On The Modeling Of Multiple Functional Outcomes With Spatially Heterogeneous Shape Characteristics
This dissertation presents an approach for analyzing functional data with multiple outcomes that exhibits spatially heterogeneous shape characteristics. An example of data of this type that motivated this study is a data from a diffusion tensor imaging (DTI) study of neuronal tract in multiple sclerosis (MS) patients. DTI is an imaging technique for measuring the diffusion of water that can be used to detect abnormalities in brain tissue. DTI tractography can be summarized by 3 functional outcomes, measuring the diffusion in different directions. One of the main and most common difficulties in functional data analysis is the large number of parameters to be estimated. This is especially challenging when multiple functional outcomes are considered. To accommodate this problem, a copula approach is adopted so that the marginal distribution and the dependence structure are estimated independently. In addition to fast computation, the two-step approach also allows flexibility in the specification of the distribution of the data as the marginal distribution and copula distribution can be specified separately. The first part of this dissertation presents an estimation algorithm using the copula approach. The marginal distribution parameters are estimated using methodology based on maximum likelihood and penalized splines. In the estimation for the dependence structure, the Karhunen-Loeve expansion and an EM algorithm are used to significantly reduce the dimension of the problem. This allows the dependence within the same outcome and across different outcomes to be captured even in the case of many functional outcomes. The second part of this dissertation demonstrates the application of the methodology to the DTI study. The goal is to identify the locations where the abnormalities occur and also explain the characteristics of the abnormalities in MS patients. The difference in the marginal distributions and structure dependence in the MS group from the healthy control group is then used to develop a method for predicting case status for patients. The last part of the dissertation explores the DTI study in longitudinal setting. A larger dataset that contains DTI data from multiple visits is studied. We adopted a multilevel approach to investigate how the DTI tractography in MS patients varies over time.
correlated functional outcomes; diffusion tensor imaging; skewed functional data
Jarrow, Robert A.; Frazier, Peter
Ph.D. of Operations Research
Doctor of Philosophy
dissertation or thesis