5.1.1 Stationary phase approximation . . . 59 5.1.2 Sparse time-frequency representation and WDM transform . . . 62 5.2 Graph computation . . . 63 5.3 Search the observational data using the graph . . . 65 5.4 Comparison study: Gaussian noise case . . . 66 5.4.1 Time-frequency graphs . . . 67 5.4.2 Simulated data sets . . . 68 5.4.3 Results . . . 68 5.5 Comparison study: real noise case . . . 71 5.5.1 cWB gating and vetoes . . . 72 5.5.2 A glitch rejection algorithm for Wavegraph . . . 73 5.5.3 Running Wavegraph with consistency test in O1 data . . . 75
In this chapter we present Wavegraph (WG), a novel clustering scheme dedicated to cWB [1]. The goal of Wavegraph is to address some of the limitations of current searches identified in Sec 4.2.4. Wavegraph is based on a pattern matching formulation of a signal detection technique performed in the TF domain. Expected patterns are computed from a template bank. Wavegraph can be viewed as a matched filtering technique performing in the TF domain. Astrophysical scenarios generally provide a range of waveforms parametrized by several physical source parameters. For example compact binary mergers phase is mainly driven by the binary components masses and spins. This leads to some variability in the expected time-frequency pattern. In Wavegraph the waveform model phase variability is encapsulated into a graph. Sec 5.1 reviews methods to compute the time-frequency pattern from a time domain waveform. We also describe a sparse signal approximation algorithm which allows to obtain a reliable description of the waveform model. Using the graph the detection problem can be reformulated as a combinatorial optimization problem for which efficient algorithms exist as explained in Sec 5.3. Along this chapter we focus on GW signals emitted from coalescing compact binaries.
5.1 Computation of time-frequency patterns
Wavegraph relies on thea priori knowledge of the targeted signal. Wavegraph establishes a mapping between the reference waveform and its representation by a finite set of pixels in the TF domain. This section reviews different methods in order to establish such a mapping.
5.1.1 Stationary phase approximation
The first pixel selection method determines which wavelets of the dictionary has the largest coupling with the targeted signal by working in the continuous limit. We use a transform where time, frequency and levels vary continuously. Sine-Gaussian (SG) basis are used to allow for analytical and hence fast computations [139]. The
SG wavelet with central timet0, central frequencyf0and level`0(or scale) is defined in the frequency domain as
˜
w0(f) = ˜g(f−f0;σ0)e−2πif t0 (5.1)
where ˜g(f;σ) = (2π)1/4√
σe−π2σ2f2is the mathematical expression of a SG atom in the frequency domain. The parameter σ0 defines the typical time-scale of the SG wavelet. It can be approximately related to the level parameter`0 in the discrete WDM transform by`0∼fsσ0(where fs is the signal sampling frequency). A SG wavelet transform is a projection onto the SG basis such that the square of the TF map coefficients are given by
In order to identify salient pixels one has to identify times t0 and levels `0 which maximise the square energy ρ0 at a given f0. However Eq (5.2) is analytically intractable in general but can be simplified for chirp-like signals. Chirp signals can be expressed in the complex domain as ˜s(f) = A(f)eiΨ(f) where we denoted the amplitude A(f) and the phase Ψ(f) of the chirp. Eq (5.2) can then be re-written in term of an oscillatory integral R
dxf(x)eig(x). Assuming slow variations of the integrand amplitude with respect to the integrand phase, Eq (5.2) can be approximated using the stationary phase approximation (SPA). As a consequence the square energy expression becomes (the derivation is detailed in Appendix 7.2):
ρ20∼ π|A(f0)|2 previous equation overt0 andσ0 at a given frequency yields
ρ(ˆt0, f0,`ˆ0)2= fs
√π
|A(f0)|2
`ˆ0Sn2(f0) (5.4)
Such a maximum is reached at ˆt0 = τ(f0) and ˆσ0 = p
|β|/π converted into ˆ`0 using the Gaussian to Meyer conversion rule stated above. Finally, the parametric curve associated to the signal ˜s(f) is
C(f0) =
Eq (5.5) provides an approximation for the wavelet transform in the continuous limit. This curve has to be discretized according to the time-frequency-level lattice adopted by cWB. It results in a finite and ordered set of pixels/waveletsCreferred to as achirp path. Four chirp paths are shown on Fig 5.1 with increasing values of total mass. We recover the fact that high mass binaries have a short signal duration. High frequency content is mostly described by short-time wavelets, i.e. low-level SG wavelets. The use of different levels enables us to obtain bright pixels above the noise level (see Fig 5.2).
Figure 5.1: Chirp paths obtained by using the SPA. Here again coalescing binary signals show an increasing frequency with time.
For the purpose of the algorithm, the cluster C is in fact discrete. It means that the cluster will in fact be described as a finite set of salient pixels (dark grey pixels in Fig 5.2). Those pixels are obtained by maximising Eq (5.4) for each frequency and level bins. An isolated cluster with this technique is displayed in Fig 5.2. Pixels follow the TF evolution trend of the GW signal and this over the decomposition levels.
Figure 5.2: Selected TF pixels by using the SPA at each TF resolution (`0= 3, ...,8). The GW signal is a chirp with associated masses 8.6 and 3.3 M. Clusters extracted at each level/scale by Wavegraph are indicated by red dots and the clusters extracted by cWB are shown with blue dots.
The previous derivation yields an efficient yet not completely accurate TF representation of the signal. Indeed noise-free chirp signals are poorly reconstructed with such a method as the low phase oscillation hypothesis on which SPA relies on is less and less verified as one gets close to the merger part of the signal. But the most important caveat is the loss of reconstructed signal energy we experienced when looking for GW events in noisy data. We realised an average 40% loss in the reconstructed SNR. We tried to extract the first neighbours of each maxima along the time axis when those neighbours at least carry a fraction of the energy of the loudest
pixel. The idea has been pushed until the third neighbours but has not proved to catch the necessary amount of energy. In the end we realised a need for a new pixel selection algorithm that conserves the phase evolution information but also gathers more energy from the noise free GW signals.
The use of wavelets to describe GW signals consists in locally fitting the waveform model. This contrasts with the matched filtering approach in the sense where a global phase accordance is needed between the data and the waveform models. Decomposing the signal with wavelets makes the search more robust with respect to poorly modeled GW signals.
5.1.2 Sparse time-frequency representation and WDM transform
Contrary to expansions performed in a Fourier basis for instance, signal expansions in redundant dictionaries are not unique. Sparse expansions present a specific interest here as they provide a complete signal representation where the power is concentrated in a small set of dominant pixels that are more likely to stand above the noise level. Those algorithms are widely used in the domain of signal processing for de-noising or signal compression purposes. Also, thanks to the small number of pixels there is a reduced probability that one or several pixels in the decomposition match noise fluctuations in GW data.
Sparse signal approximations in redundant dictionaries has been a field of research in applied mathematics for the last twenty years. Although the approximation is known to be a NP-hard problem1, algorithms exists that provide sub-optimal solutions. Given an astrophysical model y we want to obtain a sparse approximation x which satisfies
minx kxk0 subject to
y−WTx
2≤δ (5.6)
where k.kp is the Lp norm, W is the WDM transform operator from the time domain to the time-frequency domain andδis the approximation error detailed after. Wavegraph uses thematching pursuit (MP) algorithm proposed by [131] to solve the Eq 5.6 problem. Other algorithms have been proposed and tested as detailed in [3].
Procedure 1 Wavegraph matching pursuit algorithm Input: signal to be decomposedy, approximation errorδ Output: Time-frequency approximation (WDM maps)x
Initialise set of chosen pixelsx← ∅ Initialise the residuer←y
while||r||2> δ do
Compute WDM transform of the residual gn←W r Select best fitting WDM pixelp←argmaxn||gn||2
Add it to the current approximation x←x+p Update the residualr←y−WTx
returnx
In Alg (1) we detail the MP algorithm. An initial time-domain signal y is decomposed on the cWB TF grid thanks to the W operator. At each iteration one collects the best coupling WDM pixel with the signal then subtract its inverse WDM transform from the current signal called residual (as it norm keeps on decreasing with iterations). The best pixel selection and subtraction steps are repeated until a termination condition is fulfilled. Wavegraph terminates when 80 % of the original signal energy (L2 norm) is reconstructed. Fix-ing the termination condition at 80 % is a good compromise as, above this level the convergence is very slow and the approximation error reaches a plateau (see top panel of Fig 5.3). In such conditions, the algorithm selects low amplitude pixels that do not contribute significantly to the overall detection efficiency of the pipeline.
Fig (5.3) illustrates the MP algorithm with a BH signal. On the top panel is displayed the convergence of the MP algorithm tested with the same signal as on the upper panel of Fig (4.5). During the first iterations, the MP algorithm picks high amplitude pixels/wavelets which are nearly orthogonal to each other. The MP behaviour
1Non-deterministic polynomial-time (NP) problems are problems for which no algorithm exist that can solve these problems in polynomial time. A typical example of NP-hard problem is thetraveling salesman problem.
Figure 5.3: Convergence of the matching-pursuit algorithm in the case of an equal-mass non-spinning BBH waveform with 20M total mass after whitening by advancedLIGOdesign sensitivity PSD. Fixing the termi-nation condition at δ= 20%, collects 33 pixels (top). Time-frequency pixels selected by the matching pursuit algorithm (middle) and corresponding wavelets and approximation (bottom) for the very same signal. The color indicates the scale`associated to each pixel that corresponds to a wavelet timescales 2`/fswithfs= 1024 Hz.
The strain amplitude of the original waveform (black) is shown in the background. Few early pixels selected by the MP are not displayed because they fall outside the frame chosen for this figure.
mimics the one of a principal component analysis [140]. After about 20 iterations, convergence slows down until reaching a plateau. In the middle panel the TF map contains the set of WDM pixels returned by Alg (1) with δ= 20%. Clearly short (resp. long) duration in blue (resp. red) pixels describe the late (resp. early) part of a coalescing binary signal. Finally the bottom panel shows the corresponding wavelets obtained by inverse WDM transform in the time domain. Although the approximation error is relatively large the waveform reconstruction is globally satisfactory (see Appendix 7.2).
Sparse approximation techniques described here works for any signal morphologyi.e. whether they are chirp-like or not. In principle, these techniques allow to use Wavegraph on a broader class of signals. Clusters obtained with other template waveforms are illustrated in the last chapter of this thesis.