In the light of regularized dynamic time warping kernels, this paper re-considers the concept of time elastic centroid for a setof time series. We derive a new algorithm based on a probabilistic interpretation of kernel alignment matrices. This algorithm expressesthe averaging process in terms of a stochastic alignment automata. It uses an iterative agglomerative heuristic method for averagingthe aligned samples, while also averaging the times of occurrence of the aligned samples. By comparing classification accuracies for45 heterogeneous time series datasets obtained by first nearest centroid medoid classifiers we show that: i) centroid-basedapproaches significantly outperform medoid-based approaches, ii) for the considered datasets, our algorithm that combines averagingin the sample space and along the time axes, emerges as the most significantly robust model for time-elastic averaging with apromising noise reduction capability. We also demonstrate its benefit in an isolated gesture recognition experiment and its ability tosignificantly reduce the size of training instance sets. Finally we highlight its denoising capability using demonstrative synthetic data:we show that it is possible to retrieve, from few noisy instances, a signal whose components are scattered in a wide spectral band.
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_27 , @cite_21 . Although other time elastic distance measures such as the Edit Distance With Real Penalty (ERP) @cite_11 or the Time Warp Edit Distance (TWED) @cite_31 could be considered instead, without loss of generality, we remain focused throughout this paper on DTW and its kernelization.
@cite_27 @cite_21 @cite_31 @cite_11
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Experiments on the automatic recognition of 203 Russian words are described. The experimental vocabulary includes terms of the language, ALGOL -60 together with others. The logarithmic characteristics of acoustic signal in five bands are extracted as features. The measure of similarity between the words of standard and control sequences is calculated by the words maximizing a definite functional using dynamic programming. The average reliability of recognition for one speaker obtained for experiments using 5000 words is 0·95. The computational time for recognition is 2-4 sec. In a way similar to the string-to-string correction problem, we address discrete time series similarity in light of a time-series-to-time-series-correction problem for which the similarity between two time series is measured as the minimum cost sequence of edit operations needed to transform one time series into another. To define the edit operations, we use the paradigm of a graphical editing process and end up with a dynamic programming algorithm that we call time warp edit distance (TWED). TWED is slightly different in form from dynamic time warping (DTW), longest common subsequence (LCSS), or edit distance with real penalty (ERP) algorithms. In particular, it highlights a parameter that controls a kind of stiffness of the elastic measure along the time axis. We show that the similarity provided by TWED is a potentially useful metric in time series retrieval applications since it could benefit from the triangular inequality property to speed up the retrieval process while tuning the parameters of the elastic measure. In that context, a lower bound is derived to link the matching of time series into down sampled representation spaces to the matching into the original space. The empiric quality of the TWED distance is evaluated on a simple classification task. Compared to edit distance, DTW, LCSS, and ERP, TWED has proved to be quite effective on the considered experimental task. Existing studies on time series are based on two categories of distance functions. The first category consists of the Lp-norms. They are metric distance functions but cannot support local time shifting. The second category consists of distance functions which are capable of handling local time shifting but are nonmetric. The first contribution of this paper is the proposal of a new distance function, which we call ERP ("Edit distance with Real Penalty"). Representing a marriage of L1- norm and the edit distance, ERP can support local time shifting, and is a metric. The second contribution of the paper is the development of pruning strategies for large time series databases. Given that ERP is a metric, one way to prune is to apply the triangle inequality. Another way to prune is to develop a lower bound on the ERP distance. We propose such a lower bound, which has the nice computational property that it can be efficiently indexed with a standard B+- tree. Moreover, we show that these two ways of pruning can be used simultaneously for ERP distances. Specifically, the false positives obtained from the B+-tree can be further minimized by applying the triangle inequality. Based on extensive experimentation with existing benchmarks and techniques, we show that this combination delivers superb pruning power and search time performance, and dominates all existing strategies.