Functional principal component analysis of spatially correlated data

This paper focuses on the analysis of spatially correlated functional data. We propose a parametric model for spatial correlation and the between-curve correlation is modeled by correlating functional principal component scores of the functional data. Additionally, in the sparse observation framework, we propose a novel approach of spatial principal analysis by conditional expectation to explicitly estimate spatial correlations and reconstruct individual curves. Assuming spatial stationarity, empirical spatial correlations are calculated as the ratio of eigenvalues of the smoothed covariance surface Cov(Xi(s),Xi(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X_i(s),X_i(t))$$\end{document} and cross-covariance surface Cov(Xi(s),Xj(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X_i(s), X_j(t))$$\end{document} at locations indexed by i and j. Then a anisotropy Matérn spatial correlation model is fitted to empirical correlations. Finally, principal component scores are estimated to reconstruct the sparsely observed curves. This framework can naturally accommodate arbitrary covariance structures, but there is an enormous reduction in computation if one can assume the separability of temporal and spatial components. We demonstrate the consistency of our estimates and propose hypothesis tests to examine the separability as well as the isotropy effect of spatial correlation. Using simulation studies, we show that these methods have some clear advantages over existing methods of curve reconstruction and estimation of model parameters.

G (s, t): covariance function G (s, t) = Cov (X (s), X (t)) φ k (t): the kth eigenfunction of the kernel function G (s, t) λ k : the kth eigenvalue of the kernel function G (s, t) ξ ik : the fPC scores with mean 0 and variance λ k Data Assume data are collected across N spatial locations For location i, n i noise-corrupted points are sampled, denoted as is the ith realization of an underlying random function X (t) X (t) is assumed to be smooth and square integrable on a bounded and closed time interval T

Measurement errors
The eigen decomposition of the covariance kernel G (s, t) In classical functional data model The following structure is specified

Correlation Struture
The covariance between X i (s) and X j (t) can be expressed as We believe the leading fPC scores will be more prone to high correlation than trailing scores Such covariance structure is referred as separable

Independence of the correlation across curves and time is assumed
A hypothesis test is proposed to examine whether the assumption of separable covariance is justifiable For each fPC index k, the associated fPC scores at different locations can be viewed as a spatial random process The correlation between two observations at locations separated by distance d > 0 is given by Differential equation Solutions are the Bessel functions of the first kind J ±ν (z), second kind Y ν (z), third kind H (1) ν (z) and H (2) The Matérn class is isotropic which has contours of constant correlation that are circular in two-dimensional applications fPC scores present directional patterns which might be associated with geographical features The authors look at vegetation index series at Harvard Forest in Massachusetts across a 25 × 25 grid over 6 years The correlation between fPC scores at locations separated by 45 degree to the northeast and to the northwest has been calculated Figure 1 suggests the anisotropy effect in the second fPC scores Incorporate geometric anisotropy into the Matérn class Apply a linear transformation to the spatial correlation Require two additional parameters Anisotropy angle α: determines how much the axes rotate clockwise anisotropy ratio δ: specifyies how much one axis is stretched or shrunk relative to the other

Isotropy Matérn correlation
Let (x 1 , y 1 ) and (x 2 , y 2 ) be coordinates of two locations Notations: For i = j and a given spatial separation vector ∆, the local linear smoother of the cross-covariance surface G (s, t) is derived by minimizing with respect to β 0 , β 1 and β 2 κ 2 is the two-dimenstional Gaussian kernel The one-dimensional functions are evaluated over equally spaced time points t eval = (t eval 1 , ..., t eval M ) with step ĥ G ∆ (s, t) is evaluated over t eval × t eval Then we obtain λ k (∆) is the kth eigenvalue of G ∆ λ k is the kth eigenvalue ofĜ