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The existence of large volumes of time series data in many applications hasmotivated data miners to investigate specialized methods for mining time seriesdata. Clustering is a popular data mining method due to its powerfulexploratory nature and its usefulness as a preprocessing step for other datamining techniques. This article develops two novel clustering algorithms fortime series data that are extensions of a crisp c-shapes algorithm. The two newalgorithms are heuristic derivatives of fuzzy c-means (FCM). Fuzzy c-Shapesplus (FCS+) replaces the inner product norm in the FCM model with a shape-baseddistance function. Fuzzy c-Shapes double plus (FCS++) uses the shape-baseddistance, and also replaces the FCM cluster centers with shape-extractedprototypes. Numerical experiments on 48 real time series data sets show thatthe two new algorithms outperform state-of-the-art shape-based clusteringalgorithms in terms of accuracy and efficiency. Four external cluster validityindices (the Rand index, Adjusted Rand Index, Variation of Information, andNormalized Mutual Information) are used to match candidate partitions generatedby each of the studied algorithms. All four indices agree that for these finitewaveform data sets, FCS++ gives a small improvement over FCS+, and in turn,FCS+ is better than the original crisp c-shapes method. Finally, we apply twotests of statistical significance to the three algorithms. The Wilcoxon andFriedman statistics both rank the three algorithms in exactly the same way asthe four cluster validity indices.

Fuzzy c-Shape: A new algorithm for clustering finite time series waveforms