José A. Vilar

Ángel López Oriona, Christian Weiss, José Antonio Vilar

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 the 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 even though the series come from a wide variety of models. Since the dynamic 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.

Ángel López Oriona, José Antonio Vilar Fernández

The 21st century has witnessed a growing interest in the analysis of time series data. Whereas most of the literature on the topic deals with real-valued time series, ordinal time series have typically received much less attention. However, the development of specific analytical tools for the latter objects has substantially increased in recent years. The R package otsfeatures attempts to provide a set of simple functions for analyzing ordinal time series. In particular, several commands allowing the extraction of well-known statistical features and the execution of inferential tasks are available for the user. The output of several functions can be employed to perform traditional machine learning tasks including clustering, classification or outlier detection. otsfeatures also incorporates two datasets of financial time series which were used in the literature for clustering purposes, as well as three interesting synthetic databases. The main properties of the package are described and its use is illustrated through several examples. Researchers from a broad variety of disciplines could benefit from the powerful tools provided by otsfeatures.

Ángel López Oriona, Pablo Montero Manso, José Antonio Vilar Fernández

In this paper, a novel method to perform model-based clustering of time series is proposed. The procedure relies on two iterative steps: (i) K global forecasting models are fitted via pooling by considering the series pertaining to each cluster and (ii) each series is assigned to the group associated with the model producing the best forecasts according to a particular criterion. Unlike most techniques proposed in the literature, the method considers the predictive accuracy as the main element for constructing the clustering partition, which contains groups jointly minimizing the overall forecasting error. Thus, the approach leads to a new clustering paradigm where the quality of the clustering solution is measured in terms of its predictive capability. In addition, the procedure gives rise to an effective mechanism for selecting the number of clusters in a time series database and can be used in combination with any class of regression model. An extensive simulation study shows that our method outperforms several alternative techniques concerning both clustering effectiveness and predictive accuracy. The approach is also applied to perform clustering in several datasets used as standard benchmarks in the time series literature, obtaining great results.

Ángel López Oriona, José Antonio Vilar Fernández

Time series data are ubiquitous nowadays. Whereas most of the literature on the topic deals with real-valued time series, categorical time series have received much less attention. However, the development of data mining techniques for this kind of data has substantially increased in recent years. The R package ctsfeatures offers users a set of useful tools for analyzing categorical time series. In particular, several functions allowing the extraction of well-known statistical features and the construction of illustrative graphs describing underlying temporal patterns are provided in the package. The output of some functions can be employed to perform traditional machine learning tasks including clustering, classification and outlier detection. The package also includes two datasets of biological sequences introduced in the literature for clustering purposes, as well as three interesting synthetic databases. In this work, 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 valuable tools provided by ctsfeatures.

Ángel López-Oriona, Pierpaolo D'Urso, José Antonio Vilar et al.

Three robust methods for clustering multivariate time series from the point of view of generating processes are proposed. The procedures are robust versions of a fuzzy C-means model based on: (i) estimates of the quantile cross-spectral density and (ii) the classical principal component analysis. Robustness to the presence of outliers is achieved by using the so-called metric, noise and trimmed approaches. The metric approach incorporates in the objective function a distance measure 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 eliminates the most atypical series in the dataset. All the proposed techniques inherit the nice properties of the quantile cross-spectral density, as being able to uncover general types of dependence. Results from a broad simulation study including multivariate linear, nonlinear and GARCH processes indicate that the algorithms are substantially effective in coping with the presence of outlying series (i.e., series exhibiting a dependence structure different from that of the majority), clearly poutperforming alternative procedures. The usefulness of the suggested methods is highlighted by means of two specific applications regarding financial and environmental series.

Ángel López-Oriona, José A. Vilar, Pierpaolo-D'Urso

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.