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Due to the surge of data storage techniques, the need for the development ofappropriate techniques to identify patterns and to extract knowledge from theresulting enormous data sets, which can be viewed as collections of dependentfunctional data, is of increasing interest in many scientific areas. We developa similarity measure for spectral density operators of a collection offunctional time series, which is based on the aggregation of Hilbert-Schmidtdifferences of the individual time-varying spectral density operators. Underfairly general conditions, the asymptotic properties of the correspondingestimator are derived and asymptotic normality is established. The introducedstatistic lends itself naturally to quantify (dis)-similarity betweenfunctional time series, which we subsequently exploit in order to build aspectral clustering algorithm. Our algorithm is the first of its kind in theanalysis of non-stationary (functional) time series and enables to discoverparticular patterns by grouping together `similar' series into clusters,thereby reducing the complexity of the analysis considerably. The algorithm issimple to implement and computationally feasible. As a further application weprovide a simple test for the hypothesis that the second order properties oftwo non-stationary functional time series coincide.

A similarity measure for second order properties of non-stationary functional time series with applications to clustering and testing