To classify time series by nearest neighbors, we need to specify or learn one or several distance measures. We consider variations of the Mahalanobis distance measures which rely on the inverse covariance matrix of the data. Unfortunately--for time series data--the covariance matrix has often low rank. To alleviate this problem we can either use a pseudoinverse, covariance shrinking or limit the matrix to its diagonal. We review these alternatives and benchmark them against competitive methods such as the related Large Margin Nearest Neighbor Classification (LMNN) and the Dynamic Time Warping (DTW) distance. As we expected, we find that the DTW is superior, but the Mahalanobis distance measures are one to two orders of magnitude faster. To get best results with Mahalanobis distance measures, we recommend learning one distance measure per class using either covariance shrinking or the diagonal approach.
Several distance functions are used for time series classification, such as Dynamic Time Warping (DTW) , DISSIM , Threshold Queries , Edit distances , Longest Common Subsequences (LCSS) , Swale , SpADe , and Cluster, Then Classify (CTC) . @cite_8 presented an extensive comparison of these distance functions and concluded that DTW is among the best measures and that the accuracy of the Euclidean distance converges to DTW as the size of the training set increases.
@cite_8
2008348094 2182136398 2077904202 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. 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. The effectiveness of nearest neighbor search heavily relies on the definition of distance function. Unfortunately, the meaningfulness of the frequently used distance, such as Euclidean distance, fractional distance and so on, will degrade with the increasing dimensionality. This problem, which is called distance concentration or instability, makes NN method perform poorly in a proximity query. The most popular distance function for time series, dynamic time warping(DTW), also concentrates when it is used in high-dimensional space. We learn the exponent p of the norm based on nearest neighbor large margin criterion for time series classification. The distance concentration is countered by maximum discrimination instead of maximum variance of distance distribution. The empirical results we presented demonstrate that the proposed approach shows a uniformly behavior, with results comparable to classic 1NN-Euclidean and 1NN-DTW. 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.