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We investigate the fuzzy clustering of interval time series using wavelet variances and covariances; in particular, we use a fuzzy c-medoids clustering algorithm. Traditional hierarchical and non-hierarchical clustering methods lead to the identification of mutually exclusive clusters whereas fuzzy clustering methods enable the identification of overlapping clusters, implying that one or more series could belong to more than one cluster simultaneously. An interval time series (ITS) which arises when interval-valued observations are recorded over time is able to capture the variability of values within each interval at each time point. This is in contrast to single-point information available in a classical time series. Our main contribution is that by combining wavelet analysis, interval data analysis and fuzzy clustering, we are able to capture information which would otherwise have not been contemplated by the use of traditional crisp clustering methods on classical time series for which just a single value is recorded at each time point. Through simulation studies, we show that under some circumstances fuzzy c-medoids clustering performs better when applied to ITS than when it is applied to the corresponding traditional time series. Applications to exchange rates ITS and sea-level ITS show that the fuzzy clustering method reveals different and more meaningful results than when applied to associated single-point time series. [ABSTRACT FROM AUTHOR]

In this paper a Self-Organizing Map (SOM) robust to the presence of outliers, the Smoothed SOM (S-SOM), is proposed. S-SOM improves the properties of input density mapping, vector quantization, and clustering of the standard SOM in the presence of outliers by upgrading the learning rule in order to smooth the representation of outlying input vectors onto the map. The upgrade of the learning rule is based on the complementary exponential distance between the input vector and its closest codebook. The convergence of the S-SOM to a stable state is proved. Three comparative simulation studies and a suggestive application to digital innovation data show the robustness and effectiveness of the proposed S-SOM. Supplementary materials for this article are available. [ABSTRACT FROM AUTHOR]

The detection of patterns in multivariate time series is a relevant task, especially for large datasets. In this paper, four clustering models for multivariate time series are proposed, with the following characteristics. First, the Partitioning Around Medoids (PAM) framework is considered. Among the different approaches to the clustering of multivariate time series, the observation-based is adopted. To cope with the complexity of the features of each multivariate time series and the associated assignment uncertainty a fuzzy clustering approach is adopted. Finally, to neutralize the effect of possible outliers, a robust metric approach is used, i.e., the exponential transformation of dissimilarity measures. The proposed models are robust extensions of the Fuzzy C-Medoids clustering algorithm for multivariate time series. With respect to the management of the time behaviour, four variants are proposed: the Cross-Sectional Fuzzy C-Medoids clustering model with exponential transformation (CS-Exp-FCMd) classifies the multivariate time series taking into account their respective instantaneous features; the Longitudinal Fuzzy C-Medoids clustering model with exponential transformation (L-Exp-FCMd) takes into account the evolutive (longitudinal) features; the Mixed Fuzzy C-Medoids clustering model with exponential transformation (M-Exp-FCMd) which consider simultaneously both the instantaneous and the longitudinal features in the clustering process; the Dynamic Time Warping-based Fuzzy C-Medoids model with exponential transformation (DTW-Exp-FCMd) uses the Dynamic Time Warping (DTW) distance. Three simulation studies show the clustering performance of the proposed models in presence of outliers, compared to their non-robust counterparts, and to other models proposed in the literature. An application on real-world data on the concentration of three pollutants in nineteen stations in the Metropolitan City of Rome shows the relevance of the robustness to outliers in the identification of the clusters. [ABSTRACT FROM AUTHOR]

Although there are many contributions in the time series clustering literature, few studies still deal with count time series data. This paper aims to develop a fuzzy clustering procedure for count time series data. We propose an Integer GARCH-based Fuzzy C-medoids (INGARCH-FCMd) method for clustering count time series based on a Mahalanobis distance between the parameters estimated by an INGARCH model. We show how the proposed clustering method works by clustering football teams according to the number of scored goals.

Summary: In this note we deal with some admissibility conditions proved by G. B. Tranquilli to be sufficient in the class of unbiased estimators of finite population parameters and with respect to (w.r.t.) a quadratic loss function. We show that the same conditions:i) are sufficient for the admissibility of an unbiased estimator with any loss function;ii) imply hyperadmissibility with reference to a particular (critical) population of the. From this fact we deduce that, for a fixed critical population, there is at most one estimator, in the class of all unbiased estimator of a finite population parameter, which satisfies Tranquilli condition.

Four different approaches to robust fuzzy clustering of time series are presented and compared with respect to other existent approaches. These approaches are useful to cluster time series when outlying values are found in these time series, which is often the rule in most real data applications. A representation of the time series by using B-splines is considered and, later, robust fuzzy clustering methods are applied on the B-splines fitted coefficients. Feasible algorithms for implementing these methodologies are presented. A simulation study shows how these methods are useful to deal with contaminating time series and also switching time series due to fuzziness. A real data analysis example on financial data is also presented. [ABSTRACT FROM AUTHOR]