Detecting deviations from stationarity of functional time series

The advent of complex data has led to increased research in virtually all areas of statistics, including functional data analysis (FDA). Within the purview of FDA, the use of methods for serially correlated functions is often prudent. As for simpler univariate time series models, the theoretical foundations of methodology are often laid exploiting the notion of stationarity, while data analysis is often conducted on data violating this assumption. This talks looks into ways of discovering departures from stationarity in two ways. In the first part, structural breaks are considered, such that the sample is split into segments in a non-smooth fashion. The methodology to be presented does not rely on the usual dimension reduction techniques, which might be advantageous if the structural break is not sparse (that is, not concentrated within the primary modes of variation of the data). In the second part, local stationarity is introduced as a smooth deviation from stationarity. Here methods in the frequency domain are considered, based on the general result that (second-order) stationarity is equivalent to a functional version of the peridogram being uncorrelated at the Fourier frequencies. Both sets of methods are illustrated with annual Australian temperature profiles. The talk is based on joint work with Anne van Delft (Bochum), Greg Rice (Waterloo) and Ozan Sönmez (Davis).