Modeling data in time and space - studies of irregularities, dependence structure and applications
Open Access
- Author:
- Dang, Huy
- Graduate Program:
- Statistics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- August 11, 2023
- Committee Members:
- Bing Li, Professor in Charge/Director of Graduate Studies
Bharath Kumar Sriperumbudur, Major Field Member
Marzia Cremona, Special Member
Nicole Lazar, Major Field Member
Francesca Chiaromonte, Chair & Dissertation Advisor
Chaleece Sandberg (she/her), Outside Unit & Field Member - Keywords:
- functional data analysis
time series
spatio-temporal dependence
irregularities detection
EM algorithm
wavelet transform
cs-fMRI - Abstract:
- This dissertation is a compilation of three research projects on the analysis of data in time and space. The first and second projects propose approaches for the analysis of longitudinal and spatio-temporal data that comprise irregularities, and the third project proposes an integrated approach for capturing the spatio-temporal dependence structure of spatially organized time series. In the first project, we propose a method that, combining an EM algorithm with penalized smoothing, can simultaneously estimate the smooth component and detect irregular spikes in a 1-dimensional signal. Imposing some assumptions on the error distribution, we study consistency of EM updates when some or all parameters are unknown. Robustness of the proposed method to assumptions violations is ascertained via simulations. The second project is motivated by a specific application in functional magnetic resonance imaging (fMRI); namely, head motion detection. Head motion can be viewed as an abrupt change in fMRI signals that are otherwise a smooth function of brain activities in time. By transforming the data to wavelet space, and studying the decay rate of wavelet coefficients across different scales, we are able to estimate the local smoothness of data in three dimensions (a 2-dimensional brain slice and time) and identify local irregularities. In the third project, we study the complex spatio-temporal dependence structure of cortical surface fMRI data. Specifically, we model the non-stationary dependence of activation patterns across the cortical surface via a stochastic differential equation prior. Moreover, we provide evidence of varying ranges of temporal dependence across different brain regions, and model such dependence as fractional Gaussian noise of varying Hurst parameters. The result is a fully integrated, efficient framework that considers spatial and temporal dependence structure simultaneously, and is computationally viable.