Reconstructing gravitational wave core-collapse supernova signals with dynamic time warping

Core-collapse supernovae (CCSNe) are a potential source for ground-based gravitational wave detectors, as their predicted emission peaks in the detectors’ frequency band. Typical searches for gravitational wave bursts reconstruct signals using wavelets. However, as CCSN signals contain multiple complex features in the time-frequency domain, these techniques often struggle to reconstruct the entire signal. An alternative method developed in recent years involves applying principal component analysis (PCA) to a set of simulated CCSN models. This technique enables model selection between astrophysical CCSN models as well as waveform reconstruction. However, PCA faces its own difficulties, such as being unable to reconstruct signals longer than the simulations; many CCSN simulations are stopped before the emission peaks due to insufficient computational resources. In this study, we show how combining PCA with dynamic time warping improves the reconstruction of CCSN gravitational wave signals in Gaussian noise characteristic of Advanced LIGO at design sensitivity. For the waveforms used in this analysis, we find that the number of PCs needed to represent 90% of the data is reduced from nine to four by applying dynamic time warping and that the match between the original and reconstructed waveforms improves for signal-to-noise ratios in the range [0, 50].

Figure 1

An example waveform produced by Scheidegger et al. [36]. A large spike occurs at core bounce due to the rapid rotation of the progenitor.

Figure 2

The three leading PCs produced with the original waveforms (left) and time warped waveforms (right). The spike at core bounce is more pronounced in the time-warped PCs.

Figure 3

Schematic example of DTW. (a) The top and the right panels show an example of sequences $\mathbf{f}$ and $\mathbf{g}$, respectively. (b) The middle panel shows the distance matrix, in which each cell is the Euclidean distance between points from $\mathbf{f}$ (column) and $\mathbf{g}$ (row). The gray scale is shown on the left (white for distance zero, darkest shading for six). (c) The optimum warping path, computed by dynamic programming, is the path of red stars. It shows which points of $\mathbf{f}$ align with which points in $\mathbf{g}$. In this example, the optimum warping passes mostly through zero entries of the matrix (shown in white), so that the distance between the warped signals equals 1. By contrast, the Euclidean distance between original $\mathbf{f}$ and $\mathbf{g}$ is 43. (d) The bottom panel depicts $\mathbf{w}$ and $\mathbf{u}$ for the optimal warping by connecting points in $\mathbf{f}$ and $\mathbf{g}$. For reference, the warping paths are $\mathbf{w}=(1,1,1,1,1,2,3,4,4,5,6,7,8,8,9,10,11,12,13,14,15,16,17,18,19,20)$ and $\mathbf{u}=(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,17,17,17,17,17,17,18,19,20)$.

Figure 4

Illustrative example of the application of DTW to one waveform. The blue waveform is model R3E1AC. The red waveform is the template R1E3CA. The black lines map the warping of R3E1AC to R1E3CA.

Figure 5

Cumulative variance as a function of the number of PCs. Nine of the original PCs capture 90% of the variance. When DTW is applied before computing the PCs, four PCs capture 90% of the variance.

Figure 6

Example reconstructions for the three test waveforms used in this study. Top row: original waveforms. Middle row: reconstructions using the original PCs. Bottom row: reconstructions after DTW is applied. DTW clearly captures low-amplitude features in the tails of the waveforms, which are missed by the original PCs.

Figure 7

Example reconstructions for a test waveforms in noise. Left panel: original waveform (R4E1EC $h_{\times }$) plus noise ($\mathrm{SNR}=20$). Middle panel: reconstruction using the standard PCs. Right: reconstruction after DTW is applied. The standard PCA method fails to reconstruct the waveform ($M=0.098$). The DTW-PCA method reconstructs the waveform ($M=0.561$).

Figure 8

(Left) Histogram of the match score $M$ for noise-only realizations with the original (blue) and DTW (orange) PCs. (Right) Average match score for the three test signals vs SNR. Applying DTW improves the reconstructions at all SNR values and reduces the minimum SNR.