  The existence of large volumes of time series data in many applications has
motivated data miners to investigate specialized methods for mining time series
data. Clustering is a popular data mining method due to its powerful
exploratory nature and its usefulness as a preprocessing step for other data
mining techniques. This article develops two novel clustering algorithms for
time series data that are extensions of a crisp c-shapes algorithm. The two new
algorithms are heuristic derivatives of fuzzy c-means (FCM). Fuzzy c-Shapes
plus (FCS+) replaces the inner product norm in the FCM model with a shape-based
distance function. Fuzzy c-Shapes double plus (FCS++) uses the shape-based
distance, and also replaces the FCM cluster centers with shape-extracted
prototypes. Numerical experiments on 48 real time series data sets show that
the two new algorithms outperform state-of-the-art shape-based clustering
algorithms in terms of accuracy and efficiency. Four external cluster validity
indices (the Rand index, Adjusted Rand Index, Variation of Information, and
Normalized Mutual Information) are used to match candidate partitions generated
by each of the studied algorithms. All four indices agree that for these finite
waveform 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 two
tests of statistical significance to the three algorithms. The Wilcoxon and
Friedman statistics both rank the three algorithms in exactly the same way as
the four cluster validity indices.
