Sensitivity study using machine learning algorithms on simulated $r$-mode gravitational wave signals from newborn neutron stars

This is a follow-up sensitivity study on $r$-mode gravitational wave signals from newborn neutron stars illustrating the applicability of machine learning algorithms for the detection of long-lived gravitational wave transients. In this sensitivity study, we examine three machine learning algorithms (MLAs): artificial neural networks, support vector machines, and constrained subspace classifiers. The objective of this study is to compare the detection efficiencies that MLAs can achieve to the efficiency of the conventional (seedless clustering) detection algorithm discussed in an earlier paper. Comparisons are made using two distinct $r$-mode waveforms. For the training of the MLAs, we assumed that some information about the distance to the source is given so that the training was performed over distance ranges not wider than half an order of magnitude. The results of this study suggest that we can use the machine learning algorithms as part of an investigative stage in the pipeline that would be able to provide very fast and solid triggers for further, and more intense, investigation.

Figure 1

The $(f_o=1500\mathrm{Hz},\alpha =0.1)$ waveform is the most powerful waveform considered in our sensitivity studies for both the clustering and the MLAs. The second waveform we chose has an amplitude 25 times smaller than the first one. This waveform has parameters $(f_o=1100\mathrm{Hz},\alpha =0.01)$ and is approximately monochromatic for the durations for which our sensitivity studies were designed. The clustering algorithm could detect the weaker signal at distances not further than a few kiloparsecs.

Figure 2

These power evolutions correspond to the signals in Fig. 1. We see that the blue plot corresponds to a rapidly decaying signal: within 2500 s, the radiation power drops to 17% of its original value. The red plot corresponds to a power decay that dropped to only 99.9% of the initial. The power of this (red) signal is about 3 orders of magnitude lower than that of the signal plotted in blue. We chose this weak signal so that we can examine how the MLAs compare to the clustering algorithm for both powerful and weak signals.

Figure 3

These are the detection efficiencies for the $(f_o=1500\mathrm{Hz},\alpha =0.1)$ waveform binned every 0.2 kpc. This waveform was the most powerful signal we used in our previous sensitivity study [8]. The dashed-dotted black line corresponds to the SC algorithm sensitivity study at a FAP equal to 0.1%. The (blue, green, and red) dashed-dotted lines correspond to the detection efficiencies of the SC algorithm at FAPs of 4%, 5%, and 10%. The (blue, green, and red) solid lines correspond to the detection efficiencies of the ANN (at 4%), SVM (at 5%), and CSC (at 10%), respectively. This plot demonstrates that (when compared for the same FAP) the MLAs performance is at least as good as that of the SC algorithm. At the 50% FDR, the ANN shows an increase of $\sim 17\%$ in the detection distance—from $\sim 1.45$ (of the SC algorithm dashed-dotted blue line) to $\sim 1.80\mathrm{Mpc}$ (of the solid blue line). The SVM shows an increase of $\sim 23\%$—from $\sim 1.50$ (of the SC algorithm dashed-dotted green line) to $\sim 1.85\mathrm{Mpc}$ of the solid-green line. The CSC shows an increase of $\sim 13\%$—from $\sim 1.50$ (of the SC algorithm dashed-dotted red line) to $\sim 1.70\mathrm{Mpc}$ of the solid red line.

Figure 4

These are the detection efficiencies for the $(f_o=1500\mathrm{Hz},\alpha =0.1)$ waveform binned every 0.2 kpc. This waveform was the most powerful signal we used in our previous sensitivity study [8]. The dashed-dotted black line corresponds to the SC algorithm sensitivity study at a FAP equal to 0.1%. The (blue, green, and red) dashed-dotted lines correspond to the detection efficiencies of the SC algorithm at FAPs of 18%, 22%, and 36%, respectively, though the detection efficiencies at 18% and 22% do not differ, so the lines overlap. The (blue, green, and red) solid lines correspond to the detection efficiencies of the ANN (at 18%), SVM (at 22%), and CSC (at 36%), respectively. At the 50% FDR, the ANN algorithm shows an increase of $\sim 137\%$ in the detection distance—from 1.50 (of the SC algorithm dashed-dotted blue line) to 3.55 Mpc (of the solid blue line). The SVM algorithm shows an increase of $\sim 83\%$ in the detection distance—from 1.50 (of the SC algorithm dashed-dotted green line) to 2.75 Mpc (of the solid green). The CSC shows an increase of $\sim 46\%$ in the detection distance—from $\sim 2.60$ (of the SC algorithm dashed-dotted red line) to $\sim 3.40\mathrm{Mpc}$ (of the solid red line). The distance range covered in this set has a practical significance because it covers: (a) the distance of 3.5 Mpc at which the January 2014 supernova occurred in M82 and (b) the distance of 5 Mpc at which the supernova event rate in the Milky Way neighborhood is about one every one to two years.

Figure 5

These are the detection efficiencies for the $(f_o=1100\mathrm{Hz},\alpha =0.01)$ waveform binned every 0.2 kpc. In our previous sensitivity study this waveform was shown to be detectable only at distances that cover the Milky Way galaxy. This waveform is approximately monochromatic for the first 2500 s right after the signal starts getting emitted by the source. This is the time interval over which our sensitivity study was performed. The (blue, green, and red) dashed-dotted lines correspond to the detection efficiencies of the SC algorithm at FAPs of 18%, 22%, and 36%, respectively, though the detection efficiencies at 18% and 22% do not differ, so the lines overlap. The (blue, green, and red) solid lines correspond to the detection efficiencies of the ANN (at 18%), SVM (at 22%), and CSC (at 36%), respectively. At the 50% FDR, the ANN shows an increase of $\sim 18\%$ in the detection distance—from $\sim 170$ (of the SC algorithm dashed-dotted blue line) to $\sim 210\mathrm{kpc}$ (of the solid blue line). The SVM shows a decrease of $\sim 24\%$—from $\sim 170$ (of the SC algorithm dashed-dotted green line) to $\sim 140\mathrm{kpc}$ of the solid green line. The CSC shows a decrease of $\sim 3\%$—from $\sim 185\mathrm{kpc}$ (of the SC algorithm dashed-dotted red line) to $\sim 10\mathrm{kpc}$ of the solid red line.

Figure 6

This is one of the noise ft maps with the original resolution of $1000\times 5000$ pixels. The pixels along the vertical axis correspond to 1 Hz each. The pixels along the horizontal axis correspond to 0.5 s each; hence, the total duration of the map is 2500 s. The frequency cuts are well-known seismic frequency bands and suspension vibration modes.

Figure 7

The highest training efficiency for the MLAs was achieved with resolution reduction by a factor of 100 per axis, as seen in Fig. 10. This reduced $10\times 50$ resolution ft map corresponds to the full resolution noise map in Fig. 6. For the resolution reduction, we used bicubic interpolation as provided by the matlab imresize.m function. The frequency cuts were substituted with zeros before reducing the resolution.

Figure 8

This is an injection added to the noise ft map shown in Fig. 6. The waveform has parameters $\alpha =0.1$ and $f_0=1500\mathrm{Hz}$. The duration of the injection is 2500 s and corresponds to a distance to the source of 117 kpc. Injections at longer distances are harder to see by eye in the original resolution maps. The contrast between signal pixels and noise pixels is higher in the reduced resolution maps as shown in Fig. 7. This makes it easier to see the injections in the reduced resolution maps rather than the full resolution ft maps.

Figure 9

This reduced $10\times 50$ resolution ft map corresponds to the full resolution map in Fig. 8. Despite the 10 000 times reduced resolution as compared to the ft map of Fig. 8, the $r$-mode injection is still visible. It turns out that the reduced resolution ft maps increase the training efficiency for the MLAs, according to Fig. 10. However, for the parameter estimation algorithms, we use the full resolution ft maps.

Figure 10

Training efficiencies of ANN (blue), SVM (green), and CSC (red) vs the resolution reduction (per axis). For SVM and CSC, there is a clear peak at a resolution reduction factor of ${10}^{-2}$. The ANN peak seems to be a little off, but for uniformity, we used ${10}^{-2}$ for all three MLAs. The training of all three MLAs was performed using the ($\alpha =0.1,f_o=1500$) waveform. No tests have been performed to verify the validity of these plots for other waveforms or other $h$ value ranges.