Characterization of non-Gaussianity in gravitational wave detector noise

The first detection of a gravitational wave (GW) has been achieved by two detectors of the advanced LIGO. Routine detections of GW events from various GW sources are expected in the coming decades. Although the first signal was statistically significant, we expect to see numerous low signal-to-noise ratio (SNR) events with which we may be able to learn various aspects of the Universe that have yet to be unveiled. On the other hand, instrumental glitches due to nonstationarity and/or a non-Gaussian tail of detector noise distribution prevent us from confidently identifying true but low SNR GW signals out of instrumental noise. Thus, to make the best use of data from GW detectors, it is important to establish a method to safely distinguish true GW signals from false signals due to instrumental noises. For this purpose, we urgently need to understand characteristics of detector noises, since the nonstationarity and non-Gaussianity inherent in detector outputs are known to increase false detections of signals. Focusing on identifying the non-Gaussian noise components, this paper introduces a new measure for characterizing the non-Gaussian noise components using the parameter $\nu$ which characterizes the weight of tail in a Student-t distribution. A confidence interval is reported on the extent to which detector noise deviates from Gaussianity. Our method revealed stationary and transient deterioration of Gaussianity in LIGO S5 data.

Figure 1

This plot shows the 50% (red stars), 90% (green squares), 95% (blue circles), and 99% (pink triangles) quantiles of the distribution of the LIGO S5 data taken from GPS time 842 747 904 to 842 764 288. Each quantile was calculated from $2^{18}$ samples in every 16 Hz. The dashed line, dashed-dotted line, double-dotted line, and triple-dotted line represent the 50%, 90%, 95%, and 99% quantiles expected if the data follow a Gaussian distribution, respectively. This figure indicates that detector noise deviates from Gaussian distribution especially in the low-frequency band.

Figure 2

This plot shows Rayleigh distribution (black solid line) and Student-Rayleigh distribution functions with ${\sigma }_s=1$ for various $\nu$. Each color means $\nu =4$ (blue dashed line), $\nu =8$ (green dotted line), $\nu =16$ (pink double-dotted line), and $\nu =64$ (red triple-dotted line). In the small $\nu$ case, the tail of the distribution is heavy. On the other hand, the Student-Rayleigh distribution is closer to the Rayleigh distribution for large $\nu$.

Figure 3

The algorithm of estimating $\widehat{\nu }$ from the detector noise.

Figure 4

The resulting histogram of the estimated $\widehat{\nu }$ for 524,288 sets of simulated Gaussian noise data. The data length of the simulated noise is 4096 s. This figure shows that even for purely Gaussian noise, $\widehat{\nu }$ can be small with a non-negligible probability. The critical region for rejecting Gaussianity is determined by the histogram. When data length $T=4096\mathrm{s}$, ${\nu }_{\text{th}}=91.4$.

Figure 5

This plot shows the critical regions for rejecting Gaussianity with significance levels of 1% and 0.1%. In the lower region of each line, Gaussian noise hypothesis is rejected with 99% (red) and 99.9% (blue) confidence. For example, when $\nu$ is estimated from 1024 second long data, the hypothesis of Gaussianity is rejected if $\nu \le 59.0$ with 99% confidence.

Figure 6

The result of estimated $\nu$ for 524,288 sets of simulated Student-t noise in the case of $\nu =25$. Estimated $\nu$ fluctuate, and the 1% confidence upper (${\nu }_{\text{upper}}$) and lower (${\nu }_{\text{lower}}$) boundaries are ${\nu }_{\text{upper}}=48.8$ and ${\nu }_{\text{lower}}=23.4$.

Figure 7

The 98% (red line) and 99.8% (blue line) confidence interval of $\nu$ when the estimation time is 4096 s. For small $\nu$ values, resolution of $\nu$ is good. For example, when we obtain $\nu =25$ in the case of 4096 s estimation time, the 98% confidence interval of $\nu$ is $15.45\le \nu \le 27.29$.

Figure 8

Dependence of the confidence interval as a function of the data length. The precision of the estimated $\nu$ is bad for short data.

Figure 9

This plot shows the estimated $\widehat{\nu }$ for the real LIGO S5 data as a function of time. The GPS time of the data is from 842 747 904 to 842 760 192; the resolution of time $\delta t$ and frequency $\delta F$ are 128 s and 16 Hz, respectively. This result is obtained by the data length $T=1024\mathrm{s}$ with overlapped time $T_{\mathrm{lap}}=896\mathrm{s}$. The purple region means that the Gaussianity of noise is bad and there are many non-Gaussian regions, especially in the low-frequency band.

Figure 10

This plot shows the distributions of the LIGO detector noise in the time-frequency region where time from $\mathrm{GPS}=842747904$, $t$; frequency, $f$; and their widths, $dt$, $df$, are ($t=6144$, $f=128$, $dt=128\mathrm{s}$, $df=16\mathrm{Hz}$) for the red histogram and ($t=1024$, $f=32$, $dt=128\mathrm{s}$, $df=16\mathrm{Hz}$) for the blue one. The entry of each histogram is 16384.

Figure 11

These plots show the time evolution of the observed $\widehat{\nu }$ at each frequency. The GPS times of the top left, top right, bottom left, and bottom right panels are 841 449 472, 842 489 856, 864 575 488, and 870 838 272, respectively. Other parameters are the same as the ones in Fig. 9. The bottom right panel shows the bad quality data which do not belong to CAT4 in the literature of the LIGO and Virgo collaborations [17]. The other panels show good quality data which belong to CAT4.