This addresses one small error in [1]. Specifically, in the sentence just before (13), the word “ms” is incorrect. The correct words are “matrix of fuzzy coefficients.”

Time series clustering is a central machine learning task with applications in many fields. While the majority of the methods focus on real-valued time series, very few works consider series with discrete response. In this paper, the problem of clustering ordinal time series is addressed. To this aim, two novel distances between ordinal time series are introduced and used to construct fuzzy clustering procedures. Both metrics are functions of estimated cumulative probabilities, thus automatically taking advantage of the ordering inherent to the series' range. The resulting clustering algorithms are computationally efficient and able to group series generated from similar stochastic processes, reaching accurate results with series coming from a wide variety of models. Since the dynamics of the series may vary over the time, we adopt a fuzzy approach, thus enabling the procedures to locate each series into several clusters with different membership degrees. An extensive simulation study shows that the proposed methods outperform several alternative procedures. Weighted versions of the clustering algorithms are also presented and their advantages with respect to the original methods are discussed. Two specific applications involving economic time series illustrate the usefulness of the proposed approaches.

Time series data are ubiquitous nowadays. Whereas most of the literature on the topic deals with univariate time series, multivariate time series have typically received much less attention. However, the development of machine learning algorithms for the latter objects has substantially increased in recent years. The R package mlmts attempts to provide a set of widespread data mining techniques for multivariate series. Several functions allowing the execution of clustering, classification, outlier detection and forecasting methods, among others, are included in the package. mlmts also incorporates a collection of multivariate time series datasets often used to test the performance of new classification algorithms. The main characteristics of the package are described and its use is illustrated through various examples. Practitioners from a wide variety of fields could benefit from the general framework provided by mlmts.

Two novel distances between categorical time series are introduced. Both of them measure discrepancy between extracted features describing the underlying serial dependence patterns. One distance is based on well-known association measures, namely Cramer’s v and Cohen’s κ. The other one relies on the so-called binarization of a categorical process, which indicates the presence of each category by means of a canonical vector. Binarization is used to construct a set of innovative association measures, which allow to identify different types of serial dependence. The metrics are used to perform crisp and fuzzy clustering of nominal series. The proposed approaches are able to group together series generated from similar stochastic processes, achieve accurate results with series coming from a broad range of models, and are computationally efficient. Extensive simulation studies show that both hard and soft clustering algorithms outperform several alternative procedures presented in the literature. Two applications involving biological sequences from different species highlight the usefulness of the introduced techniques.

A novel procedure to perform fuzzy clustering of multivariate time series generated from different dependence models is proposed. Different amounts of dissimilarity between the generating models or changes on the dynamic behaviours over time are some arguments justifying a fuzzy approach, where each series is associated to all the clusters with specific membership levels. Our procedure considers quantile-based cross-spectral features and consists of three stages: (i) each element is characterized by a vector of proper estimates of the quantile cross-spectral densities, (ii) principal component analysis is carried out to capture the main differences reducing the effects of the noise, and (iii) the squared Euclidean distance between the first retained principal components is used to perform clustering through the standard fuzzy C-means and fuzzy C-medoids algorithms. The performance of the proposed approach is evaluated in a broad simulation study where several types of generating processes are considered, including linear, nonlinear and dynamic conditional correlation models. Assessment is done in two different ways: by directly measuring the quality of the resulting fuzzy partition and by taking into account the ability of the technique to determine the overlapping nature of series located equidistant from well-defined clusters. The procedure is compared with the few alternatives suggested in the literature, substantially outperforming all of them whatever the underlying process and the evaluation scheme. Two specific applications involving air quality and financial databases illustrate the usefulness of our approach.

Robust fuzzy clustering of multivariate time series is addressed when the clustering purpose is grouping together series generated from similar stochastic processes. Robustness to the presence of anomalous series is attained by considering three well-known robust versions of a fuzzy C-means model based on a spectral dissimilarity measure with high discriminatory power. The dissimilarity measure compares principal component scores obtained from estimates of quantile cross-spectral densities, and the robust techniques follow the so-called metric, noise and trimmed approaches. The metric approach incorporates in the objective function a distance aimed at neutralizing the effect of the outliers, the noise approach builds an artificial cluster expected to contain the outlying series, and the trimmed approach removes the most atypical series in the dataset. As result, the proposed clustering methods take advantage of both the robust nature of these techniques and the capability of the quantile cross-spectral density to identify complex dependence structures. An extensive simulation study including multivariate linear, nonlinear and GARCH processes shows that the algorithms are substantially effective in coping with the presence of outlying series, clearly outperforming other alternative procedures. Two specific applications regarding financial and environmental series illustrate the usefulness of the presented methods.

The problem of testing the equality of the generating processes of two multivariate time series is addressed in this work. To this aim, we construct four tests based on a distance measure between stochastic processes. The metric is defined in terms of the quantile cross-spectral densities of both processes. A proper estimate of this dissimilarity is the cornerstone of the proposed tests. The first test employs the asymptotic distribution of the estimate, which we derive from some standard results on complex random variables and which is useful in its own right. The bad behaviour of this test when compared with alternative ones is shown. The three remaining techniques are based on the bootstrap. Specifically, a particular bootstrap method for spectral densities and extensions of the moving blocks bootstrap and the stationary bootstrap are used for their construction. The approaches are assessed in a broad range of scenarios under the null and the alternative hypothesis. The results from the analyses show that the procedure based on the stationary bootstrap exhibits the best overall performance in terms of both size and power. The proposed techniques are used to answer the question about whether or not the dotcom bubble crash of 2000s permanently impacted the global market behavior.

In this article, a fuzzy clustering model for multivariate time series based on the quantile cross-spectral density and principal component analysis is extended by including: 1) a weighting system which assigns a weight to each principal component in accordance with its importance concerning the underlying clustering structure and 2) a penalization term allowing to take into account the spatial information. The iterative solutions of the new model, which employs the exponential distance in order to gain robustness against outlying series, are derived. A simulation study shows that the weighting system substantially enhances the effectiveness of the former approach. The behavior of the extended model in terms of the spatial penalization term is also analyzed. An application involving multivariate time series of mobility indicators concerning COVID-19 pandemic highlights the usefulness of the proposed technique.

Clustering of multivariate time series is a central problem in data mining with applications in many fields. Frequently, the clustering target is to identify groups of series generated by the same multivariate stochastic process. Most of the approaches to address this problem include a prior step of dimensionality reduction which may result in a loss of information or consider dissimilarity measures based on correlations and cross-correlations but ignoring the serial dependence structure. We propose a novel approach to measure dissimilarity between multivariate time series aimed at jointly capturing both cross dependence and serial dependence. Specifically, each series is characterized by a set of matrices of estimated quantile cross-spectral densities, where each matrix corresponds to a pair of quantile levels. Then the dissimilarity between every couple of series is evaluated by comparing their estimated quantile cross-spectral densities, and the pairwise dissimilarity matrix is taken as starting point to develop a partitioning around medoids algorithm. Since the quantile-based cross-spectra capture dependence in quantiles of the joint distribution, the proposed metric has a high capability to discriminate between high-level dependence structures. An extensive simulation study shows that our clustering procedure outperforms a wide range of alternative methods and exhibits robustness to noise distribution besides being computationally efficient. A real data application involving bivariate financial time series illustrates the usefulness of the proposed approach. The procedure is also applied to cluster nonstationary series from the UEA multivariate time series classification archive.

Financiado para publicación en acceso aberto: Universidade da Coruña/CISUG