Essence codings in functional linear regression: identification, interpretation and application

In the recent two decades, as technology for data collection and data storage advances , functional data are increasingly common in various subject areas including chemometrics, climatology, biology, medical study, finance etc. Motivated by analysis of a Lingzhi (a traditional Chinese medicine) data set, this project considers the function- on-scalar regression in which the response is a random function and the covariates is a random vector. There exist numerous models for function-on-scalar regression. We focus on functional linear regression and explore in the directions of structure identification and the subsequent estimation, inference and interpretation. That is, we assume that the vector of coefficient functions in the functional linear regression model possesses some unknown lower-dimensional structure formulated in an innovative way. The goals are to develop a data-driven algorithm to identify the unknown lower- dimensional structure and then to obtain a intrinsically parsimonious model built on it so that we can sharpen the inference and interpretation. Specifically, suppose that the vector of coefficient functions are linear combinations of some unknown smooth functions, called essence codings. We first impose some constraints on the essence codings to deal with the identification problem, and then we develop a projection-based method to estimate them. Difference-based methods will be employed to achieve smoothness of the estimated essence codings. The smoothing parameter in the difference-based curve estimation is determined by cross-validation. Theoretically there are many essence codings, yet arguably only a few of the essence codings are important. We develop a criterion to determine the number of essence codings to be included in the model. Next, we can estimate the coefficients of the essence codings, called essence effects, and interpret them. Finally we can make inference based on the estimated essence effects. Simulation studies will be conducted to examine finite sample performance of the proposed method and to compare with existing function-on-scalar regression models. We will apply the proposed methods to analyze a Lingzhi data set and a breast cancer data set.