Functional models using smoothing splines: A state space approach

With the development of modern technology, tremendous amount of data can be collected in biomedical experiments. These data can arise as curves or groups of time series, therefore it is natural to use a curve or a time series as the basic unit in the data analysis. In this dissertation, we propose two general classes of functional models for estimation and inference for such data. We first develop a new class of functional models for curve data that can incorporate the prior information about the curves. We then propose time-frequency functional linear models for time series data, in which the basic analysis unit is a time-varying spectrum. The biggest obstacle in fitting functional models is the heavy computational demand due to the curse of dimensionality. We develop O(N) computationally efficient estimation procedures for the proposed functional models by constructing equivalent state space models. The proposed methods are motivated by and applied to two data sets: (1) longitudinally-collected cortisol data from a fibromyalgia study and (2) the electroencephalogram (EEG) time series data from epilepsy patients.