Sparseness and Functional Data Analysis

In this chapter we examine two different settings in which sparseness can be important in a functional data analysis (FDA). The first setting involves sparseness in the functions. The classical assumption of FDA is that each function has been measured at all time points. However, in practice it is often the case that the functions have only been observed at a relatively small number of points. Here we discuss different general approaches that can be applied in this setting, such as basis functions, mixed effects models and local smoothing, and examine the relative merits of each. Then we briefly outline several specific methodologies that have been developed for dealing with sparse functional data in the principal components, clustering, classification and regression paradigms. The second setting involves using sparsity ideas from high dimensional regression problems, where most of the regression coefficients are assumed to be zero, to perform a dimension reduction in the functional space. We discuss two specific approaches that have been suggested in the literature.