Singular-value decomposition is a powerful technique that has been used in
the analysis of matrices in many fields. In this paper, we summarize how it has
been applied to the analysis of gravitational-wave data. These include
producing basis waveforms for matched filtering, decreasing the computational
cost of searching for many waveforms, improving parameter estimation, and
providing a method of waveform interpolation.Comment: 7 pages, Third Galileo - Xu Guangqi Conference Proceeding

Keppel, Drew

English

arXiv

Singular-value decomposition is a powerful technique that has been used in the analysis of matrices in many fields. In this paper, we summarize how it has been applied to the analysis of gravitational-wave data. These include producing basis waveforms for matched filtering, decreasing the computational cost of searching for many waveforms, improving parameter estimation, and providing a method of waveform interpolation

Keppel, D.

MPG.PuRe

September 17, 2013 11:35 WSPC/INSTRUCTION FILE 01113Proceedings of the 3rd Galileo–Xu Guangqi MeetingInternational Journal of Modern Physics: Conference SeriesVol. 23 (2013) 99–105c© World Scientiﬁc Publishing CompanyDOI: 10.1142/S2010194513011136USE OF SINGULAR-VALUE DECOMPOSITIONIN GRAVITATIONAL-WAVE DATA ANALYSISDREW KEPPELAlbert-Einstein-Institut, Max-Planck-Institut fu¨r GravitationsphysikD-30167 Hannover, GermanyLeibniz Universita¨t HannoverD-30167 Hannover, Germanydrew.keppel@ligo.orgSingular-value decomposition is a powerful technique that has been used in the analysisof matrices in many ﬁelds. In this paper, we summarize how it has been applied to theanalysis of gravitational-wave data analysis. These include producing basis waveforms formatched ﬁltering, decreasing the computational cost of searching for many waveforms,improving parameter estimation, and providing a method of waveform interpolation.1. IntroductionWithin the ﬁeld of gravitational-wave (GW) data analysis, there are several diﬃcultproblems that need to be addressed in order to extract information from the datathat will be taken by advanced GW detectors. These include determining when aGW signal occurred, where this signal came from, and what the properties of thesignal’s source are. In this paper, we summarize how singular value decomposition(SVD) has been used to ease the computational burden of answering these questions.This paper is outlined as follows: Sec. 2 gives a brief overview of SVD, Sec. 3describes how SVD can be used to approximate diﬀerent classes of waveforms, Sec. 4illustrates how SVD has be used to construct composite detection statistics, Sec. 5discusses a method of waveform interpolation based on SVD, and Sec. 6 summarizesthe use of SVD in diﬀerent aspects of parameter estimation.2. Singular Value DecompositionSVD decomposes an arbitrary matrix H into three components:H = VΣU, (1)where U is a unitary matrix of basis vectors such that the columns of U∗ are theright eigenvectors of H∗H, V is a unitary matrix of reconstruction coeﬃcients suchthat the columns of V are the left eigenvectors of HH∗, and Σ is a diagonal matrixof singular values whose non-zero components are the square root of the eigenvalues99Int. J. Mod. Phys. Conf. Ser. 2013.23:99-105. Downloaded from www.worldscientific.comby MAX PLANCK INSTITUTE FOR on 02/11/14. For personal use only.September 17, 2013 11:35 WSPC/INSTRUCTION FILE 01113100 D. Keppelassociated with H∗H or HH∗.a The eigenvectors of U and V are ordered such thatthe entries of Σ are in descending order.We will see in the following sections that the properties of U, V, and Σ havebeen used in diﬀerent contexts.3. Classes of WaveformsGW signal waveforms can be classiﬁed into two groups: those that are well mod-elled, and those that are not.b Collections of waveforms of both types can be calledwaveform catalogs, however here we restrict use of that term for non-well-modelledwaveforms. We will refer to waveforms that are well modelled as template banks. Fur-ther discussion of temaplte banks and waveform catalogs can be found in Secs. 3.1and 3.2, respectively.3.1. Waveform template banksAn example of a type of waveform that can be used to construct a template bankis a GW signal from the inspiral of a compact binary coalescence (CBC). Thesewaveforms are well described theoretically by the post-Newtonian (PN) approxima-tion. This has allowed the construction of a metric on the signal parameter spacethat characterizes the distance between two points on parameter space based onthe mismatch between the two waveforms.1 Using this metric, a template bank canbe constructed with these waveforms such that any point within a chosen region ofparameter space will be no more than a speciﬁed distance from the nearest tem-plate, commonly chosen such that maximum mismatch between any template inthe bank and any signal in the region of interest is 3%.In order to meet this requirement with the fewest number of templates, for a twodimensional parameter space with a constant metric, adjacent templates will havea mismatch of 9%, which means they have an overlap of 91%. Cannon et al. inves-tigated how the SVD could compress a matrix H whose rows are composed of thehighly overlapping waveform time-series from such a template bank.2 They derivedthe expected fractional signal-to-noise ratio (SNR) loss δρ/ρ of approximating thewaveforms in the template bank by the truncation of the SVD of H to beδρρ=12NN∑i=N ′+1σ2i , (2)where σi is the ith entry of Σ, N is the number of rows of H or twice the number oftemplates in the template bank, and N ′ is where the SVD is truncated. In addition,aThroughout this paper, M∗ denotes the complex conjugate-transpose of matrix M.bOne reason a waveform may not be able to be well modelled is because of complex and possiblychaotic physical processes at work in the production of the GW signal (e.g., the presence of matterin a GW generating system).Int. J. Mod. Phys. Conf. Ser. 2013.23:99-105. Downloaded from www.worldscientific.comby MAX PLANCK INSTITUTE FOR on 02/11/14. For personal use only.September 17, 2013 11:35 WSPC/INSTRUCTION FILE 01113Use of Singular-Value Decomposition in Gravitational-Wave Data Analysis 101the maximum phase-averaged fractional SNR loss can be bounded by(δρρ)max<12σ2N ′+1, (3)which can be derived from the unitarity of the reconstruction coeﬃcients.This approach to waveform compression has been implemented within a low-latency GW search for inspiral waveforms as one of the techniques to reduce thecomputational cost of data processing.3 Similar techniques have also been appliedto other types of GW signals using Gram-Schmidt orthogonalization.4,53.2. Waveform catalogsWaveform catalogs are necessary for waveforms that are not well modelled theo-retically. Such waveforms can arise from numerical simulations that are either toocomputationally expensive to study the parameter space thoroughly (e.g., numericalsimulations of merging spinning binary black holes), or from numerical simulationsthat include matter (e.g., numerical simulations of supernovas (SNs) or coalescingbinary neutron stars).Reference 6 ﬁrst proposed decomposing waveform catalogs of SN waveformsusing Gram-Schmidt orthogonalization. Ref. 7 expanded on this approach using theSVD and comparing it to Gram-Schmidt orthogonalization. Both approaches werefound to perform similarly, being able to extract and prioritize waveform character-istics from the SN waveform catalogs. However, in the SVD only a single decompo-sition of the catalog was needed, and additional information was obtained. Namely,features that were present in many waveforms produced basis vectors with largesingular values, whereas features present in only a few produce basis vectors withsmall singular values.4. Detection StatisticsConstructing detection statistics when searching for GW signals is diﬀerent depend-ing on what type of search you are doing. Searches can be classiﬁed into two groups,those searching for known signal waveforms and those searching for unknown signalwaveforms.The standard approach to search for any known signal within GW data from acollection of GW waveforms is to matched ﬁlter the data, producing a time-seriesof SNR, with each of the template waveforms. GW signals are then identiﬁed byidentifying which templates at certain times produce large values of SNR.8If one is only interested in answering the time component of the question, acomposite detection statistic can be used, which tells whether the data looks likeanything from a collection of template waveforms.8 Cannon et al. showed how theSVD can be used to construct such a composite detection statistic Γ associated withInt. J. Mod. Phys. Conf. Ser. 2013.23:99-105. Downloaded from www.worldscientific.comby MAX PLANCK INSTITUTE FOR on 02/11/14. For personal use only.September 17, 2013 11:35 WSPC/INSTRUCTION FILE 01113102 D. Keppela region of parameter space covered by a template bank of the formΓ =N ′∑k=1σ2kσ2k + 2N/〈A2〉ρ2k, (4)where 〈A2〉 is the expected signal amplitude squared and ρk is the SNR associatedwith the kth basis vector.9 This statistic was derived using the orthonormality of thereconstruction coeﬃcients and the basis vectors, as well as the ordered signiﬁcancegiven by the singular values. It shows promise in reducing the computational costof searching GW data for signals of known waveforms.When searching for a signal of unknown waveform, the search can be performedby looking at other aspects of how the signal interacts with the detectors. In par-ticular, when there is a network of GW detectors in operation, the data from thesedetectors can be combined in such a way to produce coherent data streams that aredirection dependent.10,11Wen outlined how the detector responses associated with a given direction canbe coherently combined and condensed using the SVD.12 The SVD of the networkdetector responses A provides the basis vectors U, singular values Σ, and recon-struction coeﬃcients V. The data from the diﬀerent detectors d can be recombinedinto new data streams d′ using the reconstruction coeﬃcientsd′ = V−1d. (5)The detection statistics for an unmodelled GW search could then be constructedusingΓ =∑kwk |d′k|2 , (6)where wk is a chosen weight associated with diﬀerent basis vectors, and the sumcan be over diﬀerent combinations of the new data streams. Diﬀerent choices foreach of these parameters results in diﬀerent optimizations of the detection statistic.5. Waveform InterpolationProducing the waveforms in the ﬁrst place can be a computational challenge, whichis the case for binary black hole coalescence waveforms produced by numericalrelativity. To help solve this problem, there have been several eﬀorts to parameterizethese waveforms, allowing waveforms from arbitrary points in parameter space tobe produced.c These eﬀorts are based on models of the waveforms that need tobe tuned. Cannon et al. have pursued a diﬀerent approach by showing that thebasis vectors from the SVD of waveforms from a template bank approximatelyenclose the space of waveforms from that region of parameter space.13 Further,they have also laid out a procedure that could be used to interpolate waveforms tocReferences for these eﬀorts can be found in Ref. 14.Int. J. Mod. Phys. Conf. Ser. 2013.23:99-105. Downloaded from www.worldscientific.comby MAX PLANCK INSTITUTE FOR on 02/11/14. For personal use only.September 17, 2013 11:35 WSPC/INSTRUCTION FILE 01113Use of Singular-Value Decomposition in Gravitational-Wave Data Analysis 103high accuracy from arbitrary points in parameter space using the components ofthe SVD.14 This is done by interpolating the reconstruction coeﬃcients. Once theinterpolated reconstruction coeﬃcients V′ are in hand, the interpolated waveformscan be generated usingH′ = V′ΣU, (7)where H′ is a matrix containing interpolated waveforms at the locations where V′was interpolated. With a suﬃcient density of numerical relativity waveforms, highlyaccurate SVD interpolated waveforms could be produced for arbitrary points inparameter space.6. Parameter EstimationOnce it is known that there exists a GW signal in a data set, the next questionthat comes up is what are the parameters of the source of the signal. Several setsof authors have used the SVD toward this goal in diﬀerent contexts.Wen identiﬁed how the SVD could be used to construct null data streams andsemi-null data streams.12 Null data streams are linear combinations of data from anetwork of GW detectors that will contain no GW signals,15 which are associatedwith zero-valued singular values from the SVD of the response of a network ofGW detectors. Similarly, semi-null data streams are associated with small-valuedsingular values from the SVD of the response of a network of GW detectors. Wen etal. showed how analyzing these semi-null data streams could improve the accuracywith which one could locate the source of a GW signal on the sky.16Wen also explained how the SVD could be used to extract a GW signal waveformfrom the data of a network of GW detectors.12 This was done by truncating the SVDof the detector response matrix in order to regularize it. The extracted waveform hTis then the data from all detectors d combined using the inverse of the regularizeddetectors response matrix,hT =(AT)−1d, (8)where ATij =∑N ′k=1 vikσkukj is the detectors’ response matrix regularized by trun-cating the SVD after the N ′th singular value.Ro¨ver et al. applied the SVD of waveform catalogs to the parameter estimationproblem.17 Using the basis vectors associated with the SVD of a SN waveform cat-alog, a parameter estimation search based on Markov chain Monte Carlo (MCMC)methods extracted the amplitudes of the signal associated with each of the basisvectors. By associating the extracted amplitudes with those of the waveforms in thecatalog, physical parameters of the simulated signal could be estimated.Extending the previously mentioned work, Logue et al. applied the SVD to mul-tiple waveform catalogs, each associated with diﬀerent SN production mechanismsor diﬀerent types of stellar cores.18 By computing the Bayes odd ratios associatedwith these diﬀerent models by nested sampling techniques, the correct model forthe SN can be determined.Int. J. Mod. Phys. Conf. Ser. 2013.23:99-105. Downloaded from www.worldscientific.comby MAX PLANCK INSTITUTE FOR on 02/11/14. For personal use only.September 17, 2013 11:35 WSPC/INSTRUCTION FILE 01113104 D. KeppelFinally, Cannon et al. discussed how the likelihood function, which is the quan-tity computed for parameter estimation, could be interpolated to decrease the com-putation cost of parameter estimation searches.14 Using the interpolated reconstruc-tion coeﬃcients, the interpolated likelihood function would be given byln (Λ′) = V′ρ2, (9)where the locations the likelihood function is estimated are given by the locationswhere the reconstruction coeﬃcients are interpolated.7. ConclusionSVD has been applied with good results in a number of contexts within GW dataanalysis. From waveform compression and interpolation, to deriving new detectionstatistics, to improving parameter estimation, the powerful properties of the SVDhave proved to be of great use.AcknowledgmentsWe would like to thank Kipp Cannon and Chad Hanna for the countless fruitfuldiscussions we have had on this topic, and Bruce Allen and Ray Frey for theiruseful comments on this manuscript. We would also like David Blair and LinqingWen for extending to us the invitation to attend the Third Galileo - Xu GuangqiMeeting. This work was supported from the Max Planck Gesellschaft and has LIGOdocument number LIGO-P1100207.References1. B. J. Owen, Phys. Rev. D 53, 6749 (1996), arXiv:gr-qc/9511032.2. K. Cannon, A. Chapman, C. Hanna, D. Keppel, A. C. Searle, and A. J. Weinstein,Phys. Rev. D 82, 044025 (2010), arXiv:1005.0012 [gr-qc].3. K. Cannon, R. Cariou, A. Chapman, M. Crisp´ın-Ortuzar, N. Fotopoulos, M. Frei,C. Hanna, E. Kara, D. Keppel, L. Liao, S. Privitera, A. Searle, L. Singer, and A. Wein-stein, Submitted to Astrophysical Journal, arXiv:1107.2664 [astro-ph.IM].4. S. E. Field, C. R. Galley, F. Herrmann, J. S. Hesthaven, E. Ochsner, and M. Tiglio,Phys. Rev. Lett. 106, 221102 (2011), arXiv:1101.3765 [gr-qc].5. S. Caudill, S. E. Field, C. R. Galley, F. Herrmann, and M. Tiglio, ArXiv e-prints,arXiv:1109.5642 [gr-qc].6. P. R. Brady and S. Ray-Majumder, Classical and Quantum Gravity, 21, 1839 (2004),arXiv:gr-qc/0405036.7. I. S. Heng, Classical and Quantum Gravity 26, 105005 (2009), arXiv:0810.5707 [gr-qc].8. L. A. Wainstein, and V. D. Zubakov, Extraction of signals from noise (Prentice-Hall,Englewood Cliﬀs, NJ, 1962).9. K. Cannon, C. Hanna, D. Keppel, and A. C. Searle, Phys. Rev. D 83, 084053 (2011),arXiv:1101.0584 [physics.data-an].10. Y. Gu¨rsel and M. Tinto, Phys. Rev. D 40, 3884 (1989).11. E´. E´. Flanagan and S. A. Hughes, Phys. Rev. D, 57, 4566 (1998), arXiv:gr-qc/9710129.Int. J. Mod. Phys. Conf. Ser. 2013.23:99-105. Downloaded from www.worldscientific.comby MAX PLANCK INSTITUTE FOR on 02/11/14. For personal use only.September 17, 2013 11:35 WSPC/INSTRUCTION FILE 01113Use of Singular-Value Decomposition in Gravitational-Wave Data Analysis 10512. L. Wen, Int. J. Mod. Phys. D 17, 1095 (2008), arXiv:gr-qc/0702096.13. K. Cannon, C. Hanna, and D. Keppel, Phys. Rev. D 84, 084003 (2011), arXiv:1101.4939 [gr-qc].14. K. Cannon, C. Hanna, and D. Keppel, Submitted to Phys. Rev. Lett., arXiv:1108.5618[gr-qc].15. L. Wen and B. F. Schutz, Classical and Quantum Gravity 22, 1321 (2005),arXiv:gr-qc/0508042.16. L. Wen, X. Fan, and Y. Chen, Journal of Physics Conference Series 122, 012038(2008), arXiv:0711.0726 [gr-qc].17. C. Ro¨ver, M.-A. Bizouard, N. Christensen, H. Dimmelmeier, I. S. Heng, and R. Meyer,Phys. Rev. D 80, 102004 (2009), arXiv:0909.1093 [gr-qc].18. J. Logue, C. D. Ott, I. S. Heng, P. Kalmus, and J. H. C. Scargill, Privatecommunication.Int. J. Mod. Phys. Conf. Ser. 2013.23:99-105. Downloaded from www.worldscientific.comby MAX PLANCK INSTITUTE FOR on 02/11/14. For personal use only.

Use of Singular-Value Decomposition in Gravitational-Wave Data Analysis

 Singular-value decomposition is a powerful technique that has been used in the analysis of matrices in many fields. In this paper, we summarize how it has been applied to the analysis of gravitational-wave data analysis. These include producing basis waveforms for matched filtering, decreasing the computational cost of searching for many waveforms, improving parameter estimation, and providing a method of waveform interpolation. </jats:p

Use of Singular-Value Decomposition in Gravitational-Wave Data Analysis

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