Joint Model for Phase and Amplitude Variation in Functional Data

Functional data analysis is a powerful statistical framework to analyze high dimensional data by viewing them as points in the function space. The classical functional data analysis is usually focused on the mean shape and amplitude variability of curve data, which are often distorted by oscillations in the time domain. In this case, the warping procedure, also known as registration, is commonly used to align the curves and remove the phase variation, prior to the classical functional data analysis. However, it has been noticed that the phase and amplitude variations may not be separable. In this thesis, we propose a joint model procedure to model phase and amplitude variations simultaneously. We define a novel metric $d_T$ on function space based on a transformed Hellinger distance and use a tuning parameter to assign different modeling emphasis on phase and amplitude features. The proposed metric $d_T$ naturally induces a Fr\\u27{e}chet mean that minimizes the mean squared distance of errors. An iterative dynamic programming algorithm is developed to estimate the induced Fr\\u27{e}chet mean and the optimal warping functions. Then the Fr\\u27{e}chet variance is decomposed into phase and amplitude variations and the warped functions are further analyzed via functional principal component analysis. This joint model procedure successfully bridges the registration problem and the classical functional data analysis, and accommodate various weights on phase and amplitude. Furthermore, we prove the strong consistency of the proposed Fr\\u27{e}chet mean estimator and show the numerical performance of the proposed joint procedure in simulations. We illustrate how the tuning parameter controls the shape distortion and discuss the separability of phase and amplitude variations. Finally, the proposed procedure is demonstrated in a real-life example of handwritings