Bias Adjusted Sign Covariance Matrix

Abstract

The spatial sign covariance matrix (SSCM), also known as the normalized sample covariance matrix (NSCM), has been widely used in signal processing as a robust alternative to the sample covariance matrix (SCM). It is well-known that the SSCM does not provide consistent estimates of the eigenvalues of the shape matrix (normalized scatter matrix). To alleviate this problem, we propose BASIC (Bias Adjusted SIgn Covariance), which performs an approximate bias correction to the eigenvalues of the SSCM under the assumption that the samples are generated from zero mean unspecified complex elliptically symmetric distributions (the real-valued case is also addressed). We then use the bias correction in order to develop a robust regularized SSCM based estimator, BASIC Shrinkage estimator (BASICS), which is suitable for high dimensional problems, where the dimension can be larger than the sample size. We assess the proposed estimator with several numerical examples as well as in a linear discriminant analysis (LDA) classification problem with real data sets. The simulations show that the proposed estimator compares well to competing robust covariance matrix estimators but has the advantage of being significantly faster to compute.

Unbiased estimators of two second-order moments of the covariance matrix

“…The main reason is that the signal estimation algorithm often involves the covariance matrix between array elements [5], [6]. In practice, the covariance matrix is often unknown [7], and it needs to be estimated with limited samples [8]. When the dimension is low and the sample size is large, the sample covariance matrix is a good estimator of the population covariance matrix.…”

Performance of subspace-based algorithms associated with the sample sign covariance matrix

“…Regarding C-CES distributed data [18], closed-form expressions of the first and second-order moments of the SSCM have been given in the particular case of different eigenvalues in [22], that were partially completed in the case of a single multiple eigenvalue in [23], [24]. Recently, an approximation of the one-dimensional integral representation [21,Proposition 3] for the eigenvalues of the expectation of the SSCM has been given in [25], making possible an approximate bias correction to its eigenvalues leading a robust regularized SSCM based estimator. Note that this SSCM was mainly used in DOA estimation with heavy-tailed noise [13] and in radar clutter modeling [9], [14]- [17] in the framework of C-CES distributed data.…”