Topics in sparse functional data analysis

Abstract

<p>This dissertation consists of three research papers that address different problems in modeling sparse functional data. The first paper (Chapter 2) focuses on the statistical inference for Analysis of Covariance (ANCOVA) models on sparse functional data. In an analysis of covariance model for sparse functional data, the treatment effects, after adjusting for the effects of subject specific covariates, are represented by functions of time. We apply the seemingly unrelated kernel estimator, which takes the within subject correlation into account, to estimate the nonparametric components of the model, and test treatment effects using a generalized quasi-likelihood ratio test. We derived the asymptotic distribution of the test statistics under both the null and some local alternative hypothesis, and show that the proposed test enjoys the Wilks property and is minimax most powerful when the within-subject correlation structure is correctly specified. The second paper (Chapter 3) develops an algorithm to impute missing values in spatiotemporal satellite images based on sparse functional data analysis methods. We model the satellite images as functional data which is both sparse in temporal domain and spatial domain and assume they are repeated measurements of a latent spatiotemporal process. We assume the latent spatiotemporal process is composed of fixed mean function, random temporal effect and random spatial effect. We propose an algorithm to estimate each component using functional principle component analysis (FPCA) techniques.</p> <p>The proposed imputation algorithm is validated on real data and shows great performance in all</p> <p>aspects compared with its competitors. The third paper (Chapter 4) proposes a hierarchical multiresolution</p> <p>imputation (HMRI) algorithm for imputation of high-resolution spatiotemporal satellite</p> <p>images, which is an extension of the second paper. HMRI is demonstrated by using the Moderate</p> <p>Resolution Imaging Spectrophotometer (MODIS) daily land surfact temperature (LST) data and</p> <p>shows satisfactory imputation results. HMRI shows large improvement in prediction accuracy</p> <p>compared with other existing methods.</p>