Stochastic Dynamic Models for Functional Data.

Abstract

Functional data arise frequently in many fields of biomedical research as sequential observations over time. The observations are generated by an unknown dynamic mechanism. This dynamic process has an unspecified mean function, and the observations can be considered as arising from this mean function plus noise. In this dissertation, we treat this unknown function as a realization or sample path of a stochastic process, using a stochastic dynamic model (SDM). This will enable us to study dynamics of the underlying process, including how the stochastic process and its derivatives evolve over time, both within the observation time (through estimation and inference) and afterwards(through forecasting).

We first introduce a new modeling strategy to estimate a smooth function for time series functional data. The proposed models and methods are illustrated on prostate specific antigen (PSA) data, where we use a Gaussian process to model the rate function of PSA and achieve more precise forecasting. We then extend the models to multi-subject functional data and consider the effect of covariates on the rate functions. We finally propose a time-varying stochastic position model, which can approximate the breakpoints in the function. The discretized model is applied to array comparative genomic hybridization (CGH) data analysis. The estimation and inference are conducted using MCMC algorithms with Euler approximation and data augmentation. Simulations and real data analysis demonstrate that our methods outperform several alternative approaches.