Functional regression analysis and variable selection for motion data

PhD ThesisModern 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

A Bayesian regression model for multivariate functional data

In this paper we present a model for the analysis of multivariate functional data with unequally spaced observation times that may differ among subjects. Our method is formulated as a Bayesian mixed-effects model in which the fixed part corresponds to the mean functions, and the random part corresponds to individual deviations from these mean functions. Covariates can be incorporated into both the fixed and the random effects. The random error term of the model is assumed to follow a multivariate Ornstein-Uhlenbeck process. For each of the response variables, both the mean and the subject-specific deviations are estimated via low-rank cubic splines using radial basis functions. Inference is performed via Markov chain Monte Carlo methods.