Inference with finite time series: Observing the gravitational Universe through windows

Time series analysis is ubiquitous in many fields of science including gravitational-wave astronomy, where strain time series are analyzed to infer the nature of gravitational-wave sources, e.g., black holes and neutron stars. It is common in gravitational-wave transient studies to apply a tapered window function to reduce the effects of spectral artifacts from the sharp edges of data segments. We show that the conventional analysis of tapered data fails to take into account covariance between frequency bins, which arises for all finite time series—no matter the choice of window function. We discuss the origin of this covariance and derive a framework that models the correlation induced by the window function. We demonstrate this solution using both simulated Gaussian noise and real Advanced LIGO/Advanced Virgo data. We show that the effect of these correlations is similar in scale to widely studied systematic errors, e.g., uncertainty in detector calibration and power spectral density estimation.

Figure 1

The estimated (top) and analytic (bottom) covariance matrix for simulated noise using a noise power spectral density estimated from LIGO Livingston data around the time of the binary black hole merger GW170814. The color bar indicates ${log}_{10}$ power spectral density in units of ${\mathrm{Hz}}^{-1}$. The estimate is obtained using approximately three months of simulated data. The off-diagonal behavior agrees well with the analytic expression. The window used is a Tukey window with $\alpha =0.1$.

Figure 2

Maximum contamination per frequency bin [Eq. (17)] for PSDs estimated near the time of binary black hole merger GW170814 [13]. We see two competing effects. First, there is broadband contamination at $\approx 10\%$ when the PSD is slowly varying. The magnitude of this contamination rises with increasing Tukey $\alpha$ (see Appendix pp4). There is also large contamination near instrumental lines due to spectral leakage that is suppressed with increasing Tukey $\alpha$.

Figure 3

Eigenvalues of an estimated noise power covariance matrix for data close to GW170814 for a Tukey window with $\alpha =0.1$. The large eigenvalues correspond to spectral lines and the low-frequency seismic wall; the slowly varying region corresponds to the smoothly varying observing band. The dashed vertical line denotes the points after which we discard the eigenmodes as determined by the window information loss. The dotted lines indicate the point at which the eigenvalue drops below the minimum of the finite-duration PSD. Both of these well approximate the turnover after which the eigenvalues rapidly decline due to information loss from windowing. The difference is most pronounced for the Virgo data, which is consistent with the increased contamination between frequency bins (see Fig. 14).

Figure 4

Eigenmodes for the covariance matrix estimated from data from the LIGO Livingston interferometer close to GW170814. This matrix encodes the correlations between physical frequencies. The horizontal axis corresponds to the physical frequencies, while the vertical axis is the order of decreasing eigenvalue. The bottommost eigenmodes are the ones that are discarded. These eigenmodes are associated with very broadband frequency content.

Figure 5

Intrinsic parameters for the binary black hole merger GW150914 [12] using our new finite-duration likelihood (blue) and the diagonal likelihood (orange). The primary and secondary mass ($m_1, m_2$) refer, respectively, to the more-massive and less-massive component masses. Including covariance between neighboring bins has no observable impact on the inferred posterior.

Figure 6

Intrinsic parameters for the binary black hole merger GW170814 [13] using our new finite-duration likelihood (blue) and the diagonal likelihood (orange). The primary and secondary mass ($m_1, m_2$) refer, respectively, to the more-massive and less-massive component masses. Including covariance between neighboring frequency bins slightly shifts the inferred posterior distribution for the mass ratio.

Figure 7

Intrinsic parameters for the binary black hole merger GW190521 [14] using our new finite-duration likelihood (blue) and the diagonal likelihood (orange). The primary and secondary mass ($m_1, m_2$) refer, respectively, to the more-massive and less-massive component masses. Including covariance between neighboring bins has no observable impact on the inferred posterior.

Figure 8

The signals considered for our population test. The four signals we consider are a Gaussian burst centered at 50 Hz and standard deviation $10\mathrm{Hz}$ (blue), a Gaussian burst centered at $500\mathrm{Hz}$ and standard deviation 10 Hz (orange), a $60M_{\odot }$ binary black hole merger (green), and a $300M_{\odot }$ binary black hole merger (red). In purple, we show the finite-duration amplitude spectral density for the LIGO Livingston interferometer at the time of binary black hole merger GW170814. The Gaussian bursts isolate the impact of specific spectral features, e.g., the large lines around $500\mathrm{Hz}$. The binary black hole mergers are broadband and accumulate most of their signal-to-noise ratio at low frequencies, near the sharp rise due to seismic noise.

Figure 9

The posterior distribution for the fraction of segments containing a signal in our toy example (Sec. 3b). We analyze $\approx 7.5$ days of simulated Gaussian noise divided into $4s$ segments ($1.6\times {10}^5$ segments), and 15 000 of these segments contain four different signals: (i) a Gaussian pulse with standard deviation $10\mathrm{Hz}$ and central frequency $50\mathrm{Hz}$, (ii) a Gaussian pulse with central frequency $500\mathrm{Hz}$, (iii) an equal-mass binary black hole merger signal with total mass $60M_{\odot }$, and (iv) an equal-mass binary black hole merger signal $300M_{\odot }$. The amplitude of the signals and the PSD are shown in Fig. 8. In blue we show the posterior distribution obtained using the standard likelihood that ignores correlations between neighboring frequency bins induced by the window function. In orange, we show the posterior distribution using a likelihood that accounts for these correlations. We see that the diagonal method gives a biased result when the signal is centered at $500\mathrm{Hz}$ and when analyzing simulated binary black hole systems.

Figure 10

The ratio of the inferred finite duration PSD ($S_i$) with a $1/4\mathrm{Hz}$ resolution when using different longer segment durations ($D$) to estimate the infinite-duration PSD (${\mathcal{S}}_{\mu }$).

Figure 11

PSD matrix (top left), regularized PSD matrix (top right), SVD eigenmatrix (bottom left), and regularized inverse PSD matrix (bottom right) for the LIGO Hanford observatory at the time of GW170814.

Figure 12

PSD matrix (top left), regularized PSD matrix (top right), SVD eigenmatrix (bottom left), and regularized inverse PSD matrix (bottom right) for the LIGO Livingston observatory at the time of GW170814.

Figure 13

PSD matrix (top left), regularized PSD matrix (top right), SVD eigenmatrix (bottom left), and regularized inverse PSD matrix (bottom right) for the Virgo observatory at the time of GW170814.

Figure 14

Fractional contamination for white noise [Eq. (20)] as a function of Tukey $\alpha$ parameter. We note that the bias is a monotonic function of $\alpha$ ranging from ${\mathrm{\Delta }}_i=0$ for $\alpha =0$ to ${\mathrm{\Delta }}_i\approx 2/3$ for $\alpha =1$.