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.
Our definition of frequent time-series motifs is similar to that of @cite_8 @cite_5 ; however they do not focus on efficiency. These approaches have quadratic @cite_8 or cubic @cite_5 time-complexity in the size of the series ( @math ) (recently, @cite_22 Source code not published. brings @cite_5 down to quadratic complexity). @cite_8 exploit symbolic representation of subsequences using the SAX scheme @cite_2 , we directly use subsequences in @math space after z-normalization and level-merging. In @cite_5 authors exploit pair motif discovery algorithm @cite_15 followed by a search for other similar subsequences in the time-series. They choose the member subsequences of a frequent motif based on bits saved through MDL encoding of the subsequences of the motif. We improve on these approaches, empirically achieving near linear performance; see Section and .
@cite_22 @cite_8 @cite_2 @cite_5 @cite_15
34835763 1513731586 2006761268 2054978951
The problem of locating motifs in real-valued, multivariate time series data involves the discovery of sets of recurring patterns embedded in the time series. Each set is composed of several non-overlapping subsequences and constitutes a motif because all of the included subsequences are similar. The ability to automatically discover such motifs allows intelligent systems to form endogenously meaningful representations of their environment through unsupervised sensor analysis. In this paper, we formulate a unifying view of motif discovery as a problem of locating regions of high density in the space of all time series subsequences. Our approach is efficient (sub-quadratic in the length of the data), requires fewer user-specified parameters than previous methods, and naturally allows variable length motif occurrences and non-linear temporal warping. We evaluate the performance of our approach using four data sets from different domains including on-body inertial sensors and speech. Time series motifs are pairs of individual time series, or subsequences of a longer time series, which are very similar to each other. As with their discrete analogues in computational biology, this similarity hints at structure which has been conserved for some reason and may therefore be of interest. Since the formalism of time series motifs in 2002, dozens of researchers have used them for diverse applications in many different domains. Because the obvious algorithm for computing motifs is quadratic in the number of items, more than a dozen approximate algorithms to discover motifs have been proposed in the literature. In this work, for the first time, we show a tractable exact algorithm to find time series motifs. As we shall show through extensive experiments, our algorithm is up to three orders of magnitude faster than brute-force search in large datasets. We further show that our algorithm is fast enough to be used as a subroutine in higher level data mining algorithms for anytime classification, near-duplicate detection and summarization, and we consider detailed case studies in domains as diverse as electroencephalograph interpretation and entomological telemetry data mining. 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. Time series motifs are repeated patterns in long and noisy time series. Motifs are typically used to understand the dynamics of the source because repeated patterns with high similarity evidentially rule out the presence of noise. Recently, time series motifs have also been used for clustering, summarization, rule discovery and compression as features. For all such purposes, many high quality motifs of various lengths are desirable and thus, originates the problem of enumerating motifs for a wide range of lengths. Existing algorithms find motifs for a given length. A trivial way to enumerate motifs is to run one of the algorithms for the whole range of lengths. However, such parameter sweep is computationally infeasible for large real datasets. In this paper, we describe an exact algorithm, called MOEN, to enumerate motifs. The algorithm is an order of magnitude faster than the naive algorithm. The algorithm frees us from re-discovering the same motif at different lengths and tuning multiple data-dependent parameters. The speedup comes from using a novel bound on the similarity function across lengths and the algorithm uses only linear space unlike other motif discovery algorithms. We describe three case studies in entomology and activity recognition where MOEN enumerates several high quality motifs.