An information criterion for detecting periodicities in functional time series

We propose an information criterion for determining an unknown number of periodic components in functional time series. Identifying the number of frequencies in large-scale time series has been a central focus. To achieve this goal, we suggest an iterative procedure, utilizing the residual process obtained through least squares fitting. This iterative approach demonstrates broad applicability. We establish the consistency of the estimated number of periodic components by minimizing the information criterion. The efficacy of the procedure is illustrated through numerical simulations. In real data analysis, we apply this information criterion to temperature data and sunspot data.