The research project has three basic goals. The first goal is to estimate the second order properties of a nonstationary process using data from multiple units. A wavelet-based functional model with stochastic subject-specific representation is proposed. A computationally efficient algorithm for estimation will be developed along with theoretical optimal properties of the estimators. While this project is focused on the frequency or spectral domain, the second project will develop methods in the time domain. The goal of this project is to accommodate structural breaks within the time series using mixtures of state-space models. These mixtures are characterized by covariates, treatment and subject-specific effects in both the mixture components and mixing weights. The third project focuses on both temporal and spatial components. In particular, this project integrates spatial information by developing models and corresponding estimators to accommodate data structures in both time and space. The goal is to assess the differences across groups and the subject-specific and covariate effects on the dynamic structure.
The research focuses on an area that has been, for the most part, neglected by the statistics community. In particular, the question to be addressed is how to best analyze data obtained from experiments when the data are taken sequentially in time and/or space. Typical examples include experiments where EEGs are taken on various subjects with different diagnoses, or functional magnetic resonance imaging (fMRI) experiments, where subjects with different conditions are requested to perform tasks while in a magnet. The difficulty in analyzing such data is that they are often irregular (nonhomogeneous or nonstationary) in nature. The techniques for analyzing regular time series are well established, but new techniques are needed to account for the irregularities over time and/or space. The fact that these data are collected in complex experiments adds another layer of difficulty onto the analysis.
