Browsing by Subject "Functional data"

Previous papers introduced a variety of extensions of the Dirichlet process to the func-

tional domain, focusing on the challenges presented by extending the stick-breaking

process. In this thesis some of these are examined in more detail for similarities

and differences in their stick-breaking extensions. Two broad classes of extensions

can be defined, differentiating by how the construction of functional mixture weights

are handled: one type of process views it as the product of a sequence of marginal

mixture weights, whereas the other specifies a joint mixture weight for an entire ob-

servation. These are termed “marginal” and “joint” labelling processes respectively,

and we show that there are significant differences in their posterior predictive perfor-

mance. Further investigation of the generalized functional Dirichlet process reveals

that a more fundamental difference exists. Whereas marginal labelling models nec-

essarily assign labels only at specific arguments, joint labelling models can allow for

the assignment of labels to random subsets of the domain of the function. This leads

naturally to the idea of a stochastic process based around a random partitioning of a

bounded domain, which we call the partitioned functional Dirichlet process. Here we

explicitly model the partitioning of the domain in a constrained manner, rather than

implicitly as happens in the generalized functional Dirichlet process. Comparisons

are made in terms of posterior predictive behaviour between this model, the general-

ized functional Dirichlet process and the functional Dirichlet process. We find that

the explicit modelling of the partitioning leads to more tractable computational and

more structured posterior predictive behaviour than in the generalized functional

Dirichlet process, while still offering increased flexibility over the functional Dirich-

let process. Finally, we extend the partitioned functional Dirichlet process to the

bivariate case.

Current frontiers in complex stochastic modeling of high-dimensional processes include major emphases on so-called functional data: problems in which the data are snapshots of curves and surfaces representing fundamentally important scientific quantities. This thesis explores new Bayesian methodologies for functional data analysis.

The first part of the thesis places emphasis on the role of factor models in functional data analysis. Data reduction becomes mandatory when dealing with such high-dimensional data, more so when data are available on a large number of individuals. In Chapter 2 we present a novel Bayesian framework which employs a latent factor construction to represent each variable by a low dimensional summary. Further, we explore the important issue of modeling and analyzing the relationship of functional data with other covariate and outcome variables simultaneously measured on the same subjects.

The second part of the thesis is concerned with the analysis of circadian data. The focus is on the identification of circadian genes that is, genes whose expression levels appear to be rhythmic through time with a period of approximately 24 hours. While addressing this goal, most of the current literature does not account for the potential dependence across genes. In Chapter 4, we propose a Bayesian approach which employs latent factors to accommodate dependence and verify patterns and relationships between genes, while representing the true gene expression trajectories in the Fourier domain allows for inference on period, phase, and amplitude of the signal.

The third part of the thesis is concerned with the statistical analysis of computer models (simulators). The heavy computational demand of these input-output maps calls for statistical techniques that quickly estimate the surface output at untried inputs given a few preliminary runs of the simulator at a set design points. In this regard, we propose a Bayesian methodology based on a non-stationary Gaussian process. Relying on a model-based assessment of uncertainty, we envision a sequential design technique which helps choosing input points where the simulator should be run to minimize the uncertainty in posterior surface estimation in an optimal way. The proposed non-stationary approach adapts well to output surfaces of unconstrained shape.