Description:

Functional Data are data in the form of functions.  The analysis of such data is termed functional data analysis (FDA). This tutorial covers basic topics in FDA, including mean and covariance estimation, functional principal component analysis and its applications, and functional regression and functional correlation. 
We begin with a brief introduction of the three basic types of functional data and the advantages of functional data over high-dimensional data, one of which is the feasibility to handle noise in functional data.  Then we focus on the fundamental problem of estimating the mean and covariance function and discuss phase transitions for functional data and optimal weighting schemes. These estimates are the basic ingredients for the most popular dimension reduction approach for functional data, functional principal component analysis (FPCA). FPCA provides the key components for the notion of modes of variation for functional data, a very useful tool for statistical practice.  We illustrate through data examples how to leverage FPCA for modeling and how to extend FPCA to adjust for covariate effects.
    We then consider the regression problem for functional data and show how to construct functional regression models for scalar as well as functional responses.  Both traditional linear models and non- or semi-parametric models can be extended to functional data and we will discuss the related challenge of inverting relevant linear operators. Another related topic of interest is the assessment of association and correlation for functional data.