Variational inference for correlated gravitational wave detector network noise

Gravitational wave detectors like the Einstein Telescope and LISA generate long multivariate time series, which pose significant challenges in spectral density estimation due to the large number of observations as well as the presence of correlated noise. Addressing both issues is crucial for accurately interpreting the signals detected by these instruments. This paper presents an application of a variational inference spectral density estimation method specifically tailored for dealing with correlated noise in the data. It is flexible in that it does not rely on any specific parametric form for the multivariate spectral density. We provide the multivariate Whittle likelihood in a form that is easy to evaluate as it depends on a low-dimensional covariance matrix. To deal with very long time series, the method employs a blocked Whittle likelihood approximation for stationary time series. It utilizes the Cholesky decomposition of the inverse spectral density matrix to ensure a positive definite estimator. A discounted regularized horseshoe prior is applied to the spline coefficients of each Cholesky factor, and the posterior distribution is computed using a stochastic gradient variational Bayes approach. This method is particularly effective in addressing correlated noise, a significant challenge in the analysis of multivariate data from colocated detectors. The method is demonstrated by analyzing 2000 s of simulated Einstein Telescope noise, which shows its ability to produce accurate spectral density estimates and quantify coherence between time series components. This makes it a powerful tool for analyzing correlated noise in gravitational wave data.