Asymptotic Dynamic Time Warping calculation with utilizing value repetition

Dynamic Time Warping (DTW) is a popular method for measuring the similarity of time series. It is widely used in various domains. A major drawback of DTW is that it has a high computational complexity. To address this problem, pruning techniques to calculate the exact DTW distance, as well as DTW approximation methods, have become important approaches. In this paper, we introduce Blocked Dynamic Time Warping (BDTW), a new similarity measure which works on run-length encoded time series representation. BDTW utilizes any repetitive values (zero and nonzero) in time series to reduce DTW computation time. BDTW closely approximates DTW distance, and it is significantly faster than traditional DTW for time series with high levels of value repetition. Moreover, BDTW can be combined with time series representation methods which provide constant segments, to serve as a close approximation method even for the time series without value repetition. Constrained BDTW, BDTW upper bound and BDTW lower bound are discussed as variations of BDTW. BDTW upper bound and BDTW lower bound are presented as a new DTW upper bound and lower bound which can be efficiently applied on time series with high levels of value repetition for pruning unhopeful alignments and matches in the exact DTW calculation. We show the effectiveness of BDTW and its variations on different applications using the following datasets: Almanac of Minutely Power, Refit Smart Homes, as well as the 85 datasets from the University of California, Riverside time series classification archive (UCR archive).     

改进DTW下界函数的距离度量方法研究

动态时间弯曲算法（DTW）是一种常见的时间序列相似性度量方法,对数据挖掘任务起着至关重要的作用。针对现有DTW算法的时间复杂度高、度量精确度一般的特征,提出一种DTW下界函数的提前终止算法（LB＿ESDTW）。引入提前终止思想,提高算法的执行效率;再在提前终止算法思想的基础上,与DTW下界函数相结合,提出一种基于提前终止DTW的下界函数算法（LB＿ESDTW）。该算法在保证高效的运行时间效率的同时,也使得算法的度量准确率得到了提升。实验结果表明,LB＿ESDTW在绝大部分时间序列数据集中,都表现出良好的适应性,针对不同类别的时间序列,都能有良好的度量性能。     

Speeding up dynamic time warping distance for sparse time series data

Dynamic time warping (DTW) distance has been effectively used in mining time series data in a multitude of domains. However, in its original formulation DTW is extremely inefficient in comparing long sparse time series, containing mostly zeros and some unevenly spaced nonzero observations. Original DTW distance does not take advantage of this sparsity, leading to redundant calculations and a prohibitively large computational cost for long time series. We derive a new time warping similarity measure (AWarp) for sparse time series that works on the run-length encoded representation of sparse time series. The complexity of AWarp is quadratic on the number of observations as opposed to the range of time of the time series. Therefore, AWarp can be several orders of magnitude faster than DTW on sparse time series. AWarp is exact for binary-valued time series and a close approximation of the original DTW distance for any-valued series. We discuss useful variants of AWarp: bounded (both upper and lower), constrained, and multidimensional. We show applications of AWarp to three data mining tasks including clustering, classification, and outlier detection, which are otherwise not feasible using classic DTW, while producing equivalent results. Potential areas of application include bot detection, human activity classification, search trend analysis, seismic analysis, and unusual review pattern mining.     

Area bound dynamic time warping based fast and accurate person authentication using a biometric pen

The paper presents a modified dynamic time warping (DTW) technique for person authentication based on time series matching obtained from handwriting. The online data has been acquired by a biometric smart pen device. The proposed method allows fast and accurate classification of human individuals based on handwritten PIN words or signature samples. Although classic DTW provides robust distance measurements essential for accurate classification of sequences, it is computationally expensive. To speed up computations we introduce area bound dynamic time warping (AB_DTW) that divides time series into several areas bounded by segments of consecutive zero crossings including local peaks and valleys. Unlike classic DTW which compares whole signals, the proposed AB_DTW warps areas bounded by the local regions. Two kinds of data abstraction formats of area bound—1 dimensional and 2 dimensional—are evaluated. Experimental results show that because of a higher-level data abstraction, the proposed approach is several times faster than classic DTW. Moreover, AB_DTW does not offer substantial loss of accuracy which is required for authentication performance using handwritten PIN words and signatures sampled by biometric pen device.     

Using dynamic time warping distances as features for improved time series classification

Dynamic time warping (DTW) has proven itself to be an exceptionally strong distance measure for time series. DTW in combination with one-nearest neighbor, one of the simplest machine learning methods, has been difficult to convincingly outperform on the time series classification task. In this paper, we present a simple technique for time series classification that exploits DTW’s strength on this task. But instead of directly using DTW as a distance measure to find nearest neighbors, the technique uses DTW to create new features which are then given to a standard machine learning method. We experimentally show that our technique improves over one-nearest neighbor DTW on 31 out of 47 UCR time series benchmark datasets. In addition, this method can be easily extended to be used in combination with other methods. In particular, we show that when combined with the symbolic aggregate approximation (SAX) method, it improves over it on 37 out of 47 UCR datasets. Thus the proposed method also provides a mechanism to combine distance-based methods like DTW with feature-based methods like SAX. We also show that combining the proposed classifiers through ensembles further improves the performance on time series classification.     

Time Warp Edit Distance with Stiffness Adjustment for Time Series Matching

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.     

Scalable time series classification

Time series classification tries to mimic the human understanding of similarity. When it comes to long or larger time series datasets, state-of-the-art classifiers reach their limits because of unreasonably high training or testing times. One representative example is the 1-nearest-neighbor dynamic time warping classifier (1-NN DTW) that is commonly used as the benchmark to compare to. It has several shortcomings: it has a quadratic time complexity in the time series length and its accuracy degenerates in the presence of noise. To reduce the computational complexity, early abandoning techniques, cascading lower bounds, or recently, a nearest centroid classifier have been introduced. Still, classification times on datasets of a few thousand time series are in the order of hours. We present our Bag-Of-SFA-Symbols in Vector Space classifier that is accurate, fast and robust to noise. We show that it is significantly more accurate than 1-NN DTW while being multiple orders of magnitude faster. Its low computational complexity combined with its good classification accuracy makes it relevant for use cases like long or large amounts of time series or real-time analytics.     

Boundary-based lower-bound functions for dynamic time warping and their indexing

Dynamic time warping (DTW) is a powerful technique in the time-series similarity search. However, its performance on large-scale data is unsatisfactory because of its high computational cost and the fact that it cannot be indexed directly. The lower bound technique for DTW is an effective solution to this problem. In this paper, we explain the existing lower-bound functions from a unified perspective and show that they are only special cases under our framework. We then propose a group of lower-bound functions for DTW and compare their performances through extensive experiments. The experimental results show that the new methods are better than the existing ones in most cases, and a theoretical explanation of the results is also given. We further implement an index structure based on the new lower-bound function. Experimental results demonstrate a similar conclusion.     

Scaling and time warping in time series querying

The last few years have seen an increasing understanding that dynamic time warping (DTW), a technique that allows local flexibility in aligning time series, is superior to the ubiquitous Euclidean distance for time series classification, clustering, and indexing. More recently, it has been shown that for some problems, uniform scaling (US), a technique that allows global scaling of time series, may just be as important for some problems. In this work, we note that for many real world problems, it is necessary to combine both DTW and US to achieve meaningful results. This is particularly true in domains where we must account for the natural variability of human actions, including biometrics, query by humming, motion-capture/animation, and handwriting recognition. We introduce the first technique which can handle both DTW and US simultaneously, our techniques involve search pruning by means of a lower bounding technique and multi-dimensional indexing to speed up the search. We demonstrate the utility and effectiveness of our method on a wide range of problems in industry, medicine, and entertainment.     

一种支持DTW距离的多元时间序列索引结构

现有的索引结构难以有效地支持DTW距离度量下的多元时间序列相似性搜索.首先给出一种将不等长多元时间序列转换为等长一元时间序列的方法,并证明这种转换满足下界距离引理;以此为基础,提出一种多元时间序列的DTW下界距离,并对其性质进行分析;然后,针对给出的下界距离,提出一种支持DTW距离度量的多元时间序列索引结构,对多元时间序列数据库进行有效组织;再给出多元时间序列相似模式搜索算法及流程,并证明该搜索方法具有非漏报性;最后,通过实验对所提方法的有效性进行验证.     

Weighted dynamic time warping for time series classification

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.     

一种新的DTW最佳弯曲窗口学习方法

时间序列相似性查询中,DTW(Dynamic Time Warping)距离是支持时间弯曲的经典度量,约束弯曲窗口的DTW是DTW最常见的实用形式。分析了传统DTW最佳弯曲窗口学习方法存在的问题,并在此基础上引入时间距离的概念,提出了新的DTW最佳弯曲窗口学习方法。由于时间距离是DTW计算的附属产物,因此该方法可以在几乎不增加运算量的情况下提高DTW的分类精度。实验证明,采用了新的学习方法后,具有最佳弯曲窗口的DTW分类精度得到明显改善,分类精度优于ERP(Edit Distance with Real Penalty)和LCSS(Longest Common SubSequence),接近TWED(Time Warp Edit Distance)的水平。     

Hierarchical clustering of time series data with parametric derivative dynamic time warping

Dynamic Time Warping (DTW) is a popular and efficient distance measure used in classification and clustering algorithms applied to time series data. By computing the DTW distance not on raw data but on the time series of the (first, discrete) derivative of the data, we obtain the so-called Derivative Dynamic Time Warping (DDTW) distance measure. DDTW, used alone, is usually inefficient, but there exist datasets on which DDTW gives good results, sometimes much better than DTW. To improve the performance of the two distance measures, we can combine them into a new single (parametric) distance function. The literature contains examples of the combining of DTW and DDTW in algorithms for supervised classification of time series data. In this paper, we demonstrate that combination of DTW and DDTW can also be applied in a method of time series clustering (unsupervised classification). In particular, we focus on a hierarchical clustering (with average linkage) of univariate (one-dimensional) time series data. We construct a new parametric distance function, combining DTW and DDTW, where a single real number parameter controls the contribution of each of the two measures to the total value of the combined distances. The parameter is tuned in the initial phase of the clustering algorithm. Using this technique in clustering methods requires a different approach (to address certain specific problems) than for supervised methods. In the clustering process we use three internal cluster validation measures (measures which do not use labels) and three external cluster validation measures (measures which do use clustering data labels). Internal measures are used to select an optimal value of the parameter of the algorithm, where external measures give information about the overall performance of the new method and enable comparison with other distance functions. Computational experiments are performed on a large real-world data base (UCR Time Series Classification Archive: 84 datasets) from a very broad range of fields, including medicine, finance, multimedia and engineering. The experimental results demonstrate the effectiveness of the proposed approach for hierarchical clustering of time series data. The method with the new parametric distance function outperforms DTW (and DDTW) on the data base used. The results are confirmed by graphical and statistical comparison.     

Fast and Exact Warping of Time Series Using Adaptive Segmental Approximations

Similarity search is a core module of many data analysis tasks, including search by example, classification, and clustering. For time series data, Dynamic Time Warping (DTW) has been proven a very effective similarity measure, since it minimizes the effects of shifting and distortion in time. However, the quadratic cost of DTW computation to the length of the matched sequences makes its direct application on databases of long time series very expensive. We propose a technique that decomposes the sequences into a number of segments and uses cheap approximations thereof to compute fast lower bounds for their warping distances. We present several, progressively tighter bounds, relying on the existence or not of warping constraints. Finally, we develop an index and a multi-step technique that uses the proposed bounds and performs two levels of filtering to efficiently process similarity queries. A thorough experimental study suggests that our method consistently outperforms state-of-the-art methods for DTW similarity search.     

弹性粗粒度动态弯曲时序相似性算法

针对动态时间弯曲(DTW)算法在提高计算速度同时不能兼顾分类正确率的问题,提出了一种基于朴素粒计算思想的弹性粗粒度动态时间弯曲(CG-DTW)算法。首先,通过计算时序方差特征的方法来获取较优的时序粒度,用粒度特征代替原始序列;其次,再代入执行DTW算法,允许动态调整被比较时序粒间的弹性大小,从而获得相对最优的时序对应粒;最后,在对应最优粒的情况下计算DTW距离。同时引入下界函数的提前终止策略进一步提高CG-DTW算法效率。实验结果表明,所提算法要比经典算法运行速率提高21.4%左右,比降维策略算法正确率提高近32.3个百分点,尤其是长序列的分类,CG-DTW能够在保持正确率的情况下兼顾较高的运行效率。CG-DTW在实际应用中能适应不确定长序列分类。     

Exact indexing of dynamic time warping

The problem of indexing time series has attracted much interest. Most algorithms used to index time series utilize the Euclidean distance or some variation thereof. However, it has been forcefully shown that the Euclidean distance is a very brittle distance measure. Dynamic time warping (DTW) is a much more robust distance measure for time series, allowing similar shapes to match even if they are out of phase in the time axis. Because of this flexibility, DTW is widely used in science, medicine, industry and finance. Unfortunately, however, DTW does not obey the triangular inequality and thus has resisted attempts at exact indexing. Instead, many researchers have introduced approximate indexing techniques or abandoned the idea of indexing and concentrated on speeding up sequential searches. In this work, we introduce a novel technique for the exact indexing of DTW. We prove that our method guarantees no false dismissals and we demonstrate its vast superiority over all competing approaches in the largest and most comprehensive set of time series indexing experiments ever undertaken.     

A Novel Similarity Measure Model for Multivariate Time Series Based on LMNN and DTW

In this paper, a novel model is proposed to measure the similarity of multivariate time series by combining large margin nearest neighbor (LMNN) and dynamic time warping (DTW). Firstly we use a Mahalanobis distance-based DTW measure for multivariable time series, which considers the relations among variables through the Mahalanobis matrix. Secondly, the LMNN algorithm is applied to learn the Mahalanobis matrix by minimizing a renewed cost function. As the cost function is non-differentiable, the minimization problem is solved from a perspective of k-means by coordinate descent method. We empirically compare the proposed model with other techniques and demonstrate its convergence and superiority in similarity measure for multivariate time series.     

Dynamic time warping constraint learning for large margin nearest neighbor classification

Nearest neighbor (NN) classifier with dynamic time warping (DTW) is considered to be an effective method for time series classification. The performance of NN-DTW is dependent on the DTW constraints because the NN classifier is sensitive to the used distance function. For time series classification, the global path constraint of DTW is learned for optimization of the alignment of time series by maximizing the nearest neighbor hypothesis margin. In addition, a reduction technique is combined with a search process to condense the prototypes. The approach is implemented and tested on UCR datasets. Experimental results show the effectiveness of the proposed method.     

Extensions and relationships of some existing lower-bound functions for dynamic time warping

Dynamic time warping (DTW) is a state-of-the-art time series similarity measure method, which warps time axes to match the same shape between two time series with different lengths. However, its quadratic time and space complexity is an obstacle to its applications in the large time series data mining. To address this problem, some lower-bound functions for DTW, fast methods to approximately measure the distance between time series, are used to prune the dissimilar objects from time series database so as to retain the candidates for further measuring their similarity with DTW. So far, the existing lower-bound functions for DTW have been widely accepted for time series similarity search and indexing. In this paper, we propose the extensions of two existing lower-bound functions and discuss the relationships among them. The extensions are improved with high tightness and without much time cost. At the same time, we theoretically prove that these extensions satisfy lower-bound requirement and are better than their old versions respectively. The experimental results demonstrate that in most cases the quality of the proposed extensions of lower-bound functions for DTW outperforms the original versions except for a slightly higher time cost.     

风电机组齿轮箱的多变量时间序列故障预警

针对风电机组故障预警中，原始动态时间规整（DTW）算法无法有效度量风电机组多变量时间序列数据之间距离的问题，提出一种基于犹豫模糊集的动态时间规整（HFS-DTW）算法。该算法是原始DTW算法的一种扩展算法，可对单变量和多变量时间序列数据进行距离度量，且精度与速度较原始DTW算法更优。以子时间序列相似度距离为目标函数，使用帝国竞争算法（ICA）优化了HFS-DTW算法中的子序列长度和步距参数。算例研究表明与仅DTW算法和非参数最优的HFS-DTW算法相对比，参数最优的HFS-DTW可挖掘更多的多维特征点信息，输出的多维特征点相似序列具有更丰富细节；且基于所提算法可提前10天预警风电机组齿轮箱故障。