Time series are ubiquitous in real world problems and computing distance between two time series is often required in several learning tasks. Computing similarity between time series by ignoring variations in speed or warping is often encountered and dynamic time warping (DTW) is the state of the art. However DTW is not applicable in algorithms which require kernel or vectors. In this paper, we propose a mechanism named WaRTEm to generate vector embeddings of time series such that distance measures in the embedding space exhibit resilience to warping. Therefore, WaRTEm is more widely applicable than DTW. WaRTEm is based on a twin auto-encoder architecture and a training strategy involving warping operators for generating warping resilient embeddings for time series datasets. We evaluate the performance of WaRTEm and observed more than @math improvement over DTW in multiple real-world datasets.
Most related to our work is a recent work @cite_8 addressing the same task as ours, that of converting a time series dataset into a set of vector embeddings. Starting from a dataset of @math time series, they build an @math matrix of pairwise similarities that is defined using DTW distances. This is done by performing @math DTW computations, and approximating the other entries in the matrix using a low-rank assumption. If a time series is of length @math , a DTW computation takes time of the order of @math . This makes just the matrix creation step a process in @math . The similarity matrix is then subject to symmetric matrix factorization so that a vector embedding matrix X is learnt such that XX @math approximates the similarity matrix well. This calls for eigen decomposition which is superlinear in @math as well. This makes it impractical for most large datasets. In contrast to this work, the method we propose is linear in both @math and @math , making our method significantly more scalable.
@cite_8
2008348094 2593719957 2759835476 58346954
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. A considerable amount of machine learning algorithms take instance-feature matrices as their inputs. As such, they cannot directly analyze time series data due to its temporal nature, usually unequal lengths, and complex properties. This is a great pity since many of these algorithms are effective, robust, efficient, and easy to use. In this paper, we bridge this gap by proposing an efficient representation learning framework that is able to convert a set of time series with equal or unequal lengths to a matrix format. In particular, we guarantee that the pairwise similarities between time series are well preserved after the transformation. The learned feature representation is particularly suitable to the class of learning problems that are sensitive to data similarities. Given a set of @math time series, we first construct an @math partially observed similarity matrix by randomly sampling @math pairs of time series and computing their pairwise similarities. We then propose an extremely efficient algorithm that solves a highly non-convex and NP-hard problem to learn new features based on the partially observed similarity matrix. We use the learned features to conduct experiments on both data classification and clustering tasks. Our extensive experimental results demonstrate that the proposed framework is both effective and efficient. Dynamic Time Warping (DTW) is one of the best similarity measures for time series, and it has extensively been used in retrieval, classification or mining applications. It is a costly measure, and applying it to numerous and or very long times series is difficult in practice. Recently, Shapelet Transform (ST) proved to enable accurate supervised classification of time series. ST learns small subsequences that well discriminate classes, and transforms the time series into vectors lying in a metric space. In this paper, we adopt the ST framework in a novel way: we focus on learning, without class label information, shapelets such that Euclidean distances in the ST-space approximate well the true DTW. Our approach leads to an ubiquitous representation of time series in a metric space, where any machine learning method (supervised or unsupervised) and indexing system can operate efficiently. 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.