Time series clustering is an important data mining task with a wide variety of applications. While most methods focus on time series taking values on the real line, very few works consider functional time series. However, functional objects frequently arise in many fields, such as actuarial science, demography or finance. Functional time series are indexed collections of infinite-dimensional curves viewed as random elements taking values in a Hilbert space. In this paper, the problem of clustering functional time series is addressed. To this aim, a distance between functional time series is introduced and used to construct a clustering procedure. The metric relies on a measure of serial dependence which can be seen as a natural extension of the classical quantile autocorrelation function to the functional setting. Since the dynamics of the series may vary over time, we adopt a fuzzy approach, which enables the procedure to locate each series into several clusters with different membership degrees. The resulting algorithm can group series generated from similar stochastic processes, reaching accurate results with series coming from a broad variety of functional models and requiring minimum hyperparameter tuning. Several simulation experiments show that the method exhibits a high clustering accuracy besides being computationally efficient. Two interesting applications involving high-frequency financial time series and age-specific mortality improvement rates illustrate the potential of the proposed approach.

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