Elastic Functional Principal Component Analysis for Modeling and Testing of Functional Data

Statistical analysis of functional data requires tools for comparing, summarizing and modeling observed functions as elements of a function space. A key issue in Functional Data Analysis (FDA) is the presence of the phase variability in the observed data. A successful statistical model of functional data has to account for the presence of phase variability. Otherwise the ensuing inferences can be inferior. Recent methods for FDA include steps for phase separation or functional alignment. For example, Elastic Functional Principal Component Analysis (Elastic FPCA) uses the strengths of Functional Principal Component Analysis (FPCA), along with the tools from Elastic FDA, to perform joint phase-amplitude separation and modeling. A related problem in FDA is to quantify and test for the amount of phase in a given data. We develop two types of hypothesis tests for testing the significance of phase variability: a metric-based approach and a model-based approach. The metric-based approach treats phase and amplitude as independent components and uses their respective metrics to apply the Friedman-Rafsky Test, Schilling's Nearest Neighbors, and Energy Test to test the differences between functions and their amplitudes. In the model-based test, we use Concordance Correlation Coefficients as a tool to quantify the agreement between functions and their reconstructions using FPCA and Elastic FPCA. We demonstrate this framework using a number of simulated and real data, including weather, tecator, and growth data.