Optimal ${\chi }^2$ discriminator against modeled noise transients in interferometric data in searches for binary black-hole mergers

A vitally important requirement for detecting gravitational-wave (GW) signals from compact binary coalescences (CBCs) with high significance is the reduction of the false-alarm rate of the matched-filter statistic. The data from GW detectors contain transient noise artifacts, or glitches, which adversely affect the performance of search algorithms, especially for finding short-lived astrophysical signals, by producing false alarms, often with high signal-to-noise ratio (SNR). These noise transients particularly affect the CBC searches, which are typically implemented by cross-correlating detector strain data with theoretically modeled waveform templates, chosen from a template bank that is densely populated to cover the source parameter ranges of interest. Owing to their large amplitudes, many of the glitches can produce detectably large peaks in the SNR time series—termed “triggers”—in spite of their small overlap with the templates. Such glitches contribute to the false alarms. Historically, the traditional ${\chi }^2$ test has proven quite useful in distinguishing triggers arising from CBC signals and those caused by glitches. In a recent paper, a unified origin for a large class of ${\chi }^2$ discriminators was formulated, along with a procedure to construct an optimal ${\chi }^2$ discriminator, especially when the glitches can be modeled. A large variety of glitches that often occur in GW detector data can be modeled as sine-Gaussians, with quality factor and central frequency, ($Q,f_0$), as parameters. An important feature of a sine-Gaussian glitch is that there is a lag between its time of occurrence in the GW data and the time of the trigger it produces in a templated search. Therefore, this time lag is the third parameter used in characterizing the glitch. The total number of sampled points in the glitch parameter space is associated with the degrees of freedom (d.o.f.) of the ${\chi }^2$. We use singular value decomposition to identify the most significant d.o.f., which helps keep the computational cost of our ${\chi }^2$ down. Finally, we utilize the above insights to construct a ${\chi }^2$ statistic that optimally discriminates between sine-Gaussian glitches and CBC signals. We also use receiver-operating characteristics to quantify the improvement in search sensitivity when it employs the optimal ${\chi }^2$ compared to the traditional ${\chi }^2$. The improvement in detection probability is by a few to several percentage points, near a false-alarm probability of a few times ${10}^{-3}$, and holds for binary black holes with component masses from several to a hundred solar masses. Moreover, the glitches that are best discriminated against are those that are like sine-Gaussians with $Q\in [25,50]$ and $f_0\in [40,80]\mathrm{Hz}$.

Figure 1

The above plots show noise transients from the LIGO-Hanford detector during its second observation run [20]. These glitches not only trigger BBH templates, with SNRs of 35–40, but also project on sine-Gaussians substantially (see Sec. 4 below). The primary sine-Gaussian that captures most of the projections for the above ones has $f_0=120\mathrm{Hz}$ and $Q=5$, whereas the loudest BBH templates triggered by them have a projection of approximately 1%. The new ${\chi }^2$ statistic introduced later in this work has an order-of-magnitude larger value on these glitches compared to simulated BBH injections with the same parameters and SNRs as the loudest templates, thereby affording a way to discriminate between them. The projections of the above glitches on the best-fit sine-Gaussian (overlaid in red, for the first one) are 78% and 51%, respectively, while they are 8.0% and 9.4% for the clipped sine-Gaussians (see Sec. 3b). Since these glitches occur with high amplitudes, they nevertheless produce high ${\chi }^2$ values in spite of their seemingly small projections.

Figure 2

The figures show uniformly sampled points in the parameter space $(f_0,Q)$ in the range $40\le f_0\le 120\mathrm{Hz}$, $5\le Q\le 50$. The minimum projection is 80%. The figure on the left is for component masses of $7M_{\odot }$ each, and the total number of sampled points is 1288 (see the inset figures to note how closely spaced the neighboring points are). The figure on the right is for component masses of $25M_{\odot }$ each, and the number of points sampled is 156.

Figure 3

The traditional and optimal “SG” ${\chi }^2$ statistics (see the legend), per degree of freedom, are plotted vs SNR for various types of triggers. These arise from injections of simulated (a) noise (Gaussian), (b) glitches (sine-Gaussians, of the high-$Q$-low-$f_0$ type defined in Table 3), and (c) BBH signals, of the category-6 type defined in Table 4, when employing targeted template bank 3, as described in Table 2. The ROC curves for these triggers are shown in the left plot in Fig. 4.

Figure 4

In the left plot, we show the ROC curves comparing performances of the same two ${\chi }^2$ statistics and triggers as in Fig. 3. The right plot is a similar comparison, for the same Gaussian noise and sine-Gaussian glitch triggers but for BBH injections of category 5 of Table 4, using targeted bank 3 of Table 2.

Figure 5

ROC curves comparing performances of the two ${\chi }^2$ statistics and triggers as in Fig. 3, except for the BBH injections, which are of categories 3 (left) and 4 (right) of Table 4, both using targeted bank 2 of Table 2. The sudden drop in the ROC for ${\chi }_{\mathrm{trad}}^2$ around a FAP of ${10}^{-3}$ most likely arises owing to the difficulty of producing enough loud triggers purely from noise and should be interpreted with care when comparing with the other ROC curve.