### Browsing by Author "Ollila, Esa, Prof., Aalto University, Department of Signal Processing and AcousticsFinland"

Now showing 1 - 1 of 1

###### Results Per Page

###### Sort Options

Item Contributions to Theory and Estimation of High-Dimensional Covariance Matrices(Aalto University, 2022) Raninen, Elias; Signaalinkäsittelyn ja akustiikan laitos; Department of Signal Processing and Acoustics; Sähkötekniikan korkeakoulu; School of Electrical Engineering; Ollila, Esa, Prof., Aalto University, Department of Signal Processing and AcousticsFinlandHigh-dimensional and low sample size problems have become increasingly common in modern data science. Generally speaking, as the dimension grows, so does the number of parameters that need to be estimated. In multivariate statistics, the covariance matrix describes the second-order associations between the variables, and it is a fundamental building block for many algorithms and statistical data analysis methods. The estimation of a high-dimensional covariance matrix is, however, a very challenging problem, not least because the number of unknown parameters increases quadratically with the dimension. A particularly difficult regime for parameter estimation problems is the case, when the dimension of the data exceeds the number of observations. In this regime, classical methods no longer work, and it becomes necessary to impose additional structure on the data or the model parameters using prior knowledge or simplifying assumptions. This thesis develops theory and methods for covariance matrix estimation in the high-dimensional low sample size regime. Different scenarios are considered, such as a single population setting and a multiple populations setting. The primary modeling tools used in this thesis are real and complex elliptically symmetric (ES) distribution theory and regularization. In this thesis, high-dimensional covariance matrix estimators are developed based on finding an optimal linear combination of the sample covariance matrix (SCM) with one or multiple target matrices. To this end, several theoretical properties of the SCM are derived under real and complex ES distributions, such as the explicit expressions for the variance-covariance matrix of the SCM and its mean squared error (MSE). In the multiple populations setting, we study different methods of pooling the class SCMs in order to reduce the overall estimation error. A coupled regularized SCM estimator and a linear pooling method are developed. The thesis also considers regularized high-dimensional robust estimation of the shape matrix (normalized covariance matrix). To this end, the spatial sign covariance matrix (SSCM) is used, which is the SCM computed from centered samples normalized to unit norm. Several properties of the SSCM under ES distributions are also derived. For example, the expectation of a complex weighted SCM is derived, which includes as a special case the expectation of the SSCM. Furthermore, an asymptotic unbiasedness result and an approximate bias correction scheme for the SSCM are developed. All of the proposed methods are shown, both via simulations and real data examples, to be computationally effective and potentially useful in many practical applications involving high-dimensional covariance matrices. Specifically, we demonstrate their usefulness in classification and portfolio optimization problems.