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Principal component analysis (PCA) is a popular method for projecting dataonto uncorrelated components in lower dimension, although the optimal number ofcomponents 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 knowna priori. The accurate estimation of the number of sources is hence a crucialissue for performance of these algorithms. In this paper, we will show thatboth PCA and MUSIC actually return the exact joint maximum-a-posteriori (MAP)estimate for uncorrelated steering vectors, although they can only compute thisMAP 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 PCAand MUSIC algorithms. Intuitively, this MAP estimate corresponds to the highestprobability that signal-plus-noise's variance still dominates projected noise'svariance on signal subspace. In simulations of overlapping multi-tone sourcesfor linear sensor array, our exact MAP estimate is far superior to theasymptotic Akaike information criterion (AIC), which is a popular method forestimating the number of components in PCA and MUSIC algorithms.

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