Joint Clustering and Registration of Functional Data

Curve registration and clustering are fundamental tools in the analysis of functional data. While several methods have been developed and explored for either task individually, limited work has been done to infer functional clusters and register curves simultaneously. We propose a hierarchical model for joint curve clustering and registration. Our proposal combines a Dirichlet process mixture model for clustering of common shapes, with a reproducing kernel representation of phase variability for registration. We show how inference can be carried out applying standard posterior simulation algorithms and compare our method to several alternatives in both engineered data and a benchmark analysis of the Berkeley growth data. We conclude our investigation with an application to time course gene expression

Identifying Finite Mixtures of Nonparametric Product Distributions and Causal Inference of Confounders

We propose a kernel method to identify finite mixtures of nonparametric product distributions. It is based on a Hilbert space embedding of the joint distribution. The rank of the constructed tensor is equal to the number of mixture components. We present an algorithm to recover the components by partitioning the data points into clusters such that the variables are jointly conditionally independent given the cluster. This method can be used to identify finite confounders.Comment: Appears in Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence (UAI2013

Probabilistic correspondence analysis for neuroimaging problems

Establecer correspondencias de forma significativas entre los objetivos como en los problemas de neuroimagen es crucial para mejorar los procesos de correspondencia. Por ejemplo, el problema de correspondencia consiste en encontrar relaciones significativas entre cualquier par de estructuras cerebrales como en el problema de registro estático, o analizar cambios temporales de una enfermedad neurodegenerativa dada a través del tiempo para un análisis dinámico de la forma del cerebro..

Automatic model construction with Gaussian processes

This thesis develops a method for automatically constructing, visualizing and describing a large class of models, useful for forecasting and finding structure in domains such as time series, geological formations, and physical dynamics. These models, based on Gaussian processes, can capture many types of statistical structure, such as periodicity, changepoints, additivity, and symmetries. Such structure can be encoded through kernels, which have historically been hand-chosen by experts. We show how to automate this task, creating a system that explores an open-ended space of models and reports the structures discovered. To automatically construct Gaussian process models, we search over sums and products of kernels, maximizing the approximate marginal likelihood. We show how any model in this class can be automatically decomposed into qualitatively different parts, and how each component can be visualized and described through text. We combine these results into a procedure that, given a dataset, automatically constructs a model along with a detailed report containing plots and generated text that illustrate the structure discovered in the data. The introductory chapters contain a tutorial showing how to express many types of structure through kernels, and how adding and multiplying different kernels combines their properties. Examples also show how symmetric kernels can produce priors over topological manifolds such as cylinders, toruses, and Möbius strips, as well as their higher-dimensional generalizations. This thesis also explores several extensions to Gaussian process models. First, building on existing work that relates Gaussian processes and neural nets, we analyze natural extensions of these models to deep kernels and deep Gaussian processes. Second, we examine additive Gaussian processes, showing their relation to the regularization method of dropout. Third, we combine Gaussian processes with the Dirichlet process to produce the warped mixture model: a Bayesian clustering model having nonparametric cluster shapes, and a corresponding latent space in which each cluster has an interpretable parametric form.This work was supported by the National Sciences and Engineering Research Council of Canada, the Cambridge Commonwealth Trust, Pembroke College, a grant from the Engineering and Physical Sciences Research Council, and a grant from Google