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In recent years, the research of statistical methods to analyze complex structures of data has increased. In particular, a lot of attention has been focused on the interval-valued data. In a classical cluster analysis framework, an interesting line of research has focused on the clustering of interval-valued data based on fuzzy approaches. Following the partitioning around medoids fuzzy approach research line, a new fuzzy clustering model for interval-valued data is suggested. In particular, we propose a new model based on the use of the entropy as a regularization function in the fuzzy clustering criterion. The model uses a robust weighted dissimilarity measure to smooth noisy data and weigh the center and radius components of the interval-valued data, respectively. To show the good performances of the proposed clustering model, we provide a simulation study and an application to the clustering of scientific journals in research evaluation.

In this paper a weighted complex network is used to detect communities of basketball players on the basis of their performances. A sparsification procedure to remove weak edges is also applied. In our proposal, at each removal of an edge the best community structure of the “giant component” is calculated, maximizing the modularity as a measure of compactness within communities and separation among communities. The “sparsification transition” is confirmed by the normalized mutual information. In this way, not only the best distribution of nodes into communities is found, but also the ideal number of communities as well. An application to community detection of basketball players for the NBA regular season 2020–2021 is presented. The proposed methodology allows a data driven decision making process in basketball.

The evaluation of the possible effects of a treatment on an outcome plays a central role in both theoretical and applied statistical and econometrical literature. This paper focuses on nonparametric tests for possible difference in the distribution of potential outcomes due to receiving or not receiving a treatment. The approach is based on weighting observed data on the basis on the estimated propensity score. Kolmogorov–Smirnov type and Wilcoxon–Mann–Whitney type tests are constructed, and their limiting distributions are studied. Rejection regions are obtained by inverting confidence intervals. This involves the study of appropriate estimators of the limiting variance of test statistics. Approximations of quantiles via subsampling are also considered. The merits of the different tests are studied by Monte Carlo simulation. An application to the construction of tests for stochastic dominance is provided.

The popularity of the cluster analysis in the tourism field has massively grown in the last decades. However, accordingly to our review, researchers are often not aware of the characteristics and limitations of the clustering algorithms adopted. An important gap in the literature emerged from our review regards the adoption of an adequate clustering algorithm for mixed data. The main purpose of this article is to overcome this gap describing, both theoretically and empirically, a suitable clustering algorithm for mixed data. Furthermore, this article contributes to the literature presenting a method to include the “Don’t know” answers in the cluster analysis. Concluding, the main issues related to cluster analysis are highlighted offering some suggestions and recommendations for future analysis.

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.

A robust fuzzy clustering model for mixed data is proposed. For each variable, or attribute, a proper dissimilarity measure is computed and the clustering procedure combines the dissimilarity matrices with weights objectively computed during the optimization process. The weights reflect the relevance of each attribute type in the clustering results. A simulation study and an empirical application to football players data are presented that show the effectiveness of the proposed clustering algorithm in finding clusters that would be hidden unless a multi-attributes approach were used.

The aim of the work is to identify a clustering structure for the 20 Italian regions according to the main variables related to COVID-19 pandemic. Data are observed over time, spanning from the last week of February 2020 to the first week of February 2021. Dealing with geographical units observed at several time occasions, the proposed fuzzy clustering model embedded both space and time information. Properly, an Exponential distance-based Fuzzy Partitioning Around Medoids algorithm with spatial penalty term has been proposed to classify the spline representation of the time trajectories. The results show that the heterogeneity among regions along with the spatial contiguity is essential to understand the spread of the pandemic and to design effective policies to mitigate the effects.

In this paper, we define clusters of homogeneous Italian regions with respect to the evolution over time of well-being. In particular we identify three partitions of the Italian regions based on the yearly time series of the economic, social and environmental dimensions of the BES (equitable and sustainable well-being), in the period 2010–2016. The partitions are obtained using a Dynamic Time Warping-based Fuzzy C-Medoids clustering model for multivariate time series. The results show a territorial gap between northern and southern regions with respect to the economic, social and environmental dimensions. The central regions have a diversified behavior with respect to the different dimensions.

Statistics in sports plays a key role in predicting winning strategies and providing objective performance indicators. Despite the growing interest in recent years in using statistical methodologies in this field, less emphasis has been given to the multivariate approach. This work aims at using the Bayesian networks to model the joint distribution of a set of indicators of players’ performances in basketball in order to discover the set of their probabilistic relationships as well as the main determinants affecting the player’s winning percentage. From a methodological point of view, the interest is to define a suitable model for non-Gaussian data, relaxing the strong assumption on normal distribution in favour of Gaussian copula. Through the estimated Bayesian network, we discovered many interesting dependence relationships, providing a scientific validation of some known results mainly based on experience. At last, some scenarios of interest have been simulated to understand the main determinants that contribute to rising in the number of won games by a player.

In this paper we propose a dashboard of indicators of territorial attractiveness at NUTS3 level in the framework of the EU Regional Competitiveness Index (RCI). Then, the Fuzzy C-Medoids Clustering model with multivariate data and contiguity constraints is applied for partitioning the Italian provinces (NUTS3). The novelty is the territorial level analized, and the identification of the elementary indicators at the basis of the construction of the eleven composite competitiveness pillars. The positioning of the Italian provinces is deeply analyzed. The clusters obtained with and without contraints are compared. The obtained partition may play an important role in the design of policies at the NUTS3 level, a route already considered by the Italian government. The analysis developed and the related set of indicators at NUTS3 level constitute an information base that could be effectively used for the implementation of the National Recovery and Resilience Plan (NRRP).

In finance, cluster analysis is a tool particularly useful for classifying stock market multivariate time series data related to daily returns, volatility daily stocks returns, commodity prices, volume trading, index, enhanced index tracking portfolio, and so on. In the literature, following different methodological approaches, several clustering methods have been proposed for clustering multivariate time series. In this paper by adopting a fuzzy approach and using the Partitioning Around Medoids strategy, we suggest to cluster multivariate financial time series by considering the dynamic time warping distance. In particular, we proposed a robust clustering method capable to neutralize the negative effects of possible outliers in the clustering process. The clustering method achieves its robustness by adopting a suitable trimming procedure to identify multivariate financial time series more distant from the bulk of data. The proposed clustering method is applied to the stocks composing the FTSE MIB index to identify common time patterns and possible outliers.

Clustering geographical units based on a set of quantitative features observed at several time occasions requires to deal with the complexity of both space and time information. In particular, one should consider (1) the spatial nature of the units to be clustered, (2) the characteristics of the space of multivariate time trajectories, and (3) the uncertainty related to the assignment of a geographical unit to a given cluster on the basis of the above complex features. This paper discusses a novel spatially constrained multivariate time series clustering for units characterized by different levels of spatial proximity. In particular, the Fuzzy Partitioning Around Medoids algorithm with Dynamic Time Warping dissimilarity measure and spatial penalization terms is applied to classify multivariate Spatial–Temporal series. The clustering method has been theoretically presented and discussed using both simulated and real data, highlighting its main features. In particular, the capability of embedding different levels of proximity among units, and the ability of considering time series with different length.

The spatial epidemic dynamics of COVID-19 outbreak in Italy were modelled by means of an Object-Oriented Bayesian Network in order to explore the dependence relationships, in a static and a dynamic way, among the weekly incidence rate, the intensive care units occupancy rate and that of deaths. Following an autoregressive approach, both spatial and time components have been embedded in the model by means of spatial and time lagged variables. The model could be a valid instrument to support or validate policy makers' decisions strategies.

The main determinants of COVID-19 spread in Italy are investigated, in this work, by means of a D-vine copula based quantile regression. The outcome is the COVID-19 cumulative infection rate registered on October 30th 2020, with reference to the 107 Italian provinces, and it is regressed on some covariates of interest accounting for medical, environmental and demographic factors. To deal with the issue of spatial autocorrelation, the D-vine copula based quantile regression also embeds a spatial autoregressive component that controls for the extent of spatial dependence. The use of vine copula enhances model flexibility accounting for non-linear relationships and tail dependencies. Moreover, the model selection procedure leads to parsimonious models providing a rank of covariates based on their explanatory power with respect to the outcome.

Air quality measurement relies on the effectiveness of a network of monitoring stations. Monitoring stations collect information about the evolution of air pollutants concentration. If more stations supplies the same information, then some of them could be deemed as redundant. Then, a clustering model for time series is useful to identify stations with similar features. Time series of pollutant concentration can be classified using the autoregressive metric in the framework of standard clustering techniques. A serious drawback is related to the lack of robustness of standard procedures. In this paper, using a partitioning around medoids approach combined with a trimming-based rule, a fuzzy model for cluster time series is proposed. The model provides a robust alternative to standard procedures. Two simulation studies are carried out to evaluate the clustering performance of the proposed clustering model. Finally, an empirical application to real time series of PM10 concentration in the Lazio region is presented and discussed showing the practical usefulness of the proposed approach.

In this paper, following the Partitioning Around Medoids (PAM) approach and the fuzzy theory, we propose a clustering model for financial time series based on the estimated cepstrum which represents the spectrum of the logarithm of the spectral density function. Selecting the optimal set of financial securities to build a portfolio that aims to maximize the risk-return tradeoff is a largely investigated topic in finance. The proposed model inherits all the advantages connected to PAM approach and fuzzy theory and it is able to compute objectively the cepstral weight associated to each cepstral coefficient by means of a suitable weighting system incorporated in the clustering model. In this way, the clustering model is able to tune objectively the different influence of each cepstral coefficient in the clustering process. The proposed clustering model performs better with respect to other clustering models. The proposed clustering model applied to each security sharpe ratio provides an efficient tool of clustering of stocks.

In this paper we propose different robust fuzzy clustering models for classifying heteroskedastic (volatility) time series, following the so-called model-based approach to time series clustering and using a partitioning around medoids procedure. The proposed models are based on a GARCH parametric modeling of the time series, i.e. the unconditional volatility and the time-varying volatility GARCH representation of the time series. We first suggest a timid robustification of the fuzzy clustering. Then, we propose three robust fuzzy clustering models belonging to the so-called metric, noise and trimmed approaches, respectively. Each model neutralizes the negative effects of the outliers in the clustering process in a different manner. In particular, the first robust model, based on the metric approach, achieves its robustness with respect to outliers by taking into account a robust distance measure; the second, based on the noise approach, achieves its robustness by introducing a noise cluster represented by a noise prototype; the third, based on the trimmed approach, achieves its robustness by trimming away a certain fraction of outlying time series. The usefulness and effectiveness of the proposed clustering models is illustrated by means of a simulation study and two applications in finance and economics.

In socio-economical clustering often the empirical information is represented by time-varying data generated by indicators observed over time on a set of subnational (regional) units. Usually among these units may exist contiguity relations, spatial but not only. In this paper we propose a fuzzy clustering model of multivariate time-varying data, the longitudinal fuzzy C-Medoids clustering with contiguity constraints. The temporal aspect is dealt with by using appropriate measures of dissimilarity between time trajectories. The contiguity among units is dealt with adding a contiguity matrix as a penalization term in the clustering model. The cross sectional fuzzy C-Medoids clustering with contiguity constraints is obtained considering one instant of time. The model is applied to the classification of the European NUTS on the basis of the observed dynamics of the Basic, Efficiency and Innovation subindexes of the Regional Competitiveness Index (RCI) 2013 and 2016. The positioning of the Italian regions is analyzed through the values of the medoids of the clusters and shows the peculiarities of the regions with respect to the subindexes either in single times or in the dynamic. Two contiguity constraints, one based on the European Western, Southern, Central and Northern geographic areas and one on the level of GDP—taken into account in the computation of the RCI—are also introduced in the models.