Entropy-based covariance determinant estimation

Abstract

An information-theoretic approach is described to estimate the determinant of the covariance matrix of a random vector sequence (a common task in a wide range of estimation and detection problems in signal processing for communications). The method is based on a prior entropy-based processing of the data using kernels and offers robustness against small-entropy contamination. The trade-off between optimality, accuracy and robustness is analyzed, along with the impact of the relative kernel bandwidth and data size.

Robust estimation of the magnitude squared coherence based on kernel signal processing

“…This rationale coincides with the previous work [9] from the authors about the robust estimation of the covariance determinant on the univariate case. However, this paper extends the idea of robust estimate for the bivariate case given its interest on studying the relation between two signals.…”

“…From the previous authors paper [9] and following a similar rationale, we can obtain an estimate of the IP based on Gaussian kernels, which yieldŝ…”

“…For simplicity, we will assume that [w] 1,1 = [w] 2,2 = w such as W 1 /W 2 = Σ 1 /Σ 2 . If the marginal variances of the original processes are known, only the kernel bandwith is needed to be estimated, for instance using the iterative method proposed in [9] by adding the previous condition. Otherwise, both kernel bandwidth and marginal variances need to be estimated by the univariate robust estimate of the covariance matrix described in the same work.…”

“…where the variance of the MIP, σ 2 U , is given by the Appendix VII-B from [9] and has the following form:…”

“…For the problem of estimating the MSC, the minimum kernel bandwidth is inversely proportional to the square root of the data size, N , in contrast with the faster decay of O N −1 given in [9]. It is also worth noting that this result applies only for the proper complex case, and that the obtained decay would instead be O(N −1 ) and O(N −2 ) for the MSC and the variance, respectively, in the bivariate real case, which would make less critical the selection of the bandwidth.…”