Maximum Entropy Spectral Analysis: an application to gravitational waves data analysis

Abstract: The Maximum Entropy Spectral Analysis (MESA) method, developed by Burg, offers a powerful tool for spectral estimation of a time-series. It relies on Jaynes' maximum entropy principle, allowing the spectrum of a stochastic process to be inferred using the coefficients of an autoregressive process AR($p$) of order $p$. A closed-form recursive solution provides estimates for both the autoregressive coefficients and the order $p$ of the process. We provide a ready-to-use implementation of this algorithm in a Python package called \texttt{memspectrum}, characterized through power spectral density (PSD) analysis on synthetic data with known PSD and comparisons of different criteria for stopping the recursion. Additionally, we compare the performance of our implementation with the ubiquitous Welch algorithm, using synthetic data generated from the GW150914 strain spectrum released by the LIGO-Virgo-Kagra collaboration. Our findings indicate that Burg's method provides PSD estimates with systematically lower variance and bias. This is particularly manifest in the case of a small (O($5000$)) number of data points, making Burg's method most suitable to work in this regime. Since this is close to the typical length of analysed gravitational waves data, improving the estimate of the PSD in this regime leads to more reliable posterior profiles for the system under study. We conclude our investigation by utilising MESA, and its particularly easy parametrisation where the only free parameter is the order $p$ of the AR process, to marginalise over the interferometers noise PSD in conjunction with inferring the parameters of GW150914.