Teste de hipóteses com estrutura combinatorial: problemas de detecção de médias e correlações

Abstract

In this dissertation, we study anomaly detection problems from the point of view of hypothesis testing. More specically, we study two detection problems in the Gaussian context, considered in (Addario-Berry et al. (2010)) and (Arias-Castro et al. (2012)), where the variables with anomalies have some geometric structure. The first problem is called detection of means. In this problem, we observe a high-dimensional vector with independent coordinates and want to decide whether all coordinates follow standard normal distribution or, alternatively, if there is a subset of coordinates (belonging to a given class of sets) for which the distribution is a normal with a non-zero mean and variance one. In this formulation, each coordinate of the random vector whose mean of the corresponding normal distribution is non-zero is called anomalous. The second problem is called detection of correlations. In this problem, we want to decide whether a high-dimensional vector has independent standard normal coordinates or, alternatively, whether there is a subset of coordinates (belonging to a given class of sets) that are correlated. In this formulation, the coordinates of the correlated random vector are called anomalous. In both problems, we seek to understand when it is possible and impossible to detect the presence of anomalous variables.