.GCG0/�0Gvec.HCH0/

�12Œvec.GCG0/�0.‰ 0 ˝ Efyg C‰ ˝ Efyg0/vec.HCH0/

C12

trŒ.GCG0/†.HCH0/†�C Efyg0.GCG0/†.HCH0/Efyg:(C52)

(d) quasi-Gauss-Laplace-normally distributed data

C fF1yC ı;F2yC ı2g D F1†F 2 (C53)

(independent from any distribution)

C fFyC ı; y0Hyg D F†.HCH0/Efyg (C54)

C fy0Gy; y0Hyg D 1

2trŒ.GCG0/˙ .HCH0/˙ �

CEfyg0.GCG0/˙ .HCH0/Efyg: (C55)

C-4 Scalar – Valued Stochastic Processes of One Parameter

We have previously reviewed the moments of a probability distribution up to orderfour. The deviation was based on integration from minus infinity to plus infinity, Adifferent concept is needed if we work with distributions on manifolds. For instance,we are confronted with the problem to find a distributions for data on a circle or asphere or on a hypersphere. As mentioned before, directional measurements, alsocalled angular observations or longitudinal data are not Gauss-Laplace distributed,but enjoy a von Mises-Fisher distribution in S

� � RC1

Example C.4.1: p=1 (von Mises, 1918)

f .�j�; �/ D Œ2I0.�/��1 expŒ� cos.� � ��/� (C56)

Example C.4.2: p=2, (Fisher, 1953)

C-4 Scalar – Valued Stochastic Processes of One Parameter 763

f .�;˚ k ��;�˚ ; �/ D �

4 sinh �expŒcos˚ cos�˚ cos.� � ��/C sin˚ sin�˚�

D �

4 sinh �exp � < �jX > (C57)

Here, we have introduced the circular normal distribution CN.�; �/ parameterizedby the mean direction �.0 � �� 2/ and by the concentration parameter�.� > 0/, the reciprocal of a dispersion measure. The spherical normal distribu-tion CN.�� ; �˚; �/ is parameterize by the “longitudinal direction, lateral meandirection .�� ; �˚/” and the concentration parameter �, the reciprocal dispersionmeasure. We note that the p-spherical normal distribution is an element of theexponential class.Most important examples of the circular normal (R. von Mises) distributions arephase observation within the Global Positioning System (“Global Problem Solver”)according to Cai et al.(2007) and the spherical normal (Fisher) distribution forspherical harmonic analysis-synthesis (Fourier-Legendre functions) describing thegravitational field within gravities, electrostatics and magnetostaties according toFisher et al. (1981), Kent (1983), Mardia (1972), Man (2005), Mc Fadden and Jones(1981) and Roberts and Ursell (1960).There have been made studies to consider an ellipsoidal reference to analyzethe proper reference field of celestial bodies like the Earth, the Moon, or anyother planetary body. The ellipsoid-of-revolution is the equilibrium figure of typeMacLaurin which represents to first order of the Earth, the Moon, Mars or otherplanets even the Jacobi triaxial reference figure is a better approximation following,for instance Chandra Sekhar (1969). We leave the question: what is the characteristicdistribution of a ellipsoid-of-revolution or a triaxial ellipsoid open!A random event X is the result of a random experiment under a given condition.If these conditions change it might have an influence on the result of randomexperiment, namely to the probability distribution. If we consider the varyingconditions on the random quantities X D X.t/, we arrive at the random effectswhich depend on the deterministic parameter t. We illustrate the new detaileddefinition of one parameter stochastic process.

Example C.4.1

We measure the temperature of a point continuously by a sensor for 3 years.Naturally we model the sensor recording by a graph shown in Fig. C.1. Actuallywe find a random variable which is function of the continuous parameter “time t”,namely X.t/. Every year we receive a temperature profile which is a deterministicfunction of time t W xD x.t/; 0 � t � 1. In this form x.t/ is understoodas realization of the random variable X.t/. Neglecting the influence of climatechanges the function x D x.t/ is cyclic. We introduce the generalized randomexperiment “continuous measurement of temperature over one year” and refer it tofX.t/j0 � t � tg. Actually one could represent climate changes by a trend function.The character of a generalized random experiment is very obvious since for closelymeasuring points ti and tiC1 range in the second up to the minute.


Fig. C.1 Graph of the function f(t) over a period of 1 year or 12 months. Two years recording:trajectories of the stochastic process

Example C.4.2

Alternatively we consider the length L=60 mm of Nylon wires and the designcondition of a variance of 0.5 mm produced by one machine under equal conditions.If we analyzed the dependence of the diameter of a function of the distance tfrom the initial point, we receive for any measurement of the wire a functionx D x.t/; 0 � t � L. These functions differ from wire to wire, but these isno significant difference between the functional different wires. Indeed we assumechanges to the constant condition of the wire. The random variations of the diameterare in the range of ˙0.02 mm and caused by the experimental conditions. Let usrefer to the random wire diameter as a function of the one parameter t , we areable to define by a “continuous measurement of the various diameter of a wire as adependence of the initial point” the generalized random experiment and formalizeit by fX.t/; 0 � t � Lg.In contrast to random experiments which resulted in real numbers, such as general-ized random experiment generates random functions, also called stochastic processor random process. We have called our parameter “t” derived from the term “time”,but it can be any parameter for instance “position”, “placement”,“pressure” etc.Finally we are able to define a stochastic process of one parameter.

Definition C.4.2: Stochastic process

C-5 Characteristic of One Parameter Stochastic Processes 765

A stochastic process of one parameter t 2 T , the parameter space, and the statez 2 Z, the state space, is defined as the set of random quantities fX.t/; t 2 T g .

If the set of parameter is finite or measurable infinite, we refer to a stochasticprocess with discrete time. Such processes are summarized as a sequence of randomquantities fX1;X2; : : : g. If the parameter t 2 T is interval, we refer to stochasticprocess with continuous time. A stochastic process fX.t/; t 2 Tg is called discrete ifthe state space z is a finite or measurable infinite. In contrast we refer to a continuousstochastic process if he state space is an interval.

Stochastic process

discrete stochasticprocess withdiscrete time

discrete stochasticprocess with

continuous time

continuous stochasticprocess withdiscrete time

continuous stochasticprocess with

continuous time

��

��

��

��

��

��

If we consider various realization of X.t/ for all t 2 T , we call them trajectory ofthe stochastic process. For a discrete stochastic process with continuous time thetrajectories are “step functions”

C-5 Characteristic of One Parameter Stochastic Processes

The most important function in connection with stochastic process is the distributionfunction of Xt ; t 2 T, namely F.x/ D P.Xt � x/. But with the set of thedistributions fFt.x/; t 2 Tg a stochastic ic process is not uniquely determined. Acomplete characterization of a stochastic process it is necessary to have informationon all n-type ft1; t2; � � � ; tn�1; tng for the discrete set N WD f1; 2; � � � ; n � 1; ngwith ti 2 T and the information of the distribution function of random vectorsfX.t1/;X.t2/; � � � ;X.tn�1/;X.tn/g, namely

Ft1;t2;��tn�1;tn .x1; x2; � � � ; xn�1; xn/ WD P.X.t1/ � x1; � � � ;X.tn/ � xn/The set of distribution functions will be introduced by Table C.5.1 as well asthe trend function, the covariance function, the variance function, the correlationfunction and the symmetry condition.


Table C.5.1 Characteristics of one parameter stochastic process

“Distribution functions of an n-dimensional random vector”

Ft1;t2;��tn�1;tn .x1; x2; � � � ; xn�1; xn/WD P fX.t1/ � x1;X.t2/ � x2; � � � ;X.tn�1/ � xn�1;X.tn/ � xng (C58)

“Trend function”EfXtg D �.t/ for t 2 T (C59)

�.t/ DC1R�1

xft .x/dx (C60)

“Covariance function and stochastic process”

Cov.s; t/ WD CovfXs;Xt gD EfŒXs �EfXsg�ŒXt � EfXtg�g

for all s; t 2 T (C61)

Cov.s; t/ D EfXs;Xt g � �.s/�.t/ (C62)

“Variance function”

VarfXsg DW Cov.t; t/ (C63)

“Correlation function”

.s; t/ DW CovfXs;Xt g � .pVarfXsg

pVarfXt g/ (C64)

“Symmetry”

Cov.s; t/ DW Cov.t; s/ (C65)lim

jt�sj!1Cov.s; t/ D lim

jt�sj.s; t/ D 0 (C66)

End of Table C.5.2

Of special importance are those stochastic processes which do not depend on theabsolute values of ti , but are functions of the distances between the points. They areconsidered as functions of the relative positions.

Definition 5.1 (Stationary in the narrow sense)

A stochastic process fXt ; t 2 Tg is stationary in the narrow sense if for all n 2f1; 2; � � � g for arbitrary subject to ti 2 T and ti C h 2 T holds

C-5 Characteristic of One Parameter Stochastic Processes 767

Ft1;t2;��tn�1;tn .x1; x2; � � � ; xn�1; xn/ D Ft1Ch;t2Ch;��tn�1Ch;tnCh.x1; x2; � � � ; xn�1; xn/(C67)

A special property is for

�.t/ D EfX.t/g D �.const/ (C68)

VarfX.t/g D .const/ (C69)

End of Definition 5.1

Special case n D 2, for instance, t1 D 0; t2 D t � s; h D s will lead us to

F0;t�s.X1x2/ D Fs;t .x1x2/ D F.�/ for � WD t � s (C70)

Cov.s; t/ D Cov.s; s C �/ D Cov.�/ for � WD t � s (C71)

“Symmetry”:Cov.�/ D Cov.��/ D Cov.j� j/ (C72)

limj� j!1

Cov.�/ D 0 (C73)

Another generalization is achieving of the treat stochastic process ”of higher order”

Definition 5.2: (stationary in the wider sense):

A stochastic process of second order is stationary in the wider sense if theconditions

1. �.t/ D const and2. Cov.j� j/ D CovfXs;Xtg


Besides the condition of stationary another support property of a stochastic processis the set of conditions which hold for increments of stochastic process fX.t/; t 2 Tgin the interval Œt1; t2� for t1 < t2, namely the random difference

Xt2 �Xt1

In general such increments may be negative. The concept of incremental stochasticprocesses will lead us later to the concept of Kolmogov’s structure functions whichis very popular in the applied sciences.

Definition 5.3: (Stationary increments):

A stochastic process fX.t/; t 2 Tg is called homogenous or stationary incremental,if the increments fX.t2 C �/g � fX.t1 C �/g for all � have identical probabilitydistributions for any fixed t1 and t2


The great advantage of the incremental stochastic process in applications has theorigin in the fact that a stationary stochastic process is not necessary.


Definition 5.4: (Stochastic processes with incremental properties):

A stochastic process fX.t/; t 2 Tg has independent increment for all n D 3; 4; � � �and values ft1; t2; � � � ; tn�1; tng subject to the order ft1 < t2 < � � � < tn�1 < tng andti 2 T with increments

Xt2 � Xt1 ;Xt3 � Xt2 ; � � � ;Xtn�1 � Xtn�2 ;Xtn � Xtn�1 (C74)

being independent.


Try to consider that the point tn�1 is placed in the future the point tn at present time,is general for n > 1 all points ft1; t2; � � � ; tn�1; tng in the past. The event at tnC1 fora Markov process depends only on the events at tn:

Definition 5.5: (Simple Markov process):

A stochastic process fX.t/jt 2 Tg has the property of a Markov process is for alln 2 f1; 2; � � � ; g if for all n 2 f1; 2; � � � g.n C 1/ tuple ft1; t2; � � � ; tn�1; tng orderedaccording to ft1 < t2 < � � � < tn�1 < tng the probability relation

P fXtnC12 ZnC1jX.tn/ 2 Zn;X.tn�1/ 2 Zn�1; � � � ;X.t1/ 2 Z1g

D P fX.tn�1/ 2 Zn�1jX.tn/ 2 Zng (C75)


Stochastic processes with the property of a simple Markov process is characterizedby independent increments.

Lemma 5.6: Markov process

A Markov process is in narrow sense stationary if and only if the one-dimensionalprobability is independent of the time such that

F.x/ D P fXt � tg D x for all t 2 T (C76)

End of Lemma 5.6

Markov process with a finite state space are called Markov chains, otherwise theyare called continuous Markov chains.

Definition 5.7: (Continuity in the squared mean):

A stochastic process of second order fX.t/jt 2 Tg located in the point t0 is in thesquared mean continuous if

C-6 Simple Examples of One Parameter Stochastic Processes 769

limh!0

E.ŒX.t0 C h/� X.t0/�2/ D 0 (C77)

The stochastic process fX.t/jt 2 Tg is in the domain T0 D T in the quadratic meancontinuous if the stochastic process has this property for all t 2 T0


Lemma 5.8: Squared mean and its covariance function

A stochastic process of second order fX.t/jt 2 Tg is continuous in the point t0 ifits covariance function Cov.s; t/ is characterized by .s; t/ D .t0t0/ and its trendfunction �.t/ is continuous in the point t D t0.

End of Lemma 5.8

C-6 Simple Examples of One Parameter Stochastic Processes

Simple examples of one parameter stochastic process are introduced divided in twotypes of stochastic processes, first with continuous time and second with discretetime:

(i) Process with linear realizations(ii) Cosine oscillations with random amplitude

(iii) Cosine oscillations with random amplitude and random phase(iv) Superposition of two uncorrelated random functions(v) Impulse modulation

(vi) Randomly retarded pulse modulation(vii) Random sequences with discrete signals

(viii) Random sequences of moving averages of order n: MA.n/(ix) Random sequences of moving averages of order n: MA.n/(x) Random sequences of unlimited order

(xi) Autoregressive sequences of first order AR.1/(xii) Autoregressive sequences of first order r: AR.r/

(xiii) ARMA (r,s) models.

We present here simple examples of stochastic processes. Statements with respectto stationarity relate always to stationarity in the wide sense.

Ex 1: Processes with Linear Scalisations

Examples of stochastic processes with continuous time


Example 6.1 Process with linear realizations

Many linear models can be characterized by the simple linear model X.t/ D V t CW , V andW are random effects with the meanEfX.t/jt 0g called trend functionsand the variance-covariance function EfŒX.s/ � EfX.s/g�ŒX.t/ � EfX.t/g� DEfŒVs CW �ŒV t C w�g � �.s/�.t/ DW ˙.s; t/.A first result is the variance-covariance function oriented to our model.

˙.s; t/ D st Var.V /C .s C t/ Cov.V;W /C Var.W / (C78)

A second result is the special caseW = constant. In such a case we find the variance-covariance function˙.s; t/ D st Var.V /

End of Example 6.1

Ex 2: Cosine Oscillations with Random Amplitudes

Example 6.2 Cosine oscillation with random amplitude

Here we assume the cosine model X.t/DA cos!t , where A is a non-negativerandom effect, the amplitude subject to EfAg < 1. Here we realize the trendfunction of oscillations of type EfX.t/g D EfAg cos!t . The variance-covariancefunction for such a simple model is given byEfŒX.s/�EfX.s/g�ŒX.t/�EfX.t/g� DEfŒA cos!s�ŒA cos!t�g � �.s/�.t/ D ŒEfAg2 �EfAg�2.cos!s/.cos!t/ DW˙.s; t/ Let us denote the variance �2 D Var.A/ such that we receive the variance-covariance function˙.s; t/, namely

˙.s; t/ D �2.cos!s/.cos!t/ (C79)

End of Example 6.2

Ex 3: Cosine Oscillations with Random Amplitudes and Phase

Example 6.3 Cosine oscillation with random amplitude and random phase

This time we assume the cosine model X.t/ D A cos.!t C ˚/, where A is a non-negative random effect and finite variance. The phase is a random effect, equallydistributed ˚ 2 Œ0; 2�, The two random effects A and ˚ are independently fromeach other in the statistical sense. Such a stochastic process fX.t/ 2 .�1;C1/gcan be considered as the outputs of the larger number of similar oscillators,arbitrarily chosen when applied at different time instances. Because of


Efcos.!t C ˚/g D 1

2

2Z

0

cos.!t C ˚/d˚

D 1

2Œsin.!t C˚/�20 D 0 (C80)

We experience a zero trend function. According the variance-covariance functionwe may represent by

˙.s; t/ D EfŒA cos.!s C ˚/�ŒA cos.!t C ˚/�gd˚

D EfA2g 12

2Z

0

cos.!s C ˚/ cos.!t C ˚/d˚

D EfA2g 12

2Z

0

1

2fcos.t � s/ cosŒ!.s C t/C 2˚�gd˚ (C81)

The first term characterizes a constant, while the second term approaches zero suchthe resulting variance-covariance function is proportional to the difference � DW t�s

˙.�/ D ˙.t � s/ D 1

2EfA2g cos!t (C82)

Such a process is stationary.

End of Example 6.3

Ex 4: Superposition of Two Uncorrelated Random Function

Example 6.4 Superposition of two uncorrelated random functions

Let A and B be two uncorrelated random functions characterized by EfAg DEfBg D 0 and var.A/ D var.B/ D �2 < 1, The derived stochastic processis defined by fX.t/; t 2 .�1;C1/g, namely by

X.t/ D A cos!t C B sin!t (C83)

Because of var.X.t// D �2 <1we have an example of random process of secondorder. Due to

EfX.t/g D EfAg cos!t C EfBg sin!t D 0 (C84)


Fig. C.2 Pulse demodulationof a binary system

We are able to compute the variance-covariance function ˙.s; t/ D EfX.t/X.s/gassuming EfA;Bg D EfAgEfBg D 0 due to uncorrelation

˙.s; t/ D EfA2 cos!s cos!t C B2 sin!s sin!tgCEfAB cos!s sin!t C sin!s cos!tg

D �2.cos!s cos!t C sin!s sin!t/

CEfABg.cos!s sin!t C sin!s cos!t/ (C85)

˙.s; t/ D �2 cos!t for � DW t � s (C86)

Again we gain a stochastic process which depends only on the difference t � s DW �which is stationary.

End of Example 6.4

Ex 5: Pulse Modulation

Example 6.5 Pulse modulation

A source generates at each time interval T time units of independent from each othera sign “1” or “0” with the probability 1-p or � . The transfer of a “1” or “0” appearsin this way that T time units produce a pulse with the amplitude A. In this way theygenerate a random signal, namely a stochastic process fX.t/; t 2 .�1;C1/g, withthis property:

X.t/ D�0 with probability �

A with probability .1 � �/:

We start from the partial sequence, for instance

� � � 1 0 1 1 0 1 � � � ,illustrated by Fig. C.2. We have chosen the initial value t D 0 in such a way that westart the transmission of a message.Finally we review the trend function of the process as a constant and the covariancefunction as a non stationary process.


Fig. C.3 Randomly retardedpulse demodulation of abinary system

EfX.t/g D AP fX.t/ D Ag COP fX.t/ D 0g D A.1� p/ (C87)

nT � s; t < .nC 1/T; n D 0;˙1;˙2; � � �EfX.s/;X.t/g D EfX.s/;X.t/jX.s/ D AgP fX.s/jX.s/ D Ag

CEfX.s/;X.t/jX.s/ D 0gP fX.s/jX.s/ D 0gD A2.1 � p/ (C88)

Form ¤ n, the process X.s/ and X.t/ are independent due tomT � s � .mC 1/Tand nT � t < .n C 1/T . Accordingly the variance-covariance function is notstationary:

˙.s; t/ D Cov.s; t/

D�A2p.1 � p/ for all nT � t < .nC 1/T; for all n D 0;˙1;˙2; � � �0 otherwise:

(C89)

End of Example 6.5

Ex 6: Random Retarded Pulse Modulation

Example 6.6: Randomly retarded pulse modulation

With regard to our previous example of a stochastic process fX.t/; t 2 .�1;C1/g,we consider another stochastic process fY.t/DfX.t � 1/g subject to t 2.�1;C1/ assuming that D 2 Œ0; T � is a random variable of a trajectoryfX.t/; t 2 .�1; C1/g shifted to the right producing the other stochasticprocess fY.t/; t 2 .�1;C1/g. We illustrate the transfer by a random number� � � 101101 � � � under the constraint DD 0. The trend function of such a process ischaracterized by EfX.t �D/g D A.1� p/ as we have seen before.X.s/ and X.t/ are independent if and only if when the inequality jt � sj > T holdsor if s and t at the points nT C D for n D 0;˙1;˙2; � � � are disjunct. Let us callthe random event by B . The complementary event is denoted by B . The related


Fig. C.4 Covariancefunction of a randomlyretarded pulse modulation

probability values are called P.B/ D .t � s/=T and P.B/ D 1 � .t � s/=T . Forall jt � sj � T to variance-covariance function is represented by

˙.s; t/ D EfX.s/;X.t/jBgP fBg C EfX.s/;X.t/jBg �EfX.s/gEfX.t/gD EfX.s/;X.t/jBgP fBg C EfŒX.s/�2gP fBg �EfX.s/gEfX.t/gD ŒA.1 � p/�2jt � sj=T CA2.1 � p/.1 � jt � sj=T / � ŒA.1 � p/�2

(C90)

˙.� WD t � s/ D�A2p.1 � p/.1 � j� j=T / forj� j � T0 otherwise:

(C91)

The random process of second order fY.t/; t 2 .�1;C1/g is therefore stationaryillustrated by its variance covariance-function of Fig. C.4

Example of stochastic processes with discrete time

Stochastic processes with discrete time scale can be understood as a randomsequence. Here we restrict us to stationary random sequences as a first order model.They play a dominant role in signal transform and signal transfer.

End of Example 6.6

Ex 7: Random Sequences with Discrete Signals

Example C6.7 Random sequences with discrete signals

Let f� � � ;X�2;X�1;X0;X1;X2; � � � g be a random sequence of uncorrelated identicaldistributed random events characterized by

EfXig D 0; VarfXi g D �2 <1; i 2 f0;˙1;˙2; � � �Due to the restricted range of variances we model a random process of second order,its trend function is zero.


EfXig D 0; for all i 2 f0;˙1;˙2; � � �

cov.�/ D��2 for all � D 0I0 for � ¤ 0: (C92)

Its covariance function with odd numbers s and t is characterized by the formCov.s; t/ D EfXsXt g such that Cov.s; t/ D EfXsgEfXtg D 0 for s ¤ t andCov.s; t/ D EfX2g D �2 for s D t . Obviously beside the property of stationarityof random sequences it is important the property of independent increments.

End of Example 6.7

Ex 8: Random Sequences of Moving Averages of Ordern W MA.n/

Example C.6.8 Random sequences of moving averages of order n: MA(n)

A random signal may be given by

Yt DnX

iD0ciXt�1for allt 2 f0;˙1;˙2; � � � g

where fXig is a random sequence of independent effects with parameter. The naturalnumber n and the sequence of real restricted real numbers c0; c1; � � � ; cn�1; cnare given. The construction of the Yt are influenced by the momentary date Xt

as well as n previous values f� � � ;X�2;X�1;X0;X1;X2; � � � g. Such a relation iscalled Principle of moving averages characterized by the process of second orderXt ; t 2 f0;˙1;˙2; � � � g.

VarfYtg D �2nX

iD0ci ci <1 for all t 2 f0;˙1;˙2; � � � g (C93)

“trend function”EfXtg D 0 for al l t 2 f0;˙1;˙2; � � � g .C 93/

“covariance function”

cov.s; t/ D EfYsYt g D E8<

:

"nX

iD0ciXs�i

#

Œ

nX

jD0cjXt�j �

9=

;

D E

8<

:

2

4X

i D 0nnX

jD0ci cjXs�iXt�j

3

5

9=

;(C94)


“if jt � sj > n , then EfXs�iXt�j g D 0 subject to s � i ¤ t � j ”,

“if s � i D t � j , then”

cov.s; t/ D EfnX

0�i�n;0�jt�sj�i�nci cjt�sjCiX2

s�i g

D �2n�jt�sjX

iD0ci cjt�sjCi (C95)

As a summary, the covariance function cov.s; t/ D cov.t/ subject to � WD t � s isas a sequence of moving average stationary

cov.�/ D��2.c0cj� j C c1cj� jC1 � � � C cn�j� jcn/ for 0 � j� j � nI0 for j� j > n: (C96)

End of Example C 6.8

Ex 9: Random Sequences with Constant Coefficients of MovingAverages of Order n W MA.n/

Example C.6.9 Random sequences of moving averages of order n: MA(n)

A special case if the coefficient ci in our previous example is the setup c D 1=.nC1/for i 2 f0; 1; � � � ; n � 1; ng. The sequence of moving average of order n can bedefined by

Yt D 1

nC 1nX

iD0Xt�i for all t 2 f0;˙1;˙2; � � � g (C97)

cov.�/ D"

�2

nC11 � j� j

nC1�for 0 � j� j � nI

0 otherwise:(C98)

“MA” summarizes “moving average” of order n.



Ex 10: Random Sequences of Unlimited Order

Example C.6.10 Random sequences of unlimited order

Let be given the random sequence of unlimited order of type

Yt D1X

iD0ciXt�i for all t 2 f0;˙1;˙2; � � � g (C99)

for real numbers ci . In order to guarantee the convergence of the series we assume

1X

iD0ci <1 (C100)

its related covariance function can be represented

Cov.�; t/ D ˙.�/ D �21X

iD0ci cj� jCi for al l � D f0;˙1;˙2; � � � g (C101)

Var.Yt / D ˙.0/ D limJ!1 �2

JX

iD0ci for al l t D f0;˙1;˙2; � � � g (C102)

Yt D ˙.0/ D limi!�1;C1 ciXt�i for al l t D f0;˙1;˙2; � � � g

Cov.s; t/ D ˙.�/ D �2 limJ2Œ�1;C1�

ci cj� jCi

The result is a two-sided sequence of moving averages


Ex 11: Autoregression Sequence of First Order: AR.1/

Example C.6.11 Autoregressive sequences of first order: AR(1)

The starting point is the autoregressive sequence of first order AR(1) of type

Yt D aYt�1 C bXt for all t D f0;˙1;˙2; � � � g (C103)


limited by jaj < 1 and b as well as the random sequence fXg. The instant state Yt

depends only of the previous state Yt�1 as well as a random disturbance Xt subjectto fXg D 0 and Ef.Xt � EfXt g/2g D b2�2. Applying n times of the series we gain

Yt D anYt�n C bn�1X

iD0aiXt�i (C104)

limn!1an D 0 .C 105/ ) Yt D b

1X

iD0aiXt�� (C105)

“Special Case”

ci DW bai W limJ!1

JX

iD0

�bai

�2 D b2 limJ!1

JX

iD0a2i D b2

1 � a2 forJ <1(C106)

For an autoregressive sequence of first order of a stationary random process wearrive a the characteristic covariance function

Cov.�/ D .b�/2 limJ!1

JX

iD0aiaj� jCi D aj� j.b�/2 lim

J!1

JX

iD0a2i (C107)

Cov.�/ D .b�/2

1 � a2 aj� j for � D 0;˙1;˙2; � � � (C108)


Ex 12: Autoregression Sequence of First Order r: AR.r/

Example C 6.12 autoregressive sequences of order r: AR(r)

An autoregressive sequence of order r called “AR(r)” is the random sequenceYt ; t D 0;˙1;˙2; � � � over the set of real numbers a1; a2; � � � ; ar�1; ar namely

Yt C a1Yt�1 C a2Yt�2 C � � � C ar�1Yt�.r�1/ C arYt�r D bXt (C109)

The set Xi is a random sequence of parameters


Is the setup Yt D limJ!1

JP

iD1ctXt�1 for

1PiD0

ci < 1 for a stationary sequence

transformable in an autoregressive sequence of order r ?

For solving this problem we have to solve for the relation of constants

c0 D bc1 C a1c0 D 0

c2 C a1c1 C a2c0 D 0cr C a1cr�1 C � � � C arc0 D 0

ci C a1ci�1 C � � � C arci�r D 0 (C110)

A non-trivial solution of the equation leads us to the algebraic equation

yr D a1yr�1 C � � � C ar�1y C ar D 0 (C111)

Example: y2 C a1y C a2 D 0For an autoregressive sequence of second order, namely r=1, we study the equationof quadratic order, namely by the eigen values �1 and �2. For the case �1 ¤ �2 wecompute the covariance function.

Cov.�/ D Cov.0/.1� �21/�j� jC1

2 � .1 � �22/�j� jC11

.�2 � �1/.1 � �1�2/ (C112)

Alternatively, for the case �1 D �2 D � we calculate the covariance function:

Cov.�/ D Cov.0/

�1C 1 � �2

1C �2 j� j��j� j for � D 0;˙1;˙2; � � � (C113)

subject to

Cov.0/ D VarfYt g

Cov.0/ D 1C a2.1 � a2/Œ.1C a2/2 � a21�

(C114)

If �1 and �2 are complex, then there exist � and ! subject to

�1 D � exp.i !/; �2 D � exp.�i !/ (C115)


Cov.�/ D Cov.0/˛�j� j sin.!j� j C ˇ/ for � D 0;˙1;˙2; � � � (C116)

˛ WD 1

sinˇ; ˇ D arctan

�1C �21 � �2 tan!

�(C117)

Example: Numerical example

Let us introduce the numerical example for the autoregressive sequence

Yt � 0:6 Yt�1 C 0:05 Yt�2 D 2Xt for t D 0;˙1;˙2; � � � (C118)

subject to

VarfXt g D 1 (C119)

Obviously the impact of the term at the time instant t-2 on Y is small, at leastcompared to term on the time instant t-1. The related algebraic equation of secondorder can be represented by

Y2 � 0:6 y C 0:05 D 0 (C120)

Their solution is �1 D 0:1 and �2 D 0:5. Indeed compared to one, they are smallthan one. The solution leads to a stationary autoregressive sequence characterizedby the covariance functions

Cov.�/ D 7:017 � .0:5/j� j � 1:063 � .0:1/j� j (C121)

subject to

� D 0;˙1;˙2; � � �Cov.0/ D VarfYt g D 5:954

More details can be taken from Andel (1984)

End of Example C.6.12

Ex 13: ARMA (r,s) Models

Example C.6.13 ARMA(r,s) models

C-7 Wiener Processes 781

As a combination of random sequences of moving averages and of auto regressivesequences we introduce the ARMA(r,s) of the order

�YtCa1Yt�1C � � � Car�1Yt�.r�1/CarYt�r D b0XtCb1Xt�1C � � � Cbs�1Xt�.s�1/CbsXt�s

�

(C122)

In practice, ARMA model we used to model time series as a sequence of realnumbers determined by observing the state variables in discrete time instants. Veryoften, time series are not realizations of stationary random sequences of typefYt ; t D 0;˙1;˙2; � � � g with constant trend functions EfYtg D �.t/, but thetransformation

Z.t/ D EffY.t/g � EfY.t/gg (C123)

reduced mean

EfZig D EŒfYt g �EfYtg� D 0 (C124)

reduced covariance matrix

DfZg D EŒfYt1g � EfYt1g� D ŒYt2EfYt2g� D cov.t1; t2/ (C125)

leads approximately to proper fit of ARMA models. They are called trend reducedmoving average interregressive sequences being “nearly stationary”. Very efficientsoftware exist nowadays.

End of Example C.6.13: ARMA(r,s)

C-7 Wiener Processes

First, we define a Wiener Process by three axioms, namely by stationary incrementsof a random function, together with path continuity. The relation to a random walkis outlined. Especially the probability density of an ordered n-dimensional WienerProcess is reviewed. Secondly, special Wiener A Process of type (a) Ornstein-Uhlenbeck (b) WIENER Process with drift and (c) integrated WIENER Processare introduced

C-71 Definition of the Wiener Processes

The English botanician R. Brown published in the year 1828 the random movementof microscopic small organic and inorganic particles suspended in liquids illustratedin Fig. C.3. We observe a totally irregular motion of particles, nowadays calledBrownian motion. Its first analytical treatment, we owe Bachelier (1900) and


Fig. C.5 Trajectory of a twodimensional Brownianmotion of 30 s distance(Perrin 1916)

Einstein (1905). Both found that a two-dimensional Gauss-Laplace distribution canmodel the typical Brownian motion. They spoke a chaotic movement of microscopicsmall particles. Wiener (1923) built up his theory of stochastic process withindependent increments to model this Brownian motion we will define first.

Definition C.7.1: Wiener Process

A stochastic process fX.t/; t 0g with continuous time and the state space Z D.�1;C1/ is called Wiener Process when it can be described as a path-continuousprocess by the following axioms.

(i) X.0/ D 0 (C 126)(ii) fX.t/; t 0g has stationary increments (C 127)

(iii) for all t > 0, X.t/ is Gauss-Laplace normally distributed (C 128)

End of Definition C.7.1: Wiener

Because of the stationarity of the increments, the difference X.t/ � X.s/ for alls; t 0 are Gauss-Laplace normally distributed with expectation EfX.t/ �X.s/g D 0 and variance �2jt � sj. Since the increments are independent theincrements are a Markov-Wiener process. If � D 1, then we call the process“standard Wiener”.

Continuity and differentiability

EfjX.t/ �X.s/j2g D VarfX.t/ � X.t/g D �2jt � sj (C129)

limh!0

EfjX.t C h/ �X.t/j2g D limh! �2jhj D 0 (C130)

“In the mean square sense the Wiener Process is continuous”

Wiener Process and the random walk


There is a intimate contact of Wiener Process and “a random walk” of a particle.The random walk of a particle can be described by

X.t/ D .X1 C X2 C � � � C XŒt=�t�/�x (C131)

“In the time interval starting by x D 0, a particle moves in the time difference �tby �x as a length unit to the left and the right with probability 1/2”.

X.t/ D�C1 if the jump is to the right�1 if the jump is to the left

(C132)

“t=�t” is the largest even number smaller or equal “t=�t”

P.Xt D 1/ D P.Xt D �1/ D 1=2 (C133)

EfXig D 0 and VarfXi g D 1)

) EfX.t/g D 0 and VarfX.t/g D .�x/2Œt=�t��t ! 0) EfX.t/g D 0, VarfX.t/g D �2t

Multidimensional distribution and conditional probability

Let fX.t/g for t 0 be a Wiener Process. We ask the question what is distributionof the sum of independent Gauss-Laplace normally distributed incremental qualitiesof type

X.ti / D X.t1/C .X.t2/X.t1//C � � � C .X.ti /X.ti�1//subject to

i D 2; 3; � � � ; n � 1; nThe result will lead us to the Special Wiener Process of type Gauss Process

Lemma C.7.2: Probability density of an n-dimensional Wiener Process

Let us assume 0 < t1 < t2 < � � � < tn�1 < tn for an ordered n-dimensional WienerProcess of the random vectors fX1;X2; � � � ;X.tn�1/;X.tn/g characterized byindependent Gauss-Laplace normally distributed increments of a Wiener Process.Then the probability density has the property


ft1;t2;�� ;tn�1;tn .X1;X2; � � � ;X.tn�1/;X.tn//

D ft1.X1/ft2�t1 .X2 � X1/ � � �ftn�1�tn�2 .Xn�1 � Xn�2/ftn�tn�1 .Xn � Xn�1/

(C134)

and has the structure

ft1;t2;�� ;tn�1;tn .X1;X2; � � � ;X.tn�1/;X.tn//

D expf�12Œx21t1C .x2 � x1/2

t2 � t1 C � � � C .xn�1 � xn�2/2

tn�1 � tn�2C .xn � xn�1/2

tn � tn�1�g

�.2/n=2�npt1.t2 � t1/ � � � .tn�1 � tn�2/.tn � tn�1/ (C135)

End of Lemma C.7.2: n-dimensional Wiener Process

Proof of Lemma 7.2

For the proof we depart from the one dimensional probability density of fXtg oftype Wiener Process.

First assumption

ft .x/ D � expŒ�x2=.2�2t/��p2t� (C136)

Second assumption

Let fs;t .x1x2/ the probability density of the random vector fX.s/X.t/g subject tothe order 0 < s < t characterized by

fs;t .x1x2/dx1dx2 D P.X.s/ D x1;X.t/ D x2/dx1dx2 (C137)

P.X.s/ D x1;X.t/ D x2/ D P.X.s/ D x1;X.t/ � X.s/ D x2 � x1/dx1dx2(C138)

fs;t .x1x2/dx1dx2 D P.X.s/ D x1/P.X.t/ �X.s/ D x2 � x1/dx1dx2 (C139)

“independent increments”

fs;t .x1x2/ D fs.x1/ft .x2 � x1/ (C140)

Third assumption

fs;t .x1x2/ D expf�Œtx21 � 2sx1x2 C sx21 �g � 2�2ps.t � s/ (C141)


“correlation coefficient, covariance function”

s; t D Cps=t (C142)

CovfX.s/;X.t/g D �2s for 0 < s � t

CovfX.s/;X.t/ �X.s/g D 0 (C143)

CovfX.s/;X.t/g D CovfX.s/;X.s/C X.t/ � X.s/gD CovfX.s/;X.s/g C CovfX.s/;X.t/ �X.s/gD CovfX.s/;X.s/g D VarfX.s/g D �2 (C144)

“conditional probability density”

Condition: X.t/ D b

fX.s/.xjX.t/ D b/ D fs;t .x; t/� ft .b/ (C145)

fX.s/.xjX.t/ D b/ D expf�.x � stb/2 �

r2s

t.t � s/�g (C146)

EfX.s/jX.t/ D bg D s

tb; VarfX.s/jX.t/ D bg D �2 s

t.t � s/ (C147)

maxVarfX.s/jX.t/ D bg , s D t=2 (C148)

Finally, a generalization from ft .t/; fs;t .s; t/ to ft1t2��tn�1tn leads us to Lemma C 7.2

End of proof for Lemma C.7.2

At this end we will define the Gauss process.

Definition C.7.3: Gauss process

A stochastic process X.t/; t 2 T is a Gauss Process of type (C135), if for arbitraryvectors .t1; t2; � � � ; tn�1; tn/ ordered as t1 < t2 < � � � < tn�1 < tn and n D 1; 2; � � �are random vectors fX.t1/;X.t2/; � � � ;X.tn�1/;X.tn/g of an n-dimensional Gauss-Laplace normal distribution.

End of Definition C.7.3: Gauss process

C-72 Special Wiener Processes: Ornstein–Uhlenbeck, WienerProcesses with Drift, Integral Wiener Processes

Here we only introduce special Wiener Process of type


(i) Ornstein-Uhlenbeck(ii) Wiener Process with drift and

(iii) integrated Wiener Process

It has to be remembered that the trajectories of a Wiener Process are nowheredifferentiable. Particles in motion have an infinite velocity when modeled a WienerProcess. In order to overcome this difficulty, Ornstein and Uhlenbeck developeda stochastic model for modeling the velocity of particles, an important issue inContinuum mechanics.

Definition 7.3: Ornstein-Uhlenbeck Process

Let fX.t/; t 0g be a Wiener Process with the parameter � . Then we call

V.t/ D Œexp.�˛t/�X.exp.˛t// (C149)

subject to ˛ > 0, a stochastic process fV.t/;�1 < t < C1g an Ornstein-Uhlenbeck Process.

End of Definition: Ornstein-Uhlenbeck Process

What are the characteristic properties of the Ornstein-Uhlenbeck Process?

Property 1: Probability density

fV.t/.x/ D Œexp.�x2=2�2/�� .p2�/ (C150)

Property 1: Expectations

EfV.t/g D 0; VarfV.t/g D �2 (C151)

Cov.s; t/ D �2Œexp.�2˛.t � s//� for s � t (C152)

Proof for the covariance function

Cov.s; t/ D Cov.V .s/; V .t// D EfV.s/V .t/gD expŒ�˛.s C t/�EfX.exp2˛s/X.exp2˛t/gD expŒ�˛.s C t/�Cov.X.exp2˛s/X.exp2˛t//

D expŒ�˛.s C t/��2Œexp.2˛.t � s//� (C153)

It has to be mentioned that the Ornstein-Uhlenbeck Process is stationary in thewider sense. As a Gauss Process it is also stationary in the narrow sense. In sum-mary, the Ornstein-Uhlenbeck Process is an instationary Wiener Process whichis stationary process due to the action of time transformation and normalization.In comparison to the Wiener Process, the Ornstein-Uhlenbeck has the followingdifferences


1. The Ornstein-Uhlenbeck Process has no independent increments2. The trajectories of an Ornstein-Uhlenbeck Process are in the quadratic

mean differentiable

Wiener Process with drift

Another example for the transformation of an instationary process to a stationaryprocess is the Wiener Process with drift.

Definition 7.4: Wiener Process with drift

A stochastic process fW.t/; t 0g with independent increments is called a WienerProcess with drift, if the following properties hold:

(i)W.0/ D 0 (C154)

(ii) Each increment has the property W.t/ � W.s/ is Gauss-Laplace normallydistributed with second order statistics

(iii)EfW.t/g D �.t � s/ (C155)

(iv)VarfW.t/g D �2jt � sj (C156)

End of Definition 7.4: drifted Wiener Process

An equivalent formulation is next:

“fW.t/; t 0g is a Wiener Process with drift, if it has the structure

W.t/ D �t C X.t/ (C157)

where fX.t/; t 0g is a Wiener Process subject to �2 D VarfX.1/g and � anarbitrary constant, called as drift parameter”.

Obviously, a Wiener Process with drift is generated by the super position of type“additive” of a Wiener Process with a deterministically growing decreasing part.The deterministic part is a trend function of type �.t/ D �t .The one dimension probability density distribution of a “Wiener Process with drift”can be characterized by

fW.t/.x/ D expŒ�.x � �t/2=.2�/�� Œp2t��; �1; x;C1 (C158)


In practice, drift parameters are daily experience. We restrict ourselves with onlyone example.

Example: Geometric Wiener Process with drift

Let us introduce a stochastic process of type fZ.t/; t 0g characterized by

Z.t/ D expW.t/ (C159)

called “geometric Wiener Process with drift”. Since the model W.t/ has bydefinition a Gauss-Laplace normal distribution the expectation of .Z.t// equals themoment generating function of W.t/. For instance,

EfexpW.t/g D expfst.�C 1=2�2s/g (C160)

s D 1 W) EfZ.t/g D expŒt.�C �2=2/� (C161)

s D 2 W

) EfZ2.t/g D expŒ2t.�C �2/� (C162)

) VarfZ.t/g D expŒt.2�C �2/�Œexp.t�2/ � 1� (C163)

Integrated Wiener Process

By transforming of a Wiener Process we receive a stochastic process which is veryimportant in the analysis. We will summarize these transformations in three casesin our

Lemma 7.5: Elementary transformations os a Wiener Process

Let fX.t/; t 0g be a standard Wiener Process such that the following stochasticprocesses are as well standard Wiener Processes:

(i)fU.t/; t 0g subject to U.t/ D cX.t=c2/; c > 0 (C164)

(ii)fV.t/; t 0g subject to V.t/ D X.t C h/� X.h/; h > 0 (C165)

(iii) fV.t/; t 0g subject to

W.t/ D�

X.t/ for all t > 00 for all t > 0

(C166)

End of Lemma 7.5

We skip the proof. Instead let fX.t/; t 0g be a Wiener Process: Accordingly “nearall” trajectories x D x.t/ are continuous in the squared mean. In consequence, there


exist the integrals u.t/ DtR

0

x.y/dy for “nearly all” trajectories x D x.t/. These

integrals are realizations of the random integral.

u.t/ DtZ

0

x.y/dy (C167)

u.t/ is the symbol for the integral acting on “nearly all” trajectories of a WienerProcess subject to the probability distribution fU.t/; t 0g defines a stochasticprocess called integrated Wiener Process.

Stochastic integration

Assume the Riemenn integration for 0 D t0 < t1 < � � � < tn�1 < tn D 1 subject to�ti D ti � ti�1 for forward computations and i D 1; 2; � � � ; n � 1; n

u.t/ D limn!1;�ti!0

fnX

iD1ŒX.ti /� X.ti�1/��ti (C168)

u.t/ is the lines of a sum of independent Gauss-Laplace normally distributedrandom variables as well Gauss-Laplace normally distributed: An integratedWiener Process is being analyzed by

.a/ Trend function:

EftZ

0

X.y/dyg DtZ

0

EfX.y/gdy D E.t/ D 0 (C169)

.b/ Covariance function:

Covfu.s/; u.t/g D EfsZ

0

X.z/d z

tZ

0

X.y/dyg DsZ

0

tZ

0

EfX.z/X.y/gdyd z

D �2

sZ

0

tZ

0

min.y; z/dyd z D �2sZ

0

yZ

0

zd zdy C �2sZ

0

tZ

y

y dz dy

D �2sZ

0

1

zy2dy C �2

sZ

0

y.t � y/dy (C170)


Covfu.s/; u.t/g D �2

6.3t � s/s2 for s � t (C171)

Varfu.t/g D �2

3t3 for s D t (C172)

An integrated wiener Process is not stationary, in general. But it can be proven thatthe stochastic process V.t/ WD u.tC�/�u.t/ for V.t/ 2 fV.t/; t 0g is stationaryfor all � > 0.

White noise

A stochastic process, Z.t/ 2 fZ.t/; t 0g subject to

Z.t/ WD dX.t/� dt D X.t/ (C173)

or

dX.t/ WD Z.t/dt (C174)

cannot be computed by standard differentiation: We have to introduce stochasticintegration, for instance

bZ

a

f .t/dX.t/ D limn!1;�ti!0

fnX

iD1f .ti�1/ŒX.ti / �X.ti�1/�g (C175)

bZ

a

f .t/dX.t/ D f .b/X.b/� f .a/X.a/ �bZ

a

X.t/df .t/ (C176)

The representation reminds us to “partial integration”. As the limit of the sumof independent Gauss-Laplace normally distributed random variables we have tocompute the expectation of type

EfbZ

a

f .t/dX.t/g D 0 (C177)


VarfnX

iD1f .ti�1/ŒX.ti / �X.ti�1/�

DnX

iD1f 2.ti�1/VarfX.ti /� X.ti�1/g

D �2nX

iD1f 2.ti�1/.ti � ti�1/ D �2

nX

iD1f 2.ti�1/�ti (C178)

The limit approaches the variance of the stochastic integral of type

VarfbZ

a

f .t/dX.t/g D �2bZ

a

f 2.t/dt (C179)

This result leads to the definition of White Noise

Definition C.7.6: White Noise

A stochastic process fZ.t/; t 0g is called “White Noise” if for any arbitraryinterval Œa; b� of continuously differentiable functions f .t/, namely

bZ

a

f .t/dZ.t/ D f .b/X.b/ � f .a/X.a/ �bZ

a

X.t/df .t/ (C180)

where fX.t/; t 0g is a Wiener Process.

End of Definition C.7.6: White Noise

?Why we call stochastic process “White Noise”?

The origin of the question is the definition of the “generalized derivative”. To thisend we define ”the Dirac Delta function” as a “generalized function”

ı.t/ WD limh!0

�1=h for � h=2 � t � Ch=20 otherwise

(C181)

or

ı.t/ WD�1 for t D 00 otherwise

(C182)


Properties of Dirac’s Delta function

(i)

C1Z

�1f .t/ı.t � t0/dt D f .t0/ (C183)

(ii) Heaviside functions

H.t/ WD�1 for t 00 for t < 0

(C184)

“The Dirac’s Delta function is the formal derivative of the Heavide function”

ı.t/ D @H.t/=@t (C185)

(iii) Covariance function

CovfZ.s/;Z.t/g D CovŒ@X.s/=@s; @X.t/=@t

D @

@s

@

@tCovfX.s/;X.t/g D @

@s

@

@tmin.s; t/

D @

@sH.s � t/ (C186)

CovfZ.s/;Z.t/g D ı.s � t/ (C187)

the covariance function is a measure of the statistically independence betweenZ.s/ and Z.t/, have a function of the small distances between the point s andthe point t, namely js � t j.

“How can we illustrate the Dirac function?”Let fN.t/; t 0g be an arbitrary numbering process following our Definition C.7.6.The instants of the process at the points T1; T2; � � � ; admit the representation in termsof the Heavide function N 0.t/ in terms of the Dirac function:

N.t/ D1X

iD1H.t � Ti/; @N.t/=@t D

1X

iD1ı.t � Ti / (C188)

Figure C.6 an illustration of the numbering process.The theory of stochastic integration has led us to the concept of generalized function,namely the Dirac function and Heaviside function, and to the notion of White Noise.More general notions like the concept of Colored Noise and finally to the notion ofits stochastic processes.

C-8 Special Analysis of One Parameter Stationary Stochastic Process 793

Fig. C.6 Dirac numbering system

C-8 Special Analysis of One Parameter StationaryStochastic Process

C.8: Spectral analysis of one parameter stationary stochastic processes

Statistic or stochastic processes are conventionally divided into three classes wehave already touched:

Class onegeneral instationaryprocess

Class twostationaryprocess

Class threeergodic stationaryprocess

In real life, of course, we have live the property that all natural process areinstationary. It is the task of modeling real world process, for instance by a firstorder trend model or by taking increments of higher order to come more closely tothe case of stationary processes for higher order increments. We will come back tothis problem.For formal reasons we introduce first real-valued and complex-valued stochasticprocess. Second we study the structure of process with a discrete spectrum. At theend, we introduce stochastic process with a continuous spectrum enriched byexamples.

C-81 Foundations: Ergodic and Stationary Processes

C.8.1: Foundations: Ergodic and stationary processes

We move on towards complex-valued stochastic processes defined by the followingaxioms:


.a/ X.t/ D Y.t/C iZ.t/ subject to i D p�1 (C189)

.b/ “Real” W fY.t/; t 2 Œ�1;C1�g

.c/ “Imaginary” W fZ.t/; t 2 Œ�1;C1�gNZ D a � b; jzj D pzNz D pa2 C b2 “absolute value” (C190)

.d/ “Trend function” W EfX.t/gEfY.t/g C iEfZ.t/g.e/ “Covariance function” W

CovfX.s/;X.t/g D EfŒX.s/ �EfX.s/g�ŒX.t/� EfX.t/g�g(C191)

.f / Stationary in the wider sense W.vi1/ EfX.t/g D �.t/ D � (constant) (C192)

.vi2/ CovfX.s/;X.t/g D Cov.oI s � t/; � WD t � s (C193)

.g/ Complex-valued stochastic process of second order:

EfjX.t/j2g <1 foral l t 2 Œ�1;C1� (C194)

(C195)

Ergodicity

A stochastic process EfX.t/; t 2 Œ�1;C1� is stationary in the narrow sense iffor all trajectories x.t/ D y.t/C iz.t/ there hold

EfX.t/g D EfX.t0/g DW limT!1

1

2

CTZ

�Tx.t/dt for t0 2 R (C196)

In this integral relation, in order to compute the expectation value Efx.t0/g weuse only all the information which is located in one realization of the stochasticprocess. In contrast to the general case, we have used up to zero the information ofall realizations of the stochastic process. In stationary ergodic process we used onlyone instant. An excellent approximation is the presentation.

EfX.t0/g D � D limN!1

1

N

NX

RD1xk.t0/ (C197)

Such a definition has a simple physical interpretation: The mean of a fixed instantequals the mean of a long time interval: The mean location is identical to the time


average. In essence, This is the property of ergodic stationary processes. Analoguewe can define the covariance function for a such processes namely

Cov.�/ D limT!1

1

2

CTZ

�TŒx.t/ � �px.t C �/ � �dt (C198)

The treatment of ergodic stationary processes is reasonable for cases in which wehave only one time like realization of the stochastic process. Of course increasingthe length of the interval Œ�T;CT � improves the estimation of EfX.t0/g D �.

Here we also take reference the Euler’s representations:

exp˙ix D cos x ˙ i sin x (C199)

sinx D 1

2iŒexp.ix/� exp.�ix/� (C200)

cos.x/ D 1

2Œexp.ix/C exp.�ix/� (C201)

C-82 Processes with Discrete Spectrum

C.8.2: Processes with discrete spectrum

We start with general structure of stationary processes with discrete spectrum: weassume that all stationary process are ergodic.

Ex 1: Special Stationary Ergodic Processes

Example 8.1: Special stationary ergodic processes

Let fX.t/; t 0g be a stochastic process structure by

lrX.t/ D X˝.t/; X a complex random variable (C202)

EfXg D 0; E.jX j2/ <1; EfX.t/X.t C �/g D ˝.t/˝.t C �/EfX2g(C203)

� D 0 W ˝.t/˝.t/ D j˝.t/j2 D constant (C204)

“ansatz” W ˝.t/ D j˝.t/ expŒi!.t/� for !.t/ real function (C205)

“ansatz” W !.t C �/� !.t/ ¤ f .t/ W dŒ!.t C �/ � !.t/�=dt D 0 (C206)


or d!.t/=dt D constant (C207)

“ansatz” W !.t/ is a function of two constants; I ! and � W (C208)

˝.t/j˝.t/j expŒi.! C �/� (C209)

End of Example 8.1: Special stationary ergodic processes

Lemma 8.1: decomposition of a complex stationary ergodic process

A stochastic process fX.t/; t 0g of the structure X.t/ D X˝.t/ is stationary inthe wider sense if and only if

X.t/ D X exp.i!t/ (C210)

subject to EfXg D 0; Efjxj2g <1; s DW EfjX j2g; Cov.�/ D s exp.�i!�/End Lemma 8.1

The real part of a complex stochastic process describes a cosine oscillation with arandom amplitude and phase

y.t/ D a cos.!t C �/ (C211)

The parameter ! is called frequency.

We generalize to two stationary processes as a linear combination: we refer to X1and X2 as two complex random variables with zero expectation and two constantparameter f!1; !ı

1 D 0; !2; !ı2 D 0g. We assume that the two complex random

functionsX1 and X2 are uncorrelated such that EfX1X2g D 0 or EfX1 �X2g D 0.Our intention is to get the form of the covariance function:

X.t/ D X1 exp.i!1t/CX2 exp.i!2t/ (C212)

covfX.t/X.t C �/gD EfŒX1 exp.i!1t/CX2 exp.i!2t/�Œ NX1 exp.�i!1t/C NX2 exp.�i!2t/�gD EfŒX1 NX2 exp.�i!1�/CX1 NX2 exp.i.!1 � !2/t � i!2�/�gCEfŒX2 NX1 exp.�i!2�/CX2 NX2 exp.i.!2 � !1/t � i!1�/�g (C213)

s1 DW EfjX1j2g; s2 DW EfjX2j2g (C214)

Cov.�/ D s1 exp.�i!1�/C s2 exp.�i!2�/ (C215)


Ex 2: Special Correlation Function of a Stationary Stochastic Processes

Example 8.2: Special correlation function of a stationary stochastic process

A special complex stationary stochastic process may be real-valued. For instancewe define

X1 D 1

2.AC iB/ and X2 DW NX1 D 1

2.A� ıB/ (C216)

A and B are two real-valued functions. In consequence of our representationX.t/ DX1 exp.i!1t/CX2 exp.i!2t/ we specialize here to

X.t/ D A cos!t � B sin!t (C217)

A, B are uncorrelated. We received the covariance function

Cov.�/ D 2s cos!� (C218)

subject to s WD EfjX1j2g D EfjX2j2gEnd of Example 8.2: Special correlation function

For pairwise uncorrelated stationary stochastic processes the general ansatz of typeX.t/ DPn

kD1 exp.i!kt/ subject to !j ¤ !k for j ¤ k and i; j D 1; 2; : : : ; n�1; nand EfXkg D 0 will lead us to the covariance function

Cov.�/ DnX

kD1sk.i!k�/ and sk D EfjXkj2g for k 2 f1; 2; : : : ; n � 1; ng

(C219)

in particular

Cov.� D 0/ D EfjX.t/j2g DnX

kD1sk (C220)

In summary, the oscillation X.t/ is an additive superposition of a harmonicoscillation. This average energy per time unit is the same of circle average energiesper time unit, an appropriate interpretation of our last representation.In order to achieve a real-valued function, we have specialize to

(a) an even number n,(b) and a pair of valuesXk being conjugate complex.

Let the set f!1; !2; : : : g define the spectrum. We have to divide the various spectruminto two classes: small band processes and broad band processes, an entry into filtertheory with many application in all sciences.


Ex 3: Dirac Representation of a Covariance Function

Example 8.3: Dirac representation of covariance function

We use the infinitely generalized covariance function for application of a Diracfunction representation.

Cov.�/ D limK!1

KX

kD1sk

C1Z

�1exp.i!�/s.! � !k/d! (C221)

Cov.�/ DC1Z

�1exp.i!�/s.!/d! (C222)

subject to

s.!/ D limK!1

KX

kD1skı.! � !k/ (C223)

The generalized function s.! is denoted by “spectral density” of a stationarystochastic process, formally equivalent to the Fourier transform.

End of Example 8.3: spectral density

C-83 Processes with Continuous Spectrum

C.8.3 Processes with continuous spectrum

Here we present first the spectral decomposition of the covariance function of astochastic process with continuous spectrum. Second we present Kolmogorov’sspectral decomposition theorem.

C8-31 Spectral decomposition of a covariance function

Let fX.t/; t 2 Rg be a complex-valued stationary stochastic process describedby its covariance function. Then there exists a real-valued function S.!/ such thatcov.�/ enjoys the representation.

Cov.�/ DC1Z

�1exp.i!�/dS.!/ (C224)

called the spectral function of the process. Note the domain t 2 Œ�1;C1�

Cov.0/ D S.1/� S.�1/ D EfjX j2g <1 (C225)


The spectral function is determined only up to an additive constant c. Mostcommonly c is chosen to S.�1/ D 0. If there exists the first derivative S.!/ DdS.!/=d!, we find the spectral density of the stochastic process: The covariancefunction of this stationary process is the Fourier transform of the spectral density.Since S.!/ has to be a decreasing function we find also its property.

Cov.�/ DC1Z

�1exp.i!�/s.!/d! (C226)

s.!/ 0; (C227)

C1Z

�1s.!/d! <1 (C228)

The set f!; s.!/ > 0g defines the continuous spectrum of the stationary stochasticprocess. The term

Cov.0/ D S.1/� S.�1/ DC1Z

�1s.!/d! (C229)

is called average power of the spectrum. If we describe the stationary stochasticprocess Xt by harmonic oscillations with random amplitude and random phase, weare able to express

exp.i!t/ D expŒi t.!C 2k/� for all k 2 f1; 2; : : : g or k 2 Œ�;C� (C230)

Cov.�/ DCZ

�exp.i�!/s.!/d! for � D 0;˙1;˙2; : : : (C231)

IfC1R�1

Cov.�/d� < 1 for stationary stochastic processes, holds, we are able to

introduce the inverse Fourier transform by

s.!/ D 1

2

C1Z

�1exp.�i!�/Cov.�/d� (C232)


s.!2/ � s.!1/ D i

2

C1Z

�1

exp.�!2�/ � exp.�i!1�/�

Cov.�/d� (C233)

Ex 4: Fourier Spectral Density

Example 8.4: Fourier spectral density

Assume real-valued constants a and cjaj < 1 . Here we will analyze the covariancefunction for an autoregressive stationary sequence of first order and compute itsspectral density.

s.!/ D 1

2

C1X

�D�1Cov.�/ exp.�i�!/

D e

2

" �1X

�D�1a�� exp.�i�!/C

1X

�D0a� exp.�i�!/

#

D c

2

" 1X

�D1a� exp.i�!/C

1X

�D0a� exp.�i�!/

#

(C234)

s.!/ D c

2

�a exp.i!/

1 � a exp.i!/C 1

1 � a exp.i!/

(C235)

End of Example: Fourier spectral density

Assume a real-valued stochastic process whose covariance function Cov.�/ DCov.��/, namely Cov.�/ D ŒCov.�/C Cov.��/�=2, can be represented by

Cov.�/ DC1Z

�1cos.!�/s.!/d! (C236)

Cov.�/ D 2C1Z

�1cos.!�/s.!/d! .cos.!�/ D cos.�!�// (C237)


s.!/ D 1

2

C1Z

�1cos.!�/Cov.�/d� (C238)

s.!/ D 1

C1Z

0

cos.!�/Cov.�/d� (C239)

Correlation time

In practice the notion of “correlation time” or “correlation length” is very appro-priate:

�0 WD 1

Cov.0/

C1Z

0

Cov.�/d� (C240)

or

�0 WD s.0/

2C1R

0

s.!/d!

(C241)

For j� j � �0 there is a unstable correlation or statistical dependence between X.t/and X.t/C � . This correlation decreases for growing j� j; j� j > �0.

Ex 5: Stationary Infinite Random Sequence

Example 8.5: Stationary infinite random sequence

The infinite random sequence f� � � ;�X1;X0;CX1; � � � g may be structuredaccording to Xt D atX subject to t D 0;˙1;˙2; � � � : The complex randomvariable is structured according to EfXg D 0 as well as VarfXg D �2. The randomsequence f� � � ;�X1;X0;CX1; � � � g is the wider sense stationary if and only if theconstants ! within the set-up.

at D exp.i t!/ for t D 0;˙1;˙2; � � � (C242)

are represented. In the stationary case, the harmonic oscillations Xt will be givenby the random amplitude and the random phase as well as a constant frequency !.For such a case a set-up of the covariance-function will be calculated.

exp.i t!/ D exp i t.! C 2k/ for t 2;˙1;˙2; � � � (C243)


Cov.�/ DCZ

�exp.i t!/s.!/d! for � 2;˙1;˙2; � � � , (C244)

,s.!/ D 1

2

C1X

�D�1exp.�i t!/Cov.�/ (C245)

If we specialize to X real-valued and uncorrelated we achieve a pure randomsequence subject to

Cov.�/ D �2 , s.!/ D �2=.2/ (C246)

End of Example 8.5: random sequence

Ex 6: Spectral Density of a Random Telegraph Signal

Example 8.6: spectral density of a random telegraph signal

Given the covariance function, we intend to find a representation of the spectraldensity for a random telegraph signal.

Cov.�/ D a exp.�bj� j/ for a > 0; b > 0 (C247)

s.!/ D 1

2

C1Z

�1exp.�i!�/a exp.�bj� j/d!

D a

2

8<

:

0Z

�1exp.�i!�/d� C

C1Z

0

expŒ�.b � i!/��d�9=

;

D a

2

�1

b � i! C1

b C i!�

(C248)

s.!/ D ab

.!2 C b2/ (C249)

The correlation time is calculated to �0 D 1=b! this result is illustrated by Figs. C.7and C.8, namely by the spectral structure of the covariance function and its spectraldensity

End of Example: spectral density telegraph signal


Fig. C.7 Covariance of Example 8.6

Fig. C.8 Spectral density of Example 8.6

Ex 7: Spectral Representation of a Pulse Modulation

Example 8.7: Spectral density of a pulse modulation

We start with the covariance function of a pulse modulation and study its spectraldensity.

Cov.�/ D�a.T � j� j/ for j� j � T0 for j� j > T (C250)

s.!/ D a

2

CTZ

�TŒexp.�i!�/�.T � j� j/d�

D a

2

8<

:T

TZ

�Texp.�i!�/d� �

CTZ

0

� exp.i!�/d� �CTZ

0

� exp.�i!�/d�9=

;

D a

2

8<

:2T

!sin.!�/ � 2

CTZ

0

� cos.!�/d�

9=

;(C251)

s.!/ D a

1 � cos!T

!2(C252)

Figures C.7 and C.8 illustrate the covariance function and the spectral density ofa pulse modulation. A slight modification of our example proves that it is notautomatic for stationarity.


Fig. C.9 Covariance ofExample 8.7

Fig. C.10 Spectral density of Example 8.7

Cov.�/ D2

4a.T � �2/ for j� j � 0

for a > 0; T > 00 for j� j < 0

(C253)

s.!/ cannot be the spectral density of a stationary stochastic process.

Alternatively, we like to resent the covariance function Cov.�/ be a Dirac delta-function ı.�/. In this case we find

ı.�/ D 1

2

C1Z

�1exp.�i!�/d� (C254)

s.!/ D 1

2

C1Z

�1exp.�i!�/ı.�/d� D 1

2(C255)

by formal inversion.

End of Example 8.7: pulse modulation


Fig. C.11 Spectral density ofa harmonic function

Ex 8: Spectral Density and Spectral Function of a Harmonic Oscillation

Example 8.8: Spectral density and spectral function of a harmonic oscillation

Given a harmonic oscillation with a given covariance function, we will derive thestructure of the spectral density function as well as the spectral function.

Cov.�/ D a cos!0� (C256)

s.!/ D a

2

C1Z

�1exp.�i!�/ cos.!0�/d�

D a

4

1Z

�1exp.�i!�/Œexp.�i!0�/ � exp.�i!0�/�d�

D a

4

8<

:

1Z

�1expŒi.!0 C !/��C

1Z

�1expŒ�i.!0 � !/��

9=

;d� (C257)

s.!/ D a

2fı.!0 � !/C ı.!0 C !/g (C258)

S.!/ D2

40 for ! < �!0a=2 for � !0 < ! � �!0 step functiona for ! > !0

(C259)

We illustrate the spectral density as well as the spectral function of a harmonicoscillation by Fig. C.11 as well as Fig. C.12. It has to be emphasized that all threefunctions Cov.�/; s.!/ and S.!/ are stationary functions.

End of Example 8.8: Spectral function of harmonic oscillation


Fig. C.12 Spectral functionof type step function for aharmonic oscillation

Ex 9: Spectral Density Function of a Damped Harmonic Oscillation

Example 8.9: Spectral density functions of a damped harmonic oscillation

Given the covariance function of a damped harmonic oscillation, we want to findthe spectral density of this stationary stochastic process,

Cov.�/ D a exp.�bj� j/ cos!0� (C260)

s.!/ D a

1Z

0

exp.�b�/ cos.!�/ cos.!0�/d�

D a

2

1Z

0

exp.�b�/Œcos.! � !0/� C cos.!0 C !/��d� (C261)

s.!/ D ab

2

�1

b2 C .! � !0/2C 1

b2 C .! C !0/2�

(C262)

!0 is called the eigen function of the stochastic process of type stationary. A typicalexample is the fading of a radio signal.

End of Example 8.9: Damped harmonic oscillation

Ex 10: Spectral Density of White Noise

Example 8.10: Spectral density of white noise

Perviously we introduced the concept of “white noise” as a real-valued stationaryprocess with continuous time of type Wiener process. Given its covariance function,we are interested in the spectral density function.

Cov.�/ D 2s0ı.�/ (C263)


s.!/ D 1

1Z

�1exp.�i!�/2s0ı.�/d� D S0 (C264)

A comparative formulation of a white noise process is the following:

White noise is a real-valued process whose spectral density is a constant

Comment

the spectral density of a white noise process has only the property s.!/ 0, but not

the property1R

�1s.!/d! <1. Accordingly, a white noise process with an average

power spectrum fulfilss.!/d! D1 (C265)

does exist only in theory not in practice. A comparison is “the mass point” or“the point mass”, which also exist only “in theory”. Nevertheless it is reasonableapproximation of any reality.A stationary process can always be considered as a excellent approximation of whitenoise if the covariance betweenZ.t/ andZ.tC�/ for growing values j� j extremelyfast goes to zero.

Example:

Let us denote the variations of the absolute value of the force by Z.t/ acting ona particle of a liquid medium at the time t and leading to a Brownian motion.Per second, the resulting force leads to approximately 1021 collisions with otherparticles. Therefore the stochastic variables Z.t/ and Z.t C �/ are stochasticallyindependent (“uncorrelated”) if j� j is of the order of 10�18 seconds. If we assume acovariance function of type Cov.�/ D exp.�bj� j/ for b > 0, the number b has tobe larger than 10�17 seconds.

End of Example 8.10: White noise

Ex 11: Band Limit of White Noise

Example 8.11: Band limited white noise

A stationary stochastic process with a constant spectral density function S.!/ D s0for ! 2 f�1;C1g does not in practice. But a stationary stochastic process withthe spectral density


Fig. C.13 Spectral density ofband limited white noiseprocess

Fig. C.14 Covariancefunction of a band limitedwhite noise process

s.!/ D�s0 for �˝=1 � ! � C˝=20 otherwise

(C266)

exists

Cov.�/ DC˝=2Z

�˝=2exp.i!�/so D 2s0 sin.˝�/=2

�(C267)

The average power of the process is proportional to Cov .0/ D s0˝ . The parameter˝ is called the band width of the process. A white noise process is generated by thelimit of a band limited white noise illustrated by Figs. C.13 and C.14.

End of Example 8.11: band limited white noise

C-84 Spectral Decomposition of the Mean andVariance-Covariance Function

C.8.4: Spectral decomposition of the expectation and the variance-covariancefunction

The decomposition of the spectrum of a stationary process fY.x/; x 2 Rg subjectto R 2 f�1;C1g applied to orthogonal increments of various order has manyrepresentations, if they do not overlap in the intervals

Œx1; x2�; Œx3; x4� or Œx1; x2; x3�; Œx4; x5; x6� etc


in the formulations of double, triple etc. differences. An example is

EfŒY.x2/ � Y.x1/�ŒY.x4/ � Y.x4/�g (C268)

EfŒY.x3/ � 2Y.x2/C Y.x1/�ŒY.x6/ � 2Y.x5/C Y.x4/�g (C269)

Accordingly, a real-valued stochastic process with independent increments whosetrend function EfY.x/g D 0 is zero, has orthogonal increments.

Example 8.12: Spatial best linear uniformly unbiased prediction: spatial BLUUP

Spatial BLUUP depends on the exact knowledge of proper covariance function. Ingeneral, the variance-covariance function depends on three parameters fı1; ı2; ı3gcalled f rugged, sill, rangeg

Cov.�/ D .kx1 � x2k/ (C270)

We assume stationarity and isotropy for the mean function as well as for thevariance-covariance function. (In the next section we define isotropy)

Model W Z.x/ � f .x/0ˇ C e.x/ (C271)

Subject to Efe.x/g D 0

Postulate one: “ Mean square continuity ”

limy!x

EfZ.y/� Z.x/g D 0 (C272)

Postulate two: Mean square differentiability

limh!0

Z.k � k C hn/�Z.k � k/hn

�!n!1

0 (C273)

If we assume “mean square differentiability”, then Z0 has a variance-covariancefunction. If “mean square differentiability” holds, it is also differentiated in anydimension. Here we introduce various classes of differential calculus applied tovariance-covariance functions.

(i) DIGGLE chaos:Cov.h/ D c exp.�ajhj� / (C274)

(ii) MATERN class:Cov.h/ D c.˛jhj�H.ajhj// (C275)

“ H” denotes the “modified Bessel function of order �”(iii) Whittle class:

spectral density of type

c.˛2 C !2/��˛=2(C276)


(iv) Stochastic process on curved spaces like the sphere or the ellipsoid: no Maternclasses, the characic functions have a discrete spectrum, for instance in termsof harmonic functions.

End of Example 8.12: spatial BLUUP

Let us give some references: Brown et al. (1994), Brus (2007), Bueso et al.(1999), Christenson (1991), Diggle and Lophaven (2006), Fedorov and Muller(2007), Groeningen et al. (1999), Kleijnen (2004), Muller et al. (2004), Mullerand Pazman (2003), Muller (2005), Omre (1987), Omre and Halvorsen (1989),Pazman and Muller (2001), Pilz (1991), Pilz et al. (1997), Pilz and Spock (2006),PilzSpock(2008), Spock (1997), Spock(2008), Stein (1999), Yaglom (1987), Zhuand Stein (2006).

Up to now we assumed a stochastic process of type fX.t/; t 2 Rg, R D f�1;C1gas a complex stationary process of second order. Then there exist a stochasticprocess of second order fU.!/; ! 2 Rg with orthogonal increments which can berepresented by

X.t/ DC1Z

�1exp.i!t/dU.!/ (C277)

The process fU.!/; ! 2 Rg of the process fX.t/; t 2 Rg is called the relatedspectral process. Its unknown additive constant we fix by the postulate

P fU.�1/ D 0g D 1 (C278)

subject to

EfU.!/g DW 0 (C279)

EfŒjU.!/j2�g � S.!/ (C280)

EfŒjdU.!/j2�g � dS.!/ (C281)

Our generalization was introduced by A. N. Kolmogorov: Compare this representa-tion with our previous model

X.t/ D limk!1

KX

kD1exp.i!t/Xk (C282)

and

C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processes 811

Xt DC1Z

�1exp.i!t/dU.!/ (C283)

The orthogonality of the Kolmogorov stochastic process fU.t/; t 0g correspondsin case of a discrete spectrum to the concept of uncorrelation. Convergence in thesquared mean of type

C1R�1

exp.i!t/dU.!/ D lima!�1b!C1

limn!C1�!k!0

nP

kD1exp.i!k�1t/ŒU.!k/� U.!k�1/�

(C284)

emphasizes the various weighting functions of frequencies: The frequency !k�1 inthe range Œ!k�1; !k� receives the random weight U.!k/ � U.!k�1/. A analogueversion of the inversion process leads to

U.!2/ � U.!1/ D i

2

C1Z

�1

exp.i!2t/ � exp.i!1t/

tX.t/dt (C285)

In practice, Kolmogorov stochastic processes are applied in linear filters for prog-nosis or prediction.

C-9 Scalar-, Vector-, and Tensor Valued Stochastic Processesof Multi-Parameter Systems

Appendix C.9: Scalar-, vector- and tensor-valued stochastic processes of multi-parameter systems

We have experienced the generalization of a scalar-valued stochastic process dueto A.N. Kolmogorov. Another generalization due to him is the concept of thestructure functions for more general vector-valued and tensor-valued for incrementsof various order as well as their power spectrum or multi-parameter systems.

First we introduce the notion of the characteristic functional replacing the notionof the characteristic function for random function. Second, the moment represen-tation for stochastic processes is outlined. Third we generalize the notion fromscalar-valued and vector-valued stochastic processes to tensor-valued stochasticprocesses of multi-point parameter systems. In particular, we introduce the notion


of homogeneous and isotropic stochastic fields and extent the Taylor-Karmanstructured criterion matrices to distributions of the sphere.

C-91 Characteristic Functional

Let us begin with the definition of the characteristic functional.

�

YxŒ�� D Efexp.iY.�//g D EfigbZ

a

�.x/Y.x/dx (C286)

�

Y.�/ as the characteristic functional elates uniquely to the random functionY.x/ of a fixed complex number. If the characteristic functional of a specificrandom function is known we are able to derive the probability density distributionfX1;:::;XN .YX ; : : : ; YXN /.

Example

If a scalar-valued function in a three dimensional field x WD .x1; x2; x3/ exists defineits characteristic functional by

�

YY Œ�� WD Efexp.iZ C1Z

�1

ZY.x1; x2; x3/�.x1; x2; x3//dx1dx2 (C287)

The domain of integration is very often chosen to be spherical coordinates ofellipsoidal coordinates .u; v;w/ for u 2 f0; 2g; v 2 f�=2 < v < C=2g andw 2 f0 < w < 1g as a first chart. A second chart is needed to cover totally theintersection surfaces, for instance by meta-longitude, meta-latitude.

End of Example

Probability distribution for random functions and the characteristic functional

As a generalization of the characteristic function for random events, the concept ofthe characteristic functionals for random functions or stochastic processes has beendeveloped. Here we aim at reviewing the theory. We apply the concept of randomfunctions for multi-functions of type scalar-valued and vector-valued functionswhich have been first applied in the statistical theory turbulence.

what is the characteristic functional?

We summarize in Table C.9.1 the detailed notion of a scalar-valued as well as avector-valued characteristic functional illustrated by an example by introducing the“Fourier-Stieljes functional”.


Table C.9.1 Probability distribution of a random function and the characteristicfunctional

“Scalar-valued characteristic functionals”

“Random function over an interval [a,b] with N-variables”

fY.x/; x 2 RN g is called a random function if the density function

fx1;x2;:::;xN�1;xN ŒY.x1/; Y.x2/; : : : ; Y.xN�1/; Y.xN /� (C288)

exist at .x1; x2; : : : ; xN�1; xN / points.

“Fourier representation of the distribution function”

�

Yx1;:::;xN .�1; : : : ; �N / WD EfexpŒi.PN

kD1 �kYk/�gD R C1

�1: : :R

expfiPNkD1 �kYkgfx1;:::;xN .Y1; : : : ; YN /dY1 : : : dYN

(C289)

subject

Y1 WD Y.x1/; : : : ; Yk WD Y.xk/ (C290)

“Existence condition”

Y Œ�1; : : : ; �N � DbR

a

�.x/dx (C291)

“Vector-valued characteristic functionals”

“at any point P of the domain D place a set of base vectors e˛.Pi / at fixedpoints Pi for ˛ D 1; 2; : : : ; n � 1; n; i D 1; 2; : : : ; N � 1;N .

Y ˛i D; Y.P � i/je˛.Pi / > (C292)

“Vector-valued characteristic functionals”

�

YYŒ�� W �.x; y; z/ DPn˛D1 �˛.Pi /Y.Pi / (C293)

End of Table C.9.1: characteristic functional


C-92 The Moment Representation of Stochastic Processes forScalar Valued and Vector Valued Quantities

The moment representation of stochastic processes is the standard tool of statisticalanalysis. If a random function is given by its probability function, then we find aunique representation of its moments. But the inverse process of a given momentrepresentation converted in its probability function is not unique. Therefore wepresent only the direct representation of moments for a given probability function.At first we derive for a given scalar-valued probability function its momentrepresentation in Table C.9.2. In Table C.9.3 we generalize this result for stochasticprocesses which are vector-valued. An example is presented for second orderstatistics, namely for the Gauss-Laplace normal distribution and its first and secondorder moments.

Table C.9.2 Moment representation of stochastic processes for scalarquantities

“scalar-valued stochastic processes”

“moment representation”

Y.x/; x 2 RN is a random function, for instance applied in turbulence theory.

Define the set of random function fx1; x2; : : : ; xN�1; xN g 3 R

with the probability distribution f .x1; x2; : : : ; xN�1; xN /.Then the moment representation is the polynomial

Mp1;p2;:::;�N�1;�N WD Efx1; x2; : : : ; xN�1; xN g

D RRC1: : :�1

RRx�11 x

�22 : : : x

�N�1

N�1 x�NN f fx1; x2; : : : ; xN�1; xN g

�dx1dx2 : : : dxN�1dxN

(C294)

(nonnegative even numbers) p1; : : : ; pN W �1; �2; : : : ; pN�1; pN 2 NN

Œp� WD �1 C p2 C � � � C pN�1 C �N (C295)

is called the order of the moment representation


Table C.9.2 Continued

“central moment representation”

fx1 � Efx1g; x2 � Efx2g; : : : ; xN�1 �EfxN�1g; xN � EfxN gg (C296)B�1�2:::pN�1;pN WD Ef.x1 �Efx1g/�1 ;

.x2 �Efx2g/�2 ; : : : ; .xN�1 � EfxN�1g/�N�1 ;

.xN � EfxN g/�N g(C297)

“Characteristic function”

˚.�1; : : : ; �N / WDRR

C1: : :�1

RRexpfiPk �kxkgf fx1; x2; : : : ; xN�1; xN gdx1dx2 : : : dxN�1dxN

(C298)

“differentiate at point �1 D 0; : : : ; �N .0/”

.i/Œ��M �1p2:::pN�1;pN Dh

@Œ��

@�p11 :::@�

pNN

˚.�1; : : : ; �N /i

�1D0;:::;�ND0(C299)

“semi-invariants”

‰ WD ln˚ (C300)

S�1;:::;�N WD.i/Œ��.�1/Œ��

h@Œ��

@��11 @�

�22 :::@�

�N�1N�1 @�

pNN

‰.�1; : : : ; �N /i

�1D0;:::;�ND0(C301)

“transformation of semi invariants to moments of order”

Si D M1; S2 D M2 �M21 D B2;

S3 DM3 � 3M2M1 C 2M31 D B3

S4 D B1 � 3B22 ; S5 D B5 � 10B3B2 (C302)

End of Table C.9.2: Moment representation of stochastic processes


Table C.9.3 Moment representation of stochastic processes for vector-valued quantities

“Vector-valued stochastic processes”

“moment representation”

“multi-linear functions of a vector field Gauss-Laplace distributed”

�

YYŒ�� W �.x/ DPn˛D1 �˛.Px/Y˛.Pi / (C303)

for a set of base vectors e˛.Pi / at fixed pointsP˛; ˛ D 1; 2; : : : ; n � 1; nI i D 1; 2; : : : ; N � 1;N

“one-point n-dimensional Gauss-Laplace distribution”

Y.P˛/ � f.P˛/

f.PN / D C exp

12

NP

i;j

gij .P˛/.Yi˛ �EfYig/.Yj

˛ �EfYj˛g/!

(C304)

“gauge”

RR C1: : :�1

RRf .P˛/dY1dY2 : : : dYN�1dYN WD 1 (C305)

“two-point n-dimensional Gauss-Laplace distribution: covariance function”

B˛;ˇ.P˛; Pˇ/ D EfŒY ˛.P˛/� EfY ˛g�ŒY ˇ.Pˇ/ �EfY ˇg�g (C306)

Y.P˛/; Y.Pˇ/ � f .P˛; Pˇ/ (C307)

f .P˛; Pˇ/ D D exp� 12

PNi;jD1 gij .P˛; Pˇ/.Y i˛ �EfY i˛g/.Y iˇ �EfY iˇg/

�

(C308)

Example : Gauss-Laplace normal distribution, character’s functional

“ansatz”


Table C.9.3 Continued

A.�/ D EfX.�/g WD EfbR

a

�.Y /Y.x/gdx DbR

a

EfY.x/gdx (C309)

B.�/ DbR

a

bR

a

B.x1; x2/�.x1/�.x2/dx1dx2 (C310)

“multi-point functions”

�

Y.Y /Œ�.x/� WD expfibR

a

�.x/EfY.x/gdx � 12

bR

a

bR

a

B.x1; x2/�.x1/�.x2/dx1dx2

(C311)

in general: multi-point functional seriesFourier-Stieljes functional series

Gauss-Laplace normal distribution functional:first and second order moments

An example is the central moment of fourth order of type two-pointssubject to a Gauss-Laplace normal distribution:

B˛ˇ;�ı .P1; P2/ D B˛;ˇ.P1/B�ı .P2/C

CB˛;� .P1; P2/Bˇı.P1; P2/C B˛ı.P1; P2/B

ˇı.P1; P2/ (C312)

general rule by JSSERLIS: on a formula for the product moment coefficientin any number of variables Biometrika 12(1918) No. 138

First and Second order moments of a probability density

and the uncertainty relation

bR

a

ŒY.x/� EfY.x/��0.x/dx D 0 (C313)

bR

a

bR

a

B.x1; x2/�.x1/�.x2/dx1dx2 D 0 (C314)


C-93 Tensor-Valued Statistical Homogeneous and IsotropicField of Multi-Point Systems

First, we introduce the ideal structure of a variance-covariance matrix of multi-pointsystems in an Euclidean space called

homogenous and isotropic in the statistical sense.

Such a notion generalized the concept of stationary functions so far introduced.Throughout we use the terminology “signal” and “signal analysis”.Let us begin with the notion of an ideal structure of the variance-covariance matrixof scalar value called signal, a multi-point function. An example is the gravity forceas a one-point function, namely the distance between two-points as a two-pointfunction or the angle between to a reference point and two targets as a three-pointfunction. Here, we are interested in a criterion matrix of vector-vector signals whichare multi-point functions. An example is the placement vector between networkpoints determined by an adjustment process, namely a vector-valued one-pointfunction proportional to the difference vector between network points, a vector-valued two-point function.Let S.X1; � � � ;X˛/ be a scalar-valued function between placement vectorsX1; � � � ;X˛ with in a three dimensional Euclidean space. The variance-covarianceof the scalar-valued signal at placement vectors X1; � � � ;X˛ and X˛C1; � � � ;Xˇ isdefined by

P.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/ WD EfŒS.X1; � � � ;X˛/ �EfS.X1; � � � ;X˛/g�� ŒS.X˛C1; � � � ;Xˇ/� EfS.X˛C1; � � � ;Xˇ/g�g (C315)

For the special case X1; � � � ;X˛;X˛C1; � � � ;Xˇ the multi-point function agrees toP.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/ to the variance in the other case to the covariance.

Definition C.9.1: Homogeneity and isotropy of a scalar-valued multi-pointfunction

The scalar-valued multi-point functionP.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/ is called

homogenous in the statistical sense. if it is translational invariant, namely

X.X1C t; � � � ;X˛C t;X˛C1C t; � � � ;XˇC t/ D

X.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/

(C316)t is called the translational vector. The function

Pis called the isotropic, if it is

rotational invariant in the sense of

X.RX1; � � � ;RX˛;RX˛C1; � � � ;RXˇ/ D

X.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/

(C317)R is called the rotational matrix.


End of Definition C.9.1: homogeneity and isotropy

The following representation theorems apply:Lemma C.9.2: homogeneity and isotropy of a scalar-valued multi-point function

The scalar-valued multi-point function is homogenous in the statistical sense if andonly if it is a function of the difference vectors X2 � X1; � � � ;Xˇ � Xˇ�1:X

.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/ DX

.X2�X1; � � � ;X˛C1 �X˛; � � � ;Xˇ �Xˇ�1/(C318)

This function is isotropic in the statistical sense,

P.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/

DX

.jX1j; < X1;X2 >;< X1;X3 >; � � � ; < Xˇ�1;Xˇ >; jXˇj/ (C319)

is a function of all scalar products including the length of the vectors X1;

� � � ;X˛;X˛C1; � � � ;Xˇ.

The functionP.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/ is homogenous and isotropic in the

statistical sense if

P.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/

DX

.jX2 �X1j; < X2 � X1jX3 � X2 >; � � � ;; � � � ; < Xˇ � Xˇ�1jXˇ�1 �Xˇ�2 >; jXˇ �Xˇ�1j/ (C320)

is a function of all scalar products including the length of the difference vectorsX2 � X1; � � � ;X˛C1 � X˛;Xˇ � Xˇ�1.End of Lemma C.9.2: homogeneity and isotropy

As a basic result we take advantage of the property that a scalar-valued functionP

of type multi-points is homogenous and isotropic in the statistical sense if

P.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/ (C321)

DX

.jX2 � X1j; jX3 � X1j; � � � ; jXˇ � Xˇ�2j; jXˇ � Xˇ�1j/

is a function of length of all difference vectors X� � Xı for all 1 � ı < � � 1. Fora proof we need the decomposition


< X;Y > D 1

2.jXj2 C jYj2 � jX � Yj2/

D 1

2.jXC Yj2 � jX � Yj2/ (C322)

Example: four-point function, var-cov function

Choose a scalar-values two-point function as the distance S.X1;X2/ between twopoints X1 and X2. The criterion matrix of a variance-covariance function under thepostulate of homogeneity and isotropy in the statistical sense

P.X1;X2;X3;X4/ as

a four-point function described by ˇ.ˇ � 1/=2 D 6 quantities, for instance

P.X1;X2;X3;X4/

DX

.jX2 � X1j; jX3 �X1j; jX4 �X1j; jX3 �X2j; jX4 � X2j; jX4 � X3j/(C323)

End of example: var-cov function, 4 points

Another example is a scalar-valued three-point function of the type angularobservation A.X1;X2;X3/ between a fixed point X1 and the two moving pointsX2 and X3.The situation of vector-valued functions between placement vectors X1; � � � ;X˛ ina three-dimensional Euclidian space based on the representation

S.X1; � � � ;X˛/ D3X

iD1Si .X1; � � � ;X˛/ei .X1; � � � ;X˛/ (C324)

is slightly more complicated. the Euclidean space is spanned by the base vectorsfe1; e2; e3g fixed to a reference point. The variance-covariance of a vector signals atplacements X1; � � � ;X˛ and X˛C1; � � � ;Xˇ is defined as the second order tensor

XWDX

i;j

.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/ei .X1; � � � ;X˛/˝ ej .X˛C1; � � � ;Xˇ/

(C325)when we have applied the sum notion convention over repeated indices.

Pi;j .X1; � � � ;X˛;X˛C1; � � � ;Xˇ/ WD EfŒSi .X1; � � � ;X˛/�EfSi.X1; � � � ;X˛/g�� ŒSj .X˛C1; � � � ;Xˇ/ �EfSj .X˛C1; � � � ;Xˇ/g�g (C326)

We remark that all Latin indices run 1; 2; 3 but all Greek indices label the numberof multi-points. For the case X1 D X˛C1; � � � ;X˛ D Xˇ we denote the multi-pointfunction

Pi;j .X1; � � � ;X˛;X˛C1; � � � ;Xˇ/ as the local variance covariance matrix

for the general case nonlocal covariance


Definition C.9.3: Homogeneity and isotropy in the statistical sense of a tensorvalued multi-point function

The multi point tensor valued functionP.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/ is called

homogenous in the statistical sense, if it is translational invariant in the sense of

X.X1C t; � � � ;X˛C t;X˛C1C t; � � � ;XˇC t/ D

X.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/

(C327)Where t denotes the translational vectorIn contrast the multi-point tensor-valued function

P.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/ is

called isotropic in the statistical sense if it is rotational invariant in the sense of

X.RX1; � � � ;RX˛;RX˛C1; � � � ;RXˇ/ D R

X.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/R

(C328)where R denotes the rotation matrix element of the SO.3/ group

End of tensor-valued multi-point functions

of key importance are the following representations for tensor-valued multi-pointhomogenous and isotropic functions

Lemma C.9.4: homogeneity and isotropy in the statistical sense of tensor-valuedsymmetric multi-point functions

A second order symmetric tensor-valued multi-point function is homogenous if,

X.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/ D

X.X2 �X1; � � � ;X˛C1 �X˛; � � � ;Xˇ �Xˇ�1/

(C329)is a function of difference vector .X2 �X1; � � � ;X˛C1 �X˛; � � � ;Xˇ �Xˇ�1/. Thisfunction is isotropic if

P.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/

D �0.jX1j; < X1;X2 >;< X1;X3 >; � � � ; < Xˇ�1;Xˇ >; jXˇj/ıij ei ˝ ej

CXˇ

�D1Xˇ

��.jX1j; < X1;X2 >;< X1;X3 >; � � � ; < Xˇ�1;Xˇ >; jXˇj/

�Œ.X� ˝X�/C .X� ˝X� /� (C330)

for a certain isotropic scalar-valued function �0 and �� for all 1 � � � � � ˇ

Finally such a function is homogenous and isotropic in the statistical sense, if andonly if


P.X1; � � � ;X˛;X˛C1; � � � ;Xˇ/

D �0.jX2 � X1j; jX3 � X1j; � � � ; jXˇ �Xˇ�2j; jXˇ �Xˇ�1j/ıij .ei ˝ ej /

CXˇ

�D2Xˇ

��.jX2 �X1j; jX3 �X1j; � � � ; jXˇ �Xˇ�1j/

�Œ.X� � X��1/˝ .X� � X��1/C .X� � X��1/˝ .X� � X��1/� (C331)

holds for a certain homogenous and isotropic scalar-valued functions �0 and �� forall 2 � � � � � ˇ

End of Lemma C.9.4: tensor-valued function var-cov functions

For the applications the Taylor-Karman structure for the case ˇD 2 is mostimportant:Corollary C.9.5: ˇ D 2, Taylor-Karman structure for two-point tensor functionhomogeneity and isotropy

A symmetric second order two-point tensor-function is homogenous and isotropicin the statistical sense if and only if

˙ .X1;X2/

D �0.jX2 � X1j/ıij .ei ˝ ej /

C �22.jX2 �X1j/Œ.jX1 �X2j/˝ .jX2 �X1j/�D �0.jX2 � X1j/ıij C �22.jX2 � X1j/ < X2 � X1; ei >< X2 �X1; ej >

D ˙ij .X1;X2/.ei ˝ ej / (C332)

(summation convention)

and

˙ij D �0.jX2 �X1j/ıij C Œ�22.jX2 � X1j/jX2 � X1j2 C �0.jX2 �X1j/�

� �0.jX2 �X1j/ �Xi�Xi

jX2 � X1j2(C333)


˙ij .X1X2/ D ˙m.jX2 � X1j/ıij (C334)

C fŒ˙l.jX2 �X1j/�˙m.jX2 � X1j/� �Xi�Xi

jX2 �X1j2

˙m and˙l are the homogenous and isotropic scalar-values functions

End of Corollary C.9.5: ˇ D 2, Taylor-Karman structure

The representation of the symmetric second-order two-point tensor-valued functionunder the postulate of homogeneity and isotropy has been analyzed by G. J. Taylor(1935) and T. Karman (1937) dealing with problems of Hydrodynamics. It wasintroduced into Geodetic sciences by E. Grafarend(1970, 1972). Most often thehomogenous and isotropic scalar-valued function˙m and˙l are called

longitudinal and lateral

correlation function. It has been C.C. Wang who wrote first the symmetric secondorder two-point tensor-valued function being homogenous and isotropic if

˙.X1X2/ D a1.X2 �X1/ıij ŒeiX1˝ ejX2�C a2.X2 �X1/Œ.X2 �X1/˝ .X2 �X1/�

(C335)holds. This representations refers to a1.X2�X1/ D b1.jX2�X1j/ and a2.X2�X1/ Db2.jX2�X1j/ according to C. C. Wang (1970 p, 214/215) as scalar-valued isotropicfunctions. In case we decompose the difference vector

X2 �X1 D < X2 �X1jei .X1/ >< X2 � X1; ej .X2/ > Œe

iX1 ˝ ejX2� DD �Xi�Xj Œe

iX1 ˝ ejX2� (C336)

˙ij .X1X2/ D b1.jX2 � X1j/ıij C b2.jX2 �X1j/�Xi�Xj

for any choice of the basis ei1 ; ej2 . A special choice is to put parallel to the basicvector X2 � X1 such that

˙11 D b1.jX2 � X1j/C b2.jX2 � X1j/jX2 �X1j2 (C337)

˙22 D ˙22 D b1.jX2 � X1j/ (C338)

The base vector e2 and e3 are orthogonal to the “longitudinal direction” X2 � X1.For such a reason

b1.jX2 � X1j/ WD ˙m.jX2 � X1j/ (C339)

and


Fig. C15 Longitudinal andlateral correlation: Two-pointfunction

b2.jX2 �X1j/ WD 1

jX2 � X1j2f˙l.jX2 � X1j/�˙m.jX2 � X1j/g (C340)

of the scalar-valued isotropic functions˙l.jX2�X1j/ and˙m.jX2�X1j/. The resultis illustrated in Fig. C.9.1A two-dimensional Euclidean example is the vector-valued one-point function of anetwork point. Let

˙ D

2

664

�x1x1 �x1y1 �x1x2 �x1y2�y1x1 �y1y1 �y1x2 �y1y2�x2x1 �x2y1 �x2x2 �x2y2�y2x1 �y2y1 �y2x2 �y2y2

3

775 (C341)

be the variance-covariance matrix of the two-dimensional network of two pointswith Cartesian coordinates. The variance-covariance matrix will be characterizedby the variance �x1x1 D �2x1 of the X-coordinate of the first point, the covariance�x1y1 of the x; y coordinates of the first point and the second point and so on. Thediscrete variance-covariance matrix

˙ D EfŒX �EfXg�ŒX � EfXg�g (C342)

or

˙ij .X1;X2/ D EfŒxi .X1/� EfxiX1g�Œxj � EfxjX2g�g (C343)

or

xi .X1/ D fx1X1; x2X1g DW fx1; y1gxj .X2/ D fx1X2; x2X2g DW fx2; y2g


Fig. C16 Homogenous and isotropic dispersion situation: (a) general, (b)homogenous and(c) homogenous-isotropic

are the estimated coordinates of the Cartesian vectors of the first network pointX1 or the second network point X2. The two-point functions ˙ij .X1;X2/ are thecoordinates labeling the tensor of second order. The local variances and covariancesare usually illustrated by local error ellipses:

˙ij .X1;X2 D X1/ (C344)

and

˙ij .X1 D X2;X2/;

��x1x1 �x1y1�y1x1 �y1y1

and

��x2x2 �x2y2�y2x2 �y2y2

(C345)

The nonlocal covariances are contained in the function ˙ij .X1;X2/ for the caseX1 ¤ X2, in particular

��x1x2 �x1y2�y1x1 �y1y2

(C346)

Figure C.9.2 illustrates the postulates of homogeneity and isotropy for variance-covariance matrices of cartesian coordinates left, we illustrate the general dis-persion situation in two-dimensional networks, of course those local dispersionellipses. In the center, the homogenous dispersion situation: At each network points,due the translation invariance all dispersion ellipses are equal. In the right side,the homogenous and isotropic dispersion situation of type Taylor-Karman is finallyillustrated.

“All dispersion ellipses are equal and circular”


Fig. C17 Correlation figures of type (a) ellipse and (b) hyperboloid

Finally we illustrate“longitudinal” and “lateral” within homogenous and isotropicnetworks by Fig. C16, namely the characteristic figures of type (a) ellipse and(b) hyperboloid. Those characteristicfunctions ˙l and ˙m are described the eigen values of the variance-covariancematrix. If the eigenvalues are positive, then the correlation figure is an ellipse.But if the two eigen values change sign (either “C” or “�” or “�” or “C”) thecharacteristic function has to be illustrated by a hyperboloid.

Ex 1: Two-Dimensional Euclidean Networks Under the Postulatesof Homogeneous and Isotropy in the Statistical Sense

Example C.9.1: Two-dimensional Euclidean networks under the postulates ofhomogeneity and isotropy in the statistical sense

Two-dimensional network in a Euclidean space are presented in terms of thepostulates homogeneity and isotropy in the statistical sense. At the beginning,we assume a variance-covariance matrix of absolute cartesian net coordinates oftype homogenous and isotropic, in particular with Taylor Karman structure. Givensuch a dispersion matrix, we derive the associated variance-covariance matrix ofcoordinate differences, a special case of the Kolmogorov structure function, anddirections, angles and distances. In order to write more simple formulae, we assumequantities of type EfYg D 0Definition C.9.6: Variance covariance matrix of Cartesian coordinate, coordi-nate differences, azimuths, angular observations and distances in two-dimensionEuclidean networks


Let ıxi .X1/ be absolute Cartesian coordinates ıxi .X1/ � ıxi .X2/ relative cartesiancoordinates or coordinate differences, ı˛.X1;X2/ the azimuths of a line observationpoint X1 to a target point X2, the difference ı˛.X1;X2/ to ı˛X1;X3/ as angularobservations or azimuth differences and ıs.X1X2/ the distances from the point X1

to the point X2. Then

˙ij .X1;X2/ D Efıxi .X1/ıxj .X2/g (C347)

˚ij .X1;X2;X3;X4/ D EfŒıxi .X1/ � ıxi .X2/�Œıxj .X3/� ıxj .X4/�g(C348)

�.X1;X2;X3/ D EfŒı˛.X1;X2/ı˛.X3;X4/�g (C349)

�.X1;X2;X3;X4;X5;X6/ D EfŒı˛.X1;X2/� ı˛.X1;X3/�

� Œı˛.X4;X5/� ı˛.X4;X6/�g˘.X1;X2;X3;X4/ D Efıs.X1;X2/� ıs.X3;X4/g (C350)

are the derived variance-covariance matrices for the increments Y.x1; � � � ; xi / �y.x1; � � � ; xi / D ıy.x1; � � � ; xi /fıxi :ıxi � ıxj ; ı˛; ı˛i � ı˛j ; ısg with respect to amodel y.

End of Definition C: derived var-cov matrices

Let the variance-covariance matrix ˙ij .X1;X2/ of absolute Cartesian coordinatesof the homogenous and isotropic type the characteristic isotropic functions ˙l and˙m given. Then we derive the variance-covariance of

Lemma C.9.7: Representation of derived variance-covariance matrices of Taylor-Karman structured variance-covariance of absolute Cartesian coordinates.

Let the variance-covariance matrix of absolute Cartesian �ij .X1;X2/ be isotropicand homogenous in terms of the Taylor-Karman structure be given. Then the derivedvariance-covariance matrices are structured according to

˚ij .X1;X2;X3;X4/ D ˙ij .X1;X3/�˙ij .X1;X4/

�˙ij .X2;X3/�˙ij .X2;X4/ (C351)

�.X1;X2;X3;X4/ D ai .X1;X2/aj .X3;X4/˚ij .X1;X2;X3;X4/

(C352)

�.X1;X2;X3;X4;X5;X6/ D �.X1;X2;X4;X5/ ��.X1;X2;X4;X6/

��.X1;X3;X4;X5/C�.X1;X3;X4;X6/ (C353)

˘.X1;X2;X3;X4/ D bi.X1;X2/bj .X3;X4/˚ij .X1;X2;X3;X4/ (C354)

subject to


˙ij .Xˇ;X˛/ D ˙m.jXˇ � X˛j/ıij C Œ˙l .jXˇ �X˛j/�˙m.jXˇ � X˛j/�bi .Xˇ;X˛/bj .Xˇ;X˛/ (C355)

a1.X˛;Xˇ/ WD �Œ Py.X˛/� Py.Xˇ/�� ŒPs.Xˇ;X˛/� (C356)

a2.X˛;Xˇ/ WD CŒ Px.X˛/� Px.Xˇ/�� ŒPs2.x˛; xˇ/� (C357)

b1.X˛;Xˇ/ WD CŒ Px.X˛/� Px.Xˇ/�� ŒPs2.x˛; xˇ/� D cos Pˇ˛ (C358)

b2.X˛;Xˇ/ WD CŒ Py.X˛/ � Py.Xˇ/�� ŒPs2.x˛; xˇ/� D sin Pˇ˛ (C359)

End of Lemma C.9.7: Taylor-Karman derived structures

For the proof, we apply summation convention with respect to two identical indices.In addition, the derivations are based on

ı˛.X1;X2/ D a1.X1;X2/Œıx.X1/� ıx.X2/�C a2.X1;X2/Œıy.X1/ � ıy.X2/�

(C360)

ıs.X1;X2/ D b1.X1;X2/Œıx.X1/ � ıx.X2/�C b2.X1;X2/Œıy.X1/� ıy.X2/�

(C361)

In order to be informed about local variance-covariance we need special resultsabout the situation X1DX3;X2DX4:

Lemma C.9.8: derived local variance-covariance matrices of a Taylor-Karmanstructure variance-covariance matrix of absolute Cartesian coordinates

˚ij .X1;X2/ D 2Œ˙m.0/ıij .jX2 � X1j/�D ˚m.jX2 � X1j/ıij C Œ˚l .jX2 �X1j/�� ˚m.jX2 � X1j/�bi .X1;X2/bj .X1;X2/ (C362)

˚l.jX2 �X1j/ D 2Œ˙l .0/�˙l.jX2 � X1j/� (C363)

˚m.jX2 �X1j/ D 2Œ˙m.0/�˙m.jX2 � X1j/� (C364)

�.X1;X2/ D 2ai .X1;X2/aj .X1;X2/Œ˙m.0/ıij �˙.X1;X2/�

D Œ˚m.jX2 � X1j/�� ŒjX2 � X1j2� (C365)


�.X1;X2;X3/ D �.X1;X2/� 2�.X1;X2;X1;X3/C�.X1;X3/

D ˚m.jX2 �X1j/jX2 �X1j2

C ˚.jX3 �X1j/jX3 �X1j2

� ai .X1;X2/aj .X1;X3/

�Œ˚ij .X1;X2/� ˚ij .X2;X3/ �˚ij .X1;X3/�

D ˚m.jX2 �X1j/ < X3 �X2;X3 � X1

jX2 �X1j2jX3 �X1j2

�˚m.jX3 � X2j/< X2 �X1;X3 � X2 >

jX2 � X1j2jX3 � X2j2

C˚m.jX3 � X2j/< X2 �X1;X3 � X1 >

jX2 � X1j2jX3 � X1j2CŒ˚m.jX2 �X1j/ �˚.jX3 �X2j/�

� ŒjX2 � X1j2jX3 � X1j2� < X2 � X1;X3 �X1 >�

ŒjX2 � X1j2jX3 � X1j2jX3 � X2j2(C366)

˘.X1;X2/� 2bi.X1;X2/j˙m.0/ıij �˙ij .X1;X2/ D ˚m.jX2 � X1j/ (C367)

End of Lemma C 9.8: Derived local var-cov matrices

The results can be interpreted as following.

First, the scalar-valued functions �.X1;X2/;�.X1;X2;X3/ and ˘.X1;X2/ arehomogenous and isotropic. Second, of the local variance-covariance matrix ofCartesian coordinate differences has the structure of nonlocal covariance matrixof absolute Cartesian coordinates as long as we depart from a Taylor-Karmanmatrix: Third, not every homogenous and isotropic local variance-covariance matrixof Cartesian coordinate differences produces derived homogenous and isotropicfunctions as we illustrated in our next example.

Of course, not every homogenous and isotropic local variance-covariance matrix oftype coordinate difference has the structure of being derived by the transformationof absolute coordinates into coordinate difference. For instance, for the special caseX1 D X2 the variance-covariance matrix ˙ij .X1;X2/ due to ˙m.0/ D ˙l.0/ canbe characterized by circles. The limits

˙ij .X1;X1/ D ˙ij .X1; limX2!X1

X2/ D limX2!X1

˙ij .X1;X2/ (C368)

are fixed for any direction of approximation. However, due to ˚m.jX2 � X1j/ ¤˚l.jX2 � X1j/ lead us to symmetry breaking in a two-dimensional EUCLIDEAN


space: Though we have started from a homogenous and isotropic variance-covariance matrix of absolute Cartesian coordinates into relative coordinates orcoordinate differences the variance-covariance matrix ˚ij .X1;X2;X3;X4/ is nothomogenous and isotropic, a result mentioned first by W. Baarda (1977): The erroreclipses are not circular. In a three dimensional Euclidean space. In contrast, wereceive rotational symmetric ellipsoids based on the difference vector X2 �X1 withrespect to the axis of rotation and circles as intersection surfaces orthogonal to thevector X2 � X1.

In order to summarize our results we present

Lemma C.9.9: Circular two-point functions

The image of the two-point function ˚ij .X1;X2/ D ˚.X1;X2;X1;X2/ is a circle ifand only if ˚l.jX2 � X1j/ D ˚m.jX2 � X1j/ or ˙l.jX2 � X1j/ D �m.jX2 � X1j/End of Lemma C.9.9: Circular two-point functions

The result is illustrated by an example: Assume network points with a distancefunction jX2 � X1j D 1. In this distance the longitudinal and the lateral correlationfunctions are assumed to the values 1/2 and 1/4. Then there holds

˙l.0/ D ˙m.0/ D 1 (C369)

˙l D 1

2; ˙m D 1

4(C370)

˚l D 1; ˚m D 3

2; �.1/ D 3

2; ˘.1/ D 1 (C371)

Lemma C.9.10: Derived local variance-covariance matrix for a variance-covariance matrix of absolute Cartesian coordinate of Special Taylor-Karmanstructure ˙l D ˙m

Let the variance-covariance matrix of absolute Cartesian coordinates with SpecialTaylor-Karman structure ˙l D ˙m for homogenous and isotropic signals given.Then the derived variance-covariance matrices are structured accordingly to

˚ij .X1;X2/ D 2Œ˙m.0/�˙m.jX2 � X1j/�ıijD ˚.jX2 � X1j/�ıij (C372)

˚l.jX2 � X1j/ D ˚m.jX2 �X1j/ D 2Œ˙m.0/�˙m.jX2 � X1j/� (C373)


�.X1;X2/ D 2

jX2 � X1j2Œ˙m.0/�˙m.jX2 �X1j/�

D 1

jX2 � X1j˚m.jX2 � X1j/ (C374)

�.X1;X2;X3/ D 1

jX2 � X1j2jX3 � X1j2�Œ˚m.jX2 �X1j/ < X3 �X2jX3 �X1 > �

� ˚m.jX3 �X1j/ < X2 �X1;X3 �X2 >

C˚m.jX3 � X2j/ < X3 �X1;X2 � X1 >� (C375)

˘.X1;X2/ D 2Œ˙m.0/�˙m.jX2 � X1j/� D ˚m.jX2 �X1j/ (C376)

End of Lemma C 9.10: Special Taylor-Karman matrix

Our example is characterized now by a homogenous and isotropic variance-covariance matrix of Cartesian absolute coordinates which are the gradient of ahomogenous and isotropic scalar-valued function. In addition, we assume that thescalar valued function is an element of a two-dimensional Markov process of firstorder. Its variance-covariance matrix is characterized by normalized longitudinaland lateral correlation function which possess the distance function s WD jX2 � X1jbetween network points d the characteristic distance,K0 andK1 the modified Besselfunction of the second kind of zero and first order. The characteristic distance d isthe only variable or degree of freedom. Within the parameter d we depend on ourexperience with geodetic networks. Due to the property ˙l.0/ D ˙l.0/ we callthe longitudinal and lateral normalized. In order to find the scale for the variance-covariance matrix we have to multiply by the variance �2. The modified Besselfunction K0 and K1 are tabulated in Table C18 in the range 0.1(0.1)8 where wefollowed M. Abramowtz and I. Stegun (1970 p. 378-379).In Fig. C18 we present the characteristic functions of type absolute coordinates˙l.X/ D �4x�2 C 2K0.X/C 4x�1K1.X/C 2xK1.X/ and ˙m.X/ D C4x�2 �2K0.X/�2K0.X/�4x�1K1.X/. Next is the representation of the derived functions˚l.X/D˘.X/ D 2˙m.0/� 2˙l.X/; ˚m.X/D 2˙m.0/� 2˙m.X/ and�.X/ D2x�2j˙m.0/ � ˙m.X/j D x�2˚m.X/. Figures C19 and C20 illustrate the variousvariance-covariance functions.An explicit example is a two-dimensional network in Fig. C21. Let point one be thecenter of a fixed (x,y) Cartesian coordinate system. In consequence, we measurenine points by distance observations assuming the distance between point one andpoint two is unity, namely

1;p2; 2;p5;p8

The related characteristic functions for a characteristic distance d=1 are approxi-mately given in Table C18. The variance-covariance matrix of absolute Cartesian


Fig. C18 Modified Bessel function K0.X/ and K1.X/ longitudinal and lateral function ofabsolute Cartesian coordinates f1.X/ DW ˙l.X/ and f2.X/ DW ˙m.X/, longitudinal and lateralcorrelation of relative Cartesian coordinates f3 WD ˚m.X/ and f4 WD ˚l.X/, variances ofdirections f5 WD �.X/ and distances ˘.X/ DW �.X/ of a two-dimensional conservation Markovprocess of first order


Fig. C19 Characteristicfunctions �l and ˙m forTaylor-Karman structuredvaraince-covariance matrix ofCartesian coordinates

coordinates is relative Cartesian coordinates or coordinate differences is computedwith respect to �2 D 1:09; �y1�y2 D 0:50; �x1�x2;y1�y2 D 0We added by purpose according to Fig. C19 the characteristic functions for thespecial homogenous and isotropic variance-covariance matrix given by ˙l D˙m. The associated variance-covariance matrix for absolute and relative Cartesiancoordinates is given in Tables C25 and C26, structured by

˙l D 4d2

s2C 2K0.

s

d/C 4d

sK1.

s

d/C 2. s

d/K1.

s

d/ (C377)

˙m D 4d2

s2C 2K0.

s

d/C 4d

sK1.

s

d/ (C378)

Finally we present the derived variance-covariance matrices for relative Cartesiancoordinates, direction s, direction differences namely angles and distances, for thespecial case ˙l D ˙m.


Fig. C20 Derivedvariance-covariance functionsfor Taylor-Karman structuredvariance-covariance matrix ofCartesian coordinates

˚l.s/ D ˘.s/ D 2˙m.0/� 2˙m.s/;

D 2C�8d2

s2

�� 4K0

sd

��8d

s

�K1

sd

�� 4

sd

�K1

sd

�

(C379)

˚m.s/ D ˘.s/ D 2˙m.0/� 2˙m.s/;

D 2 ��8d2

s2

�� 4K0

sd

�C�8d

s

�K1

sd

�; (C380)

�.s/ D ˘.s/ D 2

s2Œ˙m.0/�˙m.s/� ;

D 2

s2

�1 �

�4d2

s2

�� 2K0

sd

�C 4

sd

�K1

sd

�(C381)

Example 9.1 is based on our contribution E. Grafarend and B. Schaffrin (1979) onthe subject of criterion matrices of type homogenous and isotropic. Other statisticalproperties like anisotropic or non-isotropic correlation functions were treated byG. K. Batchelor (1946), S. Chandrasekhar (1950) and E. Grafarend (1971, 1972)on the subject of axisymmetric turbulence.


Fig. C21 Two dimensionalnetwork


Fig. C22 Characteristicfunctions of a networkexample

Fig. C23 Cartesiancoordinate data for a networkexample

Of course there is a rich list of references for “criterion matrices”. We list only fewof them.J. E. Alberta (1974), A. B. Amiri (2004), W. Baarda (1973, 1977), K. Bill (1985),R. Bill et. al. (1986), Bischoff et. al. (2006), G. Boedecker (1977), F. Bossler et. al.(1973), F. Comte et. al. (2002), P. Cross et. al. (1978), G. Gaspari et. al. (1999),E. Grafarend (1971, 1972, 1975, 1976), E. Grafarend, F. Krumm and B. Schaffrin(1985), E. Grafarend and B. Schaffrin (1979), T. Karman (1937), T. Karman et. al.(1939), R. Kelm (1976), R. M. Kerr (1985), S. Kida (1989), F. Krumm et. al (1982,1986), S Meier (1976, 1977, 1978, 1981), S. Meier and W. Keller (1990), J. vanMierlo (1982), M. Molenaar (1982), A. B. Obuchow (1958), T. Reubelt et. al. (2003),H. P. Robertson (1940), B. Schaffrin and E. Grafarend (1977, 1981), B. Schaffrinand E. Grafarend (1982), B. Schaffrin et. al. (1977), G. Schmitt (1977, 1978 i, ii),J. F. Schoeberg (1938), J. P. Shkarofky (1968), V. I. Tatarski (1961), A. I. Taylor(1935, 1938), C. C. Wang (1970), H. Wimmer (1972, 1978), M. J. Yadrenko (1983)and A. A. Yaglom (1987).

Ex 2: Criterion Matrices for Absolute Coordinates in a Space TimeHolonomic Frame of Reference, for Instance Represented in Scalar-ValuedSpherical Harmonics

Example C.9.2: Criterion matrices for absolute coordinates in a space timeholonomic frame of reference, for instance represented in scalar-valued sphericalharmonics


Fig. C24 Derivedvariance-covariance functionsfor a special Taylor-Karmanstructure variance-covariancematrix of type ˙l D ˙m forCartesian coordinates

Conventional geodetic networks have been adjusted by methods of Total LeastSquares and optimized in different reference frames. The advantage of a rigourouslythree dimensional adjustment in a uniform reference frame is obvious, especiallyunder the influence of subdtic geodetic networks within various Global PositioningSystems (GPS, Global Problem Solver). Numerical results in a three dimensional,Total Least Squares approach are presented for instance by K. Bergbauer et. al.(1979) K. Euler et. al. (1982), E. Grafarend (1991), G. Hein and H. Landau (1983),and by W. Jorge and H. G. Wenzel (1973).

The diagnosis of a geodetic network is rather complicated since the measure ofquality, namely the variance-covariance matrix of three dimensional coordinates ofCartesian type contains 3g.3gC 1/=2 independent elements. The integer number gaccounts for the number of ground stations.

Example: g = 100


Fig. C25 Homogenousisotropic normalizedvariance-covariance matrix ofabsolute coordinates from anetwork example


Fig. C26 Special homogenous isotropic normalized variance-covariance matrix of absolutecoordinates for a network example case ˙l D ˙m

Fig. C27 Special homogenous isotropic normalized variance-covariance matrix of relativecoordinates for a network example case ˙l D ˙m

The number of independent elements with in the coordinate variance-covariancematrix amounts to 45,150.

Example: g = 1,000

The number of independent elements within the coordinate variance-covariancematrix amounts to 4,501,500.

Example: g = 5,000 (GPS)


The number of independent elements within the coordinate variance-covariancematrix amounts to 125,507,500 !In a definition, the same number of unknown variance-covariance elements ofvertical deflections and gravity disturbances.

The open big problem

In light of the great number of elements of the variance-covariance matrix it is anopen question how to achieve any measure of quality of geodetic network based onGPS. We think the only way to get away from this number of millions of unknownsis to design an idealized variance-covariance matrix again called criterion matrix.Comparing the real situation of the variance-covariance with an idealized one, wereceive an objective measure of quality of the geodetic network.

Here we are aiming for designing a three-dimensional criterion matrices forabsolute coordinates in a holonomic frame of reference and for functionals ofthe “disturbing potential”. In a first section we extensively discuss the notion ofcriterion matrix on a surface of a moving frame of type homogenous and isotropicon the sphere which is a generalization of the classical Taylor-Karman structure fora vectorized stochastic process. The vector values stochastic process on the sphereis found. In detail we review the representation of the criterion matrix in scalar- andvector-valued spherical harmonics. In section two we present the criterion matricesfor functionals of the disturbing potential. The property of harmonicity is takeninto account for the case that a representation in terms of scalar-valued sphericalharmonics exists. At the end, we present a numerical example.

Example C.9.21: Criterion matrices for absolute coordinates in a space timeholonomic frame if reference

The variance-covariance matrix of absolute Cartesian coordinates in the fixedorthonormal frame of equatorial type, holonomic in spacetime, is transformed intoa moving frame, e.g. of spherical type. In the moving frame longitudinal and lateralcharacteristic functions are derived. The postulates of spherical homogenity andspherical isotropy are reviewed extending to the classical Taylor-Karman structurewith respect to three dimensional Cartesian coordinates into three dimensions.Finally the spherical harmonic representation of the coordinate criterion matrix isintroduced.

Transformation of the coordinate criterion matrix from a fixed into a moving frameof reference

F� ! e� WWe depart from the fixed orthonormal frame fF

e1�; Fe2�; Fe3�g which we transform

into a moving one fee1�; ee2�; ee3�; g by


F� ! e� D RE.�; �; 0/F� (C382)

RE.�; �; 0/ WD R3.0/R2.2��/R3.�/ is the three dimensional euler rotation matrix

including geometric longitude � and geometric latitude �. As an example assume arepresentation of a placement vector x

eby spherical coordinates .�; �; r/ such that

x D F1�r cos� cos� C F2�r sin� cos� C F3�r sin � (C383)

e1� D �x;� � kx;�k D �e�

e2� D �x;� � kx;�k D �e�

e3� D �x;r � kx;rk D �er (C384)

x;� denotes ıx=ı� etc.

e� ! f�

Now let us rotate the orthonormal frame fe1�; e2�; e3�; g into the orthonormal frameff1�; f2�; f3�; g by means of the South azimuth ˛

e� ! f� D R3.˛/e� �

2

4f1�f2�f3�

3

5

D2

4cos˛ sin˛ 0� sin ˛ cos˛ 00 0 1

3

5

2

4e1�e2�e3�

3

5 (C385)

The transformation e� ! f� on the unit sphere orientates f1� into the direction ofthe great circle connecting the points .�; �; r/ and .�0; �0; r 0/ For this reason theunit vector f1� will be called “longitudinal direction”. In contrast, f2� points indirection orthogonal to f1�, but still in the tangential plane of the unit sphere. Thuswe call the unit vector f2� “lateral direction”. Obviously f3� D e3� holds

F� ! g! f�

There is an alternative way to connect the triads F� and f�. Consider the sphericalposition of a point in the “orbit” of the great circle passing through the pointsx; x0, respectively. For a given set of “Keplerian elements” f˝; i; !g - ˝ the rightascension of the ascending node, i the inclination and ! the angle measured inthe “orbit” - we transform the fixed orthonormal frame fF1�;F2�;F3�g into theorthonormal “great circle moving frame” fg1; g2; g3g D ff3�; f1�; f2�; g by

F� ! g D R3.!/R1.i/R3.˝/F (C386)

g! F� D R3.�˝/R1.�i/R3.�!/F (C387)


Fig. C28 Moving and fixedframe of reference on thesphere : two point correlationfunction

or explicitly

2

4g1g2g3

3

5 D2

4cos! cos˝ � sin! cos i sin˝ cos! sin˝ C sin! cos i cos˝ C sin! sin i

� sin! cos˝ � cos! cos i sin˝ � sin! sin˝ C cos! cos i cos˝ cos! sin isin i sin˝ � sin i cos˝ cos i

3

5

(C388)

:

2

4F1�F2�F3�

3

5 (C389)

g3 D sin i sin˝F1� � sin i cos˝F2� C cos iF3� (C390)

represents the “great circle normal” parameterized by the longitude 2�˝ and the

latitude 2� i . g1 and g2 span the “great circle plane”, namely g1 is directed towards

the spherical point x.

Figure C28 illustrates the various fames of type “fixed” and “moving”.

fg1; g2; g3g D ff3�; f1�; f2�; g, g2 is the vector product of g3 and g1

Transformation of variance-covariance matrices

Next we compute the Cartesian coordinates variance-covariance matrix.


†.x; x0/ D †i�j�.x; x0/fFi� ˝ Fj�g†.x; x0/ D †i�j�.x; x0/ffi�.x/˝ fj�.x0/g†.x; x0/ D †ij .x; x0/fgi .x/˝ gj .x0/g (C391)

wherei; j"f1; 2; 3g holds

.˙ij 0

/ D2

4˙x;x0 ˙x;y0 ˙x;z0

˙y;x0 ˙y;y0 ˙y;z0

˙z;x0 ˙z;y0 ˙z;z0

3

5 (C392)

.˙ij 0

/ D EfŒxi �Efxi g�ŒXj 0 � EfXj 0g�g D Ef"i"j 0g (C393)

˙.x; x0/ WD Ef".x/"T .x0/g (C394)

Note that the variance-covariance matrix ˙ij 0

.x; x0/ is a two-point tensor fieldpaying attention to the fact that the coordinate errors ".x/ and ".x/ are functionsof the point x; x0, respectively. Now let us transform˙� $ ˙� $ ˙ .

˙�3�0 D RE.�; �; ˛/˙

�3�3RT

E.�0; �0; ˛0/

D R3.�˝/R1.�i/R3.�!/˙3�3R3.�!0/RT1 .�i 0/RT

3 .�˝ 0/

˙�3�0 D RE.�; �; ˛/˙

�RTE.�

0; �0; ˛0/ (C395)

D RE.�; �; ˛/˙�R.�!;�i;�˝/˙3�3RT .�!0;�i 0;�˝ 0/ � RT

E.�0; �0; ˛0/

˙3�0 D RT .�!0;�i 0;�˝ 0/˙3�3R.�!;�i;�˝/D RT .�!0;�i 0;�˝ 0/RT

E.�; �; ˛/˙�3�3RE.�

0; �0; ˛0/ � R.�!0;�i 0;�˝ 0/

A detailed write-up of the variance-covariance matrix ˙� is

.˙ij 0

/ D2

4˙ll 0 ˙lm0 ˙lr 0

˙ml 0 ˙mm0 ˙mr 0

˙rl 0 ˙rm0 ˙rr 0

3

5 (C396)

indicating the “longitudinal” base vector f1� D fl�, “lateral” base vector f2� D fm�and the “radical” base vector f3� D fr� D er . In terms of spherical coordinates˙�; ˙ , respectively, can be written

˙�3�0.x; x0/ D ˙�.�; �; r; �0; �0; r 0/ (C397)

˙3�0.x; x0/ D ˙.!; i;˝; r; !0; i 0;˝ 0; r 0/ (C398)

˙ll 0 D Ef"l.x/"l 0.x0/g;; ˙l l 0.x; x0/ffl .x/˝ fl 0.x0/g


˙lm0 D Ef"l.x/"m0.x0/g;; ˙lm0.x; x0/ffl .x/˝ fm0.x0/g

˙lr 0 D Ef"l.x/"r 0.x0/g;; ˙lr 0.x; x0/ffl .x/˝ fr 0.x0/g (C399)

The characteristic function ˙ll 0.x; x0/ is called longitudinal correlation function,˙mm0.x; x0/ lateral correlation function, ˙lm0.x; x0/ cross correlation functionbetween the longitudinal error "l.x/ at the point x and the lateral error "m0.x0/ at thepoint x0.

Spherically homogenous and spherically isotropic coordinate criterion matrices ina surface moving frame

terrestrial geodetic networks constitute points on the earths surface. Once wewant to develop an idealized variance-covariance matrix of the coordinates ofthe network points, we have to reflect the restriction given by the closed surface.A three dimensional criterion matrix of Taylor-Karman type is therefore not suitedsince it assumes extension of a network in three dimension up to infinity. a moresuitable concept of a coordinate criterion matrix starts from the representationwhere the radial part is treated separately from the longitudinal-lateral part.Again the nature of the coordinate criterion matrix as a second order two-pointtensor field will help us to construct homogenous and isotropic representations.The spherically homogeneity-isotropy-postulate reflects the intuitive notion os anidealized variance-covariance matrix.

Definition (spherical homogeneity and spherical isotropy of a tensor-valued two-point function):

The two-point second order tensor function ˙.x; x0/ is called spherically homoge-nous, if it is translational invariant in the sense of

†3�3.! C �; i;˝; r; !0 C �; i 0;˝ 0; r 0/ D †3�3.!; i;˝; r; !0; i 0;˝ 0; r 0/ (C400)

for all � 2 Œ0; 2�; i D i 0;˝ D ˝ 0; r D r 0, where !0D! C holds. It is calledspherically isotropic, if it is rotational invariant in the sense of

˙3�3.!� � !�s ; i

�;˝�; r; !�0 � !�s ; i

�0

;˝�0

; r 0/

D R3.ıs/˙3�3.! � !s; i;˝; r; !0 � !s; i 0;˝ 0; r 0/R3.ıs/ (C401)

for all ı"Œ0; 2�; r D r 0; where !0D! C , !�0 D!� C holds. R3.ıs/ indicatesa local rotation around the intersection points xs of the great circle through x; x0and through x�; x�0

, respectively. xs has the common coordinates .˝; i; !s and.˝�; i�; !s� with respect to the locally rotated frames.

End of Definition


Corollary (spherical homogeneity and spherical isotropy of a tensor-valued two-point function):

The two-point second order tensor function ˙.x; x0/ is spherically homogenous, ifit is a function of . ; i;˝; r/ only:

˙.!; i;˝; r; !0; i 0;˝ 0; r/ D ˙.!; i;˝; r; ! � !s; ; r/ (C402)

It is spherically homogenous and isotropic, if it is a function of . ; r/ only

˙.!; i;˝; r; !0; i 0;˝ 0; r/ D ˙. ; i;˝; r/ (C403)

such that

˙.x; x0/ D f˙0. ; r/ıij C˙22. ; r/ < x0 � xjfi > �

� < x0 � xjfi >g.fi ˝ fj / (C404)

holds for some spherically isotropic and homogenous scalar-valued functions˙0 and ˙22

End of Corollary

Let us interpret the definition of the corollary, especially with respect to Fig. C28.Two points x; x0 are connected to the sphere of radius r D r 0 D r0 by a great circlesuch that D 2 arcsin.kx; x0k=2r0/ is the spherical distance if these points.

Obviously a “translation” of the configuration x; x0 is generated by ! ! !C�; ! !!0 C � where the location of x; x0 in the great circle / “orbit” plane is measuredby !;!0. The “rotation” of the configuration x; x0 is more complicated. Assumetherefore an alternative configuration x�; x�0

of two point connected by anothergreat circle. Note the spherical distances of x; x0 and x�; x�0

respectively, coincide.The great circles intersect at xs, : The location of Xs within the plane x; x0 isdescribed by !s with respect to the ascending node, while within the plane x�; x�0

by !�s . the angle between the two great circles is denoted by ıs .

The configuration x; x0 and x�; x�0

is described with respect to xs by! � !s; i;˝; !0 � !s; i 0;˝ 0 and !� � !�

s ; i�;˝�; !�0 � !�

s ; i�0

;˝�0

respectively.Now it should be obvious that the two configurations under the spherical isotropypostulate coincide once we rotate around xs by the angle ıs.

We expressed˙�. given in the F �-frame in terms of the g-frame, namely by

˙ D RT3 .�!/RT1 .�i/RT3 .�˝/˙�R3.�˝/R1.�i/R3.�!/ (C405)

Introducing ! D .! � !s/C !s we compute the variance-covariance matrix ˙


˙ D RT3 .! �!s/RT

3 .!s/RT1 .�i/RT

3 .�˝/˙�R3.�˝ 0/R1.�i 0/R3.!s/R3.!0 �!s/(C406)

˙ D RT3 .! � !s/˙sR3.!

0 � !s/ (C407)

in terms of the variance-covariance matrix ˙s in the g.xs/-frame. Now we areprepared to introduce

Corollary (spherical Taylor-Karman structure):

Denote the variance-covariance matrix ˙i�j�0

in the f �- and e�- frame by

f ˙i�j�0 D

�˙ll 0. ; r/ 0

0 ˙mm0. ; r/

� f ˙2�2

(C408)

e˙i�j�

0 D�˙110 ˙120

˙210 ˙220

� e˙2�2

(C409)

and transform f ˙ ! e˙ by

e˙ D�

cos˛ � sin˛sin ˛ cos˛

f ˙ D

�cos˛0 sin˛0� sin ˛0 cos˛0

(C410)

Then

˙110 D ˙ll 0. ; r/ cos ˛ cos˛0 C˙mm0. ; r/ sin ˛ sin ˛0

˙120 D ˙ll 0. ; r/ cos ˛ sin ˛0 �˙mm0. ; r/ sin ˛ cos˛0

˙210 D ˙ll 0. ; r/ sin ˛ cos˛0 �˙mm0. ; r/ cos˛ sin ˛0

˙220 D ˙ll 0. ; r/ sin ˛ sin ˛0 C˙mm0. ; r/ cos˛ cos˛0 (C411)

˙lm0 D ˙ml 0 D 0 (C412)

holds for spherically homogenous and isotropic scalar-valued functions˙ll 0. ; r/,˙mm0. ; r/ called longitudinal and lateral correlation functions on the sphere.

End of Corollary

The corollary extends results of G.J. Taylor (1935), T. Karman (1937) andC. Wang (1970 p. 214-215) and has been given for a special variance-covariancetwo-point tensor field of potential type by H. Moritz (1972 p. 111). Here it hasbeen generalized for any vector-valued field. Note that incase of homogenity andisotropy the cross-correlation functions˙lm0 and˙ml 0 may vanish.

The classical planar Taylor-Karman structure is constrained within.


Let us put ˛ D ˛0 such that cos˛ cos˛0 D cos2 ˛ D 1� sin2 ˛; sin ˛ sin˛0 D sin2 ˛etc. can be rewritten

˙110 D ˙mm0. /C Œ˙l l 0. / �˙mm0. /� cos2 ˛

˙120 D Œ˙l l 0. / �˙mm0. /� sin ˛ cos˛ D ˙210

˙220 D ˙mm0. /C Œ˙l l 0. / �˙mm0. /� sin2 ˛ (C413)

which is the standard Taylor-Karman form

Representation of the coordinate criterion matrix in scalar spherical harmonics

Let us assume a coordinate variance-covariance matrix which is homogenous andisotropic in the sphere. With reference to (C412) we postulate a coordinate vectorof potential type, in addition, namely

" D gradR � "i D @iR .i D 1; 2; 3/ (C414)

where .@i / D . @@x; @@y; @@z / nd R the scalar potential. Now the variance-covariance

matrix of absolute coordinates can be represented by

˙ij 0 D Ef"i.x/"j 0

.x0/g D @i@j 0EfR.x/R.x0/g (C415)

where we used the commutator relation @iEDE@i of differentiation and integrationoperators. Denote the scalar-valued variance-covariance functionEfR.x/R.x0/g DWK.x; x0/. Once we postulate homogeneity and isotropy of two-point functionK.x; x0/ on the sphere we arrive at

K.x; x0/ D K.r; r 0; / (C416)

For a harmonic coordinate error potential, @i@iR D 0, we find for the covariancefunctionK.r; r 0; /

@i @iK.r; r0; / D Ef@[email protected]/R.x0/g D 0 (C417)

@j 0@j 0K.r; r 0; / D EfR.x/@j 0@j 0R.x0/g D 0 (C418)

Case 1 .kxk D kx0k/On a sphere .kxk D kx0k/ D r0 the scalar-valued variance-covariance functionK. / which fulfils (C414 - C 417) may be represented by

K. / D1X

nD0knPn.cos / (C419)


where Pn are the Legendre polynomials. The constants kn depend, of course, on thechoice of r0

Case 2 .kxk ¤ kx0k/Denote .kxkD r; kx0kD r 0/. the scalar-valued variance-covariance functionK.r; r 0; / which fulfils (C414 - C 417) may be represented by

K D K.r; r 0; /

D1X

nD01kn.

r20rr 0 /

nC1Pn.cos /C

1X

nD02kn.

rr 0

r0/

n

Pn.cos / (C420)

where the coefficients 1kn; 2kn represent the influence of the decreasing andincreasing solution, by Legendre polynomials with respect to r; r 0, respectively.We can only hope that the covariances K.r; r 0; / decrease with respect to r; r 0,respectively in order to be allowed to discriminate 2kn D 0. In the following we maethis assumption, thus representing

K.x; x0/ D K.r; r 0; / D1X

nD0kn.

r20rr 0 /

nC1Pn.cos / (C421)

Next let us assume the coordinate variance-covariance matrix (C420) is of potentialtype. Then longitudinal and lateral correlation functions on the sphere if radius rcan be represented by

˙ll 0. / D d2

d 2K. / (C422)

˙mm0. / D 1

sin

1

d K. / (C423)

˙ll 0. / D d

d .˙mm0 sin (C424)

if �.x; x0/ is homogenous and isotropic on the sphere.

End of Corollary

Note that the result ˙ll 0 D dd .sin ˙mm0/ for the sphere corresponds to ˙ll 0 D

dds.s˙mm0/ for a harmonic field in the plane.

s Dq.x0 � x/2 C .y0 � y/2 is the Euclidean distance in the plane which

corresponds to at the distance on the sphere.


Corollary (longitudinal and lateral correlation functions on the sphere of harmonictype):

Let us assume that the coordinate variance-covariance matrix id homogenous,isotropic and harmonic on the sphere in the sense of the definition. Longitudinaland lateral correlation functions on the sphere can be represented by

˙ll 0. / D1X

nD0kn

��.n2 C 3nC 2/cos2

sin2 C .nC 1/

:

�Pn.cos / � Œn2 C 3nC 2�2 cos

sin2 PnC1.cos /

�

C Œn2 C 3nC 2� 1

sin2 PnC2.cos /g (C425)

˙mm0. / D �X

nD01kn nC 1

sin2 Œcos Pn.cos /

�PnC1.cos / (C426)

End of Corollary

For the proof of the above corollary we have made successive use of the recurrencerelation .x2 � 1/dPn.x/=dx D �.nC 1/ŒxPn.x/ � PnC1.x/�;x D cos ; d=d D .dx=d /.d=dx/ D � sin d=dx

Representation of the coordinate criterion matrix in vector spherical harmonics

Previously we restricted the coordinate criterion matrix being homogenous andisotropic on the sphere to be of potential type. Here we generalize the coordinatecriterion matrix to be of potential and “turbulent” type. Thus according to the Lamelemma – for more details see A. Ben Menahem - S. J. Singh (1981) - we representthe coordinate vector

" D gradRC rotA (C427)

where we gauge the vector potential A by divA D 0. In vector analysis gradRis called the longitudinal vector field, rotA the transversal vector field. In orderto base the analysis only on scalar potentials, the source-free vector field rotA isdecomposed SCT where S is the poloidal part while T is the toroidal part such that

" D gradRC rot rot.erS/C rot.erT/ (C428)

.R; S; T / are the three scalar potentials we were looking for. The variance-covariance matrix of absolute coordinates can now be structured according to


˙.x; x0/ D Ef".x/gD grad ˝ grad 0EfR.x/R.x0/gCrot rot ˝ rot 0rot 0erer 0EfS.x/S.x0/gCrot ˝ rot 0erer 0EfT .x/T .x0/g (C429)

where we have assumedEfR.X/S.X0/g D 0;EfR.X/T .X0/g D 0;EfS.X/T .X0/gD 0 holds for a coordinate variance-covariance matrix ˙.xx0/ homogenous andisotropic on the sphere. Thus with respect to the more general representation is

˙.x; x0/ D grad ˝ grad 0K

C rot rot ˝ rot 0rot 0erer 0L

C rot ˝ rot 0erer 0M (C430)

where K WD EfR.X/R.X0/g; L WD EfS.X/S.X0/g;M WD EfT .X/T .X0/g aresupposed to be the scalar-valued two point covariance functions depending on.r; r 0; / for .˙.x; x0// homogenous and isotropic on the sphere.

The representation of the criterion matrix .˙.x; x0// in terms of vector sphericalharmonics is achieved once we introduce the spherical surface harmonics.

enm.��/ D

2

6666664

q2.2nC 1/ .n�m/Š

.nCm/ŠPnm.sin �/ cosm� 8m > 0

p2.2nC 1/Pn.sin�/ 8m D 0

q2.2nC 1/ .n�jmj/Š

.nCjmj/ŠPnjmj.sin�/ sin jmj� 8m < 0:

(C431)

where Pnm.sin�/ are the associated Legendre functions

Pnm.x/ D 1

2nnŠ.1 D x2/m=2 d

nCm

dxnCm .x2 � 1/n (C432)

Note that the quantum numbers run nD 0; 1; � � � ;1Im D �n;�nC 1; � � � ; 0; � � � ;n � 1; n. The base functions enm.�; �/ are orthonormal in the sense

1

4

2Z

0

d�

C=2Z

�=2d� cos� eklenm D ıkn ılm (C433)


Finally a vector field " can be represented in terms of vector spherical harmonicsby

" D Unmanm C Vnmbnm CWnmcnm (C434)2

666664

anm WD er enm.�; �/

bnm WD r grad enm.�; �/

cnm WD �x grad enm.�; �/:

(C435)

For more details on vector spherical harmonics we refer to standard textbooks, heree.g. to E. Grafarend (1982).

˙.x; x0/ D Ef".x/˝ ".x0/gD Unmn0m0anm ˝ an0m0

CVnmn0m0bnm ˝ bn0m0

CWnmn0m0cnm ˝ cn0m0 (C436)

which is due to < ajb >D< ajc >D< bjc >D 0. Unmn0m0 etc. are sphericalcovariance functions.

Example C 9.22: Towards criterion matrices for functionals of the distributingpotential in a space-time holonomic frame if reference

The geodetic observational equations in their linearized form contain besideunknown Cartesian coordinate corrections in an orthonormal reference frame,being holonomic in space-time, the disturbance of the potential and its derivativesas unknowns . for a complete set-up of a criterion matrix of geodetic unknownswe have therefore present thoughts about a homogenous and isotropic variance-covariance matrix of the potential disturbance and its derivatives, The postulates ofhomogeneity and isotropy will lead again to the extension of an classical Talyor-Karman structure. finally the representation of the criterion matrix of potentialrelated quantities in terms of scalar spherical harmonics is introduced. Since thepotential disturbance is harmonic function, vector spherical harmonics are notneeded.

We begin the transformation of the gravity criterion matrix from spherical intoCartesian coordinates.

With reference to the linearized observational equations the unknowns .bı�� ;cı�� ;bı�/ have to be equipped with a criterion matrix, reasonable to be homogenous and


isotropic on the sphere. In order to take advantage of lemma and corollaries of thesecond chapter, we better transform the spherical coordinates of the disturbancegravity vector ı� into Cartesian ones and then apply the variance-covariance“propagation” on the transformation formula.

ı�i WD �i � �i D2

4�� cos�� cos˚�� sin�� cos˚�� sin˚�

3

5 �2

4�� cos�� cos�� sin�� cos�� sin ��

3

5

D2

4�.� C ı�/ cos.�� C ı��/ cos.�� C ı��/�.� C ı�/ sin.�� C ı��/ cos.�� C ı��/

�.� C ı�/ sin.�� C ı��/

3

5

�2


3

5 (C437)

ı�i:D2

4�.� C ı�/.cos�� ı�� sin��/.cos�� ı�� sin��/�.� C ı�/.sin�� C ı�� cos��/.cos�� ı�� sin ��/

�.� C ı�/.sin�� C ı�� cos��/

3

5

�2


3

5 (C438)

ı�i:D2

4�ı� cos�� cos�� C ı�� sin�� cos�� C ı�� cos�� sin ��ı� sin�� cos�� ı�� cos�� cos�� ı�� sin�� sin ��

�ı� sin�� ı�� cos��

3

5

(C439)2

4ı�1

ı�2ı�3

3

5 :D2

4C� sin�� cos�� C� cos�� sin �� cos�� cos�� cos�� cos�� sin�� sin �� sin�� cos��

0 �� cos�� sin��

3

5

2

4ı��

ı��ı�

3

5

(C440)2

4ı�1

ı�2ı�3

3

5 :D

2

64��2 C�1�2.�21 C �22 /�1=2 �1��1

C�1 ��2�3.�21 C �22 /�1=2 �2��1

0 �.�21 C �22 /�1=2 �3��1

3

75

2

4ı��

ı��ı�

3

5 (C441)

The variance-covariance matrix Dfcı�i .x/;bı�j 0.x0/g as a function of the variance-

covariance matrix Df.bı�� ;cı�� ; bı�/.x/; .bı�� ;cı�� ; bı�/.x0/g based on the trans-formation ( C 438) is given. finally the Cartesian variance-covariance matrixDfcı�i .x/;bı�j 0.x0/g in the fixed frame F � should be transformed into the movingframe f � according to (C381).


Fig. C29 Normalized covariance function K. / D 5P

nD0

knPn.cos / for kn D 1nC2

Thus we refer to�˙l l 0; �

˙lm0 ; �

˙lr 0 ; �

˙ml 0 ; �

˙mm0; � � � ; �˙rr 0 (C442)

as the longitudinal correlation function �˙l l 0 , the lateral correlation function �˙mm0 etc.of the gravity disturbance vector ı�i .In order to construct a homogenous and isotropic criterion matrix for functionals ofthe disturbing potential in a surface moving frame we refer to the introduction whichapplies here, too. Again we make use of the definition of homogeneity and isotropyof the tensor-valued two-point function �˙.X;X

0/ on the sphere, the basic lemmaand the corollary for the spherical Taylor-Karman structure. Just by adding the lefthand index � on ˙ within formula (C410) (C411), we arrive at the correspondingformulae for longitudinal, lateral and cross-correlation of the gravity disturbancevector ı� .For the representation of the criterion matrix for functionals of the disturbingpotential in scalar spherical harmonics we once more refer to (C417). All resultscan be transferred, especially (C 420 - C 425) once we write K.r; r 0; / as thescalar-valued two-point function EfŒcıw � E.cıw/�xŒcıw � E.cıw/�x0g homogenousand isotropic on the sphere. Besides the first-order functionals of the potential ıwand their variance-covariance matrices, higher order functionals, e.g. for gravitygradients, and their variance-covariance matrices can be easily obtained, a projectfor the future.


Fig. C30 Normalized covariance function K. / D5P

nD0

knPn.cos / for kn D 1.nC2/.n2�2nC2/

A numerical example of the homogenous and isotropic two-point functionK.x; x0; / on the space

For demonstration purpose plots C29 - C31 show the normalized covariancefunctionsK.X;X0/ D K.r; r 0; /

D K. / D5P

nD0knPn.cos / for case 1

.kXk D kmathbf X0k D r0/ and three different choices of the coefficients kn.The coefficients are chosen in such a manner that their sum equals the unity (soas to normalize the covariance function) and the covariance function approacheszero as tends to . As an example for the construction of a criterion matrix, thecoefficients kn could be estimated for an ideal network and its variance-covariancematrix; that is derived covariance functionK.x; x0/.We have taken a part of Appendix C 9.2 from our contribution E. Grafarend,I. Krumm and B. Schaffrin (Manuscripta geodetica 10 (1988) 3-22). Sphericalhomogeneity and isotropy in tensor-valued two-point functions have been intro-duced by E. Grafarend (1976) and H. Moritz (1972, 1978).

Ex 3: Space Gravity Spectroscopy: The Benefits of Taylor–KarmanStructured Criterion Matrices

Example C.9.3. Space gravity spectroscopy: the benefits of Taylor-Karman struc-tured criterion matrices (eg, P. Marinkovice et. al. (2003))


Fig. C31 Normalized covariance function K. / D 5P

nD0

knPn.cos / for kn D 1n2C3

As soon as the space spectroscopy was successfully performed, for instance bymeans of semi-continuous ephemeris of LEO-GPS tracked satellites, the problemof data validation appeared. It is for this purpose that a stochastic model for thehomogenous isotropic analysis of measurements, obtained as “directly” measuredvalues in LEO satellite missions (CHAMP, GRACE, GOCE), is studied. Anisotropic analysis is represented by the homogenous distribution os measured valuesand the statistical properties of the model are calculated. In particular a correlationstructure function is defined by the third order tensor (Taylor-Karman tensor), forthe ensemble average of a set of incremental differences in measured components.Specifically, Taylor-Karman correlation tensor is calculated with the assumptionthat the analyzed random function is of a “potential type”. The special class ofhomogenous and isotropic correlation functions is introduced. Finally, a successfulapplication of the concept is presented in the case study CHAMP and a comparisonbetween modeled and estimated correlations is performed.

Example C.9.31. Introduction

A significant problem of LEO satellites, both in geometry and gravity space,is the association of quality standards to Cartesian ephemeris in terms ofvariance-covariance matrix valued-functions. as a pessimistic measure of thequality standards os LEO satellite ephemeris, a three dimensional Taylor-Karmanstructured criterion matrix has been proposed, named in honor of Taylor (1938) andKarman (1938), the founders of the statistical theory of turbulence.


The concept of the Taylor-Karman criterion matrices was first introduced byE. Grafarend (1979) and subsequently further developed by B. Schaffrin andE. Grafarend (1982), H. Wimmer (1982), and E. Grafarend et. al. (1985, 1986).with this contribution we extend the application of the Taylor-Karman structuredmatrices to the third-dimension, namely to the long-arc orbit analysis.

If we assume the vector-valued stochastic process to be the gradient of a randomscalar-valued potential, in particular its longitudinal and lateral correlation functionor the “correlation function along-track and across-track”, what would be the struc-ture of a three-dimensional Taylor-Karman variance-covariance matrix. In order toanswer this question, a three-dimensional correlation tensor and its decompositionin the case of homogeneity and isotropy is studied with the emphasis on R

3.Additionally, we deal with a special class of homogenous and isotropic tensor-valued correlation functions. They are derived, analyzed and applied to the datavalidation process. The theoretical concept for the application of the previouslydiscussed criterion matrix in the geometric and gravitational analysis of LEOsatellite ephemeris is presented. Finally the case study CHAMP is used for theapplication of our theory concept followed by the results and conclusion.

Example C.9.32: Homogenous and isotropic variance-covariance tensor and cor-relation functions in a three-dimensional Euclidean space

Example C.9.321: Notions of homogeneity and isotropy

The notions of homogeneity and isotropy for functions on R� are briefly explained

as the following. The general context for these two definitions involves the action ofa transitive group of motions on a homogenous and space belongs to the extensivetheory of Lie groups (F. W. Warner (1983), A. M. Yaglom (1987)). However, it isimportant to clarify the different notions of homogeneity and isotropy exist dueto the variety of homogenous spaces and transitive group actions on these spaces.The terminology introduced associates the notion of homogeneity and isotropy with

Fig. C32 Graphical representation of the longitudinal and lateral correlation functions


the functions on R� that are invariant under the translation group acting on R

� .The notion of isotropy is defined for functions on R

� that are invariant under theorthogonal group acting on R

�.

Table C.9.9: Tensor-valued correlation function of second-order“two-point correlation function”

˙.x�; x��/ D3P

i;jD1ei ej˙ij .x�; x��/ D

3P

i;jD1˙ij .x�; x��/ei ej (C443)

r WD kx�; x��k and r WD x�; x��

˙ij .r/ D ˙m.r/ıij C Œ˙l .r/ �˙m.r/��xi�xj

r2; i; j 2 f1; 2; 3g (C444)

“Cartesian versus spherical coordinates”

�x1 D �x WD .x��1 � x�

1 / D r cosˇ cos˛

�x2 D �y WD .x��2 � x�

2 / D r cosˇ sin ˛ (C445)

�x3 D �z WD .x��3 � x�

3 / D r sinˇ

“continuity of a potential type”

˙l.r/ D dŒr˙m.r/�

drD ˙m.r/C r d˙m.r/dr

(C446)

End of Table C.9.9

Example C.9.322: Homogenous and isotropic variance-covariance tensor (correla-tion tensor)

Taylor (1938) proved that he homogenous random field X.t/ correlation function˙.r/ D< X.tC r/X.t/ > depends only on the length r D krk of the vector r andnot on its direction, where < ::: > denoted the ensemble average. If the correlationfunction ˙.r/ of the homogenous random field X.t/ in R

� has the property, thenthe field X.t/ is said to be an isotropic random field in R

�. The correspondingcorrelation function ˙.r/ is then called an isotropic correlation function in R

�

( or an n-dimensional isotropic correlation function). Process, which satisfy theintroduced postulates of homogeneity and isotropy, are said to be (widely) stationary(M. J. Yadrenko (1983)). Note that for an isotropic random field in R

�, all directionsin space are obviously equivalent.

The decomposition of a homogenous and isotropic variance-covariance tensor-valued function, shown in box 1, was introduced by von Karman and Howarth


Fig. C33 Graphical representation of the correlation tensor transformation

(1938) by means of a more general and direct method than the one used byG. J. Taylor(1938). Additionally, Robertson (1940) refined and reviewed theT. Karman and L. Howard equation in the light of a classical invariant tensortheory.

The decomposition of ˙ij .kx� � x��k/, with a special emphasis on nD 3, isperformed in terms of Cartesian coordinates and with respect to the orthonormalframe of reference fe1; e2; e3j0g attached to the origin 0 of a three-dimensionalEuclidean space. kx� � x��k denotes the Euclidean distance r between the twopoints x� and x�� of E� WD fR�; ıij . Longitudinal and lateral correlation functions˙l and ˙m, are the structural elements of such a homogenous and isotropic tensor-valued variance-covariance function which appear in the spherical tensor ˙m.r/ıijas well as in the oriented tensor Œ˙l .r/�˙m.r/��xi�xj =r

2 for all i; j 2 f1; 2; 3g,see (C443) and Fig. C.32. ıij denotes the Kronecker or unit matrix, �xi theCartesian coordinate difference between the points x� and x��. These differences arealso represented in terms of relative spherical coordinates .˛; ˇ; r/, (C444). Finallythe continuity condition of a potential type is formulated by (C445) which providesthe unique relation between˙l and˙m (G. J. Taylor (1938); H. M. Obuchow (1958);E. Grafarend (1979)).Due to its complexity, it is necessary to further elaborate on the previous equationset. The correlation tensor ˙ij .r/ was transformed to a special coordinate systemO 0x0

1x02x

02 in R

� instead of the initial setOx1x2x3. The new setO 0x01x

02x

03 is selected

in such a way so that its origin O 0 is shifted by the vector x�� with respect to theorigin O , as illustrated in Fig. C15. This means that O 0 coincides with the terminalpoint of the vector x�� that refers to the initial coordinates, while the axis O 0x0

1 liesalong the vector x� � x��.


Fig. C34 The behavior of Tatarski’s correlation function for different values of shape parameter�.d D 900/ and scale parameter d.� D 3=2/

˙ij , introduced here are explanatory functions, are the components of the correla-tion tensor˙ij .r/ in the new set of coordinates. The functions˙ 0

ij .r/ clearly dependonly on the length r D krk of the vector r, since the direction of r is fixed in thenew set of coordinates. In the space R� exist a reflection which leaves the points x�and x�� D .O 0/ unmoved and it replaces the axis O 0x0

j by �O 0x0j , where j ¤ 1 is

a fixed number. However, it does not change the directions of all other coordinatesaxes O 0x0

l ; l 2 f1; 2; 3g and l ¤ j . It follows that

˙ 0ij .r/ D �˙ 0

ij .r/ D 0textrmfori ¤ j (C447)

Hence, only the diagonal elements ˙ 0ij .r/ of ˙ij .r/ can differ from zero. Further

more, if i ¤ j and j ¤ 1, then the axis O 0x0i by its rotation around the axis O 0x0

j ,can be transformed to the axis O 0x0

1. Hence

˙ 022.r/ D ˙ 0

33.r/ (C448)

The tensor ˙ 0ij and, consequently,˙ij are symmetric and their components˙ 0

ij .r/

can take at most only tow non-equal non-zero values at the already introduceslongitudinal correlation function˙ 0

i i .r/ D ˙l.r/ and the lateral correlation function˙ 022.r/ D ˙ 0

33.r/ D ˙m.r/, which specify in a unique way the correlation tensor.In order to obtain the explicit form of ˙ij .r/, as the function of ˙l.r/ and ˙m.r/,the unit vectors of the old coordinate axes Ox1;Ox2;Ox3 along the axes of thenew system O 0x0

1; O0x02; O

0x03 must be resolved and then ˙ij .r/ can be represented

as linear combination of the functions ˙ 0kl .r/; .k ¤ i; l ¤ j and k; l 2 f1; 2; 3g/,

which leads to (C442) and (C443).


Example C.1. homogenous and isotropic correlation functions

It was shown in the pervious section that for a homogenous and isotropic randomfields defined on the Euclidean space R

�, the correlation between x� and x��depends only on the Euclidean distance kx� � x��k. Therefore as shown in TableC32, we can distinguish a homogenous and isotropic correlation function on R

�

with the real valued function˙.r/ defined on Œ0;1/ and we denote by ˚n the classof all continuous permissible functions ˙.r/ W Œ0;1/ ! such that ˙.0/ D 1 (weare working in terms od correlation not covariance) and the symmetric function˙.k:k/ is a positive definite on R

�. The characterization of classes ˚n, also shownin Table C.9.10, is a well known result of J. F. Schoenberg (1938). The function˙.r/ W Œ0;1/ ! R is an element of ˚n if and only if it abmits a representationin the form of (C450), where W is a probability measure on Œ0;1/; J.n�2/=2 is theBessel function of the first kind of order .n � 2/=2 and � stands for the Gammafunction. Hence in for geodesy relevant spaces R� and R

�, (C450) reduces to theform presented in (C452).

Table C.9.10: Isotropic correlation and Schoenberg’s characterization

“homogenous and isotropic correlation function”˙.x�; x��/ D ˙.kx�; x��k/; x�; x�� 2 R

� (C449)r WD kx�; x��k

“the characterization of the ˚n class”˙n.r/ D

R

Œ0;1/

˝n.r�/dW.�/ (C450)

˝n.r�/ D � .n=2/. 2r /.n�2/=2

J.n�2/=2.r/ (C451)“reduction of the ˝n for n D 2 and n D 3”

˝2.r/ D J0.r/ and ˝3.r/ D r�1 sin r (C452)

End of Table C.9.10

Table C.9.11: Special classes of correlation functions

“Tatarski’s class”

˙Œ��.r/ D 21��

� .�/. rd/�K�.

rd/ (C 453)

“Shkarofsky’s class”

˙Œ�;ı�.r/ D . r2

d2Cı2/�=2K�.. r2

d2Cı2/1=2/

ı�K�.ı/(C454)

End of Table C.9.11


Example C.2. Tatarski’s class of homogenous and isotropic correlationfunctions

Many analytical candidate models for ˙ have been suggested in the literature (forexample, Buell, (1972); Haslett, (1989)), but we refer to Tatarski (1962) as beingthe first who elaborated on such correlation functions which fulfill all the conditionspresented in the previous section. The Tatarski’s correlation function class is shownin Table C.9.11 and illustrated by Fig. C.34. In addition to V. J. Tatarski’s class, avery general family of correlation function models due to J. P. Shkarofsky (1968)is introduced, that came as the generalization of Tatarski’s correlation functionfamily. These two classes, which have been proved to be the members of ˚3(Shkarofsky, (1968) and of ˚1 classes; Gneiting et. al. (1999)), can be applied tomany geodetic problems, (e.g. Grafarend (1979); Meier (1982); Wimmer (1982);Grafarend (1985)).

In equation of Table C.9.11, K� stands for a modified Bessel function of order�; d > 0 is a scale parameter, and ı > 0 and � are shape parameters. In the case ofı D 0 J. P. Shkarofsky’s class reduces to V. J. Tatarski’s class.

The special case of Tatarski’s class appears if the shape parameter � is sum ofa non-negative integer k and 1=2. Then the right-hand side of the equation canbe written as a product of exp.�r=d/ and a polynomial of degree k in r=d (eg.T. Gneiting, (1999)). In particular, in the case of nD 3 dimensional Markov processof the p D 1 order, shown in Fig. C34, the shape parameter is expressed as� D .2p C 1/=.n� 1/ D 3=2 and results in the flowing simplification of (C454):

˙Œ3=2�.r/ D .1C r

d/ exp.� r

d/ (C455)

Example C.3. Three dimensional Taylor-Karman criterion matrix in thegeometric and gravitational analysis of LEO satellite ephemeris

WE have so far analyzed the theoretical background and solution for design of thehomogenous and isotropic Taylor-Karman correlation tensor. The question is, howthis theoretical concept applies to the geometric and gravitational analysis of LEOsatellite ephemeris.

The basic idea is, that the errors in the position vectors of LEO satellites constitutea vector-valued stochastic process. Following this concept, a satellite orbit of LEO-GPS tracked satellites is an homogenous and anisotropic field of error vectorsand the error situation is described by the covariance function. As it is wellknown, the error situation os a newly determined position in a three-dimensionalEuclidean space is the best, when the error ellipsoid is sphere (isotropy) with the


minimal radius and if the error situation is uniform over the complete satellite orbit(homogeneity). This “ideal” situation can be explained by the three-dimensionalTaylor-Karman structured criterion matrix of W. Baarda - E. Grafarend (potential)type. Then the correlations between the vectors of pseudo-observations of satelliteephemeris described by the longitudinal and lateral correlation functions.

The characteristic correlation functions can be estimated by matching the correlationtensor with a three-dimensional Markov process of 1st order and with theintroduction of some additional information about the underlying process. Thecorrelation analysis is performed with the assumption that the vector valued three-dimensional random function is of “potential type” (E. Grafarend (1979)), i.e. thegradient of a random scalar function “signal s.x/”. The structure of the randomfunction s.x/ is outlined as an n-dimensional Markov process of the pth order.Figure C35 illustrates the case n D 3 and p D 1:

One of the simples differential equation os such a process has the form given by

.�2 � ˛2/ps.x/ D e.x/ (C456)

where e.x/ is a white noise. If the Laplace operator can be applied to thehomogenous random scalar function, then it transforms this function into a newhomogenous random function, having the spectral density that corresponds to thespectral density of the correlation function of (C453) and (C454), see P. Whittle(1954), V. Heine (1955) and P. Whittle (1963). Hence the homogenous solution ofC455 (if it exist) must have the spectral density that corresponds to the spectraldensity of the correlation function.

Table C.9.9 summarizes the representation of the homogenous and isotropic corre-lation functions of type

(i) Signal correlation function˙ ,(ii) Longitudinal correlation function˙l and

(iii) Lateral correlation function˙m

Example C.9.34: Results and conclusions

Case study: Champ

As the numerical test in this study, we processed two data sets: two three-dimensional fx.tk/; y.tk/; z.tk/g and fx.td /; y.td /; z.td /g Cartesian ephemeristime-series of CHAMP satellite orbit for the test period from day 140 to 150 of2001 (20 May to 30 May, both inclusive). We analyzed in total 27,360 tripletsof satellite positions. Both time series are indexed with a 30 s sampling rate andreferenced to a kinematic (index k) and a dynamic CHAMP orbit (index d ). Thedynamic orbit used as a reference, provides us with ephemeris differences betweenthe two orbits. The estimation of (“real”) auto and cross correlations between the


Fig. C35 The six point interaction in the grid; the three-dimensional Markov process of the 1st

order (auto regressive process); the gray rectangle represents the same process in two-dimensions

vectors of pseudo-observations as functions of time, can be performed as Priestley(1981).

According to Tables C.9.10 and C.9.11 the Taylor-Karman structured (“ideal”)correlations are computed from the three-dimensional fx.tk/; y.tk/; z.tk/g timeseries. The adopted scale parameter is d D .2=3/Rchar , where Rchar is thecharacteristic length of the process. The characteristic length is defined by thearc length of 2400 seconds (80 points for the 30 second sampling rate). The bothparameters are experimentally estimated. For further details on the scale parameterand characteristic length, please see Wimmer (1982).

The numerical results of the study are graphically presented in Fig. C37. The grayarea represents the estimated (“real”) correlation situation along the satellite arcas presupposed by Austen et al. (2001). The high auto and low cross-correlationsbetween CHAMP satellite positions for approximately 20 min of an orbit arc arevery evident. The Taylor-Karman structured correlation (black line), as theoreti-cally assumed, gives an upper bound of the “real” correlation situation along thesatellite orbit.


Fig. C36 The behavior of the longitudinal and lateral correlation functions, (C456) and (C 458),for different values of the scale parameter d.� D 3=2/

Table C.9.12 Longitudinal and lateral correlation function for a homogenousand isotropic vector-valued random field of potential type, 1st order Markovprocess

“condition for a process of potential type”

˙.x/! 2p

r2

rR

0

x˙.x/ dx !

! ˙l.r/ D ddrŒr˙m.r/�

“input”

˙.r/ D 2�1=2

� .3=2/. rd/3=2K3=2.

rd/ D .1C r

d/ exp.� r

d/ (C457)

“output”

˙l.r/ D �6. rd /�2 C exp�. rd/Œ4C 2. r

d/C 6. r

d/�1 C 6. r

d/�2� (C458)

˙m.r/ D �6. rd /�2 C exp�. rd/Œ2C 6. r

d/�1 C 6. r

d/�2� (C459)

End of Table C 9.12


Fig. C37 Graphical representation of the numerical study results (20 minute correlation length,40 points for the 30 second sampling rate)

Ex 4: Nonlinear Prediction

Example C.9.4: Nonlinear Prediction

For direct interpolation, extrapolation or filtering without trend analysis, linearprediction gives poor results, asking for a more general concept, namely nonlinearprediction. A set of functional series of Volterra type for nonlinear prediction ofscalar-, vector-, in general, tensor-valued signals is set-up. Optimizing the Helmertor Work Meister point error, a set of linear integral equations, generalizing theWeiner-Hopf integral equations. Multi-point-correlation functions (MPCF) arecharacteristic for those equations. A criterion for linear prediction, namely thevanishing of all MPCFs beside the two-point-correlation, is formulated solutionsof the discrete integral equations and the prediction error tensor of rank two arereviewed. A structure analysis of PCFs of homogenous and isotropic vector fieldsproves the number of characteristic functions to be 2.n D 2/; 4.n D 3/; 10.n D 4/and 26.n D 5/.

The methods of statistical prediction were developed relating to stochastic processesby two famous scientists, by the Russian A. N. Kolmorogov (1941) and by theAmerican N. Wiener (1941), nearly at the same time during the Second World War.


The world-wide resonance to these milestones began rather late: the work ofA. N. Kolmorogov was due to the bad documentation of Eastern publications inWestern countries nearly unknown, the great works of N. Wiener appeared asa classified report of section D2 of the American National Defence ResearchCommittee. At that time the difficult formula apparatus made the acceptancerather impossible. Nowadays, the difficulties are only history, the number ofpublications about linear filtering, interpolation and extrapolation, reaches tenthousands. For example, the American Journal JRE, Transactions on InformationTheory, New York appears regularly every year with a literature report about theState-of-the-Art. Notable are the works of R. Kalman (1960) who succeeded to solveelegant by the celebrated perdition equation, the Wiener-Hopf integral equation intransforming to a differential equation which is much easier to solve. The methodsbecame very well known as a method of data analysis applied to Navigation of theApollo Spacecraft.

In practice we begin with a two-step procedure. First, we analyze the observedvalues with respect to trend: we fit a data set to a harmonic function, to a exponentialfunction or a polynomial with respect to “position”. Second, we analyze the dataafter the trend fit to a Wiener-Kolmogorov prediction assuming a linear relationbetween the predicted signal and the reduced unknown signal. The coefficients inthis linear relation depend on the distance between the points and its direction.It is well known how to operate with the discrete approximation of the classicalWiener-Hopf integral equation.

The two step procedure accounts for the typical results that (a) the direct applicationof a linear Kolmogorov-Wiener prediction gives too inconsistent results, namely toolarge mean-square-errors of the predicted signal. The remarkable disadvantage ofthe two step procedure, in contrast is the immense computer work. Thus the centralquestion is: is the two-step-procedure necessary when a concept for nonlinearprediction exists.

For all statistical prediction concepts the correlation function between various pointsis the key quantity. In practice, the computational problems are greatly reducedif we are able to find out whether the predicted signal is subject to a statisticalhomogenous and isotropic process. It is a key information if a correlation functiondepends only on the distance between the involved points or is a function of thedifference vector, namely isotropy and homogeneity, our special topic in a specialsection.

Example C.9.41: The hierarchy for the equations in nonlinear prediction of scalar-valued signals

We review shortly the characteristical equations which are given in nonlinearprediction. Before hand we agree to the following notation: All Greek indices runf1; 2; � � � ; N � 1;N g counting the number of points P˛ with a signal s.P˛/. The


signal may be a scalar signal, a vector signal or a tensor signal. A natural assumptionis the “simple prediction model” that the predicted signal s.P / at the point P isa linear function of the observed signal s.P˛/ for ˛ 2 f1; 2; � � � ; N � 1;N g.The coefficients which describe the linear relation are dented by a.PP˛/. Theinconsistency between the predicted signal and the observed signal is denoted bythe letter “"00 leading the prediction variance Ef".P /".P /g, namely assuming that

the difference s.P / �NP

˛D1a.P; P˛/s.P˛/ is random.

Nonlinear prediction is defined by a nonlinear relation between the predicted signaland the observed signal, for instance the sum of a linear and a quadratic relationdescribed by our previous formula. The postulate os a minimum mean square errorleads to the problem to estimate the coefficients a and b. Of course, we couldcontinue with signal functions which depends of arbitrary power series. The openquestion is which is signal relation of arbitrary powers.

In order to give an answer to our key question, we have to change our notionfrom discrete to an integral concept. We arrive in this concept if we switch overto a continuous distribution of points P , namely from fP;P˛g to fr; r0g. In aone-dimensional problem we integrate over a measured line, in a two-dimensionalproblem we integrate over a surface. The integral set-up, in contrast, is reviewedby formula .C461/ � .C464/ O.s3/ expresses the order of neglected terms inthe functional series SŒs.r0�. We refer to a type of Taylor series in a generalizedHilbert space referred to V. Volterra (1959), called Volterra functional series. Thecoefficient functions given by .C462/ are called as variational derivatives in ageneralized Hilbert space.

A special criticism originates from the continuous version of the Volterra series.Help comes from the application of functional analysis, namely the Dirac functionas a generalized function illustrated by formulae (C464) and (C465). It is easy nowthe switch from the continuum to the discrete, namely for a regular grid. Nowadaysthe integral version is “simple” to end up with a discrete formulation, introducedalready by N. Wiener.

Up to now we have introduced a set-up of our prediction formulae for scalar signals.In relating our work to practical problems we have to generalize to vector-valuedfunctions.

Table C.9.13 Discrete and continuous Volterra series for scalar signals

“discrete 2nd order Volterra series”

s.P / DNP

˛D1a.P; P˛/s.P˛/C

NP

˛D1

NP

ˇD1b.P; P˛; Pˇ/s.P˛/s.Pˇ/C ".P / (C460)

“continuous 2nd order Volterra series”


s.r/ D Rdnr0a.r; r0/s.r0/C R dnr0 R dnr00a.r; r0; r00/s.r0/s.r00/CO.s3/C ".r/ .C461/

“continuous 3rd order Volterra series”s.r/ D R

dnr0a.r; r0/s.r0/C R

dnr0 R dnr00a.r; r0; r00/s.r0/s.r00/C R

dnr0 R dnr00 R dnr000a.r; r0; r00; r000/s.r0/s.r00/s.r000/CO.s4/C ".r/ .C462/

“variational derivatives”

a.r; r0/ D ıs.r/ıs.r0/

.C463/

a.r; r0; r00/ D 1

2Š

ı2s.r/ıs.r0/ıs.r00/

a.r; r0; r00; r000/ D 1

3Š

ı3s.r/ıs.r0/ıs.r00/ıs.r000/

“from continuous to discrete via Dirac functions: example”C1R�1

dnr0f .r0/ı.r; r0/ D f .r/ (C464)

ı.r; r0/ WD2

4!1 for r D r0;C1R�1

dnr0ı.r; r0/ D 10 otherwise:

s.P / D R dnr0a.P; r0/s.r0/ (C465)

DNP

˛D1K˛dnr0ı.P; r0/ D

NP

˛D1a.P; P˛/s.P˛/

End of Table C.9.12

Example C.9.42: The hierarchy of the equations in nonlinear prediction of vector-valued signals

In order to take care of practice, we will present finally the concept of predictionof vector-valued signals set-up by vector-valued Volterra series of type (C641).The partial variational derivatives of the vector-valued functions, also called signalSi.r/, are given by (C459) upto order O.s3. We let the indices of Latin type run1; 2; � � � ; n�1; n where n is the maximum dimension number. We apply the Einsteinsummation convention, namely the addition with respect to identical numberedindices.

In order to fill the deflections depend on the position r of the measurements. Inorder to determine the unknown coefficients of the nonlinear prediction “ansatz”,we choose risk function to be optimized. For instance, we may choose the traceof the variance function T1D t r�ij rDmin, called Helmert point error, or thedeterminant of the variance function T2 D det�ij r D min, called Werkmeister


point error, as two basic invariants called Hilbert invariants. Such an optimaldesign leads to a linear integral equation, a system of first kind, for the unknowntensors aij .r; r0/, aijk.r; r0; r00/ etc. If we only refer to terms up to second order,namely to quadratic signal terms, we have eliminated cubic, biquadratic etc. term,the standard concept of linear prediction.

The first nonlinear term, the quadratic terms, lead to the concept of

two-point correlation functions:˚ij .r; r0/ WD Efsi .r/sj .r0/g

three-point correlation functions:˚ijk.r; r0; r00/ WD Efsi .r/sj .r0/sk.r00/g

andfour-point correlation functions:

˚ijkl .r; r0; r00; r000/ WD Efsi .r/sj .r0/sk.r00/sl .r000/gThe term “multipoint functions” is to be understood that tensor valued functionsbetween multipoints appear as correlation functions. With respect to linearprediction newly three-point and four-point correlation functions appear. Let usformulate the discrete version of the integral equations: here run all the bar indices.1; 2; � � � ; Nn/ up to the numberNn. Indeed, we approximated the integral (C470),(C471) etc. by (C472), (C473) etc. Indeed we switched from continuum to discreteby our summation convention by neglecting the summation signs. For instance, thetwo-point and three-point functions aim.r; r1/; aimn.r; r1; r2/ are written aP�im ; b

P�ıimn

in the discrete versions.

In summary, our matrices have NnŒ.Nn/2 C .Nn/� elements. They correspond tothe correlation functions˚i;j ; ˚i;j ;k and ˚i;j ;k;l . Finally we restrict us to predictionterms of to second order when we write (C473) to (C476). Here we advantage oftwo conventions: “Extensor” calculus developed by Craig (1930–1939) abbreviates(a) as the multiindex the series i1i2; � � � i1. Again we sum up indices in brackets. Inthe new version of H. V. Craig’s multiindex we gain our characteristic equations(C476) - (C 479). The convention coefficients ai .i/ have their origin in thegeneralized Taylor series with the condition that higher order coefficients are gettingsmaller with increasing multiindex degree. If we apply the fundamental predictionequation (C475) we have to sum up the coefficients ai;i ; ai ; i1; i2; ai ; i1; i2; i3 etc.With the generalized variance-covariance matrix, function of the coordinates ofthe position vector r. The Helmert point error reduces proportional to the numberof prediction coefficients as long as the inequality (C481) holds. If we fix thecoefficient functions aij .r; r0/ for the distance zero to one, the following coefficientsaijk.r; r0; r00/; aijkl .r; r0; r00; r000/ are smaller than one.


Table C.9.44 Discrete and continuous Volterra series for vectors

continuous 2nd order Volterra series

si .r/ DnP

jD1Rdnr0aij .r; r0/sj .r0/

CnP

jD1

nP

kD1Rdnr0 R dnr00aijk.r; r0; r00/sj r0skr00 C o.s3/ (C466)

aij .r; r0/ D ısi .r/ısj .r0/

; aijk.r; r0; r00/ D ı2si .r/ısj .r0/ısk.r00/

(C467)

Variance function as a tensor of 2nd order

�ij .r/ D E˚Œsi .r/�

Rdn.r1/ai i1 .r; r1/si1 .r1/�R

dn.r1/Rdn.r2/ai i1i2 .r; r1; r2/si1.r1/si2.r2/ � � � � �

Œsj .r/ �Rdn.r1/ajj1.r; r1/sj1.r1/�R

dn.r1/Rdn.r2/ajj1j2.r; r1; r2/sj1.r1/sj2.r2/� � � � �

�

(C468)

J1 D t r�ij .r/ D min; J2 D det�ij .r/ D min (C469)

from continuous to discrete: multipoint correlation functions

˚ij .r; r0/ D R dnr1aim.r; r1/˚jm.r0; r1/CRdnr1

Rdnr2aimn.r; r1; r2/˚jmn.r; r1; r2/ (C470)

˚ijk.r; r0; r00/ D R dnr1aim.r; r1/˚jkm.r0; r00; r1/CRdnr1

Rdnr2aimn.r; r1; r2/˚jkmn.r0; r00; r1; r2/ (C471)

˚P˛jk D aP�im ˚˛�

jm C bP�ıimn˚˛�ıjmn; ˚

P˛ˇ

ijk D aP�im ˚˛ˇ�

jkm C bP�ıimn˚˛ˇ�ı

jkmn

(C472)

˚jk D aim˚jm C bimn˚jmn; ˚ijk D aim˚jkm C bimn˚jkmn(C473)

ai1 D ci1; ai2 D ci2; � � � ; aim D cim; � � � ; aiNn D ciNnbi11 D ciNnC1; bi12 D ciNnC2; � � � ; bi1n D ciNn C n; � � � ; bi1Nn D ciNn CNnbi21 D ci2NnC1; bi22 D ci2NnC2; � � � ; bi2n D ci2Nn C n; � � � ; bi2Nn D ci2Nn CNnbi31 D ci3NnC1; bi32 D ci3NnC2; � � � ; bi3n D ci3Nn C n; � � � ; bi3Nn D ci3Nn CNn

:::

bim1 D cimNnC1; bim2 D cimNnC2; � � � ; bimn D cimNn C n; � � � ;


bimNn D cimNn CNnbiNnNn D ci.Nn/2 �Nn (C474)

˚�0�0 D C�0 0˚ 0�0 ; C�0�0 D ˚�0�0˚�1�0�0 (C475)

˚i.j / � .ai.i/; ˚.j /.i// D 0 (C476)

˚ij1 � .ai i1 ; ˚j1i1 /� .ai i1i2 ; ˚j1i1i2 /� � � � D 0 (C477)

˚ij1j2 � .ai i1 ; ˚j1j2i1 / � .ai i1i2 ; ˚j1j2i1i2 /� � � � D 0 (C478)

.ai i1 ; ˚j1i1 / DRdnr1aii1 .r; r1/˚j1i1 /.r

0; r1/ (C479)

.ai i1i2 ; ˚j1j2i1i2 / DRdnr1

Rdnr2aii1i2 .r; r1; r2/˚j1j2i1i2 /.r

0; r00; r1; r2/

(C480)

�ij .r/ D ..ıi.i/ � ai.i//; ˚j.i//� ..ıi.i/ � ai.i//; aj.j /˚.i/.j // (C481)

2tr.ai.i/; ˚j.i// > tr.ai.i/; aj.j /; ˚.i/.j // (C482)

Table C.9.44: Volterra series

Example C.4. Structure analysis

Here we analyst the structure of multipoint functions of higher order with respectto statistical assumptions of homogeneity and isotropy. These assumptions madeour prediction formula much simpler. If we were not assuming this statisticalproperty, we had to compute for every point and every direction an independenthigher order correlation function. For instance, if we had to calculate two-point-correlation functions in two dimensions every 1 Gon, we had to live with 1600different correlation functions. If we would be bale to assume isotropy, the numberof characteristic functions would reduce to two functions only! In the case of three-point-functions the reductions range from 3200 non-isotropic functions to four incase of isotropy. Of course, we will prove these results.

The structure of these multi-point-functions of statistically homogenous andisotropic vector-valued signals follow from the form invariance of the functions withrespect to rotations and mirror functions of the coordinate system. Alternatively, wehave to postulate invariance against transformations of the complete orthogonalitygroup O.n/. Form invariance of a multi-point-correlation functions is to be


Fig. C38 Projections: longitudinal and lateral components for two-, three- and four pointcorrelation functions, isotropy-homogeneity assumption

understood with respect to base functions which leads to scalar quantities. Ournext section offers a better understanding of these theoretical results. As a shortintroduction to two-point-correlation os vector-valued signals we illustrate thetheory of characteristic functions for two-point-functions, for three-point-functionsand four-point-functions by Fig. C38.

In case of a vector valued functions the structure of the correlation function usedfor linear prediction is generated the coordinates si .r/ at the point r or P1 andsj .r0/ at the point r0 or P2 and the projected coordinate components along theconnection line P1P2 (along track) and orthogonal to the line P1P2 (cross track)called “longitudinal” and “lateral” sp and sn (“parallel” p, “normal” n). Thesecharacteristic functions ˚nn resp. ˚pp given by the two-point correlation functionsillustrated in Fig. C38.


Table C.15 Characteristic functions for two-point- and three-point correla-tion functions of vector-valued signals

“homogenous and isotropic two-point-function”

˚nn.r/ D Efsn.r/sn.r0/g; ˚pp.r/ D Efsp.r/sp.r0/g,˚pn.r/ D ˚np.r/ D 0 (C483)

˚j .r; r0/ D Efsi .r/sj .r0/g D ˚nnıij � Œ˚pp.r/ � ˚nn.r/��xi�xjr2(C484)

“homogenous and isotropic three-point-function”

˚nnp.r1; r2/ D Efsn.r/; sn.r0/; sp.r00/g˚npn.r1; r2/ D Efsn.r/; sp.r0/; sn.r00/g˚pnn.r1; r2/ D Efsp.r/; sn.r0/; sn.r00/g˚ppp.r1; r2/ D Efsp.r/; sp.r0/; sp.r00/g

9>>=

>>;(C485)

˚ijk.r; r0; r00/ D xi 0 �xijr�r0j ıjk˚pnn.r1; r2/C

xj 0 �xjjr�r0j ıik˚npn.r1; r2/C

D xk00 �xkx100 �x1 ıij˚nnp.r1; r2/C

.xi 0�xi /.xj 0�xj /.xk00 �xk/jr�r0j2.x100 �x1/ �

�f˚ppp.r1; r2/� ˚pnn.r1; r2/ �˚npn.r1; r2/� ˚nnp.r1; r2/g (C486)

End of Table: characteristic functions

In Table C 16, we present only results.How is the detailed proof of our results?This is here our central interest.

We begin with the definition of the popular scalar product and the norm of a Hilbertspace by equations (c 484) and (c 488), extend by the variance-covariance function(C489) by Craig calculus generalizations of a Hilbert space are given by (C490)and (C491). In terms of Craig calculus we compute Helmert’s optimal designof unknown coefficients by (C492), (C493), (C494) and (C495) and in contrastWerkmeister’s optimal design by (C496), (C497) and (C498), again of the unknowncoefficients.

Table C.16 Optimal variance function

Hilbert space: scalar product, norm

.f; g/ D R˝

f!g!dP! (C487)


jf2j Dp.f; f / D

rR

˝

jf!j2dP! (C488)

Variance function of vector-valued signals, Craig calculus

�ij .r/ D ."i .r/; "j .r// D ..ıi.i/ � ai.i//; ˚j.i//� ..ıi.i/ � ai.i//; aj.j /˚i.j //(C489)

Hilbert optimal design, Craig calculus

J1 D tr �ij .r/ D ..ıi.i/ � ai.i//; ˚j.i//� ..ıi.i/ � ai.i//; aj.j /˚i.j // D minıJ1ŒaI.i/� D 0; ı2J1ŒaI.i/� > 0

�

(C490)

.dJ1Œai.i/C"bi.i/�

d"/"D0 D 0; .

d2J1Œai.i/C"bi.i/�d"2

/"D0 > 0

J1Œai.i/ C "bi.i/� D tr ..ıi.i/ � .ai.i/ C "bi.i///; ˚j.i//�tr ..ıi.i/ � .ai.i/ C "bi.i///; .aj.j / C "bj.j //˚i.j //


d"/"D0 D 0 W tr .bi.i/; .˚j.i/ � aj.j /˚i.j // D 0

9>>>>>>>=

>>>>>>>;

(C491)

.d2J1Œai.i/C"bi.i/�

d"2/"D0 > 0 W tr .bi.k/; bj.l/˚.k/.l// > 0 (C492)

positive definiteness of the correlation function

trRdnr0 R dnr00bik.r; r0/bj i .r; r00/˚kl .r; r00/ > 0 (C493)

Werkmeister optimal design, Craig calculus

J2 D det�ij .r/ D f..ı1.i/ � a1.i//; ˚1.i// � ..ı1.i/ � a1.i//; a1.j /˚.i/.j //gf..ı2.i/ � a2.i//; ˚2.i//� ..ı2.i/ � a2.i//; a2.j /˚.i/.j //g�f..ı1.i/ � a1.i//; ˚2.i//� ..ı1.i/ � a1.i//; a2.j /˚.i/.j //2g

9=

;

(C494)ıJ2Œai.i/� D 0; ı2J2Œai.i/� > 0


d"/"D0 D 0; .

d2J1Œai.i/C"bi.i/�d"2

/"D0 > 0

J2Œai.i/ C "bi.i/� D A0 C A1"C A2"2 C A3"3 CA4"4


d"2/"D0 D 0 W A1 D 0

9>>>>>>=

>>>>>>;

(C495)



d"2/"D0 > 0 W A2 > 0 (C496)

End of Table: Craig optimal design

Finally, we study in more detail the concept of structure functions. We illustratethe form invariance of its base invariants which are invariant in terms of rotationaland mirror symmetry. Let us call the base vector ei .r/ relating to signal si .r/ at theplacement vector r; ej .r0/ relating to signal sj .r0/ at placement vector r0 etc. Theset-up of base vectors by (C449) and corresponding structure functions by (C580)are given. They lead us to the combination of base invariants ˚1;˚2 followingTaylor (1935), and Karman (1937), ˚3;˚4; ˚5 etc in Table (C501)–(C505). finally,we summarize up to order three the various isotropic and homogenous correlationfunctions in Table C.17, (C501)–(C505). They are based on the work of Karmanand Howarth (1938), and Obuchow (1958). Let us present finally the numberof characteristic functions for homogenous and isotropic correlation functions ofhigher order for vector-valued signals, collected in Table C.18.

Table C.17 Scalar products of base invariants

base invariants

xixi ; xj 0xj 0 ; xk00xk00 ; xl 000xl 000 ; etc.ei ei 0; eiei 00 ; eiei 000 ; etc.ei 0ei 00; ei 0ei 000 ; etc.etc.xiei ; xi ei 0 ; xi ei 00; xi ei 000 ; etc.xi 0ei 0; xi 0ei 00 ; xi 0ei 000 ; etc.xi 00ei 00; xi 00ei 000 ; etc.etc.

9>>>>>>>>>>>=

>>>>>>>>>>>;

(C497)

base functions

˚1.ei / D ˚iei˚2.ei ; ej 0/ D eiej 0˚ij 0.r; r0/˚3.ei ; ej 0; ek00/ D eiej 0ek00˚ij 0k00.r; r0; r00/˚4.ei ; ej 0; ek00 ; el 000/ D ei ej 0ek00el 000˚ij 0k00l 000.r; r0; r00; r000/� � �

9>>>>>=

>>>>>;

(C498)

combinations of base invariants

˚1 D ˚1.ei / D ˚1ei (C499)


˚2.ei ; ej 0/ D ei ej 0˚ij 0.r; r0/ D eiej 0f�xi�xj 0a1.r/C ıij 0a2.r/g(C500)

(Taylor (1935), Karman (1937))

˚3.ei ; ej 0; ek00/ D eiej 0ek00˚ij 0k00.r; r0; r00/ DD eiej 0ek00f�xi�xj 0�xk00a1.r1; r2/C�xiıj 0k00a2.r1; r2/C

C�xj 0ıik00a3.r1; r2/C�xk00ıij 0a4.r1; r2/g (C 501)

˚4.ei ; ej 0; ek00 ; el 000/ D eiej 0ek00el 000˚ij 0k00l 000.r; r0; r00; r000/ DD eiej 0ek00el 000f�xi�xj 0�xk00�xl 000a1.r1; r2; r3/

C�xi�xj 0ık00l 000a2.r1; r2; r3/CC�xj 0�xl 000ıik00a3.r1; r2; r3/C�xi�xk00ıj 0l 000a4.r1; r2; r3/CC�xi�xl 000ıj 0k00a5.r1; r2; r3/C�xj 0�xk00ıil 000a6.r1; r2; r3/CC�xk00�xl 000ıij 0a7.r1; r2; r3/C ıij 0ık00l 000a8.r1; r2; r3/CCıik00ıj 0l 000a9.r1; r2; r3/C ıj 0k00ıil 000a10.r1; r2; r3/g

9>>>>>>>>>=

>>>>>>>>>;

(C502)

˚4.ei ; ej 0; ek00 ; el 000 ; em0000/ D ei ej 0ek00el 000em0000˚ij 0k00l 000m0000.r; r0; r00; r000; r0000/ DD eiej 0ek00el 000em0000f�xi�xj 0�xk00�xl 000�xm0000a1.r1; r2; r3; r4/

C�xi�xj 0�xk00ıl 000m0000a2CC�xi�xj 0�xm0000ık00l 000a3 C�xi�xj 0�xl 0000ık00m0000a4C�xi�xl 000�xm0000ıj 0k00a5CC�xi�xk00�xm0000ıj 0l 000a6 C�xi�xk00�xl 000ıj 0m0000a7C�xk00�xl 000�xm0000ıij 0a8CC�xj 0�xl 000�xm0000ıik00a9 C�xj 0�xk00�xm0000ıil 000a10C�xj 0�xk00�xl 000ıim0000a11CC�xiıj 0k00ıl 000m0000a12 C�xj 0ıik00ıl 000m0000a13 C�xk00ıij 0ıl 000m0000a14CC�xl 000ıij 0ık00m0000a15 C�xm0000ıij 0ık00l 000a16 C�xiıj 0l 000ık00m0000a17CC�xj 0ıil 000ık00m0000a18 C�xk00ıil 000ıj 0m0000a19 C�xl 000ıik00ıj 0m0000a20CC�xm0000ıik00ıj 0l 000a21 C�xiık00l 000ıj 0m0000a22 C�xj 0ıim0000ık00l 000a23CC�xk00ıim0000ıj 0l 000a24 C�xl 000ıim0000ıj 0k00a25 C�xm0000ıil 000ıj 0k00a26g

9>>>>>>>>>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>>>>>>>>>;

(C503)

End of Table: Scalar products of basic invariants


Table C.18: Isotropic and homogenous correlation functions, two- andthree- point functions, normal and longitudinal components

˚11.r; r0/ D a1.r/r2 C a2.r/ D ˚pp˚12.r; r0/ D ˚12.r; r0/ D ˚pn D ˚np D 0˚22.r; r0/ D ˚33.r; r0/ D ˚nn

9=

;(C 504)

˚111 D r21 .x00

1 � x1/a1.r1; r2/C r1Œa2.r1; r2/C a3.r1; r2/�C.x00

1 � x1/a4.r1; r2/ D ˚ppp˚112 D r21 .x

00

3 � x2/a1.r1; r2/C .x00

2 � x2/a4.r1; r2/˚113 D r21 .x00

3 � x3/a1.r1; r2/C .x00

3 � x3/a4.r1; r2/˚121 D 0; ˚122 D r1a2 D ˚pnn; ˚123 D 0˚131 D 0; ˚132 D 0; ˚133 D r1a2 D ˚pnn˚211 D 0; ˚212 D r1a3 D ˚npn; ˚213 D 0˚231 D ˚232 D ˚233 D 0˚311 D ˚312 D ˚313 D r1a3 D ˚npn˚321 D ˚322 D ˚323 D 0˚331 D .x00

1 � x1/a4 D ˚nnp; ˚332 D .x00

2 � x2/a4;˚333 D .x00

3 � x3/a4

9>>>>>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>>>>>;

(C505)

a1.r/ D ˚pp�˚nnr2

; a2.r/ D ˚nn

a1.r2; r2/ D ˚ppp�˚pnn�˚npn�˚nnpr21 .x

00

1 �x1/a2.r2; r2/ D ˚pnn

r1

a3.r2; r2/ D ˚npnr1

a4.r2; r2/ D ˚nnp

.x00

1 �x1/

9>>>>=

>>>>;

(C506)

˚ij .r; r0/ D ˚nn.r/ıij C Œ˚pp.r/ �˚nn.r/��xi�xjr2(C507)

˚ijk.r; r0; r00/ D �xir1ıjk˚nnn C �xj

r1ıik˚npn C .x00

k �xk/.x00

1 �x1/ ıij˚nnpC�xi�xj .x

00

k �xk/r21 .x

00

1 �x1/ .˚ppp �˚pnn �˚npn �˚nnp/(C508)

˚ij 0k0.r; r0 D r00/ D �xi�xj 0�xk0a1.r1/C�xiıj 0k0a2.r1/

C.�xj 0ıik0 C�xk0ıij 0/a3.r1/

D ˚ppp.r1/�˚pnn.r1/�2˚nnp.r1/r31

�xi�xj 0�xk0

C˚pnn.r1/�xir1 ıj 0k0 C .�xj 0 ıik0 C�xk0 ıij 0 /

r1˚nnp.r1/

(C509)

End of Table: Isotropy and homogenity


Table C.19:Number of characteristic functions for multi-point functionsbased on vector-valued signals

multi-point correlation functions number of characteristic functions

2 23 44 105 26

End of Table: characteristic functions

Ex 5: Nonlocal Time Series Analysis

Example: Nonlocal time series analysis

Classical time series analysis, for instance Earth tide measurements, is based onlocal harmonic and disharmonic effects. Here we present a nonlocal conceptto analyze on a global scale measurements at different locations. The toolof the generalized harmonic analysis base on correlation functions of higherorder between vector-valued signals at different points and at different times.Its polyspectrum is divided into poly-coherence and poly-phase. A 3d-filder ofKolmogorov-Wiener type is designed for an array analysis. The optional distancebetween stations and the optimal registration time- Nyquist frequency- is given toavoid aliasing.

Standard procedures of signal analysis treat a registered time series of one stationonly. It is our target here to combine various observation stations: we are interestedin the global picture handled by array analysis. We like to refer to applications inseismology and tidal analysis, the great success of tomography. A 3d- or multi-channel filter makes possible to arrange an optimal signal-to-noise ratio in terms ofan optimal design of the station distance and an optimal design of the registrationtime, namely statistical accuracy of numbers which decide upon a linear or nonlinearfilter.

In the first section we introduce the classical nonlocal analysis forN D 2 stations fora vector-valued signal. We assume no correlation between the signal components.The mono-spectrum, namely the Fourier-transform of scalar-valued correlationfunctions is separated by coherence- and phase-graphs. The second section is astraight forward approach to registered vector-valued signals between N D 2

stations. Support has the origin in the analysis of coherence surface and of phasesurface of Fourier-transformed correlation-tensors. Poly-spectra of signal space-time functions form the basis of N� stations analysis in the third section. We


separate real- and imaginary-parts in terms of a poly-coherence-graph and of apoly-phase-graph, both matrix forms. An extended array analysis is based on aKolmogorov-Wiener filter in the fourth section. We give a simple analysis to predictan optimal registration and an optimal station distance. At the end we give referencesto (a) array analysis, (b) mono-spectrum and (c) bi- and poly-spectra.

Scalar signals, “classical analysis”

In terms of the “classical time series analysis” registrations at two different placesare to be combined: Which are the frequency bands in which we can detectcoherences, which is the spectral phase-dislocation between two registrations. Weare to keep in mind the Fundamental Lemma of N. Wiener that a frequency assuredby coherence has the property of maximal probability, characterized by signal time-functions at two stations. We sum up the following operations:

(i) Calculation of auto- and cross-correlation functions of scalar-valued signals,(ii) Calculation of Fourier-transforms of auto- and cross-correlation functions,

separation of mono-spectra in terms of active and passive (cos� versus sin�terms) components,

(iii) Plots of coherence graphsH2.k1; !1/ and phase graphs �2.k1; !1/

Here we depart from a scalar space-time representation of a signal, namely by asignal s.r; t/ at a placement r and at the time t and a signal s.r0; t 0/ at the placementr0 and at the time t 0 by correlation, in detail by (C512 - C 519). f Œs.r; t/; s.r0; t 0/�denotes the probability density, to meet the signals s and s0 at the placements r; r0and the times t; t 0. we intend to transform the probability integral into a space-time integral assuming homogeneity and ergodicity by (c 513) and (C514). Thespace-time complex mono-spectrum ˚.ri ; t1/ ! e.ki ; !1/ is defined by (C515),direct and inverse, (C516), e.ki ; !1/ decomposed into a real-valued part andan imaginary part. “e” denotes the Fourier transform, i W �p�1, n indicatesthe dimension. In analogy to wave propagation in electrodynamics, we defineRcfeg DW e� .ki ; !1/ active spectrum or co-spectrum, but Jmfeg DW e� quadraticor “blind spectrum”. Backward transformation ˚.ri ; �1/ cos.!1�1� < k1jr1 >/and ˚.ri ; �1/ sin.!1�1� < k1jr1 >/ lead to fe� .ki ; !1/;e�.ki ; !1/g Let usfinally H2 WD e� 2.ki ; !1/ C e�2.ki ; !1/g as the space-time coherence graph, andtan�.ki ; !1/g WD e�.ki ; !1/=e� .ki ; !1/; �.ki ; !1/ the phase graph, both veryimportant in defining “coherent light”. Figure C39 illustrates the phase graph,e� 2 C e�2 the length of the complex vector in the polar coordinates. The autocorrelation functions � .r; r/ and �.r0; r0/ clearly lead us to “angle” and “length”in the complex plane Fig. C40 illustrates the fundamental coherence- and phase-functions.

Finally the contributions of lackman (1965); Granger and Hatanaka (1964); Harris(1967); Kertz (1969, 1971); Lee (1960); Pugachev (1965); Robinson (1967),Robinson and Treitel (1964).


Table C.20: Variance-covariance function of a scalar signal, direct andinverse Fourier transform, space-time representation

EŒs.r; t/s.r0; t 0/� WD ˚.r; r0; t; t 0/ (C510)

WDC1R�1

ds.r; t/C1R�1

ds.r0; t 0/s.r; t/s.r0; t 0/f Œs.r; t/s.r0; t/� (C511)

˚.r0 � r; t 0 � t/ D limT;x;y!1

18T xy

CTR

�Tdt

CxR�xdx

CyR�ydy (C512)

s.x; y; t/sŒx C .x0 C x/; y C .y0 C y/; t C .t 0 C t/� (C513)

r0 � r WD r1; t 0 � t WD �1 (C514)

e.k1; !1/ D .2/�nC12

C1R�1

dnr1C1R�1

d�1˚.r1; �1/ expCi.!1�1 � k1r1/

(C515)

˚.r1; �1/ D .2/�nC12

C1R�1

dnk1C1R�1

d!1e.k1; !1/ exp�i.!1�1 � k1r1/

(C516)

e.k1; !1/ D ReŒe�C i ImŒe� D e� .k1; !1/C i e�.k1; !1/ (C 517)

e�.k1; !1/ WD .2/�nC12

C1R�1

dnr1C1R�1

d�1˚.r1; �1/ cos.!1�1 � k1r1/ (C518)

e� .k1; !1/ WD .2/�nC12

C1R�1

dnr1C1R�1

d�1˚.r1; �1/ sin.!1�1 � k1r1/ (C519)

H2 WD e� 2.k1; !1/C e�2.k1; !1/ (C520)

tg�.k1; !1/ WDe�2.k1; !1/e� 2.k1; !1/

(C521)

End of Table: variance-covariance functions


Fig. C39 Decompositioninto real and imaginarycomponent Gaussrepresentation fe�;e� g

Fig. C40 Scalar coherenceand phase graph

C.9.52 First generalization: vector-valued classical analysis

At first, we deal with separate single components of a general three-component tidalrecording. We develop a joint analysis of all three components observed at twopoints on the surface of the Earth, for example. Its signal functions are correlatedwith each other, in the special case here the vertical component as well as thehorizontal component. We receive a correlation tensor of second degree, namelyin the form of the coherence matrix and the phase matrix. In detail, the analysistechnique is based on the following operations:

(i) Computation of auto- and cross-correlations of the vector-valued signal,(ii) Computation of the Fourier transform of the functions separation within the

mono spectral matrix in “active” and “blind” spectra cosine- and sine-Fouriertransform,

(iii) Part of the coherenec matrix Hij .k1; !1/ as well as the phase matrix�ij .k1; !1/.

Theoretical Background


Let si .x; t/ D ıgi .x; t/ be a vector-valued space-time-signal consisting of verticaland horizontal components subject to i D 1 or x; i D 2 or y and i D 3 or z.The signal function si .x; t/ at the placement x and at the time instant t and thesignal function sj .x0; t 0/ at the placement x0 and the time instant t 0 are correlated bymeans of

Efsi .x; t/sj .x0; t 0/g DW ˚ij .x; x0; t; t 0/ for i; j 2 f1; 2; 3g (C522)

As an analogue of our previous formulae we define the coherence matrix, namelythe coherence tensor Hij , and the phase matrix �ij

Hij .k1; !1/ WD e� 2ij .k1; !1/C e�2

ij .k1; !1/ (C523)

tg �ij .k1; !1/ De�2ij .k1; !1/

e� 2ij .k1; !1/

(C524)

The coherence matrix as well as the phase matrix are represented by special surfacefor instance by coherence ellipses, coherence hyperbolae etc.

C.9.53 Second generalization: poly spectra

Up to now we analyzed only signal registrations at two different points bycorrelation, namely by coherence and phase. Here, we introduce the concept ofpoly spectra for the purpose of registration of Earth tides at all points we haveregistrations. In detail, the analysis technique is based on the following operations:

(i) Computation of multi-point correlation functions of a vector-valued tidal signal(ii) Computation of the Fourier transforms called polyspectra. The concept is able

to separate mono-, bi-, tri-, in short polyspectra in “active” and “blind” spectra,cosine- and sine-Fourier transform.

(iii) Part of generalized coherence graphs

Hi1i2��iN�1iN .k1;k2; � � � ;kN�1; !1; !2; � � � ; !N�1/ (C525)

and the generalized phase graphs

�i1i2��iN�1iN .k1;k2; � � � ;kN�1; !1; !2; � � � ; !N�1/ (C526)

at N tidal stations being functions of wave number vectors k1;k2; � � � ;kN�1 and thefrequencies !1; !2; � � � ; !N�1.


Let si1.x; t/ � ıgi.x; t/ be a tensor-valued space-time tidal signal at the first tidalstation, si2.x

0; t/ be a tensor-valued tidal signal at the second tidal station. If we


correlate all signal functions with each other, e receive a multi-point correlationfunction equal to the number of tidal stations.

Efsi1.x; t/si2 .x0; t 0/ � � � sin.xN ; tn/g WD ˚i1i2��iN�1iN (C527)

In case of space like homogeneity and the time like stationarity we are able torepresent the various correlation functions by

˚i1i2��iN�1iN .x1; x2; � � � ; xN�1; �1; �2; � � � ; �N�1/ (C528)

subject to

x1 WD x0 � x; x2 WD x00 � x; etc :�1 D t 0 � t; �2 D t 00 � t; etc (C529)

and their polyspectra

ei1i2��iN�1iN .k1;k2; � � � ;kN�1; !1; !2; � � � ; !N�1/ D

.2/�n.N�1/C.N�1/

2

C1Z

�1dn.N�1/r

C1Z

�1dn.N�1/�

˚i1i2��iN�1iN .r1; r2; � � � ; rN�1; �1; �2; � � � ; �N�1/ �expCi.!1�1 C !2�2 C � � � C !N�1�N�1 � k1r1 � k2r2 � � � � � kN�1rN�1/

(C530)

n is the dimension number, N the number of tidal stations. For the case of N D 3we use the term “bispectrum”, for the case N D 4 we use the term “trispectrum”in general “polyspectrum”. The common separation of ˚i1i2��iN�1iN in real- andimaginary part leads us to

the Polycoherence graph as well as the poly phase graph

Hi1i2��iN�1iN D Rc2Œei1i2��iN�1iN �C Jm2Œei1i2��iN�1iN � (C531)

tg�i1i2��iN�1iN DJmŒei1i2��iN�1iN �

RcŒe i1i2��iN�1iN �(C532)

Numerical results are presented in Brillinger (1964), Grafarend (1972), Hasselmannet al. (1963), Rosenblatt (1965, 1966), Sinaj (1963 i, ii), Tukey (1959), and Van Nees(1965).


Table C.9.54: Generalized array analysis on the basis of a Kolmogorov-Wiener filter (3D-filter)

We do not attempt to review “filter theory”, instead we present array analysis for thepurpose to optimally reduce noise in tidal measurements in optimal station density.The method of analysis is based in the following operations:

(i) Computation of signal correlation functions from tidal registration in multi-stations and afterwards in noise correlation functions from an analysis ofnoise characteristic multi-dimensional stochastic process, for instance of typeMarkov, homogenous and isotropic

(ii) Computation of spectra from diverse correlation functions(iii) Solving the linear matrix equation of type Wiener-Hopf equations of optimal

filter functions.

The optimal station density is achieved from the postulate < �kj�x >� 1, theoptimal registration time from the postulate �!�t � 1. �x is the distance of thetidal stations in position space, �x the distance in the Fourier space, for instancekx D 2=�x wavelength aliasing �t the discretization parameter in the timedomain and �! the corresponding frequency distance in the Fourier space, forinstance �! D 2=�T , �! the frequency aliasing.

Example

�r � 1

�k cos ^k; rD ��

2 cos ^k; r

�! � 1=�t

^k; r D 0; �� D 103 Km ; �r Š 140 Km

^k; r D 300; �� D 103 Km ; �r Š 330 Km

�t D 1 sec ; �! D 1 Hz ; �T Š 6 sec

�t D 1 msec ; �! D 103 Hz ; �T Š 6 msec


Lets us assume that the vector-valued registration signal consists of two parts, thepure signal part and the superimposed noise:

ıgi .x; t/ D si .x; t/C ni .x; t/ (C533)

The filtered signal ıg0i .x; t/ has the linear structure of type

ıg0i .x; t/ D s0

i .x; t/C n0i .x; t/ (C534)


The filter error "i .x; t/ D s0i .x; t/ � si .x; t/ C n0

i .x; t/ � ni .x; t/ is designedby I1t r�ij .x; t/ D min or I1 det�ij .x; t/ D min subject to the error tensor�ij .x; t/ WD Ef"i .x; t/"j .x; t/g D min. The filter fij .x; t/ is defined by the linearVolterra setup

n0i .x; t/ WD

C1Z

�1dnx1

Zd�1fij .x1; �1/nj .x � x1; t � �1/ (C535)

J Œ�ij .fkl .x; t//� D min lead the “necessary” generalized Wiener-Hopf equations.

˚si sj .r; r0; t; t 0/ D

C1Z

�1dnr00

C1Z

�1dt 00fik.r; r00/˚jk.r; r00/ (C536)

ef ij D esi sj .k1; !1/Œesi sk C enink �C (C537)

Œ�C identifies the generalized inverse of type minimum norm least squares.

eF Œi�i � D essŒi�i �ŒessŒi�i � C ennŒi�i �� (C538)

The result of 3D or linearized multi-channel leads to an easy solution. If we wouldcare for a nonlinear filter subject to the multi-index J WD .j1; j2; � � � ; j˛/ andK WD.k1; k2; � � � ; kˇ/ we will receive the solution

˚ij D .fik; ˚jk/ (C539)

with respect to the scalar product .:; :/ of the corresponding Hilbert space.Numericals examples you can find in C. B. Archambeau et. al. (1965), J. W. Britilland F. E. Whiteway (1965), H. W. Briscoe and P. L. Fleck (1965), J. P. Burg(1964), J. F. Claerbout (1964), P. Embree et. al. (1963), E. Grafarend (1972),A. N. Kolmogorov (1941), N. Wiener (1949, 1964, 1968).

Comments

In experimental sciences, namely geosciences like geophysics, geodesy, environ-mental sciences, namely, weather, air pollution, rainfall, oceanography, temperatureaffected sciences take action on a spatially and temporally large scale to very largescales. For this reason, they never can be considered stationary or homogenous-isotropic, but

non-stationary inhomogeneous or anisotropic

in the statistical sense Spock and Pilz (2008) if not Guttorp and Sampson (1994) asone of the attempts to model non-stationary, other examples are geodetic networks


in two- and three-dimensional Euclidean space and on two-, three- and four-dimensional manifolds including the sphere, the ellipsoid or General Relativity as anexcellent example for inhomogeneity and anisotropy presented here. C. Calder andN. Cressie (2007) review different directions in spectral and convolution based onGaussian non-stationary random modeling. Speck and Pilz (2008) present a spectraldecomposition without any restriction to the Gaussian approach. They advice thereader to apply a special Generalized Linear Mixed Model (GLMM), the topic of ourbook, namely derived from the spectral decomposition of non-stationary randomfields.

Topics

(i) Stationary and isotropic random field

versus

(ii) Non-stationary, locally isotropic random field(iii) Interpretation in the context of linear mixed models (LMMs)

Y D Fˇ C Ab C "0

a. Fˇ: design matrix of the first kind, result: regression function, trendbehaviour F fixed design matrix

b. Ab: non-deterministic random fluctuations around the trend functions, Afixed design matrix.

c. "0: independent, homoscedastic error term centered at zero

(iv) Relation to non-stationary, locally isotropic random fields(v) Generalized linear mixed models (GLMMs)

(vi) Extending generalized linear mixed models (DHGLMMs)(vii) Limear and mixed models and Hermitic polynomials

(viii) Anisotropic stochastic process axisymmetric, Chandrasekhar (1950),Grafarend (1972)

Appendix DBasics of Groebner Basis Algebra

D-1 Definitions

To enhance the understanding of the theory of Groebner bases presented inChap. 15, the following definitions supplemented with examples are presented.Three commonly used monomial ordering systems are; lexicographic ordering,graded lexicographic ordering and graded reverse lexicographic ordering. First wedefine the monomial ordering before considering the three types.

Definition D.1. (Monomial ordering):

A monomial ordering on k Œx1; : : : ; xn� is any relation> on Zn0 or equivalently any

relation on the set x˛; ˛ 2 Zn0 satisfying the following conditions:

(a) Is total (or linear) ordering on Zn0

(b) If ˛ > ˇ and � 2 Zn0, then ˛ C � > ˇ C �

(c) < is a well ordering on Zn0.

This condition is satisfied if and only if every strictly decreasing sequence in Zn0

eventually terminates.

Definition D.2. (Lexicographic ordering):

This is akin to the ordering of words used in dictionaries. If we define a polynomialin three variables as P D kŒx; y; z� and specify an ordering x > y > z, i.e., x comesbefore y and y comes before z, then any term with x will supersede that of y whichin tern supersedes that of z. If the powers of the variables for respective monomialsare given as ˛ D .˛1; : : :; ˛n/ and ˇ D .ˇ1; : : :; ˇn/; ˛; ˇ 2 Z

n0; then ˛ >lex ˇ ifin the vector difference ˛ �ˇ 2 Z

n; the most left non-zero entry is positive. For thesame variable (e.g., x) this subsequently means x˛ >lex xˇ:


887


888 D Basics of Groebner Basis Algebra

Example D.1.

x > y5z9 is an example of lexicographic ordering. As a second example, considerthe polynomial f D 2x2y8 � 3x5yz4 C xyz3 � xy4, we have the lexicographicorder; f D �3x5yz4 C 2x2y8 � xy4 C xyz3 jx > y > z .

Definition D.3. (Graded lexicographic ordering):

In this case, the total degree of the monomials is taken into account. First, oneconsiders which monomial has the highest total degree before looking at thelexicographic ordering. This ordering looks at the left most (or largest) variableof a monomial and favours the largest power. Let ˛; ˇ 2 Z

n0; then ˛ >grlex ˇ if

j˛j DnP

iD1˛i > jˇj D

nP

iD1ˇi or j˛j D jˇj, and ˛ >lex ˇ; in ˛ � ˇ 2 Z

n; the most

left non zero entry is positive.

Example D.2.

x8y3z2 >grlex x6y2z3 j.8; 3; 2/ >grlex .6; 2; 3/, since j.8; 3; 2/j D 13 > j.6; 2; 3/j D

11 and˛ � ˇ D .2; 1;�1/. Since the left most term of the difference (2) is positive,the ordering is graded lexicographic. As a second example, consider the polynomialf D 2x2y8 � 3x5yz4 C xyz3 � xy4; we have the graded lexicographic order;f D �3x5yz4 C 2x2y8 � xy4 C xyz3 jx > y > z:

Definition D.4. (Graded reverse lexicographic ordering):

In this case, the total degree of the monomials is taken into account as in the caseof graded lexicographic ordering. First, one considers which monomial has thehighest total degree before looking at the lexicographic ordering. In contrast to thegraded lexicographic ordering, one looks at the right most (or largest) variable ofa monomial and favours the smallest power. Let ˛; ˇ 2 Z

n0; then ˛ >grevlex ˇ if

j˛j DnP

iD1˛i > jˇj D

nP

iD1ˇi or j˛j D jˇj, and ˛ >grevlex ˇ, and in ˛ � ˇ 2 Z

n the

right most non zero entry is negative.

Example D.3.

x8y3z2 >grevlex x6y2z3 j.8; 3; 2/ >grevlex .6; 2; 3/ since j.8; 3; 2/j D 13 >

j.6; 2; 3/j D 11 and ˛ � ˇ D .2; 1;�1/. Since the right most term of the difference(-1) is negative, the ordering is graded reverse lexicographic. As a second example,

D-2 Buchberger Algorithm 889

consider the polynomial f D 2x2y8 � 3x5yz4 C xyz3 � xy4 , we have the gradedreverse lexicographic order: f D 2x2y8 � 3x5yz4 � xy4 C xyz3 jx > y > z .

If we consider a non-zero polynomial f D P˛ a˛x

˛ in k Œx1; : : : :; xn� and fix themonomial order, the following additional terms can be defined:

Definition D.5.

Multidegree of f : Multideg (f / D max.˛ 2 Zn0 ja˛ ¤ 0/ Leading Coefficient

of f : LC (f / D amultideg.f / 2 k Leading Monomial of f : LM (f / D xmultideg.f /

(with coefficient 1) Leading Term of f : LT .f / D LC.f / LM .f /.

Example D.4.

Consider the polynomial f D 2x2y8 � 3x5yz4 C xyz3 � xy4 with respect tolexicographic order fx > y > zg , we haveMultideg (f )=(5,1,4)LC (f )= �3LM (f )= x5yz4

LT (f )= �3x5yz4

The definitions of polynomial ordering above have been adopted from Cox et al.(1997), pp. 52–58.

D-2 Buchberger Algorithm

computes the Groebner basis using the computer algebraic systems ((CAS) ofMathematica and Maple. With Groebner basis, most nonlinear equations that areencountered in geodesy and geoinformatics can be solved. All that is required of theuser is to write algorithms that can easily run in Mathematica or Maple using stepsthat are discussed in Sects. D-21 and D-22 below.

D-21 Mathematica Computation of Groebner Basis

Groebner basis can be computed using algebraic softwares of Mathematica (e.g.,version 7 onwards). In Mathematica, Groebner basis command is executed bywriting

InŒ1� WD GroebnerBasisŒfpolynomialsg; fvariablesg�; (D.1)


where In[1]:= is the Mathematica prompt which computes the Groebner basis forthe Ideal generated by the polynomials with respect to the monomial order specifiedby monomial order options.

Example D.5. (Mathematica computation of Groebner basis):

In Example 15.3 on p. 536, the systems of polynomial equations were given as

�f1 D xy � 2yf2 D 2y2 � x2:

Groebner basis of this system would be computed by

InŒ1� WD GroebnerBasisŒff1; f2g; fx; yg�; (D.2)

leading to the same values as in (15.10) on p. 536. With this approach, however, oneobtains too many elements of Groebner basis which may not be relevant to the taskat hand. In a case where the solution of a specific variable is desired, one can avoidcomputing the undesired variables, and alleviate the need for back-substitution bysimply computing the reduced Groebner basis. In this case (D.1) modifies to

InŒ1� WD GroebnerBasisŒfpolynomialsg; fvariablesg; felimsg�; (D.3)

where elims is for elimination order. Whereas the term reduced Groebner basis iswidely used in many Groebner basis literature, we point out that within Mathematicasoftware, the concept of a reduced basis has a different technical meaning. In thisbook, as in its predecessor, and to keep with the tradition, we maintain the use ofthe term reduced Groebner basis.

Example D.6. (Mathematica computation of reduced Groebner basis):

In Example D.2.1, one would compute the reduced Groebner basis using (D.3) as

InŒ1� WD GroebnerBasisŒff1; f2g; fx; yg; fyg�; (D.4)

which will return only �2x2 C x3. Note that this form is correct only when theretained and eliminated variables are disjoint. If they are not, there is absolutely noguarantee as to which category a variable will be put in! As an example, considerthe solution of three polynomials x2 C y2 C z2 � 1, xy � zC 2, and z2 � 2x C 3y.In the approach presented in (D.4), the solution would be

InŒ1� WD GroebnerBasisŒfx2Cy2Cz2�1; xy�zC2; z2�2xC3yg; fx; y; zg; fx; yg�;(D.5)

leading to

D-2 Buchberger Algorithm 891

f1024� 832z� 215z2 C 156z3 � 25z4 C 24z5 C 13z6 C z8g:

Lichtblau (Priv. Comm.) however suggests that the retained variable, in this case .z/and the eliminated variables .x; y/ be separated, i.e.,

InŒ1� WD GroebnerBasisŒfx2C y2C z2� 1; xy � zC 2; z2 � 2xC 3yg; fzg; fx; yg�:(D.6)

The results of (D.6) are the same as those of (D.5), but with the advantage ofa vastly better speed if one uses an ordering better suited for the task at hand,e.g., elimination of variables in this example. For the problems that are solved inour books, the condition of the retained and eliminated variables being disjoint istrue, and hence the approach in (D.5) has been adopted. The univariate polynomial�2x2 C x3 is then solved for x using the roots command in Matlab (see e.g.,Hanselman and Littlefield (1997), p. 146) by

roots. Œ1 �2 0 0� /: (D.7)

The values of the row vector in (D.7) are the coefficients of the cubic polynomialx3�2x2C0xC0 D 0 obtained from (D.4). The values of y from Example (15.3) onp. 536 can equally be computed from (D.4) by replacing y in the option part withx and thus removing the need for back substitution. We leave it for the reader tocompute the values of y from Example (15.3) and also those of z in Example (15.3)using reduced Groebner basis (D.3) as an exercise. The reader should confirm thatthe solution of y leads to y3 � 2y with the roots y D 0 or y D ˙1:4142. Fromexperience, we recommend the use of reduced Groebner basis for applications ingeodesy and geoinformatics. This will; fasten the computations, save on computerspace, and alleviates the need for back-substitution.

D-22 Maple Computation of Groebner Basis

In Maple Version 10 the command is accessed by typing > with (Grobner); Basis(WL; T ) is for Groebner, gbasis (WL; T ) is for reduced Groebner basis. WL listor set of polynomials, T monomial order description, e.g. “plex” (variables) –pure lexicographic order and > is the Maple prompt and the semicolon ends theMaple command). Once the Groebner basis package has been loaded, the executioncommand then becomes > gbasis (polynomials, variables) which computes theGroebner basis for the ideal generated by the polynomials with respect to themonomial ordering specified by variables in the executable command. We shouldpoint out that at the time of writing this book, May 2012, version 16 of Maple hasbeen released.


D.3 Gauss Combinatorial Formulation

D.3 Gauss Combinatorial Formulation 893

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Index

A. Bjerhammar’s left inverse, 121Additive rank partitioning, 268, 289Adjoint form, 151Algebraic partitioning, 69Algorithms

Buchberger, 534, 536Euclidean, 534

ANOVA, 240Antisymmetry operator, 152Arithmetic mean, 146, 187, 192, 379Arnoldi method, 135Associativity, 595

Balancing variance, 335Basis, 168Bessel function, 371Best Linear Uniformly Unbiased Estimation,

207Bias matrix, 308, 332, 386Bias norms, 309Bias vector, 308, 332, 340, 386, 387, 392, 394,

395Bias weight matrix, 340, 342, 394, 395BIQUUE, 426Bounded parameters, 3Break points, 142Buchberger algorithm, 592

Canonical base, 136Canonical form, 673, 696Canonical LESS, 134Canonical MINOLESS, 297, 299, 300Canonical observation vector, 132Canonical unknown vector, 136Cartesian coordinates, 117, 642, 675

Cauchy–Green tensor, 128Cayley calculus, 236Cayley inverse, 342, 395Centering matrix, 465, 468Central point, 362Central statistical moment of second order,

189Centralized Lagrangean, 465Choleski decomposition, 617Choleski factorization, 136Circular normal distribution, 363, 374, 375Clifford algebra, 593Columns space, 141Commutativity, 595Complex algebra, 593Composition algebra, 592, 622Condition equation, 194Condition equations, 122Conformal, 380Constrained Lagrangean, 108, 196, 237, 238,

284, 311Constraint Lagrangean, 208Coordinate mapping, 265Coordinates, 152

Cartesian, 557Correlated observations, 185Covariance, 754Covariance components, 231Covariance factor, 227Covariance factorization, 215Covariances, 215Criterion matrix, 118Cross product, 150, 153, 155, 158Cumulative pdf, 679, 738Cumulative probability, 723–725Curved surfaces, 102Cyclic ordering, 152


1011


1012 Index

D. Hilbert, 538Datum defect, 265Datum problem, 495Dedekind, 538Degrees of freedom, 94, 228, 267, 682, 729Diagonal matrix, 467Differentiable manifold, 16, 102, 104Differential Geometry, 102Dilatation, 473, 674, 683Dilatation factor, 650Direct equations, 374Direct map, 641Dispersion matrix, 205, 207, 309, 394, 424,

754Dispersion vector, 217Distances, 557Division algebra, 593, 602, 622Division algorithm, 539

E. Noether, 538EIGEN, 728Eigencolumn, 300Eigenspace, 302Eigenspace analysis, 111, 130Eigenspace synthesis, 130, 142Eigensystems, 135, 301Eigenvalue, 300, 302Eigenvalue Analysis, 728Eigenvalue decomposition, 130, 132, 136Eigenvalues, 558

positive eigenvalues, 113zero eigenvalues, 113

Eigenvectors, 302Elliptic curve, 375Equations

polynomial, 538Error propagation, 470Error propagation law, 210, 402, 403, 407, 758Error vector, 210Error-in-variables, 384Euclidean, 113Euclidean length, 95Euclidean metric forms, 375Euclidean space, 112, 118Euclidian observation space, 642Euler circles, 104Euler–Lagrange strain, 128Euler–Lagrange tensor, 128Expectation operator, 757

Fibering, 95, 104, 263, 268Finite matrices, 602First central moment, 754

First derivatives, 96, 106, 191, 284, 303, 366,413, 450

First moment, 194, 754First moments, 422First order moments, 2051st Grassmann relation, 150Fisher normal distributed, 378Fisher sphere, 361Fisher-von Mises, 362Fixed effects, 187, 307, 317, 326, 329, 333,

334, 339, 384Fourier, 381Fourier–Legendre linear model, 59Fourier–Legendre space, 51Fourier-Legendre series, 381Fourth central moment, 755Fourth order moment, 196Free parameters, 3Frobenius, 311Frobenius error matrix, 462Frobenius matrix, 468Frobenius norm, 116, 311, 332, 387Frobenius norms, 334, 387Frobenius theorem, 593Fuzzy sets and systems, 128

G-inverse, 103Gain matrix, 520Gamma function, 645Gauss elimination, 110, 209, 414, 451, 534,

535, 545Gauss normal distribution, 362Gauss–Jacobi Combinatorial Algorithm, 172,

176, 178Gauss–Markov model, 208Gauss-Lapace normal distribution, 688Gauss-Laplace pdf, 677Gauß trapezoidal, 402General bases, 130General Gauss–Markov model

with mixed effects, 426General inverse Helmert transformation, 673General reciprocal, 602General Relativity, 113Generalized inverse, 103, 105, 129, 194, 315,

317Geodesic, 362Geodesy, 529Grassmann column vector, 166Grassmann coordinates, 141, 150, 152, 154,

158, 159, 166, 169direct, 170inverse, 170, 171

Index 1013

Grassmann manifold, 102Grassmann space, 141, 152, 154, 158, 164,

168, 169Grassmann vector, 159Grassmann–Plucker coordinates, 122Greatest common divisors (gcd), 534Groebner basis, 527, 539, 592

definition, 540Maple computation, 891Mathematica computation, 889

Hankel matrix, 606, 610Hat matrix, 114, 122, 146, 149, 150Helmert equation, 231Helmert matrix, 231, 609

augmented, 678extended, 678quadratic, 678

Helmert random variable, 694Helmert traces, 231Helmert transformation, 679

quadratic , 677rectangular , 677

Helmert’s ansatz, 229, 230Helmert’s distribution, 681Helmert’s polar coordinates, 733Helmert’s random variable, 708Hesse matrix, 114, 204, 403, 474, 759High leverage points, 172Hilbert basis invariants, 128Hilbert Basis Theorem, 538, 539Hilbert space, 108Hodge dualizer, 151Hodge star operator, 141, 150, 587, 640Hodge star symbol, 158Holonomity condition, 120, 122Homogeneous equations, 223Homogeneous inconsistent equations, 411Homogeneous linear prediction, 438Horizontal rank partitioning, 122, 132Hurwitz theorem, 593Hypothesis testing, 339, 394

Ideal, 536, 537Ideal membership, 540Idempotent, 113, 213Implicit Function Theorem, 263Inconsistent linear equation, 506Independent random variables, 640Inhomogeneous inconsistent equations, 411Inhomogeneous linear equations, 100Injective, 265, 267

Injective mapping, 638Injectivity, 94Injectivity defect, 302Inner product, 111, 363Interval calculus, 122Invariant quadratic estimation, 187, 221, 237Inverse equations, 374Inverse Jacobian determinant, 372Inverse map, 102, 641Inverse partitioned matrices, 109Inverse spectrum, 149Inverse transformations, 132IQUUE, 190

Jacobi determinant, 666, 673Jacobi map, 102Jacobi matrix, 69, 102, 114, 403, 641, 700, 759Jacobian, 641Jacobian determinant, 551, 647, 684Join, 150Join and meet, 141

Kalman filter, 493, 517, 519, 520Kernel, 94Khatri–Rao product, 116Killing analysis, 598Kolmogorov–Wiener prediction, 440Krige’s prediction concept, 441Kronecker, 538Kronecker matrix, 116Kronecker–Zehfuss product, 116, 338, 391,

495Kummer, 538KW prediction, 441

Lagrange function, 204Lagrange multiplier, 105, 108, 110, 191, 238,

414Lagrange multipliers, 284, 310, 413, 415, 422,

423Lagrange parameter, 449Lagrange polynomial, 124Lagrangean, 72, 336, 366, 390, 556, 557

multiplier, 536Lanczos method, 136Langevin distribution, 362Langevin sphere, 361Latent condition equations, 142, 168Latent conditions, 164Latent restrictions, 141, 168Laurent polynomials, 599

1014 Index

Leading Monomial (LM), 539Least Common Multiple (LCM), 541Least squares, 95, 180, 185, 240, 563, 602

Historical background, 180Least squares solution, 98, 103, 108, 268, 272,

411, 414Left complementary index, 267Left eigenspace, 133Left Gauss metric, 128Left inverse, 129LESS, 98Leverage point, 157, 162Lexicographic ordering, 536, 545, 887

graded reverse, 888Lie algebra, 592, 598Likelihood normal equations, 204Linear estimation, 332Linear Gauss–Markov model, 421Linear model, 108, 121, 150Linear operator, 95, 265, 268, 757Linear space, 103, 265Loss function, 361, 366LUUE, 190

Maple, 536Mathematica, 536Matlab, 536

“roots” command, 536Matrix algebra, 265, 593, 602Maximal rank, 103Maximum, 536Maximum Likelihood Estimation, 187, 202Maximum norm solution, 113Mean, 198Mean matrix, 754Mean Square Estimation Error, 307, 339, 345,

386, 392–395, 399Measurement errors, 517Median, 187, 198Median absolute deviation, 187Meet, 150, 153Minimum, 536Minimum bias, 310Minimum distance mapping, 101, 102Minimum geodesic distance, 362, 366Minimum norm, 268, 272, 618, 619Minimum variance, 207, 422MINOLESS, 266Mixed models, 240Model manifold, 265Modified Bessel function, 363Monomial ordering, 887Moore–Penrose inverse, 274, 289

Moore-Penrose inverse, 602Multideg, 540Multilinear algebra, 150Multiplicative rank partitioning, 266, 272Multipolynomial resultant, 547, 592Multipolynomial resultants, 529, 549Multivariate Gauss–Markov model, 496Multivariate LESS, 495Multivariate polynomials, 539

Necessary condition, 557Necessary conditions, 96, 106, 109, 191, 204,

208, 238, 270, 284, 304, 312, 336,337, 367, 390, 413, 448, 451

Non-symmetric matrices, 135Non-vanishing eigenvalues, 111Nondegenerate bilinear form, 111Nonlinear error propagation, 758Nonlinear model, 102Nonlinear models, 104, 525Nonlinear prediction, 438Nonlinear problem, 461Normal equation, 367Normal equations, 110, 191, 310, 337, 390,

391, 412, 414–416, 511Normal models, 196Normal space, 101Normalized column vectors, 166Normalized Grassmann vector, 159Null space, 94, 265–267

Objective functions, 113, 114Oblique normal distribution, 374, 375Observation equation, 518Observation point, 102Observation space, 94, 102, 111, 113, 130,

187, 214, 265, 266, 268, 364Observational equations, 511Observed Fischer information, 204Octonian algebra, 593One variance component, 223, 226, 230, 236Optimality conditions, 206Optimization problem, 202Orthogonal, 51Orthogonal column space, 141Orthogonal complement, 101Orthogonal matrix, 606Orthogonal projection , 104Orthogonality condition, 157, 229Orthogonality conditions, 164Orthonormal, 51, 679Orthonormal base vectors, 130, 150

Index 1015

Orthonormal frame of reference, 154Orthonormal matrix, 467, 474, 608Orthonormality, 673Outlier, 188, 198Outliers, 172, 332Overdetermined, 95, 186Overdetermined system of linear equations,

136

Parameter space, 94, 102, 175, 187, 265, 266,268, 364, 450

Partial differential equations, 540Partial inverse, 602Partial redundancies, 164PCA, 728Permutation matrix, 198Permutation operator, 151Plucker coordinate, 153Plucker coordinates, 141, 150, 154, 158, 159,

166, 169Polar coordinates, 363, 642, 675Polar decompositions, 131Polynomial, 528Polynomial coefficients, 94Polynomial equations, 540Polynomial of degree one, 94Polynomials

equivalence, 540ordering

lexicographic, 535roots, 536univariate, 536, 559

Positive definite, 106, 109, 113, 116, 130, 207,208, 296, 308, 312, 314, 332, 338,391, 416, 427, 558, 617

Positive semidefinite, 106, 108, 112, 296, 617Prediction equations, 520Prediction error, 520Prediction stage, 519Principal Component Analysis, 728Probability density function, 363, 638Probability distribution, 194Procrustes Algorithm, 461, 464, 472Projection lines, 556Projection matrices, 213Pseudo-observations, 176Pseudoinvese, 602Pullback operation, 175

Quadratic bias, 342, 396Quadratic estimation, 193Quadratic estimator, 193

Quadratic form, 189Quadratic Lagrangean, 337, 390Quaternion algebra, 593Quotient set, 104

Random, 428Random effect, 329, 339, 384, 386, 392, 399,

426, 438Random regression model, 474Random variable, 519Range, 267Range space, 94, 103, 266, 267Rank factorization, 289, 617Rank identity, 284Rank partitioning, 12, 95, 132, 157, 164, 215,

263, 266, 298Reduced Groebner basis, 537, 890Reduced Lagrangean, 465Redundancy, 147Redundancy matrix, 146, 147, 157, 164Regularization parameter, 307Relative index, 111Residual vector, 192, 316Ricci calculus, 236Ridge estimator, 342, 395Riemann manifold, 380Right complementary index, 267Right eigenspace, 133Right eigenspace analysis, 146Right Gauss metric, 128Right strain component, 128Risk function, 449, 464Robust, 332Robust approximation, 122Root mean square error, 187, 198, 679Rotation, 131, 461, 674Rotation matrix, 684Rotational invariances, 439

Sample mean, 665, 667, 676Sample median, 188, 198Sample variance, 667, 676, 683Scale change, 674Scale factor, 461Schur complement, 615Second central moment, 339, 393Second central moments, 385Second derivative, 106, 109, 271, 304, 367,

413, 450Second moment, 194, 754Second order design, 114, 115Second order moment, 196

1016 Index

Second order moments, 205Selfdual, 155Semidefinite, 111Sequential optimization problem, 312Series inversion, 659Set-theoretical partitioning, 104Similarity transformation, 521, 522Simple linear model, 200Singular value decomposition, 467, 728Skew product symbol, 158Slices, 103Slicing, 95, 263Spatial redundancies, 157Special Gauss–Markov model, 189, 205, 206,

208, 220, 223, 225, 239, 315, 318,322, 326

Special linear Gauss–Markov model, 392with datum defect, 308

Special Relativity, 113Spherical coordinates, 361Standard deviation, 188, 681Standard multivariate model, 497State differential equation, 522State equations, 519Statistical moment of first order, 189Stochastic, 205Stochastic observations, 187, 517Student’s random variable, 699Subdeterminants, 157, 162, 169Sufficiency condition, 96, 106, 109, 192, 284,

413Sufficiency conditions, 204, 208, 304, 312,

337, 390, 391Sufficient condition, 558Sufficient conditions, 451Surjective, 94, 265, 267Surjectivity defect, 94, 228, 302SVD, 728Sylvester resultant, 548Symmetric matrix, 129, 189, 224System equations, 518

Tangent space, 102Taylor series, 399, 653, 655Taylor–Karman matrix, 404Taylor–Karman structure, 118, 402Theory of Turbulence, 118Third central moment, 754Three fundamental partitionings, 95

Total degree, 888Total least squares, 108, 448Trace, 390Transformation matrix, 521Translation, 461, 674Translation vector, 473Translational invariant, 190, 193, 195, 228Trend components, 441Tykhonov–Phillips regularization, 302, 307Tykhonov–Phillips regulator, 342, 395

Unbiased estimation, 677Unconstrained Lagrangean, 96, 106,

271, 464Uniformly unbiased, 191, 193Unity, 595Univariate polynomials, 539Unknown parameters, 115Updating equations, 520

Vandermonde matrix, 606, 611Variance, 215, 231, 754Variance factor, 227Variance factorization, 215Variance-covariance, 215, 240, 403, 754Vector derivative, 106Vector of curtosis, 755Venn diagram, 104Vertical rank partitioning, 95, 98Von Mises, 362Von Mises circle, 361Von Mises distribution, 755

Weak isotropy, 439Weight factor, 342, 395Weight matrix, 115, 199, 472Weighted arithmetic mean, 175Weighted arithmetic means, 178Weighted Frobenius matrix norm, 207Weighted left inverse, 129Weighted Moore–Penrose inverse, 293Witt algebra, 592, 599

Zehfuss product, 116Zero correlation, 385

Documents

Appendix A Tensor Algebra, Linear Algebra, Matrix Algebra