A Trend-Granulation-Based Fuzzy C-Means Algorithm for Clustering Interval-Valued Time Series

Along with the abundant appearance of the interval-valued time series (ITS), the study on ITS clustering, especially shape-based ITS clustering, is becoming increasingly important. As an effective approach to extracting trend information in time series, fuzzy trend granulation addresses the needs of shape-based ITS clustering. However, when extracting trend information in ITS, unequal-size granules are inevitably produced, which makes ITS clustering difficult and challenging. Facing this issue, this article aims to generalize the widely used fuzzy C-means (FCM) algorithm to a fuzzy trend-granulation-based FCM algorithm for ITS clustering. To this end, a suite of algorithms, including ITS segmenting, segment merging, and granule building algorithms, are first developed for fuzzy trend-granulation of ITS, with which the given ITS is transformed into granular ITS, which consists of double linear fuzzy information granules (DLFIGs) and may be of different lengths. With the defined distance between DLFIGs, the distance between granular ITS is further developed through the dynamic time warping (DTW) algorithm. In designing the fuzzy trend-granulation-based FCM algorithm, the key step is to design the method for updating cluster prototypes to cope with the unequal lengths of granular ITS. The weighted DTW barycenter averaging method is a previously adopted prototype updating approach with the drawback of hardly changing the lengths of prototypes, which often makes prototypes less representative. Thus, a granule splitting and merging algorithm is designed to resolve this issue. Additionally, a prototype initialization method is also proposed to improve the clustering performance. The proposed fuzzy trend-granulation-based FCM algorithm for clustering ITS, being a typical shape-based clustering algorithm, exhibits superior performance, which is validated by the ablation experiments as well as the comparative experiments.