Deep learning for intermittent gravitational wave signals

The ensemble of unresolved compact binary coalescences is a promising source of the stochastic gravitational-wave (GW) background. For stellar-mass black hole binaries, the astrophysical stochastic GW background is expected to exhibit non-Gaussianity due to their intermittent features. We investigate the application of deep learning to detect such a non-Gaussian stochastic GW background and demonstrate it with the toy model employed by Drasco and Flanagan in 2003, in which each burst is described by a single peak concentrated at a time bin. For the detection problem, we compare three neural networks with different structures: a shallower convolutional neural network (CNN), a deeper CNN, and a residual network. We show that the residual network can achieve comparable sensitivity as the conventional non-Gaussian statistic for signals with the astrophysical duty cycle of ${\log }_{10}\xi \in [-3,-1]$. Furthermore, we apply deep learning for parameter estimation with two approaches in which the neural network (1) directly provides the duty cycle and the signal-to-noise ratio and (2) classifies the data into four classes depending on the duty cycle value. This is the first step of a deep learning application for detecting a non-Gaussian stochastic GW background and extracting information on the astrophysical duty cycle.

Figure 1

Example of the signal model. We see four bursts at times 5, 9, 13, and 18. Each burst is represented by a single peak.

Figure 2

Schematic picture of a residual block. A standard layer transforms an input $\mathit{x}$ into an output $F(\mathit{x})$, while a shortcut connection directly passes the input to the output. In total, the residual block returns their sum $F(\mathit{x})+\mathit{x}$. If the input $\mathit{x}$ and the output $F(\mathit{x})$ have different shapes, the data passing through the shortcut connection are reshaped appropriately to have the same shape.

Figure 3

Structure of the residual block we used in this work. The main path consists of the three convolutional layers and the batch normalization layer. In the shortcut connection, the convolutional layer reshapes the data so that the size of the output matches that of the output of the main path.

Figure 4

Minimum detectable SNR with 90% detection probability for the non-Gaussian statistic, two convolutional neural networks (shallower and deeper), and the residual network. The false alarm rate is set at 5%. The black squares are the non-Gaussian statistic, the blue squares are the shallower CNN, the orange circles are the deeper CNN, and the green circles are the residual network. For visibility, the dots are slightly shifted in the horizontal direction. The error bar shows the standard deviation of ${\rho }_{90\%}$ evaluated by four independent runs. The shaded area is the parameter region not used for training.

Figure 5

Parameter estimation of the duty cycle (left) and the SNR (right) by the neural network. The scatter plot shows the true value on the horizontal axis and the predicted value on the vertical axis. The diagonal line represents equal values for the predicted and true values.

Figure 6

Errors in the duty cycle (top) and the SNR (bottom) for different fiducial values of the duty cycle. The blue circles, orange squares, and green triangles, respectively, show the results of the datasets with the fiducial SNR of 10, 30, and 50. Each dot shows the average of the error, and the error bar represents the standard deviation of the errors. The left panels show the results of the neural network trained with ${\log }_{10}\xi \in [-2,0]$, and the right panels are the ones trained with ${\log }_{10}\xi \in [-4,0]$.

Figure 7

Confusion matrix for the duty cycle estimation. The row and column represent the true label and predicted label, respectively. Each class is labeled by the integer $\{1,2,3,4\}$ and they correspond to ${\log }_{10}\xi \in [-1,0)$, $[-2,-1)$, $[-3,-2)$, $[-4,-3)$, respectively. The numbers are in units of percent and represent the fraction of data classified from the true label to the predicted label.

Figure 8

Distribution of the true values of (${\log }_{10}\xi ,\rho$) of the misclassified events. The color of the dots represents the maximum value of the probability among $\{p_i{\}}_{i=1,2,3,4}$.

Figure 9

Cumulative fraction of the maximum value of the predicted probabilities in descending order. Blue and orange lines correspond to the misclassified and correctly classified events, respectively. Note that the numbers of correctly classified and misclassified events are different: 1905 events are correctly classified, and 143 events are misclassified.

Figure 10

Minimum detectable SNR as a function of the duty cycle $\xi$. Both the false alarm probability and the false dismissal probability are set to 0.1. Error bars are obtained by four independent runs. The time-series data have the length of $N={10}^4$, and the detector’s noise variances are ${\sigma }_1^2={\sigma }_2^2=1$. Note that ${\rho }_{90\%}$ represents the minimum detectable SNR with 90% detection probability and is different from that of ${\mathrm{\Omega }}_{\text{detectable}}$ in Fig. 1 of DF03.

Figure 11

The color map shows the logarithm of ${\lambda }_{\mathrm{ML}}^{\mathrm{NG}}({\alpha }^2,\xi )$. The data length is $N={10}^4$, and we set the signal parameters to $\xi =0.2$ and $\rho =40$ (indicated by the blue square). The variances of the detector noises are ${\sigma }_1^2={\sigma }_2^2=1$. The red circle indicates the parameter values of the maximum log-likelihood. Black dashed lines indicate the constant SNR for $\rho =60$, 40, 20 from top to bottom. It is clearly seen that the likelihood estimator has a degeneracy along the parameter combination that gives the same SNR value.