The proliferation and ubiquity of temporal data across many disciplines has generated substantial interest in the analysis and mining of time series. Clustering is one of the most popular data mining methods, not only due to its exploratory power, but also as a preprocessing step or subroutine for other techniques. In this paper, we present k-Shape, a novel algorithm for time-series clustering. k-Shape relies on a scalable iterative refinement procedure, which creates homogeneous and well-separated clusters. As its distance measure, k-Shape uses a normalized version of the cross-correlation measure in order to consider the shapes of time series while comparing them. Based on the properties of that distance measure, we develop a method to compute cluster centroids, which are used in every iteration to update the assignment of time series to clusters. To demonstrate the robustness of k-Shape, we perform an extensive experimental evaluation of our approach against partitional, hierarchical, and spectral clustering methods, with combinations of the most competitive distance measures. k-Shape outperforms all scalable approaches in terms of accuracy. Furthermore, k-Shape also outperforms all non-scalable (and hence impractical) combinations, with one exception that achieves similar accuracy results. However, unlike k-Shape, this combination requires tuning of its distance measure and is two orders of magnitude slower than k-Shape. Overall, k-Shape emerges as a domain-independent, highly accurate, and highly efficient clustering approach for time series with broad applications.

k-Shape: Efficient and Accurate Clustering of Time Series

Given the recent explosion of interest in streaming data and online algorithms, clustering of time-series subsequences, extracted via a sliding window, has received much attention. In this work, we make a surprising claim. Clustering of time-series subsequences is meaningless. More concretely, clusters extracted from these time series are forced to obey a certain constraint that is pathologically unlikely to be satisfied by any dataset, and because of this, the clusters extracted by any clustering algorithm are essentially random. While this constraint can be intuitively demonstrated with a simple illustration and is simple to prove, it has never appeared in the literature. We can justify calling our claim surprising because it invalidates the contribution of dozens of previously published papers. We will justify our claim with a theorem, illustrative examples, and a comprehensive set of experiments on reimplementations of previous work. Although the primary contribution of our work is to draw attention to the fact that an apparent solution to an important problem is incorrect and should no longer be used, we also introduce a novel method that, based on the concept of time-series motifs, is able to meaningfully cluster subsequences on some time-series datasets.

Clustering of time-series subsequences is meaningless: implications for previous and future research

Time series data is perhaps the most frequently encountered type of data examined by the data mining community. Clustering is perhaps the most frequently used data mining algorithm, being useful in it's own right as an exploratory technique, and also as a subroutine in more complex data mining algorithms such as rule discovery, indexing, summarization, anomaly detection, and classification. Given these two facts, it is hardly surprising that time series clustering has attracted much attention. The data to be clustered can be in one of two formats: many individual time series, or a single time series, from which individual time series are extracted with a sliding window. Given the recent explosion of interest in streaming data and online algorithms, the latter case has received much attention. We make an amazing claim. Clustering of streaming time series is completely meaningless. More concretely, clusters extracted from streaming time series are forced to obey a certain constraint that is pathologically unlikely to be satisfied by any dataset, and because of this, the clusters extracted by any clustering algorithm are essentially random. While this constraint can be intuitively demonstrated with a simple illustration and is simple to prove, it has never appeared in the literature. We can justify calling our claim surprising, since it invalidates the contribution of dozens of previously published papers. We will justify our claim with a theorem, illustrative examples, and a comprehensive set of experiments on reimplementations of previous work.

/pdf/clustering-of-time-series-subsequences-is-meaningless-1euxfe97ml.pdf

Clustering of time series subsequences is meaningless: implications for previous and future research

Recently, the research on efficient extraction of previously unknown, frequently appearing patterns in a time-series data has received much attention. These patterns are called `motifs'. Motifs are useful for various time-series data mining tasks. In this paper, we propose a motif discovery algorithm to extract a motif that represents a characteristic pattern of the given data based on Minimum Description Length (MDL) principle. In addition, the algorithm can extract motifs from multi-dimensional time-series data by using Principal Component Analysis (PCA). In experimental evaluation, we show the efficiency of the motif discovery algorithm, and the usefulness of extracted motifs to various data mining tasks.

https://link.springer.com/content/pdf/10.1007%2Fs10994-005-5829-2.pdf

Discovery of Time-Series Motif from Multi-Dimensional Data Based on MDL Principle

The proliferation and ubiquity of temporal data across many disciplines has generated substantial interest in the analysis and mining of time series. Clustering is one of the most popular data-mining methods, not only due to its exploratory power but also because it is often a preprocessing step or subroutine for other techniques. In this article, we present k-Shape and k-MultiShapes (k-MS), two novel algorithms for time-series clustering. k-Shape and k-MS rely on a scalable iterative refinement procedure. As their distance measure, k-Shape and k-MS use shape-based distance (SBD), a normalized version of the cross-correlation measure, to consider the shapes of time series while comparing them. Based on the properties of SBD, we develop two new methods, namely ShapeExtraction (SE) and MultiShapesExtraction (MSE), to compute cluster centroids that are used in every iteration to update the assignment of time series to clusters. k-Shape relies on SE to compute a single centroid per cluster based on all time series in each cluster. In contrast, k-MS relies on MSE to compute multiple centroids per cluster to account for the proximity and spatial distribution of time series in each cluster. To demonstrate the robustness of SBD, k-Shape, and k-MS, we perform an extensive experimental evaluation on 85 datasets against state-of-the-art distance measures and clustering methods for time series using rigorous statistical analysis. SBD, our efficient and parameter-free distance measure, achieves similar accuracy to Dynamic Time Warping (DTW), a highly accurate but computationally expensive distance measure that requires parameter tuning. For clustering, we compare k-Shape and k-MS against scalable and non-scalable partitional, hierarchical, spectral, density-based, and shapelet-based methods, with combinations of the most competitive distance measures. k-Shape outperforms all scalable methods in terms of accuracy. Furthermore, k-Shape also outperforms all non-scalable approaches, with one exception, namely k-medoids with DTW, which achieves similar accuracy. However, unlike k-Shape, this approach requires tuning of its distance measure and is significantly slower than k-Shape. k-MS performs similarly to k-Shape in comparison to rival methods, but k-MS is significantly more accurate than k-Shape. Beyond clustering, we demonstrate the effectiveness of k-Shape to reduce the search space of one-nearest-neighbor classifiers for time series. Overall, SBD, k-Shape, and k-MS emerge as domain-independent, highly accurate, and efficient methods for time-series comparison and clustering with broad applications.

Fast and Accurate Time-Series Clustering

In the past several years, human motion data has been used in some domains such as SFX movies, CGs and so on. Motion data is captured by a motion capture system which captures the position of sensors on body joints, so we can get motion data as 3-D time series data. Since, motion data is multi-stream data of time series for 17 body parts, the amount of data is huge. Furthermore, motion data is expensive. A motion database can help those creators to produce motion data with less cost. The database, however, requires a content-based retrieval method because it is difficult to identify the motion by using a keyword approach. Consequently we introduce association rules which represent dependency between body parts. Association rules represent the motion of body parts and can be used as visual tags. We introduce a method to discover dependency between body parts as association rules. Association rules consist of symbols uniquely representing basic patterns. We call basic patterns primitive motions. Primitive motions are extracted from motion data by using segmentation and clustering processes. Finally, we discuss some experiments to discover association rules from multi-streams of motion data.

K. Uehara

Papers

Extraction of primitive motion and discovery of association rules from motion data