GWDAW - 10

GWDAW - 10Dec. 13-18, 2005

Binary Chirp Template Banks:Tanaka-Tagoshi Parameterization for LIGO

R.P. Croce, Th. Demma, A. Fusco, V. Pierro*,I.M. Pinto, M. Principe

The Waves Group

GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMYCENTER FORCENTER FOR

*workgroup coordinator

GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY

CENTER FORCENTER FOR

Plan

Template Placement Problem

Tanaka Tagoshi Parameterization

Tanaka-Tagoshi Style Placement for LIGO [preliminary results]



ML Detection of Chirps

“Project” the (spectral data) over a certain set of templatesof the sought waveform;Take the largest projection as a detection-statistic;Compare to a false-alarm dictated detection-threshold;

If test is passed, declare detection & estimate parametersof signal detected from those of largest-projection template;

False dismissal probability is a monotonically decreasingfunction of the signal-template match;



Structure of (RedPN) Match Functional

7 / 3

7 / 3

exp 2 ( ; , )( )sup

( , )

( )

off

in

off

in

f

c h g

f

fC

f

dff f T f

fM h g

T dff

f

( ) noise PSD; , spectral windowin offf f f

Maximises over extrinsicparameter Tc

(through simple FFT)

2 1

1

( ; , ) ( ) ( ) ( )N

h g n n h n kn

f f

maximizes over extrinsicadditive phase

nntrinsic parameters



Chirp Parameters (2PN)

Free parameters: companion masses + spin-spin & spin-orbit.



m1=m2

3

(mmin,mmin,)

(mmin,mmax,)

(mmax,mmax)

8 / 3 5/ 3 10 0

2 / 35/ 3 2 / 3 1

3 0

1 21 2 2

5

256

8

;

f M

f M

m mM m m

M

Spin-Free Search Manifold



Minimal Match Prescription

The template bank {k}, aka the set {Xk} in search manifoldshould be such that for any admissible signal h there is (at least) one template g such that M(h,g)=, viz.

, : ( , )h g M h g

Corresponds roughly to allowing a fraction r = ( 1 - 3 )of potentially observable CBS sources to be missed justas an effect of insufficient parameter-space sampling

e.g., r = 10% 0.97



X0

The span S (X0),

0 0( , ) , ( )C X X X S X

0 0( , ) , ( )C X X X X

The iso-match contour-line (X0) = S (X0),

Span of a Template

X0 = par. space point (identifies template)



Building a Bank

Translate MM prescription in terms of spans:

( )i iS X (no holes)

Also try to cope with the following nice requirements: Use minimum number of templates (save on computation) Get minimum spillover across m1=m2 line (discard unphysical)

End up in a regular lattice (easier placement; interpolation)



X0

Span of a Template, contd.

At relatively large MMs, iso-match contour lines turn out to be ellipses...

You can always make them into circlesvia trivial coordinate transformations

X0 X’0

X’0



Regular Tilings of the Plane :

- Identify largest-area span-inscribed (regular) polygon : the basic tile ;- Tile the plane thereof; place templates-lattice nodes at tile vertexes. Effective span Veff of tiled-templates: Voronoi dual of basic tile.

Sparsest coverage obtained from triangular tiling (hexagonal effective span).

Basic tile

Area(Veff) : Area(Veff) : Area(Veff) = 1.3:1:0.65(6) (4) (3)

……vs. template lattices



Regular Tilings of the Plane: Triangular

R.P. Croce et al., Phys. Rev. D65 (2002) 102003.

Triangular tile

Voronoi conjugate(hexagon) of basictile: effective span.



X0


...At relatively large MMs, iso-match contour lines are ellipses : the match is well approximated by a quadratic form:

Iso-match ellipses stretch & rotate as one moves across !

( , ) 1 1r s

r srs rs p q

p q

M h g G G m mm m

2D metric has nonzero curvature!

Template placement is tricky !

6D

…and you cannot get rid of it via a global transformation…




…as a further complication,iso-match contour lines are no-longer elliptical at relatively low MMs (e.g., = 0.9).

(as required, e.g. in the firststage of hierarchical strategies)

3

The “quadratic” approximationmay underestimates the span coverage e.g. by a factor

~1.45 at =0.95 ~6.13 at =0.8



Template Placement : Mainstream

Place a string of templates along the equal mass line; [copes with minimal spillover requirement];

Lay a regular rectangular (hexagonal) tiling of the (0 ,3) plane; [pay in terms of redundancy here, to ignore iso-match contour line stretch-rotation pathology];

Do not put templates at nodes of the tiling for whichthe node is external to ; andthe node is not on a vertex of .

B. Owen and B.S. Sathyaprakash, Ph. Rev. D60 22002 (1999)

Content yourself with quadratic match approximation [OK for large , but bad underestimate of effective span at low , as in the first stage of hierarchical strategies]



Template Placement, the Virgo wayD. Buskulic et al, Class. Quantum Grav. 20, 789, 2003.

1) Triangulate par space coarsely, and compute exact (elliptical) spans at each vertex.

3) Sub-triangulate patches where interpolation fails (fifth iteration shown).

2) Compare actual span at triangle’s center with linear-interpolated one;



Progress (?) Directions

time needed to compute bank;-Reduce number of templates needed;

-Handle non-elliptical, low - iso-match contours

Our attempt : go through Tanaka Tagoshi transformation

Expected benefits:

- Work in flat manifold translation–invariant iso-math contours- Go workably beyond quadratic (elliptic contour) approximation…do not underestimated span-areas use lesser templates

…based on previous related work [ R.P. Croce et al., Ph. Rev. D64, 042005 (2001);R.P. Croce et al., Ph. Rev. D64, 042005 (2001) ]



Tanaka-Tagoshi Transformation

T.Tanaka and H.Tagoshi, Ph. Rev. D62, 82001 2000.

Linear mapping from “natural” search manifold to a globally flat one such that:

( , ) ( , )1

( , )

M h g M h g

M h g

( , ) ( )ghM h g M X X

, ,h g h g ,<<

Regular grid of nodes, globally uniform lattice in



Tanaka-Tagoshi Transformation, technical

i- Orthogonalize the

- Find a rotation which maps all three vertexes pi (i=1,2,3)

(mmin,mmin),(mmin,mmax),(mmax,mmax)

down into the (x1,x2) plane.

P and obtained from Jordan decomposition G = PT P ’ = ½

x = Q ’ such thatQ ( p’2 – p’1 ) = 21x1 + 22x2

Q ( p’3 – p’1 ) = 11x1 …not unique

In practice, form (nonsingular) Z matrix out of col. vectors

The unique (straightforward) QR decomposition of Z , Solves Q X = R , and hence the system. Hence it provides Q.

(p’3 - p’1, p’2 - p’1, ’3, ’4, ’5)^ ^ ^

Z = Q T R



LIGO200

400

x1

0.8

1

1.2

x2

0.02

0.025

0.03

0.02

0.025

0.03

x1x2

100

500

1/ 22 2 23 4 5x x x

LIGO PSD (LAL)fin=40 Hz,foff=750 Hz

1 M M1,M2 3 M

The Distance between and



TT Iso-match Contour Levels

2. 1. 0. 1. 2.x1

2.

1.

0.

1.

2.

x 2

0.70.80.90.950.970.99

LIGO PSD (LAL)fin=40 Hz,foff=750 Hz



TT Iso-match Contour Level: Exact vs Quadratic Approximation

2. 1. 0. 1. 2.x1

2.

1.

0.

1.

2.

x 2

0.2 0.1 0. 0.1 0.2x1

0.2

0.1

0.

0.1

0.2

x 2

Quadratic

Quadratic

Exact

Exact

Areaexact/Areaquadratic = 6.13 Areaexact/Areaquadratic = 1.43

=0.8 =0.95



1) Compute Tanaka-Tagoshi transformation [ from antenna PSD, (fin, foff), & PN phasing formula ] ;

2) Compute iso-match contour in TT plane [ from prescribed ] ; 3) Construct search-manifold in TT plane [ from prescribed mass-range] ;

4) Build-up “optimal” triangular-tiling [trade maximal sparsity for minimal spillover] ;

5) Create regular template lattice covering ;

6) Map TT lattice back to (0,3) or (m 1,m 2) using -1.

TT-style Bank Generation Algorithm



Optimum Tiling: Maximum Span

Draw convex hull& identify butterflyshaped region

x2

x1

(TT)

iso-match contour in TT plane



Maximal-Span Tiling

Chords through centerwithin butterfly-shaped region parameterized in angle;

Take one


x2

x1

(TT)



Maximal-Span Tiling

Chords through centerwithin butterfly-shaped region parameterize inan angle;

Take one


Slide chord to bottom (touch w/our intersect)

x2

x1

(TT)



Maximal-Span Tiling


Take one


Slide chord to bottom (touch w/out intersect)

x2

x1

(TT)



Maximal-Span Tiling


Take one


Slide chord to bottom (touch w/our intersect)

Find 3rd vertex formax triangle areaTwo-parameter (two angles) optimization

x2

x1

(TT)



Maximal-Span Tilingx2

x1

(TT)

… coverage mechanism …



Maximal-Span Tiling, contd.

x2

x1

(TT)

2



Maximal-Span vs. Minimal Spillover

Using maximal – span tilings does not guarantee Minimal template spillover across equal mass line;

Spillover across equal mass line efficiently minimizedif tile-baseline is parallel to the straight-line connecting the endpoints of the equal mass line;

This choice entails a negligible span-reduction (less than 1% at =.97), resulting in nearly-maximal sparsity, and minimal-spillover;

Resulting template-placement algorithm straightforward.



Quasi-Maximal-Span Reduced-Spillover Tiling


Take the one parallelto the line connecting the vertexes of thesearch-manifold (TT)equal mass line


One-parameter (upper vertex angle) search only

x2

x1

(TT)



Span Reduction after Tile Baseline Rotation

0. 0.5 1. 1.5 2. 2.5 3.

0.99

0.992

0.994

0.996

0.998

1.

AA tpo

0.2 0.1 0. 0.1 0.2x1

0.2

0.1

0.

0.1

0.2

x2

LIGO I Optimum triangular tiling

.97

Are

a re

duct

ion

fact

or



Template Placement Start

search manifold

equal-mass line (TT)

Vertex-joining line

base-tileshape

(downward-concave equal-mass line)

Draw basic-tile with lower-left vertex coincident with (leftmost) m1=m2=mmax TT search-manifold vertex.




search manifold


Vertex-joining line

Iso-matchcontour line

Draw associated iso-match contour line. This identifies location of first template.

Case # 1




search manifold


Vertex-joining line

Tilinglattice

Deploy tiling lattice thereof.

Case # 1




search manifold


Vertex-joining line

Deploy tiling lattice thereof.

Case # 1




search manifold


Vertex-joining line

Discard un-needed lattice points (null intersection between span and )

Case # 1



(201)

100. 200. 300. 400. 500.x1

0.8

0.9

1.

1.1

1.2

1.3

1.4

x2

Discarding Un-needed Templates

…see this at workin a practical case…

LIGOfin=40Hz, foff=730 Hz,1 M M1,M2 3 M



(201)

100. 200. 300. 400. 500.x1

0.8

0.9

1.

1.1

1.2

1.3

1.4

x2

CC’

B’ B

A


…simple constructionyields fiducial template-lineendpoints, to berefined using asimple algorithm…




Starting from X = A, B, C, B’, C’

Accept template at X. Move X one lattice node forward (if started from A,B,C)or backward (if started from B’, C’)

( )S X Y

S=0

S = 1S = 0

N

Exit

Y

Discard template at X Move X one lattice node backward (if started from A,B,C) or forward (if started from B’, C’)

N

Span of template at X Search manifold



LIGO

fin = 40Hz,

foff=730 Hz,

1 M M1,M2 3 M

= 0.97

# templates = 2331

(201)

100. 200. 300. 400. 500.x1

0.8

0.9

1.

1.1

1.2

1.3

1.4

x2

(1416)

(606)

(109)

TT-Style Template Placement for LIGO



100. 200. 300. 400. 500.x1

0.8

0.9

1.

1.1

1.2

1.3

1.4

x2

1M M1,M2 3 M

= .97

# templates = 2331

354. 355. 356. 357. 358. 359.0.6

0.7

0.8

0.9

1.

1.1

1.2

1.3

235. 236. 237. 238. 239. 240.0.6

0.8

1.

1.2

1.4

1.6

90. 92. 94. 96. 98. 100.0.6

0.7

0.8

0.9

1.

1.1

158.5 159. 159.5 160. 160.5 161. 161.5 162.0.6

0.8

1.

1.2

LIGOfin=40Hz, foff=740 Hz

Close-ups



0.2 0.4 0.6 0.8 1.0 1.20

0.06

0.08

0.10

0.12

3

LIGO PSD (LAL)

fin=40Hz,

foff=730 Hz,

1 M M1,M2 3 M

= 0.97

# templates = 2331




1. 1.5 2. 2.5 3. 3.5 4.m1

1.

1.5

2.

2.5

3.

3.5

4.

m2

FF 0.97

LIGO

fin=40Hz,

foff=730 Hz,

1 M M1,M2 3 M

= 0.97

# templates = 2331




LIGO

fin=40Hz,

foff=730 Hz,

1 M M1,M2 3 M

= 0.80

# templates = 410


100. 200. 300. 400. 500.x1

0.8

1.

1.2

1.4

x2

(401)




0.2 0.4 0.6 0.8 1.0 1.20

0.06

0.08

0.10

0.12

0.14

3

LIGO

fin=40Hz,

foff=730 Hz,

1 M M1,M2 3 M

= 0.80

# templates = 410




2000. 4000. 6000. 8000.x1

0.5

1.

1.5

2.

2.5

3.

3.5

4.

x2

The search manifoldbaseline may have a different concavity…

LIGOfin=40Hz, foff=2200 Hz,0.2 M M1,M2 1 M




2000. 4000. 6000. 8000.x1

0.5

1.

1.5

2.

2.5

3.

3.5

4.

x2

Here one starts placingbasic-tile with lower-leftvertex at the contactpoint of with a par-allel to the line joiningthe lower vertexes.

The subsequent stepsare the same…




2000. 4000. 6000. 8000.x1

0.5

1.

1.5

2.

2.5

3.

3.5

4.

x2

…one draws lines midway between thetemplate-lines., andprojecting the inter-sections with …




2000. 4000. 6000. 8000.x1

0.5

1.

1.5

2.

2.5

3.

3.5

4.

x2

…one obtains naïve first estimates of theTemplate-line end-points, to be refinedas already shown…



2000. 4000. 6000. 8000.x1

0.5

1.

1.5

2.

2.5

3.

3.5

4.

x2

LIGO

LIGO

fin=40Hz,

foff=2200 Hz,

0.2 M M1,M2 1 M

= 0.95

# templates = 56328(17088)

(19674)(8499)

(5587)

(3468)

(1721)

(201)




LIGO

fin=40Hz,

foff=2200 Hz,

0.2 M M1,M2 1 M

= 0.80

# templates = 7199

2000. 4000. 6000. 8000.x1

1.

2.

3.

4.

x2

LIGO




0.80 410

0.90 754

0.95 1203+332 = 1535

0.97 1566+646+119 = 2331

1 - 3M

40-730 Hz

# templates

0.2 - 1M

40-2200 Hz

# templates

0.80 6504+695 = 7199

0.95 17088+19674+8499+5587+3468+1721+201= 56328

TT-Style Template Budget for LIGO

98936

3624



From TT to Companion Masses or Chirp-times

X1k= 1ii(M,)X2k= 2ii(M,)

TT operator (direct)

Solve formally for (algebraic,2nd degree)

yields higher order algebraic equation in M.(solve numerically, e.g. Cholewsky)Has only one real root for which acceptable.

Pretty fast, even in MATHEMATICA pre-code.



Future Work

Write LAL compliant code & documentation

Perform blind injection benchmarks

… special thanks to Duncan Brown for providing comparative figures in “real time” !

Documents

GWDAW - 10