text: Similarity-based approaches represent a promising direction for time series analysis. However, many such methods rely on parameter tuning, and some have shortcomings if the time series are multivariate (MTS), due to dependencies between attributes, or the time series contain missing data. In this paper, we address these challenges within the powerful context of kernel methods by proposing the robust (TCK). The approach taken leverages the missing data handling properties of Gaussian mixture models (GMM) augmented with informative prior distributions. An ensemble learning approach is exploited to ensure robustness to parameters by combining the clustering results of many GMM to form the final kernel. We evaluate the TCK on synthetic and real data and compare to other state-of-the-art techniques. The experimental results demonstrate that the TCK is robust to parameter choices, provides competitive results for MTS without missing data and outstanding results for missing data.
  @cite_30: 
  @cite_18: 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 that we propose allow us 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 proof of the positive definiteness as a direct consequence of the Haussler’s convolution theorem. Furthermore, the positive definiteness is maintained when a symmetric corridor is used to reduce the search space, and thus the algorithmic complexity, which is quadratic in the worst case. The classification experiment we conducted on three classical time-warp distances (two of which are metrics), using support vector machine classifier, leads to the conclusion that when the pairwise distance matrix obtained from the training data is far from definiteness, the positive definite recursive elastic kernels outperform in general the distance substituting kernels for several classical elastic distances we have tested.
  @cite_91: 
  @cite_7: 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.
  @cite_78: During recent years much effort has been spent in incorporating problem specific a-priori knowledge into kernel methods for machine learning. A common example is a-priori knowledge given by a distance measure between objects. A simple but effective approach for kernel construction consists of substituting the Euclidean distance in ordinary kernel functions by the problem specific distance measure. We formalize this distance substitution procedure and investigate theoretical and empirical effects. In particular we state criteria for definiteness of the resulting kernels. We demonstrate the wide applicability by solving several classification tasks with SVMs. Regularization of the kernel matrices can additionally increase the recognition accuracy.
  @cite_92: 
  @cite_55: Many emerging applications formulate non-metric proximity matrices (non-positive semidefinite), and hence cannot fit into the framework of kernel machines. A popular approach to this problem is to transform the spectrum of the similarity matrix so as to generate a positive semidefinite kernel matrix. In this paper, we explore four representative transformation methods: denoise, flip, diffusion, and shift. Theoretically, we discuss a generalization problem where the test data are not available during transformation, and thus propose an efficient algorithm to address the problem of updating the cross-similarity matrix between test and training data. Extensive experiments have been conducted to evaluate the performance of these methods on several real-world (dis)similarity matrices with semantic meanings.
  @cite_96: Time series is an important class of temporal data objects and it can be easily obtained from scientific and financial applications. A time series is a collection of observations made chronologically. The nature of time series data includes: large in data size, high dimensionality and necessary to update continuously. Moreover time series data, which is characterized by its numerical and continuous nature, is always considered as a whole instead of individual numerical field. The increasing use of time series data has initiated a great deal of research and development attempts in the field of data mining. The abundant research on time series data mining in the last decade could hamper the entry of interested researchers, due to its complexity. In this paper, a comprehensive revision on the existing time series data mining research is given. They are generally categorized into representation and indexing, similarity measure, segmentation, visualization and mining. Moreover state-of-the-art research issues are also highlighted. The primary objective of this paper is to serve as a glossary for interested researchers to have an overall picture on the current time series data mining development and identify their potential research direction to further investigation.
  @cite_2: 
  @cite_46: In the research field of time series analysis, dynamic time warping distance (DTW) is a prevalent similarity measure with high precision. However, the computational complexity of DTW is high, which makes it difficult to be applied to the high dimensional time series. An effective solution is to compute DTW on the piecewise representation (PR-DTW), which employs the features of the subsequences of time series for similarity measure. However, the features that most existing piecewise representations focus on are too simple, which capture only one aspect of the fluctuation information of time series, and thus influence the precision of PR-DTW. In order to solve this problem, we propose a novel piecewise representation model, named piecewise statistic approximation (PSA), for supporting the PR-DTW measure. Rather than focusing on a single type of features, PSA extracts multiple statistical features to capture the synthetic fluctuation information for similarity measure. Besides, by taking the weighted Euclidean distance for the subsequence matching in the subroutine, PSA based DTW (PSADTW) can discriminate the expressivities of the multiple features. Comprehensive experiments over 45 real-world datasets empirically demonstrate that, PSA is well suited to support both precise and efficient PR-DTW measure.
  @cite_25: 
  @cite_82: 
summary: DTW @cite_91 is one of the most commonly used similarity measures for UTS and has become the state-of-the-art in many practical applications @cite_46 @cite_96 @cite_18 . Several formulations have been proposed to extend DTW to the multidimensional setting @cite_7 @cite_2 . Since DTW does not satisfy the triangle inequality, it is not negative definite and, therefore, one cannot obtain a psd kernel by applying an exponential function to it @cite_92 . Such an indefinite kernel may lead to a non-convex optimization problem (e.g., in an SVM), which hinders the applicability of the model @cite_78 . Several approaches have been proposed to limit this drawback at the cost of more complex and costly computations. In @cite_55 @cite_30 ad hoc spectral transformations were employed to obtain a psd matrix. @cite_25 designed a DTW-based kernel using global alignments (GAK). Marteau and Gibet proposed an approach that combines DTW and edit distances with a recursive regularizing term @cite_82 .