Fast Silhouette on Circular or Linear Data Clusters

Calculating silhouette information for clusters on circular or linear data using fast algorithms. These algorithms run in linear time on sorted data, in contrast to quadratic time by the definition of silhouette. When used together with the fast and optimal circular clustering method FOCC (Debnath & Song 2021) implemented in R package 'OptCirClust', circular silhouette can be maximized to find the optimal number of circular clusters; it can also be used to estimate the period of noisy periodical data.