Detecting changes in the mean of functional observations

Principal component analysis has become a fundamental tool of functional data analysis. It represents the functional data as X-i(t)=mu(t)+Sigma(1 < l <infinity)eta(i, l)+ v(l)(t), where mu is the common mean, v(l) are the eigenfunctions of the covariance operator and the eta(i, l) are the scores. Inferential procedures assume that the mean function mu(t) is the same for all values of i. If, in fact, the observations do not come from one population, but rather their mean changes at some point(s), the results of principal component analysis are confounded by the change(s). It is therefore important to develop a methodology to test the assumption of a common functional mean. We develop such a test using quantities which can be readily computed in the R package fda. The null distribution of the test statistic is asymptotically pivotal with a well-known asymptotic distribution. The asymptotic test has excellent finite sample performance. Its application is illustrated on temperature data from England.