Functional regression analysis and variable selection for motion data

Abstract

Modern technology o ers us highly evolved data collection devices. They allow us to observe data densely over continua such as time, distance, space and so on. The observations are normally assumed to follow certain continuous and smooth underline functions of the continua. Thus the analysis must consider two important properties of functional data: infinite dimension and the smoothness. Traditional multivariate data analysis normally works with low dimension and independent data. Therefore, we need to develop new methodology to conduct functional data analysis. In this thesis, we first study the linear relationship between a scalar variable and a group of functional variables using three di erent discrete methods. We combine this linear relationship with the idea from least angle regression to propose a new variable selection method, named as functional LARS. It is designed for functional linear regression with scalar response and a group of mixture of functional and scalar variables. We also propose two new stopping rules for the algorithm, since the conventional stopping rules may fail for functional data. The algorithm can be used when there are more variables than samples. The performance of the algorithm and the stopping rules is compared with existed algorithms by comprehensive simulation studies. The proposed algorithm is applied to analyse motion data including scalar response, more than 200 scalar covariates and 500 functional covariates. Models with or without functional variables are compared. We have achieved very accurate results for this complex data particularly the models including functional covariates. The research in functional variable selection is limited due to its complexity and onerous computational burdens. We have demonstrated that the proposed functional LARS is a very e cient method and can cope with functional data very large dimension. The methodology and the idea have the potential to be used to address other challenging problems in functional data analysis.