Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data
Yehua Li,Tailen Hsing +1 more
Reads0
Chats0
TLDR
In this paper, the authors consider nonparametric estimation of the mean and covariance functions for functional/longitudinal data and derive almost sure rates of convergence for principal component analysis using the estimated covariance function.Abstract:
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.read more
Citations
More filters
Journal ArticleDOI
Functional Data Analysis
TL;DR: In this article, the authors provide an overview of FDA, starting with simple statistical notions such as mean and covariance functions, then covering some core techniques, the most popular of which is functional principal component analysis (FPCA).
Journal ArticleDOI
A partial overview of the theory of statistics with functional data
TL;DR: The theory and practice of statistical methods in situations where the available data are functions (instead of real numbers or vectors) is often referred to as functional data analysis (FDA) as discussed by the authors.
Posted Content
Functional Regression
TL;DR: Functional data analysis (FDA) involves the analysis of data whose ideal units of observation are functions defined on some continuous domain, and the observed data consist of a sample of functions taken from some population, sampled on a discrete grid.
Journal ArticleDOI
From sparse to dense functional data and beyond
Xiaoke Zhang,Jane-Ling Wang +1 more
TL;DR: In this paper, the performance of local linear smoothers for both mean and covariance functions with a general weighing scheme, which includes two commonly used schemes, equal weight per observation (OBS), and equal weight each subject (SUBJ), as two special cases, is investigated.
Posted Content
Review of Functional Data Analysis
TL;DR: An overview of FDA is provided, starting with simple statistical notions such as mean and covariance functions, then covering some core techniques, the most popular of which is Functional Principal Component Analysis (FPCA), an important dimension reduction tool and in sparse data situations can be used to impute functional data that are sparsely observed.
References
More filters
Book
Local polynomial modelling and its applications
Jianqing Fan,Irène Gijbels +1 more
TL;DR: Applications of Local Polynomial Modeling in Nonlinear Time Series and Automatic Determination of Model Complexity and Framework for Local polynomial regression.
Book ChapterDOI
Functional Data Analysis
TL;DR: In this article, the authors introduce the concept of functional data analysis (FDA) to describe the smoothness of the process of generating functional data from a set of observed curves and images.
Journal ArticleDOI
Functional Data Analysis for Sparse Longitudinal Data
TL;DR: In this article, a nonparametric method is proposed to perform functional principal components analysis for sparse longitudinal data, where the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random (subject-specific) trajectory plus measurement errors.
Journal ArticleDOI
On Some Global Measures of the Deviations of Density Function Estimates
Peter J. Bickel,M. Rosenblatt +1 more
TL;DR: In this paper, the authors consider density estimates of the usual type generated by a weight function and obtain limit theorems for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity.
Related Papers (5)
Theoretical foundations of functional data analysis, with an introduction to linear operators
Tailen Hsing,R. L. Eubank +1 more