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Time series clustering is the act of grouping time series data without recourse to a label. Algorithms that cluster time series can be classified into two groups: those that employ a time series specific distance measure and those that derive features from time series. Both approaches usually rely on traditional clustering algorithms such as k-means. Our focus is on partitional clustering algorithms that employ elastic distance measures, i.e. distances that perform some kind of realignment whilst measuring distance. We describe nine commonly used elastic distance measures and compare their performance with k-means and k-medoids clusterer. Our findings, based on experiments using the UCR time series archive, are surprising. We find that, generally, clustering with DTW distance is not better than using Euclidean distance and that distance measures that employ editing in conjunction with warping are significantly better than other approaches. We further observe that using k-medoids clusterer rather than k-means improves the clusterings for all nine elastic distance measures. One function, the move–split–merge (MSM) distance, is the best performing algorithm of this study, with time warp edit (TWE) distance a close second. Our conclusion is that MSM or TWE with k-medoids clusterer should be considered as a good alternative to DTW for clustering time series with elastic distance measures. We provide implementations, extensive results and guidance on reproducing results on the associated GitHub repository.

The iso‐ or diachronous character of a geologic unit is scale‐dependent, especially for time‐specific facies. The Usseln Limestone is a Late Devonian time‐specific facies from Germany, occurring immediately below the Lower Kellwasser black shale. Here, we investigate whether cm‐scale rhythmical bands within the Usseln Limestone are correlatable across its depositional basin. Its facies were studied at three locations ca. 50 km apart, representing different depositional settings. Its cm‐scale alternations in lithological facies and elemental content (μXRF) form an excellent target for correlations on millennial timescales. Correlation attempts failed to converge to a solution at the cm‐scale of individual rhythmites. Dynamic Time Warping, however, provided convincing correlations at the dm‐scale, supporting its use as a high‐resolution correlation tool. The Usseln Limestone base may be diachronous, but the top is likely isochronous. This finding is in agreement with sudden basin‐wide black shale deposition at the onset of the Kellwasser Crisis. Plain Language Summary A time‐specific facies is a rock unit that can be recognized in different places and always represents the same specific moment in Earth history. Here, we study the Usseln Limestone from the Rhenish Massif in Germany. This time‐specific facies is intruiging, as it is characterized by internal cm‐ and dm‐scale rhythmites. We sampled the Usseln Limestone at three different locations and assessed how consistently individual rhythmites occur across the basin, with approximately 50 km between sites. To do this, we first obtained mm‐resolution geochemical elemental data from all samples. Subsequently, the mathemathical method “Dynamic Time Warping” was applied to attempt correlation. At the cm‐level, our correlation attempt failed. Depositional differences between the three sites were too substantial for precise correlations at the scale of individual rhythmites. However, we successfully made stratigraphic correlations at the decimeter‐scale of the rhythmite bundles. These correlations suggest that while the base of the Usseln Limestone might have formed at different times in different places, the top of this time‐specific facies likely is synchronous throughout the basin. This is important, because the top of the Usseln Limestone aligns with the sudden appearance of a widespread black shale layer, marking the start of a major biotic crisis in Earth history. Key Points This case study illustrates how the isochronony of a time‐specific facies breaks down at smaller thickness scales Dynamic Time Warping applied to an ensemble of proxies is demonstrated to be a useful high‐resolution correlation tool The inferred isochronous top of the Usseln Limestone is in line with a rapid onset of Lower Kellwasser black shale deposition

Dynamic radar cross-section (RCS) represents the radar reflectance cross-section of an aircraft at different moments during flight and is a crucial criterion for evaluating the ability of a radar to detect an aircraft. In practice, it is challenging to construct an accurate 6-DOF dynamics model for the purpose of solving its dynamic RCS. Consequently, a 3-DOF dynamics model is employed for the analysis of dynamic RCS. This paper presents a methodology for the joint simulation of the dynamic RCS under two DOF models and subsequent comparison and analysis of the resulting sequence data via the dynamic time warping algorithm. It is also concluded that the dynamic RCS values calculated using the 3-DOF dynamics model can identify the target vehicle.

The abundance of functional observations in scientific endeavors has led to a significant development in tools for functional data analysis (FDA). This kind of data comes with several challenges: infinite-dimensionality of function spaces, observation noise, and so on. However, there is another interesting phenomena that creates problems in FDA. The functional data often comes with lateral displacements/deformations in curves, a phenomenon which is different from the height or amplitude variability and is termed phase variation. The presence of phase variability artificially often inflates data variance, blurs underlying data structures, and distorts principal components. While the separation and/or removal of phase from amplitude data is desirable, this is a difficult problem. In particular, a commonly used alignment procedure, based on minimizing the 𝕃2 norm between functions, does not provide satisfactory results. In this paper we motivate the importance of dealing with the phase variability and summarize several current ideas for separating phase and amplitude components. These approaches differ in the following: (1) the definition and mathematical representation of phase variability, (2) the objective functions that are used in functional data alignment, and (3) the algorithmic tools for solving estimation/optimization problems. We use simple examples to illustrate various approaches and to provide useful contrast between them.

In recent years Dynamic Time Warping (DTW) has emerged as the distance measure of choice for virtually all time series data mining applications. For example, virtually all applications that process data from wearable devices use DTW as a core sub-routine. This is the result of significant progress in improving DTW’s efficiency, together with multiple empirical studies showing that DTW-based classifiers at least equal (and generally surpass) the accuracy of all their rivals across dozens of datasets. Thus far, most of the research has considered only the one-dimensional case, with practitioners generalizing to the multi-dimensional case in one of two ways, dependent or independent warping. In general, it appears the community believes either that the two ways are equivalent, or that the choice is irrelevant. In this work, we show that this is not the case. The two most commonly used multi-dimensional DTW methods can produce different classifications, and neither one dominates over the other. This seems to suggest that one should learn the best method for a particular application. However, we will show that this is not necessary; a simple, principled rule can be used on a case-by-case basis to predict which of the two methods we should trust at the time of classification. Our method allows us to ensure that classification results are at least as accurate as the better of the two rival methods, and, in many cases, our method is significantly more accurate. We demonstrate our ideas with the most extensive set of multi-dimensional time series classification experiments ever attempted.

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.

Sound is one of the most common communication medias used by humans. Every human has different sound characteristics. To recognize the compatibility of a sound, a special algorithm is needed, which is Dynamic Time Warping (DTW). DTW is a method to measure the similarity of a pattern with different time zones. The smaller the distance produced, the more similar between the two sound patterns. Both sound patterns are similar, thus the two voices are said to be the same. The initial data on the speech recognition process is transformed into frequency waves. Pronounce volume, pronunciation time, and noise from the sound around the recording takes place affecting the distance generated. The smaller the effect, the smaller the distance that will be generated.

Video sequences exhibit significant nuisance variations (undesired effects) of speed of actions, temporal locations, and subjects’ poses, leading to temporal-viewpoint misalignment when comparing two sets of frames or evaluating the similarity of two sequences. Thus, we propose Joint tEmporal and cAmera viewpoiNt alIgnmEnt (JEANIE) for sequence pairs. In particular, we focus on 3D skeleton sequences whose camera and subjects’ poses can be easily manipulated in 3D. We evaluate JEANIE on skeletal Few-shot Action Recognition (FSAR), where matching well temporal blocks (temporal chunks that make up a sequence) of support-query sequence pairs (by factoring out nuisance variations) is essential due to limited samples of novel classes. Given a query sequence, we create its several views by simulating several camera locations. For a support sequence, we match it with view-simulated query sequences, as in the popular Dynamic Time Warping (DTW). Specifically, each support temporal block can be matched to the query temporal block with the same or adjacent (next) temporal index, and adjacent camera views to achieve joint local temporal-viewpoint warping. JEANIE selects the smallest distance among matching paths with different temporal-viewpoint warping patterns, an advantage over DTW which only performs temporal alignment. We also propose an unsupervised FSAR akin to clustering of sequences with JEANIE as a distance measure. JEANIE achieves state-of-the-art results on NTU-60, NTU-120, Kinetics-skeleton and UWA3D Multiview Activity II on supervised and unsupervised FSAR, and their meta-learning inspired fusion.

Dynamic time warping (DTW) is one of the most important similarity measurement methods for time series analysis. In view of the high complexity and pathological alignment of DTW, a lot of variants of DTW have been proposed. However, the existing methods calculate the similarity between the original time series through dynamic programming directly, and ignore the characteristic that different components in the time series often have different degrees of importance. This paper proposes a time series similarity measurement method based on series decomposition and fast DTW, which combines time series decomposition method and DTW method. Series decomposition is an important means of time series analysis which can decompose time series into trend, seasonality, and remainder components. In this paper, after using the Seasonal-Trend decomposition using Loess (STL) method to decompose the time series, the similarity between the trend components and the similarity between seasonal components are respectively measured. The impact of the more important component is amplified, and then the comprehensive similarity measurement result will be obtained. Experimental results on 20 UCR time series datasets show that, compared with the existing fast DTW and constrained DTW and their variants, the method proposed in this paper achieves a higher classification accuracy. Simultaneously, combining the advantage of low complexity of fast DTW, the computational complexity of proposed method is still first-order linearly related to the length of time series.

Similarity search is the core procedure for several time series mining tasks. While different distance measures can be used for this purpose, there is clear evidence that the Dynamic Time Warping (DTW) is the most suitable distance function for a wide range of application domains. Despite its quadratic complexity, research efforts have proposed a significant number of pruning methods to speed up the similarity search under DTW. However, the search may still take a considerable amount of time depending on the parameters of the search, such as the length of the query and the warping window width. The main reason is that the current techniques for speeding up the similarity search focus on avoiding the costly distance calculation between as many pairs of time series as possible. Nevertheless, the few pairs of subsequences that were not discarded by the pruning techniques can represent a significant part of the entire search time. In this work, we adapt a recently proposed algorithm to improve the internal efficiency of the DTW calculation. Our method can speed up the UCR suite, considered the current fastest tool for similarity search under DTW. More important, the longer the time needed for the search, the higher the speedup ratio achieved by our method. We demonstrate that our method performs similarly to UCR suite for small queries and narrow warping constraints. However, it performs up to five times faster for long queries and large warping windows.