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

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. In many systems of array signal processing, e.g. in radar, sonar and antenna systems, linear sensor array is the most basic and universal mathematical model. Because far distant sources with different directions of arrival (DOAs) will oscillate the steering sensor array with different angular frequencies, the array's output data is then a superposition of sinusoidal signals [1]. Hence, a common problem of array systems is to detect the number of sources, as well as their tone frequencies and DOAs, from noisy sinusoidal signals. In literature, most papers only consider the case of single-tone or narrowband sources (i.e. When the number of sources is small, the DOA's line spectra are sparse and can be estimated effectively via sparse techniques like atomic norm (also known as total variation norm) [1], [2], LASSO [4], [5] and Bayesian compressed sensing [6], [7].