Statistics and Actuarial Science

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This is the collection for the University of Waterloo's Department of Statistics and Actuarial Science.

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Browsing Statistics and Actuarial Science by Author "Browne, Ryan"

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    Functional Finite Mixture Modelling and Estimation
    (University of Waterloo, 2023-09-01) Sharp, Alexander; Browne, Ryan
    Functional data analysis is a branch of statistics that studies models for information represented by functions. Meanwhile, finite mixture models serve as a conerstone in the field of cluster analysis, offering a flexible probabilisitic framework for the representation of heterogeneous data. These models posit that the observed data are drawn from a mixture of several different probability distributions from the same family, where each is conventionally thought to represent a distinct group within the overall population. However, their representation in terms of densities makes their application to function-valued random variables, the foundation of functional data analysis, difficult. Herein, we utilize density surrogates derived from the Karhunen-Loeve expansion to circumvent this discrepancy and develop functional finite mixture models for the clustering of functional data. Models developed for real-valued and vector-valued functions of a single variable. Estimation of all models is done using the expectation-maximization algorithm, and copious amounts of simulations and data examples are provided to demonstrate the properties and performance of the methodologies. Additionally, we present a new estimation approach to be used in tandem with the stochastic expectation-maximization algorithm. This estimation method offers increased precision in estimation with respect to the algorithm chain length when compared to averaging the chain. Asymptotic properties of the estimator are derived, and simulation studies are given to demonstrate its performance.
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    Methods for Merging, Parsimony and Interpretability of Finite Mixture Models
    (University of Waterloo, 2022-08-04) Kim, Nam-Hwui; Browne, Ryan
    To combat the increasing data dimensionality, parsimonious modelling for finite mixture models has risen to be an active research area. These modelling frameworks offer various constraints that can reduce the number of free parameters in a finite mixture model. However, the constraint selection process is not always clear to the user. Moreover, the relationship between the chosen constraint and the data set is often left unexplained. Such issues affect adversely the interpretability of the fitted model. That is, one may end up with a model with reduced number of free parameters, but how it was selected, and what the parameter-reducing constraints mean, remain mysterious. Over-estimation of the mixture component count is another way in which the model interpretability may suffer. When the individual components of a mixture model fail to capture adequately the underlying clusters of a data set, the model may compensate by introducing extra components, thereby representing a single cluster with multiple components. This reality challenges the common assumption that a single component represents a cluster. Addressing the interpretability-related issues can improve the informativeness of model-based clustering, thereby better assisting the user during the exploratory analysis and/or data segmentation step.