Bayesian autoregressive spectral estimation

TL;DR

This paper introduces Bayesian autoregressive spectral estimation (BASE), which quantifies uncertainty in spectral estimates by deriving the full posterior distribution of AR parameters, improving upon traditional point estimates.

Contribution

The paper proposes a Bayesian framework for AR spectral estimation that accounts for uncertainty by integrating over AR parameter posteriors, unlike traditional methods.

Findings

BASE provides error bars alongside point estimates.

BASE closely matches ASE in point estimates.

Validated on synthetic and real signals.

Abstract

Autoregressive (AR) time series models are widely used in parametric spectral estimation (SE), where the power spectral density (PSD) of the time series is approximated by that of the \emph{best-fit} AR model, which is available in closed form. Since AR parameters are usually found via maximum-likelihood, least squares or the method of moments, AR-based SE fails to account for the uncertainty of the approximate PSD, and thus only yields point estimates. We propose to handle the uncertainty related to the AR approximation by finding the full posterior distribution of the AR parameters to then propagate this uncertainty to the PSD approximation by \emph{integrating out the AR parameters}; we implement this concept by assuming two different priors over the model noise. Through practical experiments, we show that the proposed Bayesian autoregressive spectral estimation (BASE) provides point estimates that follow closely those of standard autoregressive spectral estimation (ASE), while also providing error bars. BASE is validated against ASE and the Periodogram on both synthetic and real-world signals.