Time series clustering is the process of grouping time series with respect to their similarity or characteristics. Previous approaches usually combine a specific distance measure for time series and a standard clustering method. However, these approaches do not take the similarity of the different subsequences of each time series into account, which can be used to better compare the time series objects of the dataset. In this paper, we propose a novel technique of time series clustering based on two clustering stages. In a first step, a least squares polynomial segmentation procedure is applied to each time series, which is based on a growing window technique that returns different-length segments. Then, all the segments are projected into same dimensional space, based on the coefficients of the model that approximates the segment and a set of statistical features. After mapping, a first hierarchical clustering phase is applied to all mapped segments, returning groups of segments for each time series. These clusters are used to represent all time series in the same dimensional space, after defining another specific mapping process. In a second and final clustering stage, all the time series objects are grouped. We consider internal clustering quality to automatically adjust the main parameter of the algorithm, which is an error threshold for the segmenta- tion. The results obtained on 84 datasets from the UCR Time Series Classification Archive have been compared against two state-of-the-art methods, showing that the performance of this methodology is very promising.
Many of the proposals for time series clustering are based on the combination of a distance measure and a clustering algorithm. First, we will analyse the most important distance measures proposed for time series comparison, and then we will introduce the clustering methods that can be applied based on them Further information about time series clustering can be found in @cite_37 or @cite_56 .
@cite_37 @cite_56
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Sequential time series clustering is a technique used to extract important features from time series data. The method can be shown to be the process of clustering in the delay-vector space formalism used in the Dynamical Systems literature. Recently, the startling claim was made that sequential time series clustering is meaningless. This has important consequences for a significant amount of work in the literature, since such a claim invalidates these work’s contribution. In this paper, we show that sequential time series clustering is not meaningless, and that the problem highlighted in these works stem from their use of the Euclidean distance metric as the distance measure in the delay-vector space. As a solution, we consider quite a general class of time series, and propose a regime based on two types of similarity that can exist between delay vectors, giving rise naturally to an alternative distance measure to Euclidean distance in the delay-vector space. We show that, using this alternative distance measure, sequential time series clustering can indeed be meaningful. We repeat a key experiment in the work on which the “meaningless” claim was based, and show that our method leads to a successful clustering outcome. Time series clustering has become an increasingly important research topic over the past decade. Most existing methods for time series clustering rely on distances calculated from the entire raw data using the Euclidean distance or Dynamic Time Warping distance as the distance measure. However, the presence of significant noise, dropouts, or extraneous data can greatly limit the accuracy of clustering in this domain. Moreover, for most real world problems, we cannot expect objects from the same class to be equal in length. As a consequence, most work on time series clustering only considers the clustering of individual time series "behaviors," e.g., individual heart beats or individual gait cycles, and contrives the time series in some way to make them all equal in length. However, contriving the data in such a way is often a harder problem than the clustering itself. In this work, we show that by using only some local patterns and deliberately ignoring the rest of the data, we can mitigate the above problems and cluster time series of different lengths, i.e., cluster one heartbeat with multiple heartbeats. To achieve this we exploit and extend a recently introduced concept in time series data mining called shapelets. Unlike existing work, our work demonstrates for the first time the unintuitive fact that shapelets can be learned from unlabeled time series. We show, with extensive empirical evaluation in diverse domains, that our method is more accurate than existing methods. Moreover, in addition to accurate clustering results, we show that our work also has the potential to give insights into the domains to which it is applied. The startling claim was made that sequential time series clustering is meaningless. This has important consequences for a significant amount of work in the literature, since such a claim invalidates this work's contribution. In this paper, we show that sequential time series clustering is not meaningless, and that the problem highlighted in these works stem from their use of the Euclidean distance metric as the distance measure in the subsequence vector space. As a solution, we consider quite a general class of time series, and propose a regime based on two types of similarity that can exist between subsequence vectors, which give rise naturally to an alternative distance measure to Euclidean distance in the subsequence vector space. We show that, using this alternative distance measure, sequential time series clustering can indeed be meaningful. We repeat a key experiment in the work on which the "meaningless" claim was based, and show that our method leads to a successful clustering outcome. Recent papers have claimed that the result of K-means clustering for time series subsequences (STS clustering) is independent of the time series that created it. Our paper revisits this claim. In particular, we consider the following question: Given several time series sequences and a set of STS cluster centroids from one of them (generated by the K-means algorithm), is it possible to reliably determine which of the sequences produced these cluster centroids? While recent results suggest that the answer should be NO, we answer this question in the affirmative.We present cluster shape distance, an alternate distance measure for time series subsequence clusters, based on cluster shapes. Given a set of clusters, its shape is the sorted list of the pairwise Euclidean distances between their centroids. We then present two algorithms based on this distance measure, which match a set of STS cluster centroids with the time series that produced it. While the first algorithm creates DQG reuse this term more smaller "fingerprints" for the sequences, the second is more accurate. In our experiments with a dataset of 10 sequences, it produced a correct match 100 of the time.Furthermore, we offer an analysis that explains why our cluster shape distance provides a reliable way to match STS clusters to the original sequences, whereas cluster set distance fails to do so. Our work establishes for the first time a strong relation between the result of K-means STS clustering and the time series sequence that created it, despite earlier predictions that this is not possible. Dynamic time warping (DTW), which finds the minimum path by providing non-linear alignments between two time series, has been widely used as a distance measure for time series classification and clustering. However, DTW does not account for the relative importance regarding the phase difference between a reference point and a testing point. This may lead to misclassification especially in applications where the shape similarity between two sequences is a major consideration for an accurate recognition. Therefore, we propose a novel distance measure, called a weighted DTW (WDTW), which is a penalty-based DTW. Our approach penalizes points with higher phase difference between a reference point and a testing point in order to prevent minimum distance distortion caused by outliers. The rationale underlying the proposed distance measure is demonstrated with some illustrative examples. A new weight function, called the modified logistic weight function (MLWF), is also proposed to systematically assign weights as a function of the phase difference between a reference point and a testing point. By applying different weights to adjacent points, the proposed algorithm can enhance the detection of similarity between two time series. We show that some popular distance measures such as DTW and Euclidean distance are special cases of our proposed WDTW measure. We extend the proposed idea to other variants of DTW such as derivative dynamic time warping (DDTW) and propose the weighted version of DDTW. We have compared the performances of our proposed procedures with other popular approaches using public data sets available through the UCR Time Series Data Mining Archive for both time series classification and clustering problems. The experimental results indicate that the proposed approaches can achieve improved accuracy for time series classification and clustering problems. Many high level representations of time series have been proposed for data mining, including Fourier transforms, wavelets, eigenwaves, piecewise polynomial models, etc. Many researchers have also considered symbolic representations of time series, noting that such representations would potentiality allow researchers to avail of the wealth of data structures and algorithms from the text processing and bioinformatics communities. While many symbolic representations of time series have been introduced over the past decades, they all suffer from two fatal flaws. First, the dimensionality of the symbolic representation is the same as the original data, and virtually all data mining algorithms scale poorly with dimensionality. Second, although distance measures can be defined on the symbolic approaches, these distance measures have little correlation with distance measures defined on the original time series. In this work we formulate a new symbolic representation of time series. Our representation is unique in that it allows dimensionality numerosity reduction, and it also allows distance measures to be defined on the symbolic approach that lower bound corresponding distance measures defined on the original series. As we shall demonstrate, this latter feature is particularly exciting because it allows one to run certain data mining algorithms on the efficiently manipulated symbolic representation, while producing identical results to the algorithms that operate on the original data. In particular, we will demonstrate the utility of our representation on various data mining tasks of clustering, classification, query by content, anomaly detection, motif discovery, and visualization. Time series clustering has been shown effective in providing useful information in various domains. There seems to be an increased interest in time series clustering as part of the effort in temporal data mining research. To provide an overview, this paper surveys and summarizes previous works that investigated the clustering of time series data in various application domains. The basics of time series clustering are presented, including general-purpose clustering algorithms commonly used in time series clustering studies, the criteria for evaluating the performance of the clustering results, and the measures to determine the similarity dissimilarity between two time series being compared, either in the forms of raw data, extracted features, or some model parameters. The past researchs are organized into three groups depending upon whether they work directly with the raw data either in the time or frequency domain, indirectly with features extracted from the raw data, or indirectly with models built from the raw data. The uniqueness and limitation of previous research are discussed and several possible topics for future research are identified. Moreover, the areas that time series clustering have been applied to are also summarized, including the sources of data used. It is hoped that this review will serve as the steppingstone for those interested in advancing this area of research. Similarity measure between time series is a key issue in data mining of time series database. Euclidean distance measure is typically used init. However, the measure is an extremely brittle distance measure. Dynamic Time Warping (DTW) is proposed to deal with this case, but its expensive computation limits its application in massive datasets. In this paper, we present a new distance measure algorithm, called local segmented dynamic time warping (LSDTW), which is based on viewing the local DTW measure at the segment level. The DTW measure between the two segments is the product of the square of the distance between their mean times the number of points of the longer segment. Experiments about cluster analysis on the basis of this algorithm were implemented on a synthetic and a real world dataset comparing with Euclidean and classical DTW measure. The experiment results show that the new algorithm gives better computational performance in comparison to classical DTW with no loss of accuracy. We introduce a framework for defining a distance on the (non-Euclidean) space of Linear Dynamical Systems (LDSs). The proposed distance is induced by the action of the group of orthogonal matrices on the space of statespace realizations of LDSs. This distance can be efficiently computed for large-scale problems, hence it is suitable for applications in the analysis of dynamic visual scenes and other high dimensional time series. Based on this distance we devise a simple LDS averaging algorithm, which can be used for classification and clustering of time-series data. We test the validity as well as the performance of our group-action based distance on synthetic as well as real data and provide comparison with state-of-the-art methods. The parallel explosions of interest in streaming data, and data mining of time series have had surprisingly little intersection. This is in spite of the fact that time series data are typically streaming data. The main reason for this apparent paradox is the fact that the vast majority of work on streaming data explicitly assumes that the data is discrete, whereas the vast majority of time series data is real valued.Many researchers have also considered transforming real valued time series into symbolic representations, nothing that such representations would potentially allow researchers to avail of the wealth of data structures and algorithms from the text processing and bioinformatics communities, in addition to allowing formerly "batch-only" problems to be tackled by the streaming community. While many symbolic representations of time series have been introduced over the past decades, they all suffer from three fatal flaws. Firstly, the dimensionality of the symbolic representation is the same as the original data, and virtually all data mining algorithms scale poorly with dimensionality. Secondly, although distance measures can be defined on the symbolic approaches, these distance measures have little correlation with distance measures defined on the original time series. Finally, most of these symbolic approaches require one to have access to all the data, before creating the symbolic representation. This last feature explicitly thwarts efforts to use the representations with streaming algorithms.In this work we introduce a new symbolic representation of time series. Our representation is unique in that it allows dimensionality numerosity reduction, and it also allows distance measures to be defined on the symbolic approach that lower bound corresponding distance measures defined on the original series. As we shall demonstrate, this latter feature is particularly exciting because it allows one to run certain data mining algorithms on the efficiently manipulated symbolic representation, while producing identical results to the algorithms that operate on the original data. Finally, our representation allows the real valued data to be converted in a streaming fashion, with only an infinitesimal time and space overhead.We will demonstrate the utility of our representation on the classic data mining tasks of clustering, classification, query by content and anomaly detection. Time series clustering is an active research area with applications in a wide range of fields. One key component in cluster analysis is determining a proper dissimilarity measure between two data objects, and many criteria have been proposed in the literature to assess dissimilarity between two time series. The R package TSclust is aimed to implement a large set of well-established peer-reviewed time series dissimilarity measures, including measures based on raw data, extracted features, underlying parametric models, complexity levels, and forecast behaviors. Computation of these measures allows the user to perform clustering by using conventional clustering algorithms. TSclust also includes a clustering procedure based on p values from checking the equality of generating models, and some utilities to evaluate cluster solutions. The implemented dissimilarity functions are accessible individually for an easier extension and possible use out of the clustering context. The main features of TSclust are described and examples of its use are presented. Given the pervasiveness of time series data in all human endeavors, and the ubiquity of clustering as a data mining application, it is somewhat surprising that the problem of time series clustering from a single stream remains largely unsolved. Most work on time series clustering considers the clustering of individual time series, e.g., gene expression profiles, individual heartbeats or individual gait cycles. The few attempts at clustering time series streams have been shown to be objectively incorrect in some cases, and in other cases shown to work only on the most contrived datasets by carefully adjusting a large set of parameters. In this work, we make two fundamental contributions. First, we show that the problem definition for time series clustering from streams currently used is inherently flawed, and a new definition is necessary. Second, we show that the Minimum Description Length (MDL) framework offers an efficient, effective and essentially parameter-free method for time series clustering. We show that our method produces objectively correct results on a wide variety of datasets from medicine, zoology and industrial process analyses. We explore computational topology for clustering.We focus on time series and spatial data.Shape features are extracted based on n-dimensional holes.We run experiments with synthetic and real-world data.Results show improvements when compared to traditional clustering. Topology is the branch of mathematics that studies how objects relate to one another for their qualitative structural properties, such as connectivity and shape. In this paper, we present an approach for data clustering based on topological features computed over the persistence diagram, estimated using the theory of persistent homology. The features indicate topological properties such as Betti numbers, i.e., the number of n-dimensional holes in the discretized data space. The main contribution of our approach is enabling the clustering of time series that have similar recurrent behavior characterized by their attractors in phase space and spatial data that have similar scale-invariant spatial distributions, as traditional clustering techniques ignore that information as they rely on point-to-point dissimilarity measures such as Euclidean distance or elastic measures. We present experiments that confirm the usefulness of our approach with time series and spatial data applications in the fields of biology, medicine and ecology. Clustering is a solution for classifying enormous data when there is not any early knowledge about classes. With emerging new concepts like cloud computing and big data and their vast applications in recent years, research works have been increased on unsupervised solutions like clustering algorithms to extract knowledge from this avalanche of data. Clustering time-series data has been used in diverse scientific areas to discover patterns which empower data analysts to extract valuable information from complex and massive datasets. In case of huge datasets, using supervised classification solutions is almost impossible, while clustering can solve this problem using un-supervised approaches. In this research work, the focus is on time-series data, which is one of the popular data types in clustering problems and is broadly used from gene expression data in biology to stock market analysis in finance. This review will expose four main components of time-series clustering and is aimed to represent an updated investigation on the trend of improvements in efficiency, quality and complexity of clustering time-series approaches during the last decade and enlighten new paths for future works. Anatomy of time-series clustering is revealed by introducing its 4 main component.Research works in each of the four main components are reviewed in detail and compared.Analysis of research works published in the last decade.Enlighten new paths for future works for time-series clustering and its components. It has long been known that Dynamic Time Warping (DTW) is superior to Euclidean distance for classification and clustering of time series. However, until lately, most research has utilized Euclidean distance because it is more efficiently calculated. A recently introduced technique that greatly mitigates DTWs demanding CPU time has sparked a flurry of research activity. However, the technique and its many extensions still only allow DTW to be applied to moderately large datasets. In addition, almost all of the research on DTW has focused exclusively on speeding up its calculation; there has been little work done on improving its accuracy. In this work, we target the accuracy aspect of DTW performance and introduce a new framework that learns arbitrary constraints on the warping path of the DTW calculation. Apart from improving the accuracy of classification, our technique as a side effect speeds up DTW by a wide margin as well. We show the utility of our approach on datasets from diverse domains and demonstrate significant gains in accuracy and efficiency. E u clid ean D istan c e D yn am ic T im e W arp in g D is tan ce Figure 1: Note that while the two time series have an overall similar shape, they are not aligned in the time axis. Euclidean distance, which assumes the i point in one sequence is aligned with the i point in the other, will produce a pessimistic dissimilarity measure. The non-linear Dynamic Time Warped alignment allows a more intuitive distance measure to be calculated. 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. Most time series data mining algorithms use similarity search as a core subroutine, and thus the time taken for similarity search is the bottleneck for virtually all time series data mining algorithms. The difficulty of scaling search to large datasets largely explains why most academic work on time series data mining has plateaued at considering a few millions of time series objects, while much of industry and science sits on billions of time series objects waiting to be explored. In this work we show that by using a combination of four novel ideas we can search and mine truly massive time series for the first time. We demonstrate the following extremely unintuitive fact; in large datasets we can exactly search under DTW much more quickly than the current state-of-the-art Euclidean distance search algorithms. We demonstrate our work on the largest set of time series experiments ever attempted. In particular, the largest dataset we consider is larger than the combined size of all of the time series datasets considered in all data mining papers ever published. We show that our ideas allow us to solve higher-level time series data mining problem such as motif discovery and clustering at scales that would otherwise be untenable. In addition to mining massive datasets, we will show that our ideas also have implications for real-time monitoring of data streams, allowing us to handle much faster arrival rates and or use cheaper and lower powered devices than are currently possible. Comparison of time series is a key issue in data mining of time series database. Variation or extension of Euclidean distance is generally used. However Euclidean distance will vary much when time series is to be stretched or compressed along the time-axis. Dynamic time warping distance has been proposed to deal with this case, but its expensive computation limits its application. In this paper, a novel distance based on a new linear segmentation method of time series is proposed to avoid such drawbacks. Experiment results in this paper show that the proposed method achieves significant speed up to about 20 times than dynamic time warping distance without accuracy decrease. For more than a decade, time series similarity search has been given a great deal of attention by data mining researchers. As a result, many time series representations and distance measures have been proposed. However, most existing work on time series similarity search relies on shape-based similarity matching. While some of the existing approaches work well for short time series data, they typically fail to produce satisfactory results when the sequence is long. For long sequences, it is more appropriate to consider the similarity based on the higher-level structures. In this work, we present a histogram-based representation for time series data, similar to the "bag of words" approach that is widely accepted by the text mining and information retrieval communities. We performed extensive experiments and show that our approach outperforms the leading existing methods in clustering, classification, and anomaly detection on dozens of real datasets. We further demonstrate that the representation allows rotation-invariant matching in shape datasets. 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.