A Method for Projecting Functional Data Onto a Low-Dimensional Space

Abstract

Functional data are intrinsically infinite-dimensional, and so dimension reduction is an attractive precursor to functional data analysis. Methods based on principal components analysis are widely used, but their value is limited by the awkwardness of interpreting the components, which are defined in a relatively abstract way. Sometimes, quantities that are nonlinear functions of the data curves are easier to interpret, and convey more information for classification and discrimination, than relatively conventional principal components. Here we illustrate this property in a particular case, by introducing four easily understood variables associated with each member of a functional dataset. These quantities represent the mean and variance of the frequency of each curve, and the mean and variance of the curve's amplitude. In fact, just two of these variables, the means, convey much of the information. These quantities can be used very effectively for clustering and classification, often with greater success than the first two principal components. A simulation study, and an application to Ramsay's classic weather-station dataset, illustrate this point.