Time Warp Edit Distance with Stiffness Adjustment for Time Series Matching

In a way similar to the string-to-string correction problem we address time series similarity in the 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, Longest Common Subsequence or Edit Distance with Real Penalty algorithms. In particular, it highlights a parameter which drives a kind of stiffness of the elastic measure along the time axis. We show that the similarity provided by TWED is a metric potentially useful 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 relate the matching of time series into down sampled representation spaces to the matching into the original space. Empiric quality of the TWED distance is evaluated on a simple classification task. Compared to Edit Distance, Dynamic Time Warping, Longest Common Subsequnce and Edit Distance with Real Penalty, TWED has proven to be quite effective on the considered experimental task

Time series classification with ensembles of elastic distance measures

Several alternative distance measures for comparing time series have recently been proposed and evaluated on time series classification (TSC) problems. These include variants of dynamic time warping (DTW), such as weighted and derivative DTW, and edit distance-based measures, including longest common subsequence, edit distance with real penalty, time warp with edit, and move–split–merge. These measures have the common characteristic that they operate in the time domain and compensate for potential localised misalignment through some elastic adjustment. Our aim is to experimentally test two hypotheses related to these distance measures. Firstly, we test whether there is any significant difference in accuracy for TSC problems between nearest neighbour classifiers using these distance measures. Secondly, we test whether combining these elastic distance measures through simple ensemble schemes gives significantly better accuracy. We test these hypotheses by carrying out one of the largest experimental studies ever conducted into time series classification. Our first key finding is that there is no significant difference between the elastic distance measures in terms of classification accuracy on our data sets. Our second finding, and the major contribution of this work, is to define an ensemble classifier that significantly outperforms the individual classifiers. We also demonstrate that the ensemble is more accurate than approaches not based in the time domain. Nearly all TSC papers in the data mining literature cite DTW (with warping window set through cross validation) as the benchmark for comparison. We believe that our ensemble is the first ever classifier to significantly outperform DTW and as such raises the bar for future work in this area

On Recursive Edit Distance Kernels with Application to Time Series Classification

This paper proposes some extensions to the work on kernels dedicated to string or time series global alignment based on the aggregation of scores obtained by local alignments. The extensions we propose allow to construct, from classical recursive definition of elastic distances, recursive edit distance (or time-warp) kernels that are positive definite if some sufficient conditions are satisfied. The sufficient conditions we end-up with are original and weaker than those proposed in earlier works, although a recursive regularizing term is required to get the proof of the positive definiteness as a direct consequence of the Haussler's convolution theorem. The classification experiment we conducted on three classical time warp distances (two of which being metrics), using Support Vector Machine classifier, leads to conclude that, when the pairwise distance matrix obtained from the training data is \textit{far} from definiteness, the positive definite recursive elastic kernels outperform in general the distance substituting kernels for the classical elastic distances we have tested.Comment: 14 page

Time Warp Edit Distance

This technical report details a family of time warp distances on the set of discrete time series. This family is constructed as an editing distance whose elementary operations apply on linear segments. A specific parameter allows controlling the stiffness of the elastic matching. It is well suited for the processing of event data for which each data sample is associated with a timestamp, not necessarily obtained according to a constant sampling rate. Some properties verified by these distances are proposed and proved in this report.Comment: Pattern Recognition - Clustering - Algorithms - Similarity Measure

Discrete Elastic Inner Vector Spaces with Application in Time Series and Sequence Mining

This paper proposes a framework dedicated to the construction of what we call discrete elastic inner product allowing one to embed sets of non-uniformly sampled multivariate time series or sequences of varying lengths into inner product space structures. This framework is based on a recursive definition that covers the case of multiple embedded time elastic dimensions. We prove that such inner products exist in our general framework and show how a simple instance of this inner product class operates on some prospective applications, while generalizing the Euclidean inner product. Classification experimentations on time series and symbolic sequences datasets demonstrate the benefits that we can expect by embedding time series or sequences into elastic inner spaces rather than into classical Euclidean spaces. These experiments show good accuracy when compared to the euclidean distance or even dynamic programming algorithms while maintaining a linear algorithmic complexity at exploitation stage, although a quadratic indexing phase beforehand is required.Comment: arXiv admin note: substantial text overlap with arXiv:1101.431

Times series averaging from a probabilistic interpretation of time-elastic kernel

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

Comparing Dynamic Hand Rehabilitation Gestures in Leap Motion Using Multi dimensional Dynamic Time Warping

We propose and evaluate the use of Multi-dimensional Dynamic Time Warping (MDTW) for comparing dynamic hand rehabilitation gestures that would be performed by a patient (query) relative to hand gestures prepared by a physiotherapist (reference). MDTW enables us to determine how similar or different a query dynamic hand gesture is to a reference one whilst filtering out unwanted sources of error resulting from positional, rotational or speed differences between the query and the reference actions. It produces a minimum-distance value of a warp path after aligning a query dynamic hand gesture with a reference one. A low minimum-distance value implies the two gestures being compared are similar and high minimum-distance value implies the two gestures vary to a greater extent. When we deliberately compare a specific hand gesture with itself, we obtain a minimum-distance value of 0° indicating the similarity is 100%. Furthermore, when we compare two closely similar hand gestures i.e. gesture 1 and gesture 4, a minimum-distance value of 35.9° is obtained. However, when we compare two quite different gestures i.e. gesture 2 and gesture 3, a minimum-distance value of 248.5° is obtained. Therefore, a physiotherapist can establish whether a patient performs hand rehabilitation gestures satisfactorily or an adjustment is required based on the minimum-distance values of the warp paths