Vicente, Sergio
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Faculty of Arts and Science - Department of Mathematics and Statistics
André-Aisenstadt
Courriels
Research area
Student supervision Expand all Collapse all
Apprentissage statistique avec le processus ponctuel déterminantal
Theses and supervised dissertations / 2021-02
Vicente, Sergio
Abstract
Abstract
This thesis presents the determinantal point process, a probabilistic model that captures
repulsion between points of a certain space. This repulsion is encompassed by a similarity
matrix, the kernel matrix, which selects which points are more similar and then less likely to
appear in the same subset. This point process gives more weight to subsets characterized by
a larger diversity of its elements, which is not the case with the traditional uniform random
sampling. Diversity has become a key concept in domains such as medicine, sociology,
forensic sciences and behavioral sciences. The determinantal point process is considered
a promising alternative to traditional sampling methods, since it takes into account the
diversity of selected elements. It is already actively used in machine learning as a subset
selection method. Its application in statistics is illustrated with three papers. The first
paper presents the consensus clustering, which consists in running a clustering algorithm
on the same data, a large number of times. To sample the initials points of the algorithm,
we propose the determinantal point process as a sampling method instead of a uniform
random sampling and show that the former option produces better clustering results. The
second paper extends the methodology developed in the first paper to large-data. Such
datasets impose a computational burden since sampling with the determinantal point process
is based on the spectral decomposition of the large kernel matrix. We introduce two methods
to deal with this issue. These methods also produce better clustering results than consensus
clustering based on a uniform sampling of initial points. The third paper addresses the
problem of variable selection for the linear model and the logistic regression, when the
number of predictors is large. A Bayesian approach is adopted, using Markov Chain Monte
Carlo methods with Metropolis-Hasting algorithm. We show that setting the determinantal
point process as the prior distribution for the model space selects a better final model than
the model selected by a uniform prior on the model space.