We consider the problem of estimating a high-dimensional (HD) covariance matrix when the sample size is smaller, or not much larger, than the dimensionality of the data, which could potentially be very large. In this talk we address both the conventional single-class and multiple-class covariance estimation problem. The latter problem commonly occurs for example in classification applications, where the class covariance matrices needs to be learned from the training data consisting of observations from multiple classes or populations.
For both problems, we develop a regularized sample covariance matrix (RSCM) estimator(s) that is optimal (in minimum mean squared error sense) when the data is sampled from an unspecified elliptically symmetric distribution. The proposed covariance estimators are then used in portfolio optimization problems in finance and classification problems using regularized discriminant analysis framework. In portfolio optimization problem, we use our estimator(s) for optimally allocating the total wealth to a large number of assets, where optimality means that the risk (i.e., variance of the portfolio returns) is minimized. Our analysis results on real stock market data and classification data sets illustrate that the proposed approaches are often able to outperform the current benchmark methods.
Esa Ollila received the M.Sc. degree in mathematics from the University of Oulu, in 1998, Ph.D. degree in statistics with honors from the University of Jyvaskyla, in 2002, and the D.Sc.(Tech) degree with honors in signal processing from Aalto University, in 2010. From 2004 to 2007 he was a post-doctoral fellow of the Academy of Finland. He has also been a Senior Researcher and a Senior Lecturer at the University of Oulu, respectively. From August 2010 to June 2015, he was an Academy Research Fellow (a prestigious research fellowship position nominated by the Academy of Finland) at Aalto University. Currently, from June 2015 he is an Associate Professor of Signal Processing at Aalto University, School of Electrical Engineering. He is also an adjunct Professor (statistics) of the University of Oulu.
Prof. Ollila has had several long-term research visits abroad. Fall-term 2001 he was a Visiting Researcher with the Department of Statistics, Pennsylvania State University, while the academic year 2010-2011 he spent as a Visiting Post-doctoral Research Associate with the Department of Electrical Engineering, Princeton University. He is a member of the EURASIP SAT in Theoretical and Methodological Trends in Signal Processing (TMTSP) and frequent reviewer of many journals in signal processing, statistics and machine learning. He has co-authored a book, Robust Statistics for Signal Processing, published in 2018 by Cambridge University Press.