The research group in Mathematical Statistics and Data Science studies advanced methods and models for analysing and representing data. We employ probability theory and stochastic processes to rigorously model uncertainty and randomness, and abstract and linear algebra to understand the structure of statistical models and the relationships between their parameters.
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Pauliina Ilmonen Associate Professor Multivariate extreme values, functional data analysis, cancer epidemiology |
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Kaie Kubjas Associate Professor Algebraic statistics |
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Lasse Leskelä Associate Professor Mathematical statistics, network analysis, probability theory |
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Vanni Noferini Associate Professor Network analysis, random matrix theory |
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Jukka Kohonen University Lecturer Statistics, combinatorics |
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Pekka Pere University Lecturer Statistics |
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Jonas Tölle University Lecturer Stochastic processes, probability theory |
Scatter matrices generalize the covariance matrix and are useful in many multivariate data analysis methods, including principal component analysis, which is usually based on the diagonalization of the covariance matrix. The simultaneous diagonalization of two or more scatter matrices goes beyond PCA and is used more and more often. In this talk, we offer an overview of many methods that are based on joint diagonalization. These methods range from the unsupervised context with invariant coordinate selection and blind source separation, which includes independent component analysis, to the supervised context with linear discriminant analysis and sliced inverse regression. They also encompass methods that handle dependent data such as time series or spatial data.
It is known that, in Gaussian two-group separation, the optimally discriminating projection direction can be estimated without any knowledge on the group labels. In this presentation, we (a) motivate this estimation problem, and (b) gather several unsupervised estimators based on skewness and derive their limiting distributions. As one of our main results, we show that all affine equivariant estimators of the optimal direction have proportional asymptotic covariance matrices, making their comparison straightforward. We use simulations to verify our results and to inspect the finite-sample behaviors of the estimators.
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