  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.
