Over the last decade, time series motif discovery has emerged as a useful primitive for many downstream analytical tasks, including clustering, classification, rule discovery, segmentation, and summarization. In parallel, there has been an increased understanding that Dynamic Time Warping (DTW) is the best time series similarity measure in a host of settings. Surprisingly however, there has been virtually no work on using DTW to discover motifs. The most obvious explanation of this is the fact that both motif discovery and the use of DTW can be computationally challenging, and the current best mechanisms to address their lethargy are mutually incompatible. In this work, we present the first scalable exact method to discover time series motifs under DTW. Our method automatically performs the best trade-off between time-to-compute and tightness-of-lower-bounds for a novel hierarchy of lower bounds representation we introduce. We show that under realistic settings, our algorithm can admissibly prune up to 99.99% of the DTW computations.

The discovery of conserved (repeated) patterns in time series is arguably the most important primitive in time series data mining. Called time series motifs, these primitive patterns are useful in their own right, and are also used as inputs into classification, clustering, segmentation, visualization, and anomaly detection algorithms. Recently the Matrix Profile has emerged as a promising representation to allow the efficient exact computation of the top-k motifs in a time series. State-of-the-art algorithms for computing the Matrix Profile are fast enough for many tasks. However, in a handful of domains, including astronomy and seismology, there is an insatiable appetite to consider ever larger datasets. In this work we show that with several novel insights we can push the motif discovery envelope using a novel scalable framework in conjunction with a deployment to commercial GPU clusters in the cloud. We demonstrate the utility of our ideas with detailed case studies in seismology, demonstrating that the efficiency of our algorithm allows us to exhaustively consider datasets that are currently only approximately searchable, allowing us to find subtle precursor earthquakes that had previously escaped attention, and other novel seismic regularities.

In recent years the data mining community has largely coalesced around the idea that many problems in time series analytics essentially reduce to finding and then reasoning about repeated structure in time series. Existing tools can find conserved structure within a single time series (motifs) and between pairs of time series (joins). However, to date there are no tools to find repeated structure in sets of time series, an idea we call time series consensus motifs in recognition of their similarity to their discrete analogs in DNA strings. In this work we introduce a definition of time series consensus motifs and a scalable algorithm to discover them in large data collections. We further show that given this new primitive, we can solve multiple higherlevel problems in time series data mining. We demonstrate the utility of our ideas with case studies in diverse domains.

Time series motif discovery is an important primitive for time series analytics, and is used in domains as diverse as neuroscience, music and sports analytics. In recent years, algorithmic advances (coupled with hardware improvements) have greatly expanded the purview of motif discovery. Nevertheless, we argue that there is an insatiable need for further scalability. This is because more than most types of analytics, motif discovery benefits from interactivity. The two state-of-the-art algorithms to find motifs are STOMP, which requires O(n2) time, and STAMP, which, despite being an O(logn) factor slower, is the preferred solution for most applications, as it is a fast converging anytime algorithm. In favorable scenarios STAMP needs only to be run to a small fraction of completion to provide a very accurate approximation of the top-k motifs. In this work we introduce SCRIMP++, an O(n2) time algorithm that is also an anytime algorithm, combining the best features of STOMP and STAMP. As we shall show, SCRIMP++ maintains all the desirable properties of the original algorithms, but converges much faster, in almost all scenarios producing the correct output after spending a tiny fraction of the full computation time. We argue that for many end-users, this allows motif discovery to be performed in interactive sessions. Moreover, this interactivity can be game changing in terms of the analytics that can be performed.

Unsupervised semantic segmentation in the time series domain is a much-studied problem due to its potential to detect unexpected regularities and regimes in poorly understood data. However, the current techniques have several shortcomings, which have limited the adoption of time series semantic segmentation beyond academic settings for three primary reasons. First, most methods require setting/learning many parameters and thus may have problems generalizing to novel situations. Second, most methods implicitly assume that all the data is segmentable, and have difficulty when that assumption is unwarranted. Finally, most research efforts have been confined to the batch case, but online segmentation is clearly more useful and actionable. To address these issues, we present an algorithm which is domain agnostic, has only one easily determined parameter, and can handle data streaming at a high rate. In this context, we test our algorithm on the largest and most diverse collection of time series datasets ever considered, and demonstrate our algorithm's superiority over current solutions. Furthermore, we are the first to show that semantic segmentation may be possible at superhuman performance levels.