Quantile autocovariances: a powerful tool for hard and soft partitional clustering of time series

A key issue in cluster analysis is determining a proper dissimilarity measure between two data objects, and many pairwise dissimilarities have been proposed to deal with time series. Assuming that the clustering purpose is to group series according to the underlying dependence structures, a detailed study of the behavior in clustering of a dissimilarity based on comparing estimated quantile autocovariance functions (QAF) is carried out. Quantile autocovariances provide information about the serial dependence structure that other conventional features are not able to capture, which suggests great potential to perform clustering of series. The asymptotic behavior of the sample quantile autocovariances is studied and an algorithm to determine optimal combinations of lags and pairs of quantile levels to perform clustering is introduced. The proposed metric is used to perform hard and soft partitioning-based clustering. First, a broad simulation study examines the behavior of the proposed metric in crisp clustering with the PAM procedure. A novel fuzzy C-medoids algorithm based on the QAF-dissimilarity is then proposed and compared with other fuzzy procedures in a new simulation study conducted to cluster fuzzy scenarios involving AR and GARCH models. In all cases, the QAF-based procedures outperform or are highly competitive with a range of dissimilarities reported in the literature, particularly exhibiting high capability to cluster conditionally heteroskedastic time series and robustness to the distributional form of the errors. Two specific applications involving air quality data and financial time series illustrate the usefulness of the proposed procedures.