As discussed before, the Australian Temperature data at Gayndah Station was de-meaned by segmenting the data based on the change points in the mean function with keeping the base-line mean. Then the test for change in the joint eigenvalue is appiled and the year 1952 (or the change point estimate \(\hat{k}_n=60\), \(n=115\)) is detected as the change point location. Then the question of interest was whether the eigenfunctions were changing or not.
Assume that \(X_1, \dots, X_n\) is the demeaned data with the change happenning at \(\hat{k}_n\). The for fixed dimension \(d\) consider
\[ Y_i(t) = \begin{cases} \sum_{j=1}^d\frac{\langle X_i, \hat{\varphi_j}\rangle}{\sqrt{\hat{\lambda}^{b}_j}}\hat{\varphi_j} & i\leq\hat{k}_n \\ \sum_{j=1}^d\frac{\langle X_i, \hat{\varphi_j}\rangle}{\sqrt{\hat{\lambda}^{a}_j}}\hat{\varphi_j} & i>\hat{k}_n \\ \end{cases} \] where \(\hat{\lambda}^{b}_j\) and \(\hat{\lambda}^{a}_j\) are the estimated eigenvalues before and after the change, respectively. Then the test is constructed to as a two sample problem to test whether the covariance functions are the same for the two samples (before and after the change).