Principal Component Analysis of Two-Dimensional Functional Data with Serial Correlation

Abstract

Functional principal component analysis is a popular technique used for analyzing the intrinsically infinite-dimensional functions. The functional principal components help explore the variation patterns of functions and achieve dimension reduction. Some functional data are sequentially observed on two-dimensional domains. How to analyze the serial correlated two-dimensional functional data is an important issue. This dissertation consists of two projects developing the dynamic two-dimensional functional principal component analysis to analyze the serial correlated functional data with Gaussian distribution or generally, with a distribution from exponential family.

The first project proposes a novel model to analyze serial correlated two-dimensional functional data observed sparsely and irregularly on a domain which may not be a rectangle. The approach employs a mixed effects model that specifies the principal component functions as bi-variate splines on triangulations and the component scores as random effects which follow an auto-regressive model. We apply the roughness penalty for regularizing the function estimation and develop an effective EM algorithm along with Kalman filter and smoother for calculating the penalized likelihood estimates of the parameters. This approach was applied on simulated datasets and on Texas monthly average temperature data of 49 weather stations from January year 1915 to December year 2014.

The second project proposes the approach to analyze data which follow a distribution from exponential family and are observed over time on two-dimensional domain. Assuming that the natural parameter is a dynamic smooth function of the two-dimensional location, we propose a functional principal component model which models the natural parameter through the combination of smooth principal component functions on two-dimensional domain and principal component scores modeled by autoregressive processes. To address the problem of scalability of large data which is often seen in practice, a variational EM algorithm is proposed for fitting the model. Numerical results on simulated data and the motivating Arctic sea-ice-extent data demonstrate the good performance of the proposed approach.