While analyzing vehicular sensor data, we found that frequently occurring waveforms could serve as features for further analysis, such as rule mining, classification, and anomaly detection. The discovery of waveform patterns, also known as time-series motifs, has been studied extensively; however, available techniques for discovering frequently occurring time-series motifs were found lacking in either efficiency or quality: Standard subsequence clustering results in poor quality, to the extent that it has even been termed 'meaningless'. Variants of hierarchical clustering using techniques for efficient discovery of 'exact pair motifs' find high-quality frequent motifs, but at the cost of high computational complexity, making such techniques unusable for our voluminous vehicular sensor data. We show that good quality frequent motifs can be discovered using bounded spherical clustering of time-series subsequences, which we refer to as COIN clustering, with near linear complexity in time-series size. COIN clustering addresses many of the challenges that previously led to subsequence clustering being viewed as meaningless. We describe an end-to-end motif-discovery procedure using a sequence of pre and post-processing techniques that remove trivial-matches and shifted-motifs, which also plagued previous subsequence-clustering approaches. We demonstrate that our technique efficiently discovers frequent motifs in voluminous vehicular sensor data as well as in publicly available data sets.
The problem of trivially matching subsequences has been identified in the research literature related to STS clustering @cite_4 @cite_23 @cite_12 @cite_8 @cite_0 @cite_1 . Most of these approaches @cite_4 @cite_23 @cite_1 focus on non-overlapping subsequences at the outset therefore such approaches may altogether miss some of the motifs due to their lower support. Further, Chen has also argued in @cite_12 that removing the subsequences before clustering also does not completely solve the issue of smoothing of subsequence clusters. Not enough attention has been given to an approach for removal of such subsequences after clustering, primarily because of the absence of suitable clustering method itself. Similar to above publications we also remove the trivially matching subsequences before clustering, however one of the key-contributions of our work is removal of trivially matching subsequences through post-processing, see , as well as highlighting the importance of level-splitting so that the discovered motifs are useful in practice.
@cite_4 @cite_8 @cite_1 @cite_0 @cite_23 @cite_12
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In time series mining, one of the interesting tasks that attract many researchers is time series clustering which is classified into two main categories. Whole time series clustering considers how to cluster multiple time series, and the other one is Subsequence Time Series (STS) clustering, a clustering of subparts or subsequences within a single time series. Deplorably, STS clustering is not preferable even though it had widely been used as a subroutine in various mining tasks, e.g., rule discovery, anomaly detection, or classification, due to the recent finding a decade ago that STS clustering problem can produce meaningless results. There have been numerous attempts to resolve this problem but seemed to be unsuccessful. Until the two most recent attempts, they seem to accomplish in producing meaningful results; however, their approaches do need some predefined constraint values, such as the width of the subsequences that are in fact quite subjective and sensitive. Thus, we propose a novel parameter-free clustering technique to eliminate this problem by utilizing a motif discovery algorithm and some statistical principles to properly determine these parameters. Our experimental results from well-known datasets demonstrate the effectiveness of the proposed algorithm in selecting the proper subsequence width, and in turn leading to meaningful and highly accurate results. Clustering time series data using the popular subsequence (STS) technique has been widely used in the data mining and wider communities. Recently the conclusion was made that it is meaningless, based on the findings that it produces (a) clustering outcomes for distinct time series that are not distinguishable from one another, and (b) cluster centroids that are smoothed. More recent work has since showed that (a) could be solved by introducing a lag in the subsequence vector construction process, however we show in this paper that such an approach does not solve (b). Motivating the terminology that a clustering method which overcomes (a) is meaningful, while one which overcomes (a) and (b) is useful, we propose an approach that produces useful time series clustering. The approach is based on restricting the clustering space to extend only over the region visited by the time series in the subsequence vector space. We test the approach on a set of 12 diverse real-world and synthetic data sets and find that (a) one can distinguish between the clusterings of these time series, and (b) that the centroids produced in each case retain the character of the underlying series from which they came. Several important time series data mining problems reduce to the core task of finding approximately repeated subsequences in a longer time series. In an earlier work, we formalized the idea of approximately repeated subsequences by introducing the notion of time series motifs. Two limitations of this work were the poor scalability of the motif discovery algorithm, and the inability to discover motifs in the presence of noise.Here we address these limitations by introducing a novel algorithm inspired by recent advances in the problem of pattern discovery in biosequences. Our algorithm is probabilistic in nature, but as we show empirically and theoretically, it can find time series motifs with very high probability even in the presence of noise or "don't care" symbols. Not only is the algorithm fast, but it is an anytime algorithm, producing likely candidate motifs almost immediately, and gradually improving the quality of results over time. We consider two closely related problems: planted clustering and submatrix localization. The planted clustering problem assumes that a random graph is generated based on some underlying clusters of the nodes; the task is to recover these clusters given the graph. The submatrix localization problem concerns locating hidden submatrices with elevated means inside a large real-valued random matrix. Of particular interest is the setting where the number of clusters submatrices is allowed to grow unbounded with the problem size. These formulations cover several classical models such as planted clique, planted densest subgraph, planted partition, planted coloring, and stochastic block model, which are widely used for studying community detection and clustering bi-clustering. For both problems, we show that the space of the model parameters (cluster submatrix size, cluster density, and submatrix mean) can be partitioned into four disjoint regions corresponding to decreasing statistical and computational complexities: (1) the regime, where all algorithms fail; (2) the regime, where the computationally expensive Maximum Likelihood Estimator (MLE) succeeds; (3) the regime, where the polynomial-time convexified MLE succeeds; (4) the regime, where a simple counting thresholding procedure succeeds. Moreover, we show that each of these algorithms provably fails in the previous harder regimes. Our theorems establish the minimax recovery limit, which are tight up to constants and hold with a growing number of clusters submatrices, and provide a stronger performance guarantee than previously known for polynomial-time algorithms. Our study demonstrates the tradeoffs between statistical and computational considerations, and suggests that the minimax recovery limit may not be achievable by polynomial-time algorithms.