Aalto Stochastics and Statistics Seminar is organized by Kalle Kytölä, Lasse Leskelä, and Pauliina Ilmonen. Feel free to contact one of us if you are interested in giving a talk. You may also earn credit points by active participation.
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.
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.
Multiway clustering is a clustering problem with multidimensional data arrays. Such data can be used to represent higher-order interactions, hypergraphs and multilayer networks. This has various applications such as gene clustering from multitissue gene expression data or higher-order gene interactions, and personalized web search from clickthrough data. The main objective of this thesis is to determine when an underlying true cluster structure can be recovered from large and noisy data. Specifically, assuming a statistical model (tensor block model), how sparse a data array can be for a fast algorithm to recover the underlying clusters with high probability. This thesis develops a spectral clustering algorithm to solve this statistical problem, proves weak consistency with mathematical rigor and demonstrates it with numerical simulations. The weak consistency is proved by developing concentration inequalities for certain random matrices. The obtained weak consistency regime improves existing results.
Over the centuries, many claims have been made of numerical patterns of miraculous nature hidden within the text of sacred writings, including the Jewish, Christian and Islamic scriptures. Usually the patterns involve counting of letters and words, or calculations involving numerical equivalents of the letters. Until recently, all such claims were made by people with little mathematical understanding and were easily explained. This situation changed when a highly respected Israeli mathematician Eliyahu Rips and two others published a paper in the academic journal Statistical Science claiming to prove that information about medieval Jewish rabbis was encoded in the Hebrew text of the Book of Genesis. The journal reported that its reviewers were "baffled". The paper in Statistical Science spawned a huge "Bible Codes" industry, complete with best selling books, TV documentaries, and even a romance movie. The talk will reveal the inside story of the Codes and the people behind them, from their inception through to their refutation.
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