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

Optimal Bayes Classifiers for Functional Data and Density Ratios

Bayes classifiers for functional data pose a challenge. This is because probability density functions do not exist for functional data. As a consequence, the classical Bayes classifier using density quotients needs to be modified. We propose to use density ratios of projections on a sequence of eigenfunctions that are common to the groups to be classified. The density ratios can then be factored into density ratios of individual functional principal components whence the classification problem is reduced to a sequence of nonparametric one-dimensional density estimates. This is an extension to functional data of some of the very earliest nonparametric Bayes classifiers that were based on simple density ratios in the one-dimensional case. By means of the factorization of the density quotients the curse of dimensionality that would otherwise severely affect Bayes classifiers for functional data can be avoided. We demonstrate that in the case of Gaussian functional data, the proposed functional Bayes classifier reduces to a functional version of the classical quadratic discriminant. A study of the asymptotic behavior of the proposed classifiers in the large sample limit shows that under certain conditions the misclassification rate converges to zero, a phenomenon that has been referred to as "perfect classification". The proposed classifiers also perform favorably in finite sample applications, as we demonstrate in comparisons with other functional classifiers in simulations and various data applications, including wine spectral data, functional magnetic resonance imaging (fMRI) data for attention deficit hyperactivity disorder (ADHD) patients, and yeast gene expression data

Regularized Halfspace Depth for Functional Data

Data depth is a powerful nonparametric tool originally proposed to rank multivariate data from center outward. In this context, one of the most archetypical depth notions is Tukey's halfspace depth. In the last few decades notions of depth have also been proposed for functional data. However, Tukey's depth cannot be extended to handle functional data because of its degeneracy. Here, we propose a new halfspace depth for functional data which avoids degeneracy by regularization. The halfspace projection directions are constrained to have a small reproducing kernel Hilbert space norm. Desirable theoretical properties of the proposed depth, such as isometry invariance, maximality at center, monotonicity relative to a deepest point, upper semi-continuity, and consistency are established. Moreover, the regularized halfspace depth can rank functional data with varying emphasis in shape or magnitude, depending on the regularization. A new outlier detection approach is also proposed, which is capable of detecting both shape and magnitude outliers. It is applicable to trajectories in L2, a very general space of functions that include non-smooth trajectories. Based on extensive numerical studies, our methods are shown to perform well in terms of detecting outliers of different types. Three real data examples showcase the proposed depth notion

Bootstrap inference in functional linear regression models with scalar response under heteroscedasticity

Inference for functional linear models in the presence of heteroscedastic errors has received insufficient attention given its practical importance; in fact, even a central limit theorem has not been studied in this case. At issue, conditional mean (projection of the slope function) estimates have complicated sampling distributions due to the infinite dimensional regressors, which create truncation bias and scaling problems that are compounded by non-constant variance under heteroscedasticity. As a foundation for distributional inference, we establish a central limit theorem for the estimated projection under general dependent errors, and subsequently we develop a paired bootstrap method to approximate sampling distributions. The proposed paired bootstrap does not follow the standard bootstrap algorithm for finite dimensional regressors, as this version fails outside of a narrow window for implementation with functional regressors. The reason owes to a bias with functional regressors in a naive bootstrap construction. Our bootstrap proposal incorporates debiasing and thereby attains much broader validity and flexibility with truncation parameters for inference under heteroscedasticity; even when the naive approach may be valid, the proposed bootstrap method performs better numerically. The bootstrap is applied to construct confidence intervals for projections and for conducting hypothesis tests for the slope function. Our theoretical results on bootstrap consistency are demonstrated through simulation studies and also illustrated with real data examples