Abstract
In recent years, time series motif discovery has emerged as perhaps the most important primitive for many analytical tasks, including clustering, classification, rule discovery, segmentation, and summarization. In parallel, it has long been known that Dynamic Time Warping (DTW) is superior to other similarity measures such as Euclidean Distance under most settings. However, due to the computational complexity of both DTW and motif discovery, virtually no research efforts have been directed at combining these two ideas. The current best mechanisms to address their lethargy appear to be mutually incompatible. In this work, we present the first efficient, scalable and exact method to find time series motifs under DTW. Our method automatically performs the best tradeoff of timetocompute versus tightnessoflowerbounds for a novel hierarchy of lower bounds that we introduce. As we shall show through extensive experiments, our algorithm prunes up to 99.99% of the DTW computations under realistic settings and is up to three to four orders of magnitude faster than the brute force search, and two orders of magnitude faster than the only other competitor algorithm. This allows us to discover DTW motifs in massive datasets for the first time. As we will show, in many domains, DTWbased motifs represent semantically meaningful conserved behavior that would escape our attention using all existing Euclidean distancebased methods.
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Notes
 1.
In brief, the argument is this: Recall that cDTW is constrained by a parameter w, the maximum amount of warping allowed, and that as w approaches zero, cDTW degenerates to the Euclidean distance. It has been shown that the best setting for w decreases as the number of comparisons increase (see Fig. 6 of (Mueen et al. 2009)). For similarity search, there are \(O(n)\) comparisons, but for motif search there are \(O({n}^{2})\) comparisons, favoring a small value for w, perhaps approaching zero.
 2.
German for “timegiver”, Zeitgeber is normally only used for biological processes. Here we extend the meaning to social and cultural processes.
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Acknowledgements
We thank all the creators of the data sets used in this work.
Funding
Funding was provided by National Science Foundation (Grant No. 1631776)
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Responsible editor: Johannes Fürnkranz.
Appendix
Appendix
Reproducibility
We have taken extraordinary steps to make sure that every experiment (including the figures and samples that proceed the official experimental section) are easy to reproduce. To this end:

For experiments that have a stochastic element, we initialize with the same random number generator seed before each iteration. This ensures that a reader can exactly reproduce our output, independent of their platform.

Every data used in each figure or table is explicitly labeled with the name of the figure/table and archived at Alaee (2020) in a universally readable ASCII plain text format, in addition to the.mat format that we use internally.

We have created a presentation that gives additional information about anything we did to create our final figures. For example, purely for aesthetic reasons, we “flipped” one of the dendrograms shown in Fig. 3 upside down (without changing its topology or distances). The presentation reconciles the slight differences between the output of the code, and the final figures.

In addition to the main code, we have included all the minor code, including the code to produce dendrograms, etc.
For many experiments we choose to use time series and query lengths that are powers of two. This is not required for SWAMP but is a consideration for future researchers who may try to improve on our results with either DFT or DWT methods, both of which have their best cases when the data lengths are powers of two.
As noted in the paper but reiterated here, in many works, the size of the warping window is often given as a percentage of the length of the time series (Keogh and Ratanamahatana 2005; Ratanamahatana and Keogh 2005), in this work we give it as an absolute number. One reason for this is because a given percentage may not evenly divide a time series length, and different rounding policies may affect the results.
Where warranted, we presented some details in the paper very tersely. For example, we noted in the main text:
Finally, we compared to Silva and Batista (2018), which is the only other exact algorithm for finding DTW motifs. On the three datasets above this algorithm was 17,274%, 185,511% and 13,857% respectively.
The details are a little sparse in that text. However:

The differences are so large that we hope the reader will understand our decision not to spend too much of the page limits here.

The full detailed results are available at Alaee (2020), together with the full code and data needed to reproduce the results.
Here we note that this comparison was completely fair. We used the exact same computer, same datasets, and same implementations of all common subroutines, including the various lower bounds, ED and DTW comparison algorithms, etc. Moreover, we further optimized the original algorithm extensively. The original algorithm finds both discords and motifs under DTW, but we made it faster by removing the need to find discords, and only requiring it to find the top1 motif.
Likewise, our comparison to bruteforce search was rigorously fair. There are many ways to make a DTWbased algorithm perform poorly. For example, one could implement the rival method using the recursive version of DTW instead of the iterative version. The recursive version of DTW is one to two orders of magnitude slower than the iterative version. However, here we again used the exact same computer, same datasets, and most importantly same implementations of all common subroutines, including the various lower bounds, ED and DTW comparison algorithms.
A reproducibility “ROSETTA STONE”
As noted above, we have made all our code publicly available in perpetuity (Alaee 2020). However, a reader may wish to implement and test our ideas on another platform. If we both agree on all distance measures, including the Euclidean distance, cDTW distance and parametrized lower bounds, then we can be virtually assured that all other steps will be in agreement. It may seem unlikely that we could disagree on such matters. However, our experience suggests otherwise. For example, we have seen the w parameter in cDTW interpreted as the total freedom to wander off the diagonal. In essence, that (mis)understanding will give only half the w value that we mean to communicate (and is more commonly understood (Rakthanmanon et al. 2013)). Likewise, by default, some DTW programs normalize the distance by the path length. This makes only a very subtle difference when w is small, nevertheless it could cause our lower bounds to no longer be admissible. Thus, in order to make sure we agree on all measures, in Table 5 we will create a pair of time series that the interested reader can literally cutandpaste into their framework and compare results on all measures.
Note that after we znormalized these time series, we rounded them to have just two significant digits, in order to further facilitate a detailed forensic tracing of the computation. However, this rounding means that the two time series are no longer exactly znormalized. All subsequent analysis assumes the exact values in Table 5.
In Fig. 43 we show a visual intuition for the various measures that are key to this work. The Euclidean distance ED(Q,T) is 7.88098.
Recall that in our implementation we perform the optimization of not using the squared root function (see Sect. 4.1.1 of Rakthanmanon et al. 2013). However, we ignore that optimization here. Using a value of eight for the warping parameter w, cDTW(Q,T) is 2.4240. The value of Keogh’s classic lower bound, in our notation LB_{Keogh}1:1(Q,T), is 1.5865. It is important to recall that this function is not symmetric, in general LB_{Keogh}1:1(Q,T) ≠ LB_{Keogh}1:1(T,Q). Finally, Fig. 43.bottom illustrates the fourfold reduced lower bound, LB_{Keogh}4:1(Q,T), which has a value of 0.4999.
Note that LB_{Keogh}4:1(Q,T) ≤ LB_{Keogh}1:1(Q,T) ≤ cDTW(Q,T) ≤ ED(Q,T) as we should expect.
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Alaee, S., Mercer, R., Kamgar, K. et al. Time series motifs discovery under DTW allows more robust discovery of conserved structure. Data Min Knowl Disc (2021). https://doi.org/10.1007/s10618021007400
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Keywords
 Time series
 Motifs
 Dynamic time warping