Explicit Inverse of the Covariance Matrix of Random Variables with Power-Law Covariance

Abstract

This paper presents the explicit inverses of a special class of symmetric matrices with power-law elements, that is, the element on the m-th row and the n-th column is${\rho ^{\left| {{l_m} - {l_n}} \right|}}$, where ρ ∈ [0, 1) is the power-law coefficient and lm is a real number. We derive the explicit inverse matrix and find that it follows a tridiagonal structure. The complexity of the inverse operation scales with$\mathcal{O}\left( N \right)$, with N being the size of the square matrix. The matrix can be considered as the covariance matrix of random variables sampled from a linear wide-sense stationary (WSS) random field, with lm being the coordinate or time stamp of the samples. With the inverse covariance matrix, the discrete random samples are used to reconstruct the continuous random field by following the minimum mean squared error (MMSE) criterion. It is discovered that the MMSE estimation demonstrates a Markovian property, that is, the estimation of any given point in the field using the two discrete samples immediately adjacent to the point of interest yields the same results as using all the N discrete samples.