Local functional principal component analysis

Covariance operators of random functions are crucial tools to study the way random elements concentrate over their support. The principal component analysis of a random function X is well-known from a theoretical viewpoint and extensively used in practical situations. In this work we focus on local covariance operators. They provide some pieces of information about the distribution of X around a fixed point of the space x₀. A description of the asymptotic behaviour of the theoretical and empirical counterparts is carried out. Asymptotic developments are given under assumptions on the location of x₀ and on the distributions of projections of the data on the eigenspaces of the (non-local) covariance operator

Multilevel functional principal component analysis

The Sleep Heart Health Study (SHHS) is a comprehensive landmark study of sleep and its impacts on health outcomes. A primary metric of the SHHS is the in-home polysomnogram, which includes two electroencephalographic (EEG) channels for each subject, at two visits. The volume and importance of this data presents enormous challenges for analysis. To address these challenges, we introduce multilevel functional principal component analysis (MFPCA), a novel statistical methodology designed to extract core intra- and inter-subject geometric components of multilevel functional data. Though motivated by the SHHS, the proposed methodology is generally applicable, with potential relevance to many modern scientific studies of hierarchical or longitudinal functional outcomes. Notably, using MFPCA, we identify and quantify associations between EEG activity during sleep and adverse cardiovascular outcomes.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS206 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

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))(Xi(s),Xi(t)) and cross-covariance surface Cov (Xi(s),Xj(t))(Xi(s),Xj(t)) 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

Principal Component Analysis for Functional Data on Riemannian Manifolds and Spheres

Functional data analysis on nonlinear manifolds has drawn recent interest. Sphere-valued functional data, which are encountered for example as movement trajectories on the surface of the earth, are an important special case. We consider an intrinsic principal component analysis for smooth Riemannian manifold-valued functional data and study its asymptotic properties. Riemannian functional principal component analysis (RFPCA) is carried out by first mapping the manifold-valued data through Riemannian logarithm maps to tangent spaces around the time-varying Fr\'echet mean function, and then performing a classical multivariate functional principal component analysis on the linear tangent spaces. Representations of the Riemannian manifold-valued functions and the eigenfunctions on the original manifold are then obtained with exponential maps. The tangent-space approximation through functional principal component analysis is shown to be well-behaved in terms of controlling the residual variation if the Riemannian manifold has nonnegative curvature. Specifically, we derive a central limit theorem for the mean function, as well as root-$n$ uniform convergence rates for other model components, including the covariance function, eigenfunctions, and functional principal component scores. Our applications include a novel framework for the analysis of longitudinal compositional data, achieved by mapping longitudinal compositional data to trajectories on the sphere, illustrated with longitudinal fruit fly behavior patterns. RFPCA is shown to be superior in terms of trajectory recovery in comparison to an unrestricted functional principal component analysis in applications and simulations and is also found to produce principal component scores that are better predictors for classification compared to traditional functional functional principal component scores

Principal Component Analysis for Functional Data

In functional principal component analysis (PCA), we treat the data that consist of functions not of vectors (Ramsay and Silverman, 1997). It is an attractive methodology, because we often meet the cases where we wish to apply PCA to such data. But, to make this method widely useful, it is desirable to study advantages and disadvantages in actual applications. As alternatives to functional PCA, we may consider multivariate PCA applied to 1) original observation data, 2) sampled functional data with appropriate intervals, and 3) coefficients of basis function expansion. Theoretical and numerical comparison is made among ordinary functional PCA, penalized functional PCA and the above three multivariate PCA

Functional Principal Component Analysis for Non-stationary Dynamic Time Series

Motivated by a highly dynamic hydrological high-frequency time series, we propose time-varying Functional Principal Component Analysis (FPCA) as a novel approach for the analysis of non-stationary Functional Time Series (FTS) in the frequency domain. Traditional FPCA does not take into account (i) the temporal dependence between the functional observations and (ii) the changes in the covariance/variability structure over time, which could result in inadequate dimension reduction. The novel time-varying FPCA proposed adapts to the changes in the auto-covariance structure and varies smoothly over frequency and time to allow investigation of whether and how the variability structure in an FTS changes over time. Based on the (smooth) time-varying dynamic FPCs, a bootstrap inference procedure is proposed to detect significant changes in the covariance structure over time. Although this time-varying dynamic FPCA can be applied to any dynamic FTS, it has been applied here to study the daily processes of partial pressure of CO2 in a small river catchment in Scotland

MULTILEVEL SPARSE FUNCTIONAL PRINCIPAL COMPONENT ANALYSIS

The basic observational unit in this paper is a function. Data are assumed to have a natural hierarchy of basic units. A simple example is when functions are recorded at multiple visits for the same subject. Di et al. (2009) proposed Multilevel Functional Principal Component Analysis (MFPCA) for this type of data structure when functions are densely sampled. Here we consider the case when functions are sparsely sampled and may contain as few as 2 or 3 observations per function. As with MFPCA, we exploit the multilevel structure of covariance operators and data reduction induced by the use of principal component bases. However, we address inherent methodological differences in the sparse sampling context to: 1) estimate the covariance operators; 2) estimate the functional scores and predict the underlying curves. We show that in the sparse context 1) is harder and propose an algorithm to circumvent the problem. Surprisingly, we show that 2) is easier via new BLUP calculations. Using simulations and real data analysis we show that the ability of our method to reconstruct underlying curves with few observations is stunning. This approach is illustrated by an application to the Sleep Heart Health Study, which contains two electroencephalographic (EEG) series at two visits for each subject

Robust principal component analysis for functional data.

dispersion matrices;