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    "text": "COBRASTS: A new approach to\nSemi-Supervised Clustering of Time Series Toon Van Craenendonck, Wannes Meert, Sebastijan Dumanˇci´c and\nHendrik Blockeel KU Leuven, Department of Computer Science\n{firstname.lastname}@kuleuven.be2018",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
    "authors": [
      "Toon Van Craenendonck",
      "Wannes Meert",
      "Sebastijan Dumancic",
      "Hendrik Blockeel"
    ],
    "published_date": "2018-05-02",
    "primary_category": "stat.ML",
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    "text": "Clustering is ubiquitous in data analysis, including analysis of time series. It is inherently subjective: different users may preferMay different clusterings for a particular dataset. Semi-supervised clustering\naddresses this by allowing the user to provide examples of instances that2\nshould (not) be in the same cluster. This paper studies semi-supervised\nclustering in the context of time series. We show that COBRAS, a stateof-the-art semi-supervised clustering method, can be adapted to this setting. We refer to this approach as COBRASTS. An extensive experimental evaluation supports the following claims: (1) COBRASTS far outperforms the current state of the art in semi-supervised clustering for time\nseries, and thus presents a new baseline for the field; (2) COBRASTS can\nidentify clusters with separated components; (3) COBRASTS can iden-[stat.ML]\ntify clusters that are characterized by small local patterns; (4) a small\namount of semi-supervision can greatly improve clustering quality for\ntime series; (5) the choice of the clustering algorithm matters (contrary\nto earlier claims in the literature). Clustering is ubiquitous in data analysis. There is a large diversity in algorithms,\nloss functions, similarity measures, etc. This is partly due to the fact that clustering is inherently subjective: in many cases, there is no single correct clustering,\nand different users may prefer different clusterings, depending on their goals and\nprior knowledge [5,20]. Depending on their preference, they should use the right\nalgorithm, similarity measure, loss function, hyperparameter settings, etc. This\nsolutions that the user finds interesting. Many such systems obtain these constraints by asking the user to answer queries of the following type: should these\ntwo elements be in the same cluster? A must-link constraint is obtained if the\nanswer is yes, a cannot-link otherwise. In many situations, answering this type of\nquestions is much easier for the user than selecting the right algorithm, defining\nthe similarity measure, etc.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
    "authors": [
      "Toon Van Craenendonck",
      "Wannes Meert",
      "Sebastijan Dumancic",
      "Hendrik Blockeel"
    ],
    "published_date": "2018-05-02",
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    "text": "In the context of clustering time series, the subjectiveness of clustering is even\nmore prominent. In some contexts, the time scale matters, in other contexts it\ndoes not. Similarly, the response scale may (not) matter. One may want to cluster\ntime series based on certain types of qualitative behavior (monotonic, periodic,\n. . . ), local patterns that occur in them, etc. Despite this variability, and although\nthere is a plethora of work on time series clustering, semi-supervised clustering\nof time series has only very recently started receiving attention. The cDTWSS\nmethod developed by Dau et al. [7] is to our knowledge the only attempt to date\nto address this task. In this paper, we show that COBRAS, an existing semi-supervised clustering system, can be used practically \"as-is\" for time series clustering. The only\nadaptation that is needed, is the plugging in of a suitable similarity measure and\na corresponding (unsupervised) clustering approach for time series. Two plug-in\nmethods are considered for this: spectral clustering using dynamic time warping\n(DTW), and k-Shape [13]. We refer to COBRAS with one of these plugged in as\nCOBRASTS (COBRAS for Time Series).",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
    "authors": [
      "Toon Van Craenendonck",
      "Wannes Meert",
      "Sebastijan Dumancic",
      "Hendrik Blockeel"
    ],
    "published_date": "2018-05-02",
    "primary_category": "stat.ML",
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    "text": "We perform an extensive experimental\nevaluation of this approach. The main contributions of the paper are twofold. First, it contributes a novel\napproach to semi-supervised clustering of time series, and two concrete, freely\ndownloadable and ready-to-use implementations of it. Second, the paper provides extensive evidence for the following claims: (1) COBRASTS outperforms\ncDTWSS (the current state of the art) by a large margin; (2) COBRASTS can\nidentify clusters with separated components, and this is one reason why it performs well; (3) COBRASTS can identify clusters that are characterized by small\nlocal patterns; (4) a small amount of supervision can greatly improve results\nin time series clustering; (5) the choice of clustering algorithm matters, it is\nnot negligible compared to the choice of similarity. Except for claim 4, all these\nclaims are novel, and some are at variance with the current literature. Claim 4\nhas been made before, but with much weaker empirical support.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
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    "text": "Semi-supervised clustering has been studied extensively for clustering attributevalue data, starting with COP-KMeans [21]. Most semi-supervised methods extend unsupervised ones by adapting their clustering procedure [21], their similarity measure [23], or both [2]. Alternatively, constraints can also be used to\nselect and tune an unsupervised clustering algorithm [16]. Traditional methods assume that a set of pairwise queries is given prior to\nrunning the clustering algorithm, and in practice, pairs are often queried randomly. Active semi-supervised clustering methods try to query the most informative pairs first, instead of random ones [10]. Typically, this results in better\nclusterings for an equal number of queries. COBRAS [18] is a recently proposed\nactive semi-supervised clustering method that was shown to be effective for clustering attribute-value data. In this paper, we show that it can be used to cluster time series with little modification. We describe COBRAS in more detail in the\nnext section.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
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    "text": "In contrast to the wealth of papers on semi-supervised clustering of attributevalue data, only one method has been proposed specifically for semi-supervised\ntime series clustering. cDTWSS [7] uses pairwise constraints to tune the warping\nwidth parameter w in constrained DTW. cDTWSS is also an active clustering\nmethod, as it comes with a strategy to select the pairwise queries that are most\ninformative for tuning w. We compare COBRASTS to this method in the experiments. Zhou et al. [25] introduce a method that uses different distance measures\nto generate pairwise constraints, and then uses these constraints in a semisupervised variant of spectral clustering [9]. While related, this is not a semisupervised method, as it does not exploit supervision by the user. Rather, it\nmakes it possible to use semi-supervised algorithms in an unsupervised setting. In contrast to semi-supervised time series clustering, semi-supervised time\nseries classification has received significant attention [22]. Note that these two\nsettings are quite different: in semi-supervised classification, the set of classes is\nknown beforehand, and at least one labeled example of each class is provided. In\nsemi-supervised clustering, it is not known in advance how many classes (clusters) there are, and a class may be identified correctly even if none of its instances\nhave been involved in the pairwise constraints. 3 Clustering time series with COBRAS We describe COBRAS only to the extent necessary to follow the remainder of\nthe paper; for more information, see Van Craenendonck et al. [17,18]. COBRAS is based on two key ideas. The first [17] is that of super-instances:\nsets of instances that are temporarily assumed to belong to the same cluster in\nthe unknown target clustering. In COBRAS, a clustering is a set of clusters, each\ncluster is a set of super-instances, and each super-instance is a set of instances. This intermediate level of super-instances makes it possible to exploit constraints\nmuch more efficiently: querying is performed at the level of super-instances,\nwhich means that each instance does not have to be considered individually in the\nquerying process. The second key idea in COBRAS [18] is that of the automatic\ndetection of the right level at which these super-instances are constructed.",
    "paper_id": "1805.00779",
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    "text": "COBRAS starts with a single superinstance that contains all the examples, and a single cluster containing that\nsuper-instance. In each iteration the largest super-instance is taken out of its\ncluster, split into smaller super-instances, and the latter are reassigned to (new\nor existing) clusters. Thus, COBRAS constructs a clustering of super-instances\nat an increasingly fine-grained level of granularity. The clustering process stops\nwhen the query budget is exhausted.",
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    "text": "We illustrate this procedure using the example in Figure 1. Panel A shows\na toy dataset that can be clustered according to several criteria. An illustration of the COBRAS clustering procedure. differentiability and monotonicity as relevant properties. Initially, all instances\nbelong to a single super-instance (S0), which constitutes the only cluster (C0).",
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    "text": "The second and third rows of Figure 1 show two iterations of COBRAS. In the first step of iteration 1, COBRAS refines S0 into 4 new super-instances,\nwhich are each put in their own cluster (panel B). The refinement procedure uses\nk-means, and the number of super-instances in which to split is determined based\non constraints; for details, see [18]. In the second step of iteration 1, COBRAS\ndetermines the relation between new and existing clusters. To determine the\nrelation between two clusters, COBRAS queries the pairwise relation between\nthe medoids of their closest super-instances. In this example, we assume that the\nuser is interested in a clustering based on differentiability. The relation between\nC1 = {S1} and C2 = {S2} is determined by posing the following query to the",
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    "text": "user: should and be in the same cluster? The user answers with a mustlink constraint, resulting in C1 and C2 being merged into C5. Similarly, COBRAS\ndetermines the other pairwise relations between clusters. It does not need to\nquery all of them, as many can be derived through transitivity or entailment\n[18]. The first iteration ends once all pairwise relations between clusters are\nknown. This is the situation depicted in panel C.",
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    "text": "Note that COBRAS has not\nproduced a perfect clustering at this point, as S2 contains both differentiable\nand non-differentiable instances. In the second iteration, COBRAS again starts by refining its largest superinstance. In this case, S2 is refined into S5 and S6, as illustrated in panel D. A new\ncluster is created for each of these super-instances, and the relation between new\nand existing clusters is determined by querying pairwise constraints. A must-link\nconstraint between S5 and S1 results in the creation of C9 = {S1, S5}. Similarly,\na must-link between S6 and S3 results in the creation of C10 = {S3, S4, S6}. At\nthis point, the second iteration ends as all pairwise relations between clusters\nare known. In general, COBRAS keeps repeating its two steps (refining super-instances\nand querying their pairwise relations) until the query budget is exhausted. Separated components A noteworthy property of COBRAS is that, by interleaving splitting and merging, it can split offa subcluster from a cluster\nand reassign it to another cluster. In this way, it can construct clusters that\ncontain separated components (different dense regions that are separated by a\ndense region belonging to another cluster). It may, at first, seem strange to call\nsuch a structure a \"cluster\", as clusters are usually considered to be coherent\nhigh-density areas. However, note that a coherent cluster may become incoherent when projected onto a subspace. Figure 2 illustrates this. Two clusters are\nclearly visible in the XY-space, yet projection on the X-axis yields a trimodal\ndistribution where the outer modes belong to one cluster and the middle mode\nto another. In semi-supervised clustering, it is realistic that the user evaluates\nsimilarity on the basis of more complete information than explicitly present in\nthe data; coherence in the user's mind may therefore not translate to coherence\nin the data space.",
    "paper_id": "1805.00779",
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    "authors": [
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    "text": "The need for handling clusters with multi-modal distributions has been mentioned repeatedly in work on time series anomaly detection [?], on unsupervised\ntime series clustering [14], and on attribute-value semi-supervised constrained\nclustering [?]. Note, however, a subtle difference between having a multi-modal\ndistribution and containing separated components: the first assumes that the\ncomponents are separated by a low-density area, whereas the second allows them\nto be separated by a dense region of instances from another cluster. 3.2 COBRASDTW and COBRASk-Shape COBRAS is not suited out-of-the-box for time series clustering, for two reasons. First, it defines the super-instance medoids w.r.t. the Euclidean distance, which User view x Data view x x User feedback cannotProject link\nmustlink",
    "paper_id": "1805.00779",
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    "text": "Clusters may contain separated components when projected on a lowerdimensional subspace. is well-known to be suboptimal for time series. Second, it uses k-means to refine super-instances, which is known to be sub-state-of-the-art for time series\nclustering [13]. Both of these issues can easily be resolved by plugging in distance measures\nand clustering methods that are developed specifically for time series.",
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    "text": "We refer\nto this approach as COBRASTS. We now present two concrete instantiations of\nit: COBRASDTW and COBRASk-Shape. COBRASDTW uses DTW as its distance measure, and spectral clustering\nto refine super-instances. It is described in Algorithm 1. DTW is commonly\naccepted to be a competitive distance measure for time series analysis [1], and\nspectral clustering is well-known to be an effective clustering method [19]. We use\nthe constrained variant of DTW, cDTW, which restricts the amount by which\nthe warping path can deviate from the diagonal in the warping matrix. cDTW\noffers benefits over DTW in terms of both runtime and solution quality [13,7],\nif run with an appropriate window width. Algorithm 1 COBRASDTW\nInput: A dataset, the DTW warping window width w, the γ parameter used in converting distances to similarities and access to an oracle answering pairwise queries\nOutput: A clustering\n1: Compute the full pairwise DTW distance matrix\n2: Convert each distance d to an affinity a: ai,j = e−γdi,j\n3: Run COBRAS, substituting k-means for splitting super-instances with spectral\nclustering on the previously computed affinity matrix COBRASk-Shape uses the shape-based distance (SBD, [13]) as its distance\nmeasure, and the corresponding k-Shape clustering algorithm [13] to refine superinstances. k-Shape can be seen as a k-means variant developed specifically for\ntime series. It uses SBD instead of the Euclidean distance, and comes with a method of computing cluster centroids that is tailored to time series. k-Shape\nwas shown to be an effective and scalable method for time series clustering in\n[13]. Instead of the medoid, COBRASk-Shape uses the instance that is closest to\nthe SBD centroid as a super-instance representative. In our experiments we evaluate COBRASDTW and COBRASk-Shape in terms\nof both clustering quality and runtime, and compare them to state-of-the-art\nsemi-supervised (cDTWSS and COBS) and unsupervised (k-Shape and k-MS)\ncompetitors. The experiments presented in this paper are fully reproducible: we\nprovide code for COBRASTS in a public git repository1, and a separate git repository that contains our scripts for running the experiments2. The experiments\nare performed on the public UCR time series collection [6].",
    "paper_id": "1805.00779",
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    "text": "4.1 Methods\nCOBRASTS COBRASk-Shape has no parameters (the number of clusters used\nin k-Shape to refine super-instances is chosen based on the constraints in COBRAS). We use a publicly available Python implementation3 to obtain the kShape clusterings. COBRASDTW has two parameters: γ (used in converting\ndistances to affinities) and w (the warping window width). We use a publicly\navailable C implementation to construct the DTW distance matrices [12]. In our\nexperiments, γ is set to 0.5 and w to 10% of the time series length. The value\nw = 10% was chosen as Dau et al. [7] report that most datasets do not require\nw greater than 10%. We note that γ and w could in principle also be tuned for\nCOBRASDTW. There is, however, no well-defined way of doing this. We cannot\nuse the constraints for this, as they are actively selected during the execution\nof the algorithm (which of course requires the affinity matrix to already be constructed). We did not do any tuning on these parameters, as this is also hard in\na practical clustering scenario, but observed that the chosen parameter values\nalready performed very well in the experiments. We performed a parameter sensitivity analysis, illustrated in Figure 3, which shows that the influence of these\nparameters is highly dataset-dependent: for many datasets their values do not\nmatter much, for some they result in large differences. cDTWSS cDTWSS uses pairwise constraints to tune the w parameter in cDTW. In principle, the resulting tuned cDTW measure can be used with any clustering\nalgorithm.",
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    "text": "The authors in [7] use it in combination with TADPole [4], and we\ndo the same here. We use the code that is publicly available on the authors'\nwebsite4. The cutoffdistances used in TADPole were obtained from the authors\nin personal communication.",
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    "text": "1 https://bitbucket.org/toon_vc/cobras_ts or using pip install cobras ts\n2 https://bitbucket.org/toon_vc/cobras_ts_experiments\n3 https://github.com/Mic92/kshape\n4 https://sites.google.com/site/dtwclustering/ 50 constraints, w = 10 50 constraints, = 0.5 CBF 1 CBF 1\nTwoLeadECG\nItalyPowerDemand ItalyPowerDemand\nTwoLeadECG ARI ARI\nMoteStrain\nECG200\nECG200\nWordsSynonyms\nMoteStrain\nWordsSynonyms 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0 Sensitivity to γ and w for several datasets. COBS COBS [16] uses constraints to select and tune an unsupervised clustering algorithm. It was originally proposed for attribute-value data, but it\ncan trivially be modified to work with time series data as follows. First, the\nfull pairwise distance matrix is generated with cDTW using w = 10% of the\ntime series length. Next, COBS generates clusterings by varying the hyperparameters of several standard unsupervised clustering methods, and selects\nthe resulting clustering that satisfies the most pairwise queries. We use the\nactive variant of COBS, as described in [16]. Note that COBS is conceptually similar to cDTWSS, as both methods use constraints for hyperparameter\nselection. The important difference is that COBS uses a fixed distance measure and selects and tunes the clustering algorithm, whereas cDTWSS tunes the\nsimilarity measure and uses a fixed clustering algorithm. We use the following\nunsupervised clustering methods and corresponding hyperparameter ranges in\nCOBS: spectral clustering (K ∈[max(2, Ktrue −5), Ktrue + 5]), hierarchical\nclustering (K ∈[max(2, Ktrue −5), Ktrue + 5], with both average and complete linkage), affinity propagation (damping ∈[0.5, 1.0]) and DBSCAN (ϵ ∈\n[min pairwise dist., max. pairwise dist], min samples ∈[2, 21]). For the\ncontinuous parameters, clusterings were generated for 20 evenly spaced values\nin the specified intervals. Additionally, the γ parameter in converting distances\nto affinities was varied in [0, 2.0] for clustering methods that take affinities as\ninput, which are all of them except DBSCAN, which works with distances. We\ndid not vary the warping window width w for generating clusterings in COBS.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
    "authors": [
      "Toon Van Craenendonck",
      "Wannes Meert",
      "Sebastijan Dumancic",
      "Hendrik Blockeel"
    ],
    "published_date": "2018-05-02",
    "primary_category": "stat.ML",
    "arxiv_url": "http://arxiv.org/abs/1805.00779v1",
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    "text": "This would mean a significant further increase in computation time, both for\ngenerating the DTW distance matrices, and for generating clusterings with all\nmethods and parameter settings for each value of w. k-Shape and k-MS Besides the three previous semi-supervised methods, we\nalso include k-Shape [13] and k-MultiShape (k-MS) [14] in our experiments as\nunsupervised baselines. k-MS [14] is similar to k-Shape, but uses multiple centroids, instead of one, to represent each cluster. It was found to be the most\naccurate method in an extensive experimental study that compares a large number of unsupervised time series clustering methods on the UCR collection [14]. The number of centroids that k-MS uses to represent a cluster is a parameter; following the original paper we set it to 5 for all datasets. The k-MS code was\nobtained from the authors. We perform experiments on the entire UCR time series classification collection\n[6], which is the largest public collection of time series datasets. It consists of\n85 datasets from a wide variety of domains. The UCR datasets come with a\npredefined training and test set. We use the test sets as our datasets as they are\noften much bigger than the training sets. This means that whenever we refer to\na dataset in the remainder of this text, we refer to the test set of that dataset\nas defined in [6]. This procedure was also followed by Dau et al. [7]. As is typically done in evaluating semi-supervised clustering methods, the\nclasses are assumed to represent the clusterings of interests.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
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    "text": "When computing\nrankings and average ARIs, we ignored results from 21 datasets where cDTWSS\neither crashed or timed out after 24h.5 We use 10-fold cross-validation, as is common in evaluating semi-supervised clustering methods [3,10]. The full dataset is clustered in each run, but the methods\ncan only query pairs of which both instances are in the training set. The result\nof a run is evaluated by computing the Adjusted Rand Index (ARI) [8] on the\ninstances of the test set. The ARI measures the similarity between the generated\nclusterings and the ground-truth clustering, as indicated by the class labels. It\nis 0 for a random clustering, and 1 for a perfect one. The final ARI scores that\nare reported are the average ARIs over the 10 folds.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
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    "text": "We ensure that cDTWSS and COBS do not query pairs that contain instances\nfrom the test set by simply excluding such candidates from the list of constraints\nthat they consider. For COBRASTS, we do this by only using training instances\nto compute the super-instance representatives. COBRASTS and COBS do not require the number of clusters as an input\nparameter, whereas cDTWSS, k-Shape and k-MS do. The latter three were given\nthe correct number of clusters, as indicated by the class labels. Note that this is\na significant advantage for these algorithms, and that in many practical applications the number of clusters is not known beforehand.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
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    "text": "Clustering quality Figure 4(a) shows the average ranks of the compared methods over all datasets. Figure 4(b) shows the average ARIs. Both plots clearly show\nthat, on average, COBRASTS outperforms all the competitors by a large margin. 5 These datasets are listed at https://bitbucket.org/toon_vc/cobras_ts_\nexperiments Only when the number of queries is small (roughly < 15), is it outperformed by\nCOBS and k-MS.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
    "authors": [
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    "text": "These observations are confirmed by Table 1, which reports the number of\ntimes COBRASDTW wins and loses against the alternatives. The differences\nwith cDTWSS and k-Shape are significant for all the considered numbers of\nqueries (Wilcoxon test, p < 0.05). The difference between COBRASDTW and\nCOBS is significant for 50 and 100 queries, but not for 25. The same holds\nfor COBRASDTW vs. k-MS. This confirms the observation from Figure 4(a),\nwhich showed that the performance gap between COBRASDTW and the competitors becomes larger as more queries are answered. The difference between\nCOBRASDTW and COBRASk-Shape is only statistically significant for 100 queries. Rank Average ARI cDTWSS 0.5\nCOBRASDTW\nkShape 0.4\n4 COBRASkShape\nk-MS 0.3 COBS\nk-MS\n3 COBS kShape\nCOBRASkShape 0.2 cDTWss COBRASDTW\n2 0.1 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100\nNumber of queries Number of queries Fig. 4. (a) Average rank for all methods over all clustering tasks. Lower is better. (b)\nAverage ARI. Wins and losses over the 64 datasets. An asterisk indicates that the difference\nis significant according to the Wilcoxon test with p < 0.05.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
    "authors": [
      "Toon Van Craenendonck",
      "Wannes Meert",
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    "published_date": "2018-05-02",
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    "text": "25 queries 50 queries 100 queries\nwin loss win loss win loss COBRASk-Shape 35 29 37 27 41* 23\nCOBRASDTW vs. k-MS 35 29 40* 24 47* 14\nCOBRASDTW vs. COBS 37 27 42* 22 45* 19\nCOBRASDTW vs. cDTWSS 62* 2 53* 11 55* 9\nCOBRASDTW vs. k-Shape 40* 24 46* 18 50* 14 It is surprising to see that the unsupervised baselines significantly outperform\nthe semi-supervised cDTWSS. This conclusion is at variance with the claim that\nthe choice of w dwarfs any improvements by the k-Shape algorithm [7]. To ensure\nthat this is not an effect of the evaluation strategy (10-fold CV using the ARI, compared to no CV and the Rand index (RI) in [7]), we have also computed the\nRIs for all of the clusterings generated by k-Shape and compared them directly\nto the values provided by the authors of cDTWSS on their webpage6. In this\nexperiment k-Shape attained an average RI of 0.68, whereas cDTWSS had an\naverage RI of 0.67. We note that the claim in [7] was based on a comparison\non two datasets. Our experiments clearly indicate that it does not generalize\ntowards all datasets. Thus, contrary to earlier suggestions, our results indicate that constraints\nare better used to select and tune the algorithm (i.e. COBS) than to tune the\nsimilarity measure (i.e. cDTWSS). Runtime COBRASDTW, cDTWSS and COBS require the construction of the\npairwise DTW distance matrix. This becomes infeasible for large datasets. For\nexample, computing one distance matrix for the ECG5000 dataset took ca. 30h\nin our experiments, using an optimized C implementation of DTW.\nk-Shape and k-MS are much more scalable [13], as they do not require computing a similarity matrix.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
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    "text": "COBRASk-Shape inherits this scalability, as it uses\nk-Shape to refine super-instances. In our experiments, COBRASk-Shape was on\naverage 28 times faster than COBRASDTW.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
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    "text": "5 Case studies: CBF, TwoLeadECG and MoteStrain In this section, we investigate why COBRASTS significantly outperforms its\ncompetitors. Our main claim is that COBRASTS is able to deal with the inherent\ncomplexity of time series clustering by repeatedly refining super-instances. To support this claim, we inspect the clusterings that are generated for three\nUCR datasets in more detail. CBF and TwoLeadECG are examples for which\nCOBRASDTW and COBRASk-Shape significantly outperform their competitors,\nwhereas MoteStrain is one of the few datasets for which they are significantly\noutperformed by unsupervised k-Shape clustering. These three datasets illustrate\ndifferent reasons why time series clustering may be difficult. Clustering CBF is\ndifficult because of the fact that one of the clusters comprises two separated\nsubclusters; TwoLeadECG, because only limited subsequences of the time series\nare relevant for the clustering at hand, and the remaining parts obfuscate the\ndistance measurements; and MoteStrain, because of noise. The first column of Figure 5 shows the \"true\" clusters as they are indicated by\nthe class labels. It is clear that the classes correspond to three distinct patterns\n(horizontal, upward and downward). The next columns show the clusterings that\nare produced by each of the competitors. Semi-supervised approaches are given\na budget of 50 queries.",
    "paper_id": "1805.00779",
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    "text": "COBRASDTW and COBRASk-Shape are the only methods 6 https://sites.google.com/site/dtwclustering/ that provide a near perfect solution (ARI = 0.96). cDTWSS mixes patterns of\ndifferent types in each cluster. COBS find pure clusters, but too many: the plot\nonly shows the largest three of 15 clusters for COBS. k-Shape and k-MS mix\nhorizontal and downward patterns in their third cluster. To clarify this mixing of\npatterns, the figure shows the instances in the third k-Shape and k-MS clusters\nagain, but separated according to their true class. The first column shows the true clustering of CBF. The remaining columns show\nthe clusterings that are produced by all considered methods. For COBS, only the three\nlargest of 15 clusters are shown.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
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    "text": "The prototypes are shown in red. For COBRASDTW,\ncDTWSS and COBS the prototypes are selected as the medoids w.r.t. For COBRASk-Shape, k-Shape and k-MS the prototypes are the medoids w.r.t. the SBD\ndistance. Figure 6 illustrates how repeated refinement of super-instances helps COBRASTS\ndeal with the complexities of clustering CBF. It shows a super-instance in the\nroot, with its subsequent refinements attached as children. The super-instance\nin the root of Figure 6 (which is itself a result of a previous super-instance\nsplit) contains time series showing horizontal and upward patterns. Clustering\nit into two new super-instances does not yield a clean separation of these two\ntypes: a perfectly pure cluster with upward patterns is created, but the other\nsuper-instance still mixes horizontal and upward patterns. This is not a problem\nfor COBRASTS, as it simply refines the latter super-instance again. This time\nthe remaining time series are split into nearly pure super-instances separating\nhorizontal from upward patterns. Note that the two super-instances containing\nupward patterns correspond to two distinct subclusters: some upward patterns\ndrop down very close to the end of the time series, whereas the drop in the\nother subcluster occurs much earlier. Typically, patterns in the latter subcluster\nincrease with a steeper slope. The clustering process just mentioned illustrates the point made earlier, in\nSection 3.1, about COBRAS's ability to construct clusters with separated components. It is clear that this ability is advantageous in the CBF dataset. Note\nthat being able to deal with separated components is key here; k-MS, which is\nable to find multi-modal clusters, but not clusters with modes that are separated\nby a mode from another cluster, produces a clustering that is far from perfect\nfor CBF. A super-instance that is generated while clustering CBF, and its refinements.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
    "authors": [
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    "published_date": "2018-05-02",
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    "text": "The green line indicates a must-link constraint, and illustrates that these two superinstances will be part of the same multi-modal cluster (corresponding to upward patterns). The red lines between super-instances indicate cannot-link constraints. The purity of a super-instance is computed as the ratio of the occurrence of the most frequent\nclass in the super-instance, over the total number of elements in the super-instance. The first column in Figure 7 shows the \"true\" clusters for TwoLeadECG. Cluster\n1 is defined by a large peak before the drop, and a slight bump in the upward\ncurve after the drop. Instances in cluster 2 typically only show a small peak\nbefore the drop, and no bump in the upward curve after the drop. For the\nremainder of the discussion we focus on the peak as the defining pattern, simply\nbecause it is easier to see than the more subtle bump. The second column in Figure 7 shows the clustering that is produced by\nCOBRASDTW; the one produced by COBRASk-Shape is highly similar. They are\nthe only methods able to recover these characteristic patterns.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
    "authors": [
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    "text": "The last column\nin Figure 7 shows the clustering that is produced by COBS, which is the best of\nthe competitors. This clustering has an ARI of 0.12, which is not much better\nthan random. From the zoomed insets in Figure 7, it is clear that this clustering\ndoes not recover the defining patterns: the small peak that is characteristic for\ncluster 2 is hard to distinguish. This example illustrates that by using COBRASTS for semi-supervised clustering, a domain expert can discover more accurate explanatory patterns than\nwith competing methods. None of the alternatives is able to recover the characteristic patterns in this case, potentially leaving the domain expert with an",
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    "text": "The first column shows the \"true\" clustering of TwoLeadECG. The second\ncolumn shows the clustering produced by COBRASDTW. The third column shows the\nclustering produced by COBS, which is the best competitor for this dataset. Prototypes\nare shown in red, and are the medoids w.r.t. the DTW distance. incorrect interpretation of the data.",
    "paper_id": "1805.00779",
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    "text": "Obtaining these patterns comes with relatively little additional effort, as with a good visualizer answering 50 queries only\ntakes a few minutes. This time would probably be insignificant compared to the\ntime that was needed to collect the 1139 instances in the TwoLeadECG dataset. Two super-instances generated by COBRASDTW. The super-instances are\nbased on the location of the noise. In our third case study we discuss an example for which COBRASTS does not\nwork well, as this provides insight into its limitations.",
    "paper_id": "1805.00779",
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    "text": "We consider the MoteStrain dataset, for which the unsupervised methods perform best. k-MS attains\nan ARI of 0.62, and k-Shape of 0.61. COBRASk-Shape ranks third with an ARI\nof 0.51, and COBRASDTW fourth with an ARI of 0.48. These results are surprising, as the COBRAS algorithms have access to more information than the\nunsupervised k-Shape and k-MS. Figure 8 gives a reason for this outcome; it\nshows that COBRASTS creates super-instances that are based on the location\nof the noise. The poor performance of the COBRASTS variants can in this case\nbe explained by their large variance. The process of super-instance refinement\nis much more flexible than the clustering procedure of k-Shape, which has a\nstronger bias. For most datasets, COBRASTS's weaker bias led to performance\nimprovements in our experiments, but in this case it has a detrimental effect due\nto the large magnitude of the noise. In practice, the issue could be alleviated\nhere by simply applying a low-pass filter to remove noise prior to clustering.",
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    "text": "Time series arise in virtually all disciplines. Consequently, there is substantial\ninterest in methods that are able to obtain insights from them. One of the most\nprominent ways of doing this, is by using clustering. In this paper we have presented COBRASTS, an novel approach to time series clustering. COBRASTS is\nsemi-supervised: it uses small amounts of supervision in the form of must-link and cannot-link constraints.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
    "authors": [
      "Toon Van Craenendonck",
      "Wannes Meert",
      "Sebastijan Dumancic",
      "Hendrik Blockeel"
    ],
    "published_date": "2018-05-02",
    "primary_category": "stat.ML",
    "arxiv_url": "http://arxiv.org/abs/1805.00779v1",
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    "text": "This sets it apart from the large majority of existing methods, which are unsupervised. An extensive experimental evaluation\nshows that COBRASTS is able to effectively exploit this supervision; it outperforms unsupervised and semi-supervised competitors by a large margin. As our\nimplementation is readily available, COBRASTS offers a valuable new tool for\npractitioners that are interested in analyzing time series data. Besides the contribution of the COBRASTS approach itself, we have also\nprovided insight into why it works well. A key factor in its success is its ability\nto handle clusters with separated components.",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
    "authors": [
      "Toon Van Craenendonck",
      "Wannes Meert",
      "Sebastijan Dumancic",
      "Hendrik Blockeel"
    ],
    "published_date": "2018-05-02",
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    "text": "We thank Hoang Anh Dau for help with setting up the cDTWSS experiments. Toon Van Craenendonck is supported by the Agency for Innovation by Science\nand Technology in Flanders (IWT). This research is supported by Research\nFund KU Leuven (GOA/13/010), FWO (G079416N) and FWO-SBO (HYMOP-\n150033).",
    "paper_id": "1805.00779",
    "title": "COBRAS-TS: A new approach to Semi-Supervised Clustering of Time Series",
    "authors": [
      "Toon Van Craenendonck",
      "Wannes Meert",
      "Sebastijan Dumancic",
      "Hendrik Blockeel"
    ],
    "published_date": "2018-05-02",
    "primary_category": "stat.ML",
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