Bayesian inference for PCA and MUSIC algorithms with unknown number of sources

TL;DR

This paper introduces a Bayesian approach to accurately estimate the number of sources in PCA and MUSIC algorithms, improving upon traditional methods like AIC by providing exact MAP estimates for source count in signal processing.

Contribution

It is the first to apply Bayesian methods for computing the MAP estimate of the number of sources in PCA and MUSIC algorithms, enhancing their accuracy.

Findings

Bayesian MAP estimate outperforms AIC in simulations

Exact MAP estimates are derived for uncorrelated steering vectors

Proposed method improves source number estimation accuracy

Abstract

Principal component analysis (PCA) is a popular method for projecting data onto uncorrelated components in lower dimension, although the optimal number of components is not specified. Likewise, multiple signal classification (MUSIC) algorithm is a popular PCA-based method for estimating directions of arrival (DOAs) of sinusoidal sources, yet it requires the number of sources to be known a priori. The accurate estimation of the number of sources is hence a crucial issue for performance of these algorithms. In this paper, we will show that both PCA and MUSIC actually return the exact joint maximum-a-posteriori (MAP) estimate for uncorrelated steering vectors, although they can only compute this MAP estimate approximately in correlated case. We then use Bayesian method to, for the first time, compute the MAP estimate for the number of sources in PCA and MUSIC algorithms. Intuitively, this MAP estimate corresponds to the highest probability that signal-plus-noise's variance still dominates projected noise's variance on signal subspace. In simulations of overlapping multi-tone sources for linear sensor array, our exact MAP estimate is far superior to the asymptotic Akaike information criterion (AIC), which is a popular method for estimating the number of components in PCA and MUSIC algorithms.