Bayesian inference for spectral estimation of gravitational wave detector noise

Gravitational wave data from ground-based detectors is dominated by instrument noise. Signals will be comparatively weak, and our understanding of the noise will influence detection confidence and signal characterization. Mismodeled noise can produce large systematic biases in both model selection and parameter estimation. Here we introduce a multicomponent, variable dimension, parametrized model to describe the Gaussian-noise power spectrum for data from ground-based gravitational wave interferometers. Called BayesLine, the algorithm models the noise power spectral density using cubic splines for smoothly varying broadband noise and Lorentzians for narrow-band line features in the spectrum. We describe the algorithm and demonstrate its performance on data from the fifth and sixth LIGO science runs. Once fully integrated into LIGO/Virgo data analysis software, BayesLine will produce accurate spectral estimation and provide a means for marginalizing inferences drawn from the data over all plausible noise spectra.

Figure 1

Example strain spectral density (square root of the Fourier power; gray, dashed) and $\sqrt{2S_n(f)/T}$ (red, solid) during LIGO’s 5th science run, compared to the design sensitivity (blue, dotted). Notice that the design sensitive consists strictly of broadband features due to seismic, thermal, and quantum noise, while the actual data show high-amplitude narrow-band spectral lines. The design sensitivity is the type of noise curve used for studies in simulated data, while the red (solid) line is needed for analyzing the actual data.

Figure 2

Demonstration of the time variability for the broadband Gaussian noise. Top panel: PSD over 1024 s of S6 data at 100 Hz estimated using BayesLine over intervals of 16 (red), 32 (green), and 64 (blue) seconds of data. The thick line is the median PSD from each segment’s chain and the shaded region spans the 90% credible interval. Bottom panel: Legendre polynomial fit to the data, where a RJMCMC was used to determine the best-fit degree ($n$) for the polynomial ($n=10$ for 16 s, $n=7$ for 32 s, and $n=11$ for 64 s).

Figure 3

The 60 Hz power line shows nonstationary frequency (left axis, red solid line) and amplitude (right axis, blue dashed line) over timescales shorter than the amount of data needed to characterize binary in spiral signals. Higher frequency harmonics are practically 100% correlated with the main power line.

Figure 4

Example S6 data power spectrum (gray, dashed) with BayesLine PSD shown separated into the cubic spline (red, solid) and Lorentzian (blue, dotted) components.

Figure 5

Initial LIGO example distribution of whitened Fourier domain data using the standard PSD estimation in LALInference (blue, dotted) and the BayesLine PSD (red, solid), compared to a normal distribution with zero mean and unit variance (gray, dashed) which is assumed by the likelihood function.

Figure 6

Example distribution of whitened Fourier domain data made from the amount of data consistent with what is needed for Advanced LIGO BNS analyses. PSDs were determined via the standard PSD estimation in LALInference using 10 (magenta, dotted) and 30 (blue, short dashed) 1024 s segments of data for averaging and the BayesLine PSD (red, solid). All three distributions are compared to $N[0,1]$ (gray, long dashed).

Figure 7

Comparison of marginalized posterior distribution functions of the mass ratio $q=m_2/m_1$, chirp mass $M_{\text{chirp}}=(m_1{m}_2{)}^{3/5}(m_1+m_2{)}^{-1/5}$, and luminosity distance $D_L$ for two different simulated compact binary signals in 16 s of real detector noise using LALInterference. The x-axes are shifted by the true value of each parameter. Results obtained with the BayesLine PSD (red, solid) are compared to posteriors using off-source PSDs determined from 256 s of data before the signal entered the LIGO band (gray, solid), 256 s after the merger time (dark blue, dashed) and 512 s after the merger time (blue, dotted). When the noise spectrum is stable the BayesLine PSD agrees well with off-source estimates (bottom row) and does not “harm” parameter estimation. Posteriors in the top row are sensitive to the start time of the data used for spectral estimation. The BayesLine posterior is most consistent with the results using data immediately after the trigger time. This result suggests that BayesLine will enable parameter estimates that are more robust to long-term nonstationary noise.

Figure 8

Comparison of reconstructed waveforms in simulated data using the true PSD (red) and the BayesLine PSD (blue). The top panels are the whitened data (grey) and waveforms. Solid lines indicate the median reconstructed signal while dashed lines span the $1\sigma$ errors. The bottom panel shows the whitened residual between the two recovered waveforms, which is consistent with zero. The figure demonstrates how including the BayesLine PSD parameters does not degrade signal recovery.

Figure 9

The power spectrum of the data [red, solid], the coherent 60 Hz line model [green, dashed] and the residual [blue, dotted] after the line is subtracted.

Figure 10

The frequency [left axis; red, solid] and amplitude [right axis; blue short-dashed] of the data as a function of time as estimated by taking FFTs of each 16 s interval. The best fit model for $f(t)$ and $A(t)$ derived from the full 1024 s of data are shown in the green long-dashed curve and the magenta dotted curve, respectively.