Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains

Existing approaches for multivariate functional principal component analysis are restricted to data on the same one-dimensional interval. The presented approach focuses on multivariate functional data on different domains that may differ in dimension, e.g. functions and images. The theoretical basis for multivariate functional principal component analysis is given in terms of a Karhunen-Lo\`eve Theorem. For the practically relevant case of a finite Karhunen-Lo\`eve representation, a relationship between univariate and multivariate functional principal component analysis is established. This offers an estimation strategy to calculate multivariate functional principal components and scores based on their univariate counterparts. For the resulting estimators, asymptotic results are derived. The approach can be extended to finite univariate expansions in general, not necessarily orthonormal bases. It is also applicable for sparse functional data or data with measurement error. A flexible R-implementation is available on CRAN. The new method is shown to be competitive to existing approaches for data observed on a common one-dimensional domain. The motivating application is a neuroimaging study, where the goal is to explore how longitudinal trajectories of a neuropsychological test score covary with FDG-PET brain scans at baseline. Supplementary material, including detailed proofs, additional simulation results and software is available online.Comment: Revised Version. R-Code for the online appendix is available in the .zip file associated with this article in subdirectory "/Software". The software associated with this article is available on CRAN (packages funData and MFPCA

Functional linear regression analysis for longitudinal data

We propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data, where both the predictor and response are functions of a covariate such as time. Predictor and response processes have smooth random trajectories, and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects. In longitudinal studies, the number of repeated measurements per subject is often small and may be modeled as a discrete random number and, accordingly, only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit. We propose a functional regression approach for this situation, using functional principal component analysis, where we estimate the functional principal component scores through conditional expectations. This allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory. The resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories. Asymptotic properties for a sample of $n$ subjects are investigated under mild conditions, as $n\to \infty$, and we obtain consistent estimation for the regression function. Besides convergence results for the components of functional linear regression, such as the regression parameter function, we construct asymptotic pointwise confidence bands for the predicted trajectories. A functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced, extending the standard $R^2$ to the functional case. The proposed methods are illustrated with a simulation study, longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index.Comment: Published at http://dx.doi.org/10.1214/009053605000000660 in the Annals of Statistics (http://www.imstat.org/aos/) 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

Detecting and handling outlying trajectories in irregularly sampled functional datasets

Outlying curves often occur in functional or longitudinal datasets, and can be very influential on parameter estimators and very hard to detect visually. In this article we introduce estimators of the mean and the principal components that are resistant to, and then can be used for detection of, outlying sample trajectories. The estimators are based on reduced-rank t-models and are specifically aimed at sparse and irregularly sampled functional data. The outlier-resistance properties of the estimators and their relative efficiency for noncontaminated data are studied theoretically and by simulation. Applications to the analysis of Internet traffic data and glycated hemoglobin levels in diabetic children are presented.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS257 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Multi-Rank Sparse and Functional PCA: Manifold Optimization and Iterative Deflation Techniques

We consider the problem of estimating multiple principal components using the recently-proposed Sparse and Functional Principal Components Analysis (SFPCA) estimator. We first propose an extension of SFPCA which estimates several principal components simultaneously using manifold optimization techniques to enforce orthogonality constraints. While effective, this approach is computationally burdensome so we also consider iterative deflation approaches which take advantage of existing fast algorithms for rank-one SFPCA. We show that alternative deflation schemes can more efficiently extract signal from the data, in turn improving estimation of subsequent components. Finally, we compare the performance of our manifold optimization and deflation techniques in a scenario where orthogonality does not hold and find that they still lead to significantly improved performance.Comment: To appear in IEEE CAMSAP 201

Covariance Estimation and Principal Component Analysis for Mixed-Type Functional Data with application to mHealth in Mood Disorders

Mobile digital health (mHealth) studies often collect multiple within-day self-reported assessments of participants' behaviour and health. Indexed by time of day, these assessments can be treated as functional observations of continuous, truncated, ordinal, and binary type. We develop covariance estimation and principal component analysis for mixed-type functional data like that. We propose a semiparametric Gaussian copula model that assumes a generalized latent non-paranormal process generating observed mixed-type functional data and defining temporal dependence via a latent covariance. The smooth estimate of latent covariance is constructed via Kendall's Tau bridging method that incorporates smoothness within the bridging step. The approach is then extended with methods for handling both dense and sparse sampling designs, calculating subject-specific latent representations of observed data, latent principal components and principal component scores. Importantly, the proposed framework handles all four mixed types in a unified way. Simulation studies show a competitive performance of the proposed method under both dense and sparse sampling designs. The method is applied to data from 497 participants of National Institute of Mental Health Family Study of the Mood Disorder Spectrum to characterize the differences in within-day temporal patterns of mood in individuals with the major mood disorder subtypes including Major Depressive Disorder, and Type 1 and 2 Bipolar Disorder

Covariance Function Estimation for High-Dimensional Functional Time Series with Dual Factor Structures

We propose a flexible dual functional factor model for modelling high-dimensional functional time series. In this model, a high-dimensional fully functional factor parametrisation is imposed on the observed functional processes, whereas a low-dimensional version (via series approximation) is assumed for the latent functional factors. We extend the classic principal component analysis technique for the estimation of a low-rank structure to the estimation of a large covariance matrix of random functions that satisfies a notion of (approximate) functional "low-rank plus sparse" structure; and generalise the matrix shrinkage method to functional shrinkage in order to estimate the sparse structure of functional idiosyncratic components. Under appropriate regularity conditions, we derive the large sample theory of the developed estimators, including the consistency of the estimated factors and functional factor loadings and the convergence rates of the estimated matrices of covariance functions measured by various (functional) matrix norms. Consistent selection of the number of factors and a data-driven rule to choose the shrinkage parameter are discussed. Simulation and empirical studies are provided to demonstrate the finite-sample performance of the developed model and estimation methodology