Bayesian nonparametric spectral density estimation using B-spline priors

TL;DR

This paper introduces a Bayesian nonparametric method using B-spline priors for spectral density estimation of stationary time series, improving accuracy for complex spectral features and allowing data-driven model complexity.

Contribution

It proposes a novel B-spline based prior for spectral density estimation, generalizing previous Bernstein polynomial approaches, with an efficient MCMC sampling scheme and demonstrated superior performance.

Findings

More accurate spectral density estimates for complex signals

Effective estimation of spectral features with sharp changes

Successful application to real-world data including sunspot and gravitational wave data

Abstract

We present a new Bayesian nonparametric approach to estimating the spectral density of a stationary time series. A nonparametric prior based on a mixture of B-spline distributions is specified and can be regarded as a generalization of the Bernstein polynomial prior of Petrone (1999a,b) and Choudhuri et al. (2004). Whittle's likelihood approximation is used to obtain the pseudo-posterior distribution. This method allows for a data-driven choice of the number of mixture components and the location of knots. Posterior samples are obtained using a Metropolis-within-Gibbs Markov chain Monte Carlo algorithm, and mixing is improved using parallel tempering. We conduct a simulation study to demonstrate that for complicated spectral densities, the B-spline prior provides more accurate Monte Carlo estimates in terms of $L_1$-error and uniform coverage probabilities than the Bernstein polynomial prior. We apply the algorithm to annual mean sunspot data to estimate the solar cycle. Finally, we demonstrate the algorithm's ability to estimate a spectral density with sharp features, using real gravitational wave detector data from LIGO's sixth science run, recoloured to match the Advanced LIGO target sensitivity.