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Register a new userUniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data
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Yehua Li
Publication date
2012
fields
Mathematical Statistics
and research's language is
English
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We consider nonparametric estimation of the mean and covariance functions for functional/longitudinal data. Strong uniform convergence rates are developed for estimators that are local-linear smoothers. Our results are obtained in a unified framework in which the number of observations within each curve/cluster can be of any rate relative to the sample size. We show that the convergence rates for the procedures depend on both the number of sample curves and the number of observations on each curve. For sparse functional data, these rates are equivalent to the optimal rates in nonparametric regression. For dense functional data, root-n rates of convergence can be achieved with proper choices of bandwidths. We further derive almost sure rates of convergence for principal component analysis using the estimated covariance function. The results are illustrated with simulation studies.
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In functional linear regression, the slope ``parameter is a function. Therefore, in a nonparametric context, it is determined by an infinite number of unknowns. Its estimation involves solving an ill-posed problem and has points of contact with a range of methodologies, including statistical smoothing and deconvolution. The standard approach to estimating the slope function is based explicitly on functional principal components analysis and, consequently, on spectral decomposition in terms of eigenvalues and eigenfunctions. We discuss this approach in detail and show that in certain circumstances, optimal convergence rates are achieved by the PCA technique. An alternative approach based on quadratic regularisation is suggested and shown to have advantages from some points of view.
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 Frechet 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.
Functional principal component analysis is essential in functional data analysis, but the inferences will become unconvincing when some non-Gaussian characteristics occur, such as heavy tail and skewness. The focus of this paper is to develop a robust functional principal component analysis methodology in dealing with non-Gaussian longitudinal data, for which sparsity and irregularity along with non-negligible measurement errors must be considered. We introduce a Kendalls $tau$ function whose particular properties make it a nice proxy for the covariance function in the eigenequation when handling non-Gaussian cases. Moreover, the estimation procedure is presented and the asymptotic theory is also established. We further demonstrate the superiority and robustness of our method through simulation studies and apply the method to the longitudinal CD4 cell count data in an AIDS study.
Let $X$ be a mean zero Gaussian random vector in a separable Hilbert space ${mathbb H}$ with covariance operator $Sigma:={mathbb E}(Xotimes X).$ Let $Sigma=sum_{rgeq 1}mu_r P_r$ be the spectral decomposition of $Sigma$ with distinct eigenvalues $mu_1>mu_2> dots$ and the corresponding spectral projectors $P_1, P_2, dots.$ Given a sample $X_1,dots, X_n$ of size $n$ of i.i.d. copies of $X,$ the sample covariance operator is defined as $hat Sigma_n := n^{-1}sum_{j=1}^n X_jotimes X_j.$ The main goal of principal component analysis is to estimate spectral projectors $P_1, P_2, dots$ by their empirical counterparts $hat P_1, hat P_2, dots$ properly defined in terms of spectral decomposition of the sample covariance operator $hat Sigma_n.$ The aim of this paper is to study asymptotic distributions of important statistics related to this problem, in particular, of statistic $|hat P_r-P_r|_2^2,$ where $|cdot|_2^2$ is the squared Hilbert--Schmidt norm. This is done in a high-complexity asymptotic framework in which the so called effective rank ${bf r}(Sigma):=frac{{rm tr}(Sigma)}{|Sigma|_{infty}}$ (${rm tr}(cdot)$ being the trace and $|cdot|_{infty}$ being the operator norm) of the true covariance $Sigma$ is becoming large simultaneously with the sample size $n,$ but ${bf r}(Sigma)=o(n)$ as $ntoinfty.$ In this setting, we prove that, in the case of one-dimensional spectral projector $P_r,$ the properly centered and normalized statistic $|hat P_r-P_r|_2^2$ with {it data-dependent} centering and normalization converges in distribution to a Cauchy type limit. The proofs of this and other related results rely on perturbation analysis and Gaussian concentration.
Component-wise MCMC algorithms, including Gibbs and conditional Metropolis-Hastings samplers, are commonly used for sampling from multivariate probability distributions. A long-standing question regarding Gibbs algorithms is whether a deterministic-scan (systematic-scan) sampler converges faster than its random-scan counterpart. We answer this question when the samplers involve two components by establishing an exact quantitative relationship between the $L^2$ convergence rates of the two samplers. The relationship shows that the deterministic-scan sampler converges faster. We also establish qualitative relations among the convergence rates of two-component Gibbs samplers and some conditional Metropolis-Hastings variants. For instance, it is shown that if some two-component conditional Metropolis-Hastings samplers are geometrically ergodic, then so are the associated Gibbs samplers.
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