Abstract
With the recent advances in data-collection technology, it becomes much easier to collect a large amount of data. When the values of a variable can be frequently observed and recorded at different, say times or locations, the so-called ``functional data'' emerges. Suppose that a variable $Y(t)$ can be observed at many different $t_i$'s (e.g., $t$ can be time or location). When the number of $t_i$'s is large and $Y(t)$ is a smooth curve of $t$, we prefer to regard the values of $Y(t_i)$'s as the realization of a random function of $t$ in the analysis. The collection of the values of $Y(t_i)$'s is then referred to as a functional data. Functional data usually have a rather large dimensionality and high correlations. The functional data analysis has been extensively theoretically studied in recent decades, and widely applied in the analysis of data across many areas (such as industry, finance, biology). In the thesis, we focus on the functional linear model with a functional response and several scalar predictors. In the model, the effects of the predictors on the response are considered as random effects, and every random effect corresponds to a known function of the predictors and an unknown functional basis. The product of the random effect, the known function of the predictors, and the unknown functional basis is an explanatory term in the model. In the thesis, we study and address the problems of identifying the most important random effects, and estimating their corresponding functional bases. We treat a functional basis as a direction on which the functional response can be projected. By using the idea of projection, we transform the functional linear model into several linear mixed-effect models (LMEMs), each corresponding to a random effect. Inspired by the connection between the statistic for testing a random effect in an LMEM and the criterion in the linear discriminant analysis for finding the best discriminant direction, we propose a method for estimating the functional bases. Because the functional bases are assumed to be mutually orthogonal in the model, the method is modified so as to give a sequential procedure, in each step of which the most significant random effect among the remaining random effects is identified under the restriction that its corresponding functional basis must be orthogonal to the previously identified ones. We also compare the difference between the functional bases, the bases identified by functional principal component analysis, and the bases obtained by applying the concept of least squares on the functional linear model. A discussion is given for a generalization of the functional linear model and its impact on the estimation of the functional bases. In the end, the method developed in the thesis is applied on a functional data of wafer thickness profile to identify the most important random effects and estimate their functional bases.