Robust Methods for Multivariate Functional Data Analysis
Type of Degreedissertation
DepartmentMathematics and Statistics
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In the present work we have proposed a robust multivariate functional principal component analysis (RMFPCA) method, that is efficient in estimation and fast in computation, to achieve dimension reduction of dataset and to develop tools for detection of functional outliers. We intend to develop smooth principal functions as M-type smoothing spline estimators by using penalized M-regression with a bounded loss function. The proposed method is more efficient since it makes maximal use of the normally observed measurements by separately downweighing abnormally observed measurements in a single curve. Using natural cubic splines formulation the computation of the proposed method becomes fast for functional data. We have described accompanying diagnostic plots that can be used to detect possible outliers. Simulations are conducted to investigate the effectiveness of the proposed robust multivariate functional principal component analysis (PCA) based on MM estimation, in which we compare our proposed methodology with classical multivariate functional PCA. The estimation of a functional coefficient in a regression setting where the response is a scalar and explanatory variables are sampling points of a continuous process is considered. Multivariate linear model is considered for several applications, for instance chemometrics where some chemical variables have to be predicted by a digitized signal such as the Near Infrared Reflectance (NIR) spectroscopic information. These methods do not really take into account the functional nature of the data. Further it is typical to have outlying observations in such datasets. Fitting functional parameter by using functional regression is vulnerable to unusual data. Therefore from functional point of view a robust functional principal component regression (RFPCR) is proposed for regressing scalar response on the space, spanned by small number of eigenfunctions of the functional predictor. Before running a regression the outlying trajectories in this space are down weighted by using re-weighted least squares approach. Several simulation results and the analysis of a real data set indicate the robustness of the proposed method.