This academic paper was submitted for the degree of Master of Science at the University of Cape Town (South Africa). The study was conducted under the supervision of A. Prof Sugnet Lubbe in the Department of Mathematical Statistics, University of Cape Town. Some of my R-scripts were written with the support of Dr. Shuichi Kawano, Department of Mathematical Sciences, Osaka Prefecture University, Japan.
Functional data analysis, commonly known as FDA, refers to the analysis of information on curves of functions. Key aspects of FDA include the choice of smoothing techniques, data reduction, model evaluation, functional linear modelling and forecasting methods. FDA is applicable in numerous applications such as Bioscience, Geology, Psychology, Sports Science, Econometrics, Meteorology, etc.
This dissertation main objective is to focus more specifically on Functional Linear Regression Modeling (FLRM), which is an extension of Multivariate Linear Regression Modeling. The problem of constructing a Functional Linear Regression modeling with functional predictors and functional response variable is considered in great details. Discretely observed data for each variable involved in the modeling are expressed as smooth functions using: Fourier Basis, B-Splines Basis and Gaussian Basis. The Functional Linear Regression Model is estimated by the Least Square method, Maximum Likelihood method and more thoroughly by Penalized Maximum Likelihood method. A central issue when modeling Functional Regression models is the choice of a suitable model criterion as well as the number of basis functions and an appropriate smoothing parameter. Four different types of model criteria are reviewed: the Generalized Cross-Validation, the Generalized Information Criterion, the modified Akaike Information Criterion and Generalized Bayesian Information Criterion. Each of these aforementioned methods are applied to a dataset and contrasted based on their respective results.
Functional Data Analysis, Basis Expansion, Functional Regression, Smoothing Techniques.
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