As an ordered extension of evidence theory, Random permutation set (RPS) theory has received increasing attention due to its advantage in dealing with order-structured uncertain information. However, a significant research gap remains in the current literature concerning the construction of RPS. Building on the interpretation of RPS as a layer-2 belief structure, this paper proposes a data-driven hierarchical method for generating RPS, called DDHRPS. Specifically, DDHRPS first generates BPA from statistical features of data on the layer-1 belief structure, and then refines them with propensity information derived from distance analysis between samples to single classes, ultimately forming RPS on the layer-2 belief structure. Moreover, a DDHRPS-based classification algorithm (DDHRPSCA) is presented. Experimental comparisons involving two kinds of classifiers, namely, two uncertainty-based classifiers and seven machine learning classifiers validate the effectiveness and superiority of DDHRPSCA in handling uncertain information in classification tasks.

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.”

Long-term forecasting of multivariate time series has been an important research issue in the field of data mining and knowledge discovery. Fuzzy information granularity is used as an effective tool to handle long-term forecasting of time series. On account of its good interpretability and effect, it has received the attention of more and more scholars. However, although the method has been universally used in univariate time series, its application for multivariate time series has received little attention. In order to utilize the advantages of fuzzy information granularity and fill its gap in solving multivariate time-series forecasting problems. In view of this purpose, we design a long-term multivariate time-series forecasting modeling framework in light of Gaussian fuzzy information granules. The model includes a granulation method of under multivariate time series, as well as a neural network model that combines the backpropagation neural network, long short-term memory neural network, and transformer for long-term prediction, in which there is a fuzzy information granule segmentation method with polynomials as the core line and a new representation method for fuzzy information granules. We carry out experimental evaluations using eight publicly available time-series data, and the results show that our model is able to perform long-term forecasting of multivariate time series with a high satisfactory accuracy.

In many cases, data are not expressed as individual values on a timeline, but are a collection of values obtained at certain moments in time - they are time series. In these cases, traditional clustering models for one-time data are unable to properly account for the time-variability of the data. In this paper, by considering the partitioning around medoids approach in a fuzzy framework, we propose fuzzy clustering models for multivariate time series. In order to neutralize the negative effects of outlier time series in the clustering process, we proposed robust fuzzy c-medoids clustering models for time series based on the combination of Huber's M-estimators and Yager's OWA operators. The proposed models are able to smooth the influence of anomalous time series by means of the so-called typicality parameter, capable to tune the influence of the outliers. The performance of the proposed models has been shown by means of a simulation and real-data sets study: (i) two-dimensional dataset of time series, (ii) the average daily time series of temperatures, and (iii) the pregnancy dataset of time series. The comparison made with the robust clustering models known from the literature indicates the competitiveness of the introduced model to others.

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.

International audience

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.

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.

In this work, a new approach to cluster large sets of time series is presented. The proposed methodology takes into account the dependency among the time series to obtain a fuzzy partition of the set of observations. A two-step procedure to accomplish this is presented. First, the cophenetic distances, based on a time series linear cross-dependency measure, are obtained. Second, these distances are used as an input of a non-Euclidean fuzzy relational clustering algorithm. As a result, we obtain a robust fuzzy procedure capable of detecting groups of time series with different types of cross-dependency. We illustrate the usefulness of the stated methodology through some Monte Carlo experiments and a real data example. Our results show that the methodology proposed in this work substantially improves the hard partitioning clustering alternative.

Clustering of space-time series should consider: 1) the spatial nature of the objects to be clustered; 2) the characteristics of the feature space, namely the space of multivariate time trajectories; 3) the uncertainty associated to the assignment of a spatial unit to a given cluster on the basis of the above complex features. The last aspect is dealt with by using the Fuzzy C-Means objective function, based on an appropriate measure of dissimilarity between time trajectories. In order to take into account the spatial nature of the statistical units, a spatial penalization term is added to the above function, depending on a suitable spatial proximity/contiguity matrix. A tuning coefficient takes care of the balance between, on one side, discriminating according to the pattern of the time trajectories and, on the other side, ensuring an approximate spatial homogeneity of the clusters. A technique for determining an optimal value of this coefficient is proposed, based on an appropriate spatial autocorrelation measure. Finally, an application is discussed.

In many knowledge discovery and data mining tasks, fuzzy clustering is one of the most common tools for data partitioning. In this paper dynamic fuzzy clustering models for classifying a set of multivariate time trajectories (time series, sequences) are developed. In particular, by adopting an exploratory approach, based on a geometric-algebraic formulation of the data time array, different kinds of dynamic fuzzy clustering models, based on cross sectional and longitudinal aspects, are suggested. Furthermore, a modified version of the previous clustering models, that can be seen as a generalization of these models, is proposed. By utilizing these models we can obtain beneficial effects in the clustering process when anomalous trajectories (trajectories with anomalous positions and slopes) are present in the dataset; in fact the models are suitable for detecting structures of time trajectories with anomalous patterns that are not uniformly distributed over the structure's domains and are characterized by strange slopes. In these models, the disruptive effect of the anomalous trajectories is neutralized and smoothed and the information on the influence of individual time trajectories on the detected groups is given. Furthermore, some remarks on dynamic three-way extensions of a few robust fuzzy clustering models for two-way data are suggested. Demonstrative examples are shown and a comparison assessment based on artificial multivariate time-varying data is carried out

The classification of multivariate time-varying data finds application in several fields, such as economics, finance, marketing research, psychometrics, bioinformatics, medicine, signal processing, pattern recognition, etc. In this paper, by considering an exploratory formalization, we propose different unsupervised clustering models for multivariate data time arrays (objects×quantitative variables×times). These models can be classified in two different approaches: the cross sectional and the longitudinal approach. In the first case, after the objects, observed at each time, have been classified, comparison among the classifications made in different time instants will be done. In the second approach, we cluster the time trajectories of the objects; then, we obtain only one classification by comparing the instantaneous and evolutive features of the trajectories of the objects. In particular, in this work, the second approach is analyzed in detail, with reference to the so-called single and double step procedures. Geometric, correlative, instantaneous, evolutive and trend characteristics of the multivariate time arrays are taken into account in the different proposed clustering models. Furthermore, the fuzzy approach, that is particularly suitable in the dynamic classification problem, has been considered. Extensions of a cluster-validity criterion for the proposed fuzzy dynamic clustering models are also suggested. A socio-economic example concludes the paper.