At the light of regularized dynamic time warping kernels, this paper reconsider the concept of time elastic centroid (TEC) for a set of time series. From this perspective, we show first how TEC can easily be addressed as a preimage problem. Unfortunately this preimage problem is ill-posed, may suffer from over-fitting especially for long time series and getting a sub-optimal solution involves heavy computational costs. We then derive two new algorithms based on a probabilistic interpretation of kernel alignment matrices that expresses in terms of probabilistic distributions over sets of alignment paths. The first algorithm is an iterative agglomerative heuristics inspired from the state of the art DTW barycenter averaging (DBA) algorithm proposed specifically for the Dynamic Time Warping measure. The second proposed algorithm achieves a classical averaging of the aligned samples but also implements an averaging of the time of occurrences of the aligned samples. It exploits a straightforward progressive agglomerative heuristics. An experimentation that compares for 45 time series datasets classification error rates obtained by first near neighbors classifiers exploiting a single medoid or centroid estimate to represent each categories show that: i) centroids based approaches significantly outperform medoids based approaches, ii) on the considered experience, the two proposed algorithms outperform the state of the art DBA algorithm, and iii) the second proposed algorithm that implements an averaging jointly in the sample space and along the time axes emerges as the most significantly robust time elastic averaging heuristic with an interesting noise reduction capability. Index Terms—Time series averaging Time elastic kernel Dynamic Time Warping Time series clustering and classification.
Time series averaging in the context of (multiple) time elastic distance alignments has been mainly addressed in the scope of the Dynamic Time Warping (DTW) measure @cite_26 , @cite_21 . Although other time elastic distance measures such as the Edit Distance With Real Penalty (ERP) @cite_12 or the Time Warp Edit Distance (TWED) @cite_28 could be considered instead, without loss of generality, we remain focused throughout this paper on DTW and its kernelization.
@cite_28 @cite_26 @cite_21 @cite_12
2586337417 2008348094 58346954 2107092366
Dynamic Time Warping (DTW) is probably the most popular distance measure for time series data, because it captures flexible similarities under time distortions. However, DTW has long been suffering from the pathological alignment problem, and most existing solutions, which essentially impose rigid constraints on the warping path, are likely to miss the correct alignments. A crucial observation on pathological alignment is that it always leads to an abnormally large number of links between two sequences. Based on this new observation, we propose a novel variant of DTW called LDTW, which limits the total number of links during the optimization process of DTW. LDTW not only oppresses the pathological alignment effectively, but also allows more flexibilities when measuring similarities. It is a softer constraint because we still let the optimization process of DTW decide how many links to allocate to each data point and where to put these links. In this paper, we introduce the motivation and algorithm of LDTW and we conduct a nearest neighbor classification experiment on UCR time series archive to show its performance. 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. 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. Dynamic time warping is a popular technique for comparing time series, providing both a distance measure that is insensitive to local compression and stretches and the warping which optimally deforms one of the two input series onto the other. A variety of algorithms and constraints have been discussed in the literature. The dtw package provides an unification of them; it allows R users to compute time series alignments mixing freely a variety of continuity constraints, restriction windows, endpoints, local distance definitions, and so on. The package also provides functions for visualizing alignments and constraints using several classic diagram types.