  A novel procedure to perform fuzzy clustering of multivariate time series
generated from different dependence models is proposed. Different amounts of
dissimilarity between the generating models or changes on the dynamic
behaviours over time are some arguments justifying a fuzzy approach, where each
series is associated to all the clusters with specific membership levels. Our
procedure considers quantile-based cross-spectral features and consists of
three stages: (i) each element is characterized by a vector of proper estimates
of the quantile cross-spectral densities, (ii) principal component analysis is
carried out to capture the main differences reducing the effects of the noise,
and (iii) the squared Euclidean distance between the first retained principal
components is used to perform clustering through the standard fuzzy C-means and
fuzzy C-medoids algorithms. The performance of the proposed approach is
evaluated in a broad simulation study where several types of generating
processes are considered, including linear, nonlinear and dynamic conditional
correlation models. Assessment is done in two different ways: by directly
measuring the quality of the resulting fuzzy partition and by taking into
account the ability of the technique to determine the overlapping nature of
series located equidistant from well-defined clusters. The procedure is
compared with the few alternatives suggested in the literature, substantially
outperforming all of them whatever the underlying process and the evaluation
scheme. Two specific applications involving air quality and financial databases
illustrate the usefulness of our approach.
