We address the problem of estimating high-dimensional covariance matrices of multiple classes (populations) in the small sample size setting. We propose to estimate each class covariance matrix as a weighted linear combination of sample covariance matrices of individual classes. The optimal weights that yield the minimum mean squared error (MSE) are then derived. The proposed estimator based on optimal weights is shown to attain significant reduction in estimation error when sample sizes are limited and/or when at least some of the true class covariance matrices share similar structure. We develop an effective method for estimating the optimal weighs in the case that the true unknown populations are members of elliptically symmetric distributions with finite fourth-order moments. To this end, we utilize the spatial sign covariance matrix, which we show (under rather general conditions) to be an unbiased estimator of the normalized covariance matrix as the dimension grow to infinity. We also show how the proposed approach can be used in choosing the regularization parameters for multiple target matrices in a single class covariance matrix estimation problem. Our approach is general, and is applicable also for complex-valued data. We illustrate the effectiveness of the proposed covariance matrix estimators via numerical simulation studies and portfolio optimization problem using real historical stock data wherein the proposed approach is able to outperform several state-of-the-art methods. Finally, an application in radar signal processing of complex-valued data is also discussed. This is a joint work with Elias Raninen (Aalto University) and David E. Tyler (Rutgers University, USA)
Esa Ollila (M'03) received the M.Sc. degree in mathematics from the University of Oulu, Oulu, Finland, in 1998, Ph.D. degree in statistics with honours from the University of Jyvaskyla, Jyvaskyla, Finland, in 2002, and the D.Sc. (Tech) degree with honours in signal processing from Aalto University, Aalto, Finland, in 2010. From 2004 to 2007, he was a Postdoctoral Fellow and from August 2010 to May 2015 an Academy Research Fellow of the Academy of Finland. He has also been a Senior Lecturer with the University of Oulu. Currently, since June 2015, he has been an Associate Professor of Signal Processing, Aalto University, Finland. He is also an Adjunct Professor (Statistics) of Oulu University. During the Fall-term 2001, he was a Visiting Researcher with the Department of Statistics, Pennsylvania State University, State College, PA while the academic year 2010-2011 he spent as a Visiting Postdoctoral Research Associate with the Department of Electrical Engineering, Princeton University, Princeton, NJ, USA. He is a co-author of a recent textbook, “Robust Statistics for Signal Processing,”published by Cambridge University Press in 2018. His research interests lie in the intersection of the fields of statistical signal processing, high-dimensional statistics, bioinformatics and machine learning with emphasis on robust statistical methods and modeling.
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