Wavelet-based Fuzzy Clustering of Time Series

Abstract

Traditional procedures for clustering time series are based mostly on crisp hierarchical or partitioning methods. Given that the dynamics of a time series may change over time, a time series might display patterns that may enable it to belong to one cluster over one period while over another period, its pattern may be more consistent with those in another cluster. The traditional clustering procedures are unable to identify the changing patterns over time. However, clustering based on fuzzy logic will be able to detect the switching patterns from one time period to another thus enabling some time series to simultaneously belong to more than one cluster. In particular, this paper proposes a fuzzy approach to the clustering of time series based on their variances through wavelet decomposition. We will show that this approach will distinguish between time series with different patterns in variability as well identifying time series with switching patterns in variability

Fuzzy clustering of time series in the frequency domain

“…In the introduction, we have remarked the two most important motivations justifying the utilization of a fuzzy approach for clustering time series are the sensitivity and the adaptivity [26,30,32,54]. In particular, with respect to the first aspect, we mean the sensitivity in capturing the information connected to the dynamic behavior characterizing the pattern of the time series.…”

“…In fact, a non-fuzzy definition of clusters contrasts, for example, with the ambiguities presented when switching time series may occur. Notice that the switching time series are time series showing, for instance, a pattern typical of a given cluster during a certain time period and a completely different pattern (characteristic of another cluster) over a another time period [26,30,32,54]). An example of switching time series is shown in Fig.…”

“…By contrast, there are just a few studies on the fuzzy clustering of time series; notable those by D'Urso and Maharaj [32] who used autocorrelation coefficients, Maharaj et al [54] who used wavelet variances, Zhang et al [85] and Chen and Fu [20] who used wavelets coefficients. Coppi et al [26] and D'Urso [30] in particular, succinctly put forward the two main motivations justifying the utilization of a fuzzy approach for the clustering time series data, namely, (1) Greater sensitivity, in capturing the details characterizing the time pattern of the time series.…”

Time series clustering by a robust autoregressive metric with application to air pollution

“…To identify the switching time series, we have set the membership degrees in the interval (0.3, 0.7) in the scenarios with two clusters, and in the interval (0.3, 0.6) in those with three clusters, so as to obtain fuzzy membership degrees across clusters. Note that the selected cut-off values are compatible with those suggested in literature: for simulation studies, see [28,29,83], while for empirical applications see [84].…”

Calibrating wavelet neural networks by distance orientation similarity fuzzy C-means for approximation problems

“…Apart from that, it could also be interesting to investigate the application of the DOS-FCM in clustering the stationary and non-stationary time series data [52][53][54][55] and anomalous data [56].…”