Interrelationships between sensitivity and selectivity measures for spectroscopic analysis

ELSEVIER Chemometrics and Intelligent Laboratory Systems 32 (1996) 135-149

Chemometrics and intelligent laboratory systems

Tutorial

Interrelationships between sensitivity and selectivity measures for spectroscopic analysis

John H. Kalivas a3 *, Patrick M. Lang b a Department of Chemistry, Idaho State Unioersiry, Pocatello, ID 83209, USA

b Department of Mathematics, Idaho State University, Pocatello ID 83209, USA

Received 16 February 1995; accepted 18 May 1995

Abstract

Quantitative analysis based on spectroscopic data often uses the matrix equation R = CK + E, which is the K-matrix form of the Beer-Lambert law. This equation’s utility depends on the character of K’s numbers. The quality of these numbers is often discussed by referring to how strongly components respond to measured wavelengths and the extent one component responds toward measured wavelengths compared to other components, i.e. by referring to the sensitivity and selectivity of the used wavelengths. Over the years, various measures for quantifying sensitivity and selectivity at the local (analyte specific) and global (all analytes simultaneously) levels have been put forth with varied degrees of success. This tutorial introduces and discusses many of the commonly used local and global measures and establishes interrelationships between them. It is hoped that this presentation will give the user a better understanding of their utility and drawbacks, as well as provide a background for future work in the assessment of spectroscopic information.

Keywords: Sensitivity; Selectivity; Spectroscopic analysis

Contents

1. Introduction. ................................................... 136 2. Variance and error considerations ........................................ 137 3. Sensitivity measures ............................................... 140

3.1. Local sensitivity. .............................................. 140 3.2. Global sensitivity .............................................. 142

4. Selectivity measures ............................................... 143 4.1. Local selectivity. .............................................. 143 4.2. Global selectivity .............................................. 145

5. Combined sensitivity and selectivity measures. ................................. 146 5.1. Local sensitivity and selectivity ....................................... 146 5.2. Global sensitivity and selectivity ...................................... 147

6. Discussion. .................................................... 148 References.. .................................................... 149

* Corresponding author.

0169-7439/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0169-7439(95)00051-8

136 J.H. Kaliuas, P.M. Lang/Chemometrics and Intelligent Laboratory Systems 32 (1996) 135-149

1. Introduction

In spectroscopic applications involving the Beer-Lambert law researchers have found that there is a connection between spectral sensitivity and selectivity and concentration prediction performance, where, sensitivity refers to how strongly the components under consideration respond to specific measured wavelengths and selectivity refers to the extent one component responds toward measured wavelengths compared to other components, i.e. the amount of spectral overlap. In particular when sensitivity and selectivity are good, i.e. when responses are large and spectral overlap is minimal, concentration predictions tend to be reasonable. Similarly, when sensitivity and selectivity are bad, i.e. responses are small and spectral overlap is considerable, concentration predictions are typically poor. In this tutorial these connections are investigated with regards to the K-matrix form of Beer’s law.

The K-matrix form of Beer’s law is R = CK + E, where R represents a m X p matrix of spectral calibration responses for m calibration samples measured at p wavelengths, C denotes a m X n matrix of calibration sample concentrations for n analytes, K signifies a n Xp matrix of calibration coefficients, and E represents a m X p matrix of errors. In practice, K is either known (from knowledge of pure- component spectra) or is approximated by ‘solving’ the above equation for K, e.g. by using the method of least squares. Let k denote either K or its approximation. In all that follows, fi is assumed to be the result of a good calibration design and, as a consequence, has rank n.

Sensitivity and selectivity are independent concepts intimately related to the numbers making up K. By way of illustration, consider the following matrices, which could represent calibration matrices for the same three analytes determined at different wavelengths selected from pure-component spectra (as- sume constant noise for all wavelengths):

0.02 0 0

10 0 0 cc\ = 0 10 0 , and

i I 0 0 10

(

0.02 0.01 0.01 Iz1, = 0.01 0.02 0.01 .

0.01 0.01 0.02 1

The mathematical orthogonality of the columns of $; indicates that the wavelengths used to produce K’, exhibit good component-wise selectivity. However, the smallness of numbers making up ki also indicates that the components respond poorly to these wavelengths, i.e. the wavelengths exhibit poor overall sensitivity. By way of a contrast, the relative largeness of the numbers in @ indicates that the components have good overall sensitivity to the wavelengths being used, but the near mathematical dependence of the columns making up fci indicates that the wavelengths have poor global selectivity. The matrix Ki has both good overall wavelength sensitivity and selectivity, while the matrix e(1, has bad global wavelength sensitivity and selectivity.

These example rank 3 matrices are particularly nice in the sense that qualitatively, their sensitivities and selectivities are relatively easy to assess. Unfor- tunately, most fir’s are not like these four matrices, but instead show signs of both good and bad sensitivity and selectivity, e.g.

For matrices like 8:, it is hard to qualitatively assess overall wavelength sensitivity and selectivity.

Two approaches can be taken to quantitatively measure sensitivity or s,electivity. One approach assesses sensitivity or selectivity locally (analyte specific). The other assesses at the global level (all analytes simultaneously). Over the years various researchers have tried to quantify sensitivity and selectivity at the local and global levels through the use of various quantifiers [l-11]. For example, the absolute value of the ‘determinant’ of & has been put forth as a global measure of sensitivity and the condition number of K has been put forth as a measure of global selectivity in &. As will be shown in this tutorial, these two measures and others sometimes adequately quantify sensitivity and selectivity, but at other times

J.H. Kalioas, P.M. Lang/ Chemometrics and Intelligent Laboratory Systems 32 (1996) 135-149 137

they fail miserably. A study of why they sometimes fail to adequately quantify and assess sensitivity and selectivity has lead to the considerations put forth in this work.

The numbers making up & not only play a role in sensitivity or selectivity quantifications, but also in quantifications of error and variance. Thus, sensitivity, selectivity, error, and variance quantifications are related. For purposes of completeness, the next section of this tutorial puts forth error and variance considerations associated with the use of e in prediction. These developments will then be used periodi- cally throughout the rest of this work to illustrate specific relationships between discussed sensitivity and selectivity measures and error and variance.

In the sections following the error and variance discussion, separate and combined local and global sensitivity and selectivity measures are introduced. Following their introduction, relationships and problems associated with them are explored and presented.

Before proceeding, it should be noted that much of the material that follows relies heavily on linear algebraic concepts, in particular, singular value decompositions. This reliance is necessarily dictated by the nature of the subject at hand. Briefly, a singular value decomposition of e’ can be written as U2 V’, where U = (u, u2 . . . u,)is pXp, V=(v, vz...vn) is n X n, and 2 = (gij) is a p X n matrix whose en- tries are all zero except for uii, i = 1, 2, . . . , II. The values aii are the singular values of &’ and are usu- ally denoted by g;. The values are the square roots of non-zero eigenvalues of &‘e and are ordered, by convention, so that they satisfy (T, 2 u2 2 . . . 2 cn. The columns u, and vi of U and V are orthonormal eigenvectors of ft’fc and 6&I, respectively. For a more detailed discussion of singular value decompositions and their use in analytical chemistry see [lo].

2. Variance and error considerations

Once k has been acquired, the equation r = kcilc + e is used to obtain an approximation E of c, where c denotes the unknown n X 1 vector of component concentrations present in an analysis sample, r represents the corresponding p X 1 vector of spectral responses for the analysis sample, and e is a p X 1 er-

ror vector. In what follows, the elements of e are assumed to obey uncorrelated normal distributions with mean zero and constant variance a’, i.e. E(e) = 0 and V(e) = (+*I,, where E and V denote the standard multivariate expectation and variance-covari- ante operators and I, denotes the p Xp identity matrix. This assumption of error being normally dis- tributed implies that the method of least squares is appropriate for solving r = kfc + e. When this method is applied, the approximation of c is found to be i? = (k’)+r, where (K’)+ is the Moore-Penrose generalized inverse of k’. Since it has been assumed that rank 2’ is n, it follows that an alternate representation for &I>+ is <i2;il’>- ‘ii.

Having obtained 2 by the method of least squares, the questions of, ‘How well does e approximate c?’ and ‘How do sensitivity, selectivity, and noise effect the approximation?‘, need to be answered. One natural theoretical means of assisting in answering the first question is to determine the absolute error in the approximation, i.e. 116 - c]12, where I( . iI2 indicates the Euclidean norm. In general this number is diffi- cult, if not impossible, to determine. Consequently, estimates of its value are used in practice. One popu- lar estimate of (It - c(12, is the expectation E(lliZ - ~11:). This number represents the mean square error of C and is often denoted by MSE(i2). In essence, this value is the mean of ]]e - cl];. To further assist in answering the first question, the variance-covariance matrix V(6) is also computed. In the next few paragraphs, various ways for computing MSE(@) and V(e) are explicitly exhibited.

In [12] it is shown that

MSE(C) = ~*tr((‘cic~)-~), where tr denotes the standard matrix trace function. Alternatively, since the trace of a square matrix is just the sum of its eigenvalues, it follows that

where ol*, * a*, . . . . u,,* denote both the eigenvalues of fifir’ and the squares of the singular values of &‘. As

i: -$ = II(B’) i=l “i


[12], i.e. the sum of the eigenvalues of (s’>- ’ is equal to the Frobenious norm (defined in the next section) of the Moore-Penrose generalized inverse of ii;‘, it follows that

MSE(C) = a211(iZ;f) + 11;

is also true. The variance-covariance matrix V(e) can be ex-

pressed by

v(e) = a2(tit)-1

[10,12]. Using a singular value decomposition of &(‘, it is straightforward to show that another representation for the variance-covariance matrix of 8 is given by

i=l wi

If Zj denotes the jth component of C, j = 1, 2, . . . , n, and if uji, denotes the jth element in vi, then the last representation of V(t) implies

i=l “i

Since the matrix V appearing in the singular value decomposition of &’ is orthogonal, it follows that C;= 1u,: = 1, and hence,

v( tj) I uz/C$.

The equality l/an2 = ]I<~‘>+]]~, derived in [121, yields

V(Zj) I cr211(8t)+ll:. It should be noted that had the numbers making up

Table 1 Comparison of Iz’ sensitivity measures

ltt GSEN(k’) De&‘)

&’ been scaled so that c(c(’ is a correlation matrix, then the expression Cl=, u,:/ui2 would be the variance inflation factor for the jth component (VZFj) [13]. Since the numbers in 1T(’ have not been modi- fied, this expression is termed the unscaled variance inflation factor for the jth component. It is denoted and computed by

uvq = ig, -$ 1

Using the facts that the trace of a square matrix is the sum of its diagonal elements and that V<2j> is the jth diagonal element of V(2), it follows that another representation for the mean square error is

MSE(C) = u2 k UVIF;

and for

V(tj) = u2UVZFj.

In summary the following relationships hold for MSE(C):

MSE(C) = c~~tr((i(i(‘)-‘) = u2jcl -$ I

nu2 = g211(&)+ll; = u2 i UVZFj I -

j=l a,2 *

(1) For V(e), the following holds:

LSENi a UVIFi a Xl/q’ = IKi<‘)+ 11’ F l/s = II(P’ a,

it; 0.020 O.OOOOO8 0.020 2500 7500 50.00 0.020

i<: 28.00 28.00 16.19 0.667 2.00 1.00 1.00

ik; 10.00 1000 10.00 0.010 0.030 0.100 10.00

Pr, 0.040 0.000004 0.024 6.88 20 625 100.0 0.010

k; 2.10 1.70 1.42 0.899 2.70 1.11 0.90

a Value is the same for all three components.

J.H. Kalivas, P.M. Lang/ Chemometrics and Intelligent Laboratory Systems 32 (1996) 135-149 139

0 5 10 15 20 25 30 35 40 45 50 Wnnlvlovl

0.0 -

0.8-

b _

0.7 -

0.6 -

lO.5

0.4 -

0.3-

0.2 -

0.1 -

0 0 5 10 15 20 25 3-Y 35 40 45 50

WllwkT@h

1

0.9 -

0.8 - 1

0.7 -

~

\

\

c

3

0.6 -

10.5. 0.4 - i

\\\

01 ’ ” 0 5 10 15 20 W6ZQlh 30 35 40 45 50

Fig. 1. Rows of K using simulated Gaussian pure-component spectra for three components (1, 2 and 3): (a) Case la; (b) Case lb; (c) Case lc.

with the variance for the jth component satisfying

v( 2j) = a2(KKf),;1 = a*lJVz~

and finally,

These relatiofships show that a,, the smallest singular value of K, plays a key role in the upper bound- ing of Z&X(@, V(tj), and UVZFj. The importance of this is described in Section 5.

As M%(Z), V(e), and vlFj values are dependent upon values making up K, and as sensitivity a?d selectivity are dependent upon values making up K, it follows that the former values should be related to the latter concepts. This observation provides the im- petus for trying to assess sensitivity and selectivity by an ‘a priori’ determination of sensitivity and selectivity effects on variance and error for a specific chemi- cal analysis.

For example, since l/o,,* 5 CT= r1/oi2 I n/cr,‘, it follows that the values Cl= rl/o;* and l/a,,* used by M%(B) in Eq. (1) and the upper estimate of V(Ej) and UVIFj, used in Eqs. (2) and (3) respectively, are equivalent, i.e. they are both small or large together. Listed in Table 1 are C~=rl/~~* = ll(K’>‘ll~ and l/a;, = Il(Kt)‘l12 for the matrices K:, Ki, . . . , k\

A ” IO 15 20 WA 30 35 40 45 50

Fig. 2. Rows of K for Case 2 using simulated Gaussian pure-component spectra for three components (1, 2 and 3).


OO ,\ i

5 10 15 20 2.3 30 35 40

Wanknpth No.

Fig. 3. Rows of K for RNA nucleotides over the wavelength range 220 to 290 nm: A = adenylic, C = cytidylic, G = guanylic, and U = uridylic [20].

introduced earlier. As this table shows, these measures clearly identify &i as the one that should (theoretically) lead to small errors and variances, and hence, the matrix with good sensitivity and selectivity.

Shown in Figs. 1 and 2 are simulated pure-component spectra at unit concentration. Table 2 lists the values for these situations. Notice the effect of sensitivity in going from Fig. la to Fig. lb, and the effect of selectivity in going from Fig. la to Fig. lc. In Figs.

3 and 4 are displayed the spectra of four RNA con- stituents and three infrared spectra, respectively. Table 2 lists the corresponding values. Notice how the WZFj values describe the combined effects of sensitivity and selectivity for each analyte.

3. Sensitivity measures

3.1. Local sensitiuity

Earlier in this tutorial sensitivity was said to indi- cate how strongly an analyte responds to a given wavelength. Since the sji element of &’ indicates how strongly the ith component responds to the jth wavelength, it makes sense to sum the values in the ith column in order to obtain a local measure of sensitivity for the ith analyte. Before doing this it should be noted that, in general, the numbers Lji are non- negative. However, noise and instrument error may make some values negative. As a consequence, it may be possible for a sum of the kji values to have value zero, even though none of the iji values were zero. To g$ around this potential problem, the sum C,P_ 1 1 kjil is put forth as a prfliminary local sensitivity measure instead of C& Ikji.

Table 2 Comparison of sensitivity measures for systems shown in Figs. l-4

System GSEN(K’) De&‘) LSE4 a lJVIFi a Cl/q* = IKii’)+ll* F l/r” = IKCi')+llz Fig. la 2.73 15.40 2.50 0.163 0.491 0.444

2.50 0.166 2.50 0.163

Fig. lb 2.50 0.154 0.250 16.26 32.69 4.07 2.50 0.166 0.250 16.26

Fig. lc 4.28 0.171 2.50 52.93 305.5 17.42 2.50 199.6 2.50 52.93

Fig. 2 2.84 1.50 2.50 4.28 21.77 4.63 1.25 17.31 2.50 0.171

Fig. 3 84.63 4.40 x lo4 48.68 0.071 0.209 0.439 47.64 0.003 46.44 0.012 35.06 0.123

Fig. 4 6.72 13.65 5.06 0.959 2.34 1.49 4.38 1.27 3.15 0.108

a For components 1,2 and 3, respectively. For Fig 3, the order is A, C, G and U. For Fig. 4, the order is 1-butanol, 1-pentanol and l-hexanone.

J.H. Kalivas, P.M. Lang/Chemometrics andlntelligent Laboratory Systems 32 (1996) 135-149 141

0.9-

0.6 -

0.7-

0.6-

EOS- iz

0.4-

0.3-

0.2-

0.1 -

‘7

1

*

b

&6-

0.4.

0.3-

0.2.

1OW 15w 2Ow 2m 2000 Wwhcm(an")

u 4oa

,

Fig. 4. Rows of K for normalized infrared spectra: (a) 1-butanol (b) 1-pentanol (c) 1-hexanone.

If k is a general p-vector whose components are indicated by kj, then a measure of how large k is provided by a norm. Three commonly used norms are

the l-norm, 2-norm, and m-norm which are computed as follows.

llkh = ; IkJ,

IIkII2 = i Ikj12 i I

l/2

7 j=l

i With this in mind, it is seen that the preliminary measure introduced above is nothing more than the l-norm of the ith column of R’. A basic fact from linear algebra is that all finite dimensional vector space norms are equivalent. This means that if k is deemed large by use of one norm, it will also be deemed large by use of any other norm. A similar statement holds for k being deemed small. A direct implication of this is that the preliminary local sensitivity measure is equivalent to any other norm mea- surement of the ith column. As the 2-norm is the most commonly used norm for measuring p-vectors in chemistry, the number

LSEN, = ll”ki112

is set forth as a local sensifiuity measure for the ith component, where ki indicates the ith column of R*. It should be noted that, as defined, this measure is the same as the local sensitivity measure set forth in [3]. The larger the value LSENi has, the greater the sensitivity. Tables 1 and 2 list LSEl( values for the matrices R1 through R, and for the spectra plotted in Figs. l-4.

In [7], the number

1

SEN, = I(R+cil12

was recently introduced as a measure of sensitivity for the ith analyte. In this expression, R+ denotes the Moore-Penrose generalized inverse of R, and ci denotes the ith column of C. When m = n and pure- component spectra are used to form R, then this expression can be computed by

1 SEN, =

Il(icl),:,,,ll2 ’

where <k’>&, represents the ith row of <R’>+. Ref.

142 J.H. Kalivas, P.M. Lang/Chemometrics and Intelligent Laboratory Systems 32 (1996) 135-149

[lo] shows Il(K*)if_row]l~ = (UV~~i)1’2. (This relationship actually holds any time K’ has rank n. The requirement m = n is not necessary.) Thus, EN, can, in this special case, be determined by

1 SENi =

(uwy2 *

It should be noted that the parenthetical statement above, implies this last equation could be used as a definition for local sensitivity anytime the rank of K’ is equal to IZ. However, in Section 4 of this work, the number uvIIi will be shown to be a function of both sensitivity and selectivity. Consequently, SENi cannot be a strict measure of local sensitivity in the special or more general situations.

3.2. Global sensitivity

Just as the absolute value of the elements making up a column of K’ can be summed to form a measure of local sensitivity, so can a sum CT= 1 C: _ 1 I iji I of all lkjil values be used to form a global sensitivity measure. This sum is a finite-dimensional norm of K’ [12]. As all finite-dimensional norms are equivalent, it follows that any other norm of Kr could be used as a global sensistivity measure. Four norms see com- mon usage. They are the l-, 2-, and m-matrix norms, and the Frobenius norm. With regard to Kf, they are computed, respectively, as follows:

Il~‘ll~ = mx 5 Ikji19 i j=l

Ilf(‘ll2 = (+I,

lliifllm = max k Iljil, j i=l

where ur, m2, . . . . a, are the singular values of K’. Of these equivalent norms the ‘easiest’ to work with is the 2-matrix norm. For this reason, global sensitivity of K’ is denoted and defined by

GSEN(iz’) = ulr

where by way of a reminder, u1 denotes the largest singular value of Kf. Clearly, the larger g1 is, the better the global sensitivity is in K’ and, by the same reasoning, the closer to zero or is in value, the poorer the global sensitivity is in K’.

Tables 1 and 2 display how this measure per- forms on the matrices Ki through 8: and the spectra plotted in Figs. 1-4, respectively. As Table 1 indicates, GSEN identifies Ki and K(: as having better global sensitivity than Kg and matrix Ki as having better global sensitivity than Ki and bi. Clearly the global sensitivity in K(I and K$ is bad.

In Table 2 in can be seen that the value for GSEN for Fig. la is approximately half that of GSEN for Fig. lc. Since the only difference between these two figures is the amount of spectral overlap, it would seem to follow that GSEN is not a global measure of sensitivity. This thinking is flawed, because it fails to recognize the difference between global and local measures. Global measures quantify results that are dependent upon all local input, whereas local measures quantify individual inputs, independent of other inputs. Fig. la and Fig. lc are made up of spectra having the same local sensitivity measures. When the spectra for Fig. la are combined they produce a global spectrum whose maximum height is signifi- cantly smaller than that produced by the combination of spectra in Fig. lc. This fact is picked up by the values of GSEN. When GSEN is large, it is not saying that all analytes are responding well to the wavelengths, instead it is saying that the resultant summed spectrum has a large peak. This peak could be the result of one good local response or the result of many local responses concentrated in one region.

With the comments of the last paragraph in mind, one might conjecture that a better measure of global sensitivity would be one that uses some combination of local sensitivities, for example a sum or an average. This thinking is good, but unfortunately incor- rect, because when such a measure is produced, it will end up being equivalent to GSEN. To see an example of this phenomena, suppose the average of the squares of the local sensitivities was put forth as a measure of global sensitivity. Then since

J.H. Kalivas, P.M. Lang/Chemomenics and Intelligent Laboratory Systems 32 (1996) 135-149 143

it follows that this measure is equivalent to GSEN because

Thus, both of these measures are large and small together, implying that they both identify the same type of behavior at the same time.

The following should be noted. Fix a > 0 and let b be such that a 2 b r 0. Set

Then GS_JZN(&s) = a. This implies that a whole fam- ily of matrices indexed by b have the same global sensitivity measure. This is a problem in that intu- itively

ii;=,” ,“. ( 1 has better global sensitivity than say

*‘I K 0

a/2 = ( 1 ; a/2 *

This problem is intrinsic to all global sensitivity measures. For example, if the sum of all components making up &’ is set forth as a global sensitivity measure, then

(::: K)and( lb99 OP,l)

would have the same global sensitivity measure. By way of another example, suppose a global sensitivity measure, call it M, was made up of a linear combination of local sensitivity measures, say mi, i.e. M = Cairni, where CY~ are non-negative weights. Then for the given oi, there exist an infinite number of combinations of values for mi that yield the same sum M. Similar statements can be made about products of local measures. These observations help to explain why global sensitivity measures cannot be expected to identify local behavior in most cases.

Investigators have often used De&‘) = u1u2 . . . gn, the absolute determinant of d’, as a global sensitivity measure. Unfortunately, this measure suffers from the same problems alluded to in the last paragraph as well as others. To see that this is the

case, let a, b and c be numbers such that a and c are positive and b is non-negative. Set

For this matrix, De&‘> = Idet(8’) I= UC, a number independent of 6. This implies that it is possible for many different &“s to have the same Det value. For example, let a = b = 100 and c = 0.0001. In this case, De&‘) = 0.01. Should a = 0.1, b = 0, and c = 0.1, then De&‘) = 0.01 as well. Thus, different values for k’ can produce the same Det value. Note that in this example, the first &’ has large sensitivity values, while the other has only small sensitivity values. Yet, in both cases the value of Det is the same. This shows the Det is incapable of accurately assessing global sensitivity. For this example, GSEN has the values 141 and 0.1, for the former and latter k”s, showing that it does, in this case, accurately assess the global sensitivity situation.

In summary, for GSEN, or any equivalent global measure, a small value indicates that the ‘sum’ of the magnitude of all analyte responses to a given wavelength set is small, i.e. individual analyte responses, across the board are small indicating a bad sensitivity situation. A large value indicates the sum of responses to a given wavelength set is large. This is good in the sense that it indicates the wavelength set causes ‘useful’ responses, however, it cannot tell if the large value is due to pn good local responses of just 1 good local response and pn - 1 poor local responses. Thus, the measure GSEN can be used to produce local information in certain situations, namely when its value is near zero.

4. Selectivity measures

4.1. Local selectivity

Numerous measures have been put forth to quanti- tate analyte specific selectivity for a given iz;’ [3-111. Early on, Morgan [9], and later, Lorber 171, looked to measure local selectivity by determining how an analyte’s spectrum in k’ differed from the others making up K’. They accomplished this by finding the sine of the angle between the column in ef associated


/ .‘& / / /

Fig. 5. Geometrical representation of LSEL,.

with the specified analyte and the space spanned by the remaining columns. The details follow.

Let ki denote the ith column of K’ and let f: denote the matrix obtained by removing ki from K’. Set

P=@(Q)+.

Then P is the orthogonal projection of RP onto the range, R(K:), of K:, i.e. the column space of K:. The matrix P can also be obtained by P = U,Uf, where Ui is obtained from the singular value decomposition of K(:: Ui &Vi’. The vectors Pk, *and (I - P)k, represent, respectively, that part of k, in R&i) and that part orthogonal to R&i). Fig. 5 illustrates the situation. From the geometry displayed in this figure, it is clear that the angle 8, between ki and R&i) can be determined from either of the following two expressions:

lIpkill cm&= -

sin 8, _ I!(’ - ‘J’ill2

Ikill ’ - Ikill ’

The values of these expressions are related by the fundamental trigonometric identity cos20 + sin28 = 1. Elementary linear algebraic considerations show that alternate expressions for the above cosine and sine values are

k:pk,

cos @ = I~,l1211Pkil12

sin ei = &I - P)k,

l~il1211(1 - p)^kil12 ’

Furthermore, considerations in [4] imply that 1 1

sin e, = Il(fcr)i~~~~l121rk~l12 = II(B’)i~,,ll2~SEN, *

Lorber and Morgan set forth LSELi = sin lIi

as a local selectivity measure for the ith component. Clearly, LSEL, satisfies 0 I LSEL, I 1. The closer in value LSELi is to zero, the worse the selectivity and the closer in value LSEL, is to one, the better the selectivity, with perfect local selectivity occuring when LSEL, = 1.

Using the result Il(Kt)i+-row)l~ = (WIFi)1’2 and the definition of local sensitivity, it follows after simple algebraic manipulations that

1 UV.IFi =

( LSELJ’( LSEIVJ’ *

This equation shows that uvIF,‘s are functions of both local sensitivity and selectivity as noted earlier in Section 3 of this tutorial. Additionally, it implies that the product (LSEJV~,XUVIF,)‘/~ could be a measure of local selectivity. In this case, the values would range from one to infinity with better selectivity oc- curring for values near 1 and worse selectivity occur- ring with large values. Work in [3] defines local selectivity to be the reciprocal of this value, which is just LSEL,. Finally, it should be noted that when local selectivity is perfect, i.e. LSEL, = 1, the number UVIF, is a measure of local sensitivity.

Another equivalent local selectivity measure which also uses angles is described next. It is based on considerations presented in [14], where similari- ties between two data sets were quantified by use of singular value decompositions. Let ni denote ^ki normalized to unit Euclidean length, i.e. ni = kJLSEyi, and Ui Xiv; be a singular value decomposition of Ki. The angle $i between ni and Ui is defined implicitly by cos qi = yi, where yi is the largest singular value associated with n:U,. The value of Jli determines this local selectivity measure. It should be noted that elementary linear algebraic considerations can be used to show that the value of cos I,!+ is the same as cos Bi described earlier in this section.

Tables 3 and 4 list LSEL, values computed for matrices Ki through K: and for the data associated with Figs. 1-4, respectively.

In [15], the values of the correlation matrix associated with fc’ are shown to be the cosines of the angles between respective columns and in references [16,17] the diagonal elements from the inverse of this correlation matrix are shown to be the VZFi values noted in Section 3. This implies that WFi values can


Table 3 Comparison of ic’ selectivity measures

k’ GSEL(k’) K&‘) LSEL, a

k; 1.00 1.00 1.00

k; 0.007 28.00 0.076

i(; 1.00 1.00 1.00

kt, 0.272 4.00 0.492

Iz; 0.597 2.33 0.744

a Value is the same for all three components.

be thought of as local selectivity measures. This is in stark contrast with WIFi values which are functions of both sensitivity and selectivity. The reason for the difference is that the normalization process required to create the correlation matrix, effectively elimi- nates sensitivity effects.

Ref. [4] has also introduced a local selectivity measure. This measure can be shown to be a function of UVIF, values and, hence, is not a selectivity measure, but instead a combination sensitivity/ selectivity measure. More will be said about such measures in Section 5.

Table 4 Comparison of selectivity measures for systems shown in Figs. l-4

System GSEL&‘)

Fig. la 0.982

Fig. lb 0.982

Fig. lc 0.011

Fig. 2 0.190

Fig. 3 0.012

Fig. 4 0.196

K@) LSEL:

1.21 0.991 0.982 0.991

10.18 0.991 0.982 0.991

74.56 0.055 0.028 0.055

13.15 0.193 0.192 0.966

37.15 0.077 0.371 0.194 0.081

9.99 0.202 0.202 0.968

’ For 1, 2 components and 3, respectively. For Fig. 3, the order is A, C, G and U. For Fig. 4, the order is I-butanol, 1-pentanol and 1-hexanone.

Variance decomposition proportions, discussed in Ref. 11, are measures related to selectivity considerations. They identify the columns in k’ responsible for an analyte’s poor selectivity. With regard to measures developed in this section, another way to obtain this sort of information would be to first com- pute LSEL, and then, using dot products 1121, find the angles between ^ki and the remaining columns in f(’ to determine which columns of k(t are responsible for a less than desirable LSEL, value.

4.2. Global selectivity

Global selectivity measures are sensitivity free measures that look to assess the general spectral overlap state of a given 2’. Mathematically, these measures assess how close the column vectors are to being an orthogonal set of vectors or, in slightly different words, how much multicollinearity is present.

It should be noted that global selectivity measures, like global sensitivity measures, cannot be expected to provide local information, even if the global measure is a linear combination of local measures. Additionally, a variety of situations can produce the same value. Thus, global measures cannot be expected, except in a few special cases, to completely classify selectivity for a given Iz’.

In Ref. [12], the following global selectivity measure is developed. It is effective in that it is entirely free of sensitivity effects and accurately identifies highly multicollinear and orthogonal situations. It is defined as follows:

By way of reminder, the values oi are the singular values of A’,

If GSEL(K’) is 1 or close to 1 in value, then the columns of kr are orthogonal or are very close to being an orthogonal set of vectors. In this situation, fc’ is said to have good global selectivity. If GSEL(k;‘) is close to 0 in value, then the columns of k’ are nearly a dependent set of vectors. In this case, kr is said to have poor global selectivity. It should be noted that GSEL(K’) is the absolute determinant of the pure-component spectra after they have been normalized to unit Euclidean length.

146 J.H. Kaliuas, P.M. Lang/ Chemometrics and Intelligent Laboratory Systems 32 (1996) 135-149

A value of GSEL near 1 indicates that locally there is good selectivity for each analyte under consideration. However, a value of GSEL near zero does not say that there is bad selectivity for each analyte under consideration. Instead, it indicates that a large amount of spectral overlap exists between some of the components under consideration. This implies that GSEL can be used to make statements about local selectivity behavior only when its value is near one. Tables 3 and 4 list values of GSEL for the indicated k”s and data sets.

Another equivalent global selectivity measure can be obtained by averaging the values associated with any one of the previously described local selectivity measures. Such a process is clearly more involved than that associated with GSEL, but the resultant number will assess the overall situation as well as GSEL. Because of its greater complexity, this type of measure will not be mentioned any further.

The condition number, K.@‘) = cr,/o,, of &* has been used as a global measure of selectivity. Un- fortunately, it does not work as it suffers from a problem that is analogous to a problem De&‘) suf- fered when used as a global assessor of sensitivity. Namely, a &’ can have good global selectivity that is not detected by the value of K.@?. This problem results from the fact that condition number is a function of both selectivity and sensitivity.

For example, let a and b be numbers such that a and b are both positive and satisfy u 2 b. Set

The condition number for this et is a/b. Note that e”s selectivity is perfect, yet the value of I&‘) can be anything between one and infinity. For example, when a = b = 1, the condition number has value 1, whereas when a = 100 and b = 1, the condition number is 100. In these two cases, the actual selectivities are the same, yet they have radically different condition numbers. This is a consequence of the different sensitivities. This observation might lead one to conjecture that condition number might be a sensitivity measure. This is not the case. To see this, suppose a = 100 and b = 10. Then the condition number of &’ is 10. Should a = 0.1 and b = 0.01, then the condition number is still 10. Thus, it is seen that radically different sensitivities have lead to the

same condition number, implying that K&~) cannot be a global sensitivity measure either. In general, spectral situations are combinations of various levels of selectivity and sensitivity. For such situations, the correspondence between spectral state and condition number is irregular at best.

Even though De&‘) and K@‘) are not recom- mended for use as measures of global sensitivity and selectivity, the following interrelationship between them, GSEN and GSEL has been established [12].

1 Det( iz’)

K@)” 5 GSEN( iz’) n I GSEL(iZ’) I 1.

Note that the relationship between the condition number and absolute determinant is reciprocal and polynomial in nature.

5. Combined sensitivity and selectivity measures

The measures LSENi, GSENi, LSEL,, and GSEL adequately quantify sensitivity and selectivity at the local and global levels. The next two sections look at quantifiers that are related to combinations of them.

5.1. Local sensitivity and selectivity

In [3], an effort was made to develop a relationship between the variance of concentration predictions, selectivity, sensitivity, and noise. Using the no tation of this tutorial, the following expression re- sulted:

v( t;) = f9(WZFi) = (+2 i

1

( LSE&)2( LSELJ~ I .

This expression provides a alternate way to view a2(UVZFi). Namely, it can be thought of as a resolvability index for the ith analyte. Since MSE(C) = u 2Cyz lUVZFi, it follows that this sum could serve as a global resolvability index. Although these numbers provide a wealth of information that can be used in variety of ways (any review of the statistical litera- ture will attest to this claim), they only include calibration information and not information that describes the analysis sample. Ref. [lo] shows that UVZFi values do not correlate with actual prediction


errors when an analyte’s spectrum composing part of the prediction sample differs from the pure-component spectrum generating K*. This implies that the utility of V(2,) as a reliable resolvability index is limited.

The comments above imply that a reliable resolvability index should include information about the prediction sample. In [7] and [Ml, Lorber introduces a measure referred to as net analyte signal. It can be defined by

r.+ = (Kf):_,,,r Ei

I II(Izt):-,,112 = w-eY2 ’

where r denotes the prediction sample spectrum and Ej is the concentration prediction of analyte i. Ref. [lo] shows that r-ret does correlate with prediction errors when the analyte concentrations vary from sample to sample. If the net analyte signal is ratioed to the standard deviation, (+, a signal to noise measure results. This measure could be used as a general resolvability index. Being that it incorporates prediction sample information, it should perform better than V(&).

5.2. Global sensitiuity and selectivity

In Section 2, it was noted that MsE(i?), V(e), and UVIFi are upper bounded by l/o,‘, This implies that a,,, the smallest singular value of K’, should be able to play a role as a global sensitivity and selectivity measure.

The number a, quantifies the distance 2’ is from rank deficient p X n matrices. When a;l is small, ft’ is near rank deficiency and when a;, is large, k’ is far removed from rank deficiency.

Consider

To 2 decimal places, these matrices have an = o2 = 0.05. They show that a, can be small because of poor selectivity, as in the case of @; because of poor sensitivity, as in the case of g(:; or because of large differences in sensitivities, as in the case of.&;. In general, various combinations of poor selectivity,

poor sensitivity, and sensitivity differences lead to situations where an is small.

The examples given in the last paragraph imply that a;, cannot in general be a separate measure of global selectivity or sensitivity. However, in certain situations, it can provide quantitative local sensitivity information. To see that this is the case, suppose GsEL(~;‘) = 1, i.e. &’ exhibits perfect global selectivity. Then the columns of k’ form an orthogonal set of vectors, which implies that &&’ is a IZ X n diagonal matrix whose diagonal elements are the squared Euclidean lengths of the columns making up K’. Since the diagonal elements of a diagonal matrix are its $genvalues, it follows that the diagonal elements of KK’ are its eigenvalues, i.e. they are of, 2 a,, ..-, un2. Thus, in this case, the singular values ut, u2, . ..) an represent the Euclidean lengths of each column in &’ and as a consequence, are quantitative measures of local sensitivity. (Note that ut does not necessarily correspond to the first column of &‘, but it does correspond to one of the columns of g’. Sim- ilar statements hold for the other ui.) When GSEL(k’) is near one in value, an analogous argu- ment shows that uI, a,, . . . , an provide good ap- proximations to the lengths of the columns making up &I, and hence, reasonable quantitative local measures of sensitivity. In particular, an corresponds to the column of the weakest responding component. Thus, for this situation, a- provides a measure of local sensitivity which implies that on should be large if local sensitivity is to be good. As an aside it should be noted that when GSEL is near one in value, then K~(IZ’) = ut/u, measures local sensistivity differences. That is to say, it expresses the order of magnitude difference between the response of the best and worst responding components. Similar statements can be made about the ratios at/u,, i = 2, 3, . . . , n - 1, which are often referred to as condition indices of &I.

If Z’ is used to denote the squared Euclidean length of the ith column of &I, then the fact that C:= 1 ui2 = Cl= 1 1; implies

1 n u,2r- CZ’IU:.

This string of inequalities shows that a,,’ provides a lower bound for the average of the squared lengths of the columns of k’ and for global sensitivity. Thus,

148 J.H. Kalivas, P.M. Lang/Chemometrics and Intelligent Laboratory Systems 32 (1996) 135-149

even for the cases when GSEL(Ki’) is not close to one in value, a, can provide a ‘feel’ for local sensitivity in K’. The above inequality also implies that global sensitivity is ‘good’ when a,, is large.

Statistics provides three design criteria that one might believe to be useful in determining sensitivity and selectivity in K-matrix analysis [19]. They are (1) A-op$mal design, which looks to minimize tr((KK’)- ‘> = Cl= ll/a,2, (2) D-optimal design, which looks to minimize D&(&K’>- ‘) = l/(~f~t . . . g * and (3) E-optimal design, which looks to minim&e the maximum eigenvalue of <KK’>- ‘, which is l/u,,*. Since

Il(iz’) + 11; = -$ 5 igI -$ = Il(cif) + 11°F 5 -$ I

= A(@ + II:, it follows that A and E optimal designs are equivalent. Moreover, minimizing l/a,* is equivalent to maximizing a,. This means that A and E optimal designs look to maximize K”s distance from non-full rank matrices and minimize the upper bounds for M%?(C) and V(i?,>. Note that these designs do not look to specifically optimize selectivity or sensitivity. As for D-fpiimal design, it is clear that minimizing Det((KK ‘)- ‘) is equivalent to maximizing Det(K.K’) = a&r; . . . a,*. If all the columns of K’ have Euclidean length 1, then this design process maximizes selectivity in K’, otherwise, it just provides a number that is related to the hyper-volume of the corresponding hyper-ellipse treated in [12]. This means that in certain special cases, this design process looks to maximize selectivity. Note that none of thy design processes directly deal with sensitivity in K.

6. Discussion

Results presented in this tutorial suggest a simple qualitative guideline for assessing sensitivity and selectivity in K*: Maximize (+1 and a, subject to the condition that the ‘difference’ between crl and a;, be kept small. The following looks to quantify this guideline. It uses the condition number, the value a,,, and ‘a priori’ information about the assumed error variance u *.

Note that

V(I?~) < a*/~. 5zMSE(t) I u*n/u,2.

If this string of inequalities is to be upper bounded by a positive number p, then a, must be greater than or equal to am. Let 6 be a positive number between 0 and 1. If S I l/rc2(Kf)” is to be satisfied (thereby ensuring some degree of selectivity), then K~(K~) must be less than or equal to l/S’/“. These two observations yield a guideline for quantization of the above qualitative guideline: maximize u1 and o;, subject to the conditions

a;, 2 u\lnlp and a,/~,, I l/6’/“,

where u is the standard deviation associated with error.

If Kr satisfies these criteria, then it would be deemed as having acceptable sensitivity and selectivity. The use of this guideline requires an ‘a priori’ determination of /3 and S. This implies that some ‘feel’ for the magnitude of u is required. For small p and 8, i.e. close to zero, large a,, are needed and large K~(K’) values are allowed, whereas, large /3 and 6 values allow small u,, values, but require that K~(K’) values to be near 1.

To illustrate these ideas, suppose u = lo-‘, /Zl= 10e3, n = 3, and 6 = l/2. Then, according to this guideline, a Kjr with u3 and u1 satisfying a3 2 udn//3 = 0.547723 and u1/u3 I 1/S’13 = 1.25992 would be considered acceptable. Should S = l/30, then u1/u3 I 30113 = 3.10723 must be satisfied. Table 1 lists a,, i.e. an and a,/~,, i.e. K*(K’) for the initial five 3 X 3 matrices considered in this paper. A moments inspection shows that only Kf, satisfies the first set of criteria on p and 6, whereas, both Ki and Ki;: satisfy the second set of criteria on p and 6.

These last three paragraphs theoretically address and solve the problems associated with just using the condition number as an assessment tool. It provides selected upper bounds on the theoretical variances and errors, it provides control on sensitivity and selectivity, and it keeps the ‘differences’ between a, and u1 small. However, it is prediction sample independent. This matter is of primary concern when one must use K’ over a wide range of analysis sample concentrations. Ref. [lo] describes the problem in greater detail.

J.H. Kalivas, P.M. Lang/Chemometrics and Intelligent Laboratory Systems 32 (1996) 135-149 149

By way of a final comment, it should be noted that the assessments discussed are not directly applicable to P-matrix analysis where the fundamental equation is C = FW + E. For more discussion along this line, see [lo].

References

[l] H. Kaiser, Spectrochim. Acta, 33B (1978) 551. [2] P.J. Brown, J. Chemom., 7 (1993) 255. [3] G. Bergmann, B. von Oepen and P. Zinn, Anal. Chem., 59

(1987) 2522. [4] G. Bauer, W. Wegscheider and H.M. Ortner, Spectrochim.

Acta, 146B (1991) 1185. [5] G. Bauer, W. Wegscheider and H.M. Ortner, Spectrochim.

Acta, 147B (1992) 179. [6] M. Otto and W. Wegscheider, Anal. Chim. Acta, 180 (1986)

[7] ?&be, Anal. Chem., 58 (1986) 1167.

Dl

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Interrelationships between sensitivity and selectivity measures for spectroscopic analysis