Exploring Variation: Functional Principal and Canonical Components Analysis

Abstract

Now we look at how observations vary from one replication or sampled value to the next. There is, of course, also variation within observations, but we focused on that type of variation when considering data smoothing in Chapter 5.

Principal components analysis, or PCA, is often the first method that we turn to after descriptive statistics and plots. We want to see what primary modes of variation are in the data, and how many of them seem to be substantial. As in multivariate statistics, eigenvalues of the bivariate variance-covariance function v(s; t) are indicators of the importance of these principal components, and plotting eigenvalues is a method for determining how many principal components are required to produce a reasonable summary of the data.