An introduction to functional data analysis and a principal component approach for testing the equality of mean curves

We give an introduction to functional data analysis, with examples, and provide a brief review of the literature. We explain how principal component analysis (PCA) can be used to transform curves into finite dimensional data. An application of PCA is developed to test for the equality of the means of several populations (functional analysis of variance). Asymptotics are derived under the null hypothesis that the populations have the same mean curves. The selection of the basis for the projections and the power of the test is discussed for simple random samples and stationary time series samples of curves. We review the part of the literature which is needed to establish the validity of the PCA method. Two data sets, magnetogram records and stock returns, are used to illustrate the applicability of our limit results.