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## Über dieses Buch

This proceedings volume is a collection of peer reviewed papers presented at the 8th International Conference on Soft Methods in Probability and Statistics (SMPS 2016) held in Rome (Italy).

The book is dedicated to Data science which aims at developing automated methods to analyze massive amounts of data and to extract knowledge from them. It shows how Data science employs various programming techniques and methods of data wrangling, data visualization, machine learning, probability and statistics. The soft methods proposed in this volume represent a collection of tools in these fields that can also be useful for data science.

## Inhaltsverzeichnis

### Mean Value and Variance of Fuzzy Numbers with Non-continuous Membership Functions

We propose a definition of mean valueAnzilli, Luca and varianceFacchinetti, Gisella for fuzzy numbers whose membership functions are upper-semicontinuous but are not necessarily continuous. Our proposal uses the total variation of bounded variation functions.

Luca Anzilli, Gisella Facchinetti

### On the Construction of Radially Symmetric Trivariate Copulas

We proposeArias García, José De Jesús a method to construct a 3-dimensional symmetric functionDe Baets, Bernard that is radially symmetric, using two symmetric 2-copulas, with one of them being also radially symmetric. We study the properties of the presented construction in some specific cases and provide several examples for different families of copulas.

José De Jesús Arias García, Hans De Meyer, Bernard De Baets

### Simulation of the Night Shift Solid Waste Collection System of Phuket Municipality

This research wasBanditvilai, Somsri conducted in orderNiraso, Mantira to simulate the night shift solid waste collection system of Phuket Municipality, Thailand. The Phuket Municipality faced the problems of residual waste and an unbalanced load for solid waste collection teams. The waste management committee of Phuket Municipality wanted to improve the solid waste collection system to run more efficiently. This research analyzed the volume of solid waste collection instead of the weight, and has separated the solid waste collection points into 11 “types”. The data was collected from the survey form. Minitab 16.1 was used to analyze and test the data distribution, and then used them to build the model. Microsoft Visual C++ was used to build the simulation model, which was then verified and validated extensively. The model represented the actual night shift solid waste collection system of Phuket Municipality. The heuristic approach was then employed to apply new assigned zones and routings. The results from the study of the new system of night shift solid waste collection system of Phuket Municipality shows that there is no residual waste and no unbalanced load between solid waste collection teams. The new system works effectively and can decrease the total number of trips for solid waste collection by 9.1 % and the average distance and time for the solid waste collection system are decreased by 7.42 % and 7.10 % respectively.

Somsri Banditvilai, Mantira Niraso

### Updating Context in the Equation: An Experimental Argument with Eye Tracking

The Bayesian model was recentlyBaratgin, JenaproposedBessaa, Hamid as a normativeOcak, Brian reference for psychology studiesStilgenbauer, Jean-Louis in deductive reasoning. This new paradigm supports that individuals evaluate the probability of an indicative conditional if A then C in the natural language as the conditional probability $$P(\textit{C given A})$$P(C givenA) (P(C|A) according to Bayes’ rule). In this paper, we show applying an eye-tracking methodology that if the cognitive process for both probability assessments ($$P(\textit{if A then C})$$P(if Athen C) and P(C|A)) is really identical, it actually doesn’t match the traditional focusing situation of revision corresponding to Bayes’ rule (change of reference class in a static universe). Individuals appear to revise their probability as if the universe was evolving. They use a minimal rule in mentally removing the elements of the worlds that are not A. This situation, called updating, actually seems to be the natural frame for individuals to evaluate the probability of indicative conditional and the conditional probability.

Jean Baratgin, Brian Ocak, Hamid Bessaa, Jean-Louis Stilgenbauer

### Black-Litterman Model with Multiple Experts’ Linguistic Views

This paper presents fuzzyBartkowiak, MarcinextensionsRutkowska, Aleksandra of the Black-Litterman portfolio selection model. Black and Litterman identified two sources of information about expected returns and combined these two sources of information into one expected return formula. The first source of information is the expected returns that follow from the Capital Asset Pricing Model and thus should hold if the market is in equilibrium. The second source of information is comprised of the views held by investors. The presented extension, owing to the use of fuzzy random variables, includes two elements that are important from the point of view of practice: linguistic information and the views of multiple experts. The paper introduces the model extension step-by-step and presents an empirical example.

Marcin Bartkowiak, Aleksandra Rutkowska

### Representing Lightweight Ontologies in a Product-Based Possibility Theory Framework

This paper investigates an extensionBenferhat, Salem of lightweightBoutouhami, Khaoula ontologies, encoded here in DL-Lite languagesHaned, Faiza, to the product-based possibilityNouioua, Farid theory framework. We first introduce the language (and its associated semantics) used for representing uncertainty in lightweight ontologies. We show that, contrarily to a min-based possibilistic DL-Lite, query answering in a product-based possibility theory is a hard task. We provide equivalent transformations between the problem of computing an inconsistency degree (the key notion in reasoning from a possibilistic DL-Lite knowledge base) and the weighted maximum 2-Horn SAT problem.

Salem Benferhat, Khaoula Boutouhami, Faiza Khellaf, Farid Nouioua

### Asymptotics of Predictive Distributions

Let $$(X_n)$$(Xn) be a sequenceBerti, Patrizia of randomRigo, Pietro variables, adaptedPratelli, Luca to a filtration $$(\mathcal {G}_n)$$(Gn), and let $$\mu _n=(1/n)\,\sum _{i=1}^n\delta _{X_i}$$μn=(1/n)∑i=1nδXi and $$a_n(\cdot )=P(X_{n+1}\in \cdot \mid \mathcal {G}_n)$$an(·)=P(Xn+1∈·∣Gn) be the empirical and the predictive measures. We focus on $$||\mu _n-a_n||=\sup _{B\in \mathcal {D}}\,|\mu _n(B)-a_n(B)|$$||μn-an||=supB∈D|μn(B)-an(B)|, where $$\mathcal {D}$$D is a class of measurable sets. Conditions for $$||\mu _n-a_n||\rightarrow 0$$||μn-an||→0, almost surely or in probability, are given. Also, to determine the rate of convergence, the asymptotic behavior of $$r_n\,||\mu _n-a_n||$$rn||μn-an|| is investigated for suitable constants $$r_n$$rn. Special attention is paid to $$r_n=\sqrt{n}$$rn=n. The sequence $$(X_n)$$(Xn) is exchangeable or, more generally, conditionally identically distributed.

Patrizia Berti, Luca Pratelli, Pietro Rigo

### Independent k-Sample Equality Distribution Test Based on the Fuzzy Representation

ClassicalRamos-Guajardo, Ana B. tests for the equalityBlanco-Fernández, Angela of distributions of real-valued random variables are widely applied in Statistics. When the normality assumption for the variables fails, non-parametric techniques are to be considered; Mann-Whitney, Wilcoxon, Kruskal-Wallis, Friedman tests, among other alternatives. Fuzzy representations of real-valued random variables have been recently shown to describe in an effective way the statistical behaviour of the variables. Indeed, the expected value of certain fuzzy representations fully characterizes the distribution of the variable. The aim of this paper is to use this characterization to test the equality of distribution for two or more real-valued random variables, as an alternative to classical procedures. The inferential problem is solved through a parametric test for the equality of expectations of fuzzy-valued random variables. Theoretical results on inferences for fuzzy random variables support the validity of the test. Besides, simulation studies and practical applications show the empirical goodness of the method.

Angela Blanco-Fernández, Ana B. Ramos-Guajardo

### Agglomerative Fuzzy Clustering

The term fuzzy clusteringusuallyBorgelt, Christian refers to prototype-basedKruse, Rudolf methods that optimize an objective function in order to find a (fuzzy) partition of a given data set and are inspired by the classical c-means clustering algorithm. Possible transfers of other classical approaches, particularly hierarchical agglomerative clustering, received much less attention as starting points for developing fuzzy clustering methods. In this chapter we strive to improve this situation by presenting a (hierarchical) agglomerative fuzzy clustering algorithm. We report experimental results on two well-known data sets on which we compare our method to classical hierarchical agglomerative clustering.

Christian Borgelt, Rudolf Kruse

### Bayesian Inference for a Finite Population Total Using Linked Data

We consider the problem of estimatingBriscolini, Dario the totalLiseo, Brunero (or the mean) of a continuous variableTancredi, Andrea in a finite population setting, using the auxiliary information provided by a covariate which is available in a different file. However the matching steps between the two files is uncertain due to a lack of identification code for the single unit. We propose a fully Bayesian approach which merges the record linkage step with the subsequent estimation procedure.

Dario Briscolini, Brunero Liseo, Andrea Tancredi

### The Extension of Imprecise Probabilities Based on Generalized Credal Sets

In the paper we continue investigationsBronevich, Andrey started in the paper presentedRozenberg, Igor at ISIPTA’15, where the notions of lower and upper generalized credal sets has been introduced. Generalized credal sets are models of imprecise probabilities, where it is possible to describe contradiction in information, when the avoiding sure loss condition is not satisfied. The paper contains the basic principles of approximate reasoning: models of uncertainty based on upper previsions and generalized credal sets, natural extension, and coherence principles.

Andrey G. Bronevich, Igor N. Rozenberg

### A Generalized SMART Fuzzy Disjunction of Volatility Indicators Applied to Option Pricing in a Binomial Model

In this paper we extend our previousCapotorti, AndreacontributionsFigà-Talamanca, Gianna on the elicitation of the fuzzy volatility membership function in option pricing models. More specifically we generalize the SMART disjunction for a multi-model volatility behavior (Uniform, LogNormal, Gamma, ...) and within a double-source (direct vs. indirect) information set. The whole procedure is then applied to the Cox-Ross-Rubinstein framework for option pricing on the S&P500 Index where the historical volatility, computed from the Index returns’ time series, and the VIX Index observed data are respectively considered as the direct and indirect sources of knowledge. A suitable distance among the resulting fuzzy option prices and the market bid-ask spread make us appreciate the proposed procedure against the classical fuzzy mean.

Andrea Capotorti, Gianna Figà-Talamanca

### The Representation of Conglomerative Functionals

We prove results concerningCassese, Gianluca the representation of certain linear functionals based on the notion of conglomerability, originally introduced by Dubins and de Finetti. We show that this property has some applications in probability and in statistics.

Gianluca Cassese

### The Likelihood Interpretation of Fuzzy Data

The interpretation of degreesCattaneo, Marco of membership as statistical likelihood is probably the oldest interpretation of fuzzy sets. It allows in particular to easily incorporate fuzzy data and fuzzy inferences in statistical methods, and sheds some light on the central role played by extension principle and $$\alpha$$α-cuts in fuzzy set theory.

Marco E. G. V. Cattaneo

### Combining the Information of Multiple Ranker in Ranked Set Sampling with Fuzzy Set Approach

Ranked set sampling (RSS) is a usefulCetintav, BekiralternativeDemirel, NeslihansamplingGurler, Selma method for parameter estimation. ComparedUlutagay, Gozde to other sampling methods, it uses the ranking information of the units in the ranking mechanism before the actual measurement. The ranking mechanism can be described as a visual inspection of an expert or a highly-correlated concomitant variable. Accuracy for ranking of the sample units affects the precision of the estimation. This study proposes an alternative approach, called Fuzzy-weighted Ranked Set Sampling (FwRSS), to RSS for dealing with the uncertainty in ranking using fuzzy set. It assumes that there are K ($$K>1$$K>1) rankers for rank decisions and uses three different fuzzy norm operators to combine the decisions of all rankers in order to provide the accuracy of ranking. A simulation study is constructed to see the performance of the mean estimators based on RSS and FwRSS.

Bekir Cetintav, Selma Gurler, Neslihan Demirel, Gozde Ulutagay

### A Savage-Like Representation Theorem for Preferences on Multi-acts

We deal with a Savage-like decisionColetti, GiulianellaproblemPetturiti, Davide under uncertaintyVantaggi, Barbara where, for every state of the world, the consequence of each decision (multi-act) is generally uncertain: the decision maker only knows the set of possible alternatives where it can range (multi-consequence). A Choquet expected utility representation theorem for a preference relation on multi-acts is provided, relying on a state-independent cardinal utility function defined on the (finite) set of all alternatives.

Giulianella Coletti, Davide Petturiti, Barbara Vantaggi

### On Some Functional Characterizations of (Fuzzy) Set-Valued Random Elements

One of the most commonGonzález Rodríguez, Gil spaces to model imprecise dataColubi, Ana through (fuzzy) sets is that of convex and compact (fuzzy) subsets in $$\mathbb {R}^p$$Rp. The properties of compactness and convexity allow the identification of such elements by means of the so-called support function, through an embedding into a functional space. This embedding satisfies certain valuable properties, however it is not always intuitive. Recently, an alternative functional representation has been considered for the analysis of imprecise data based on the star-shaped sets theory. The alternative representation admits an easier interpretation in terms of ‘location’ and ‘imprecision’, as a generalized idea of the concepts of mid-point and spread of an interval. A comparative study of both functional representations is made, with an emphasis on the structures required for a meaningful statistical analysis from the ontic perspective.

Ana Colubi, Gil Gonzalez-Rodriguez

### Maximum Likelihood Under Incomplete Information: Toward a Comparison of Criteria

Maximum likelihoodCouso, Inés is a standard approachDubois, Didier to computing a probability distribution that best fits a given dataset. However, when datasets are incomplete or contain imprecise data, depending on the purpose, a major issue is to properly define the likelihood function to be maximized. This paper compares several proposals in terms of their intuitive appeal, showing their anomalous behavior on examples.

Inés Couso, Didier Dubois

### The Use of Uncertainty to Choose Matching Variables in Statistical Matching

Statistical matching aimsD’Orazio, Marcello at combiningScanu, MauroinformationDi Zio, Marco available in distinct sample surveys referred to the same target population. The matching is usually based on a set of common variables shared by the available data sources. For matching purposes just a subset of all the common variables should be used, the so called matching variables. The paper presents a novel method for selecting the matching variables based on the analysis of the uncertainty characterizing the matching framework. The uncertainty is caused by unavailability of data for estimating parameters describing the association/correlation between variables not jointly observed in a single data source. The paper focuses on the case of categorical variables and presents a sequential procedure for identifying the most effective subset of common variables in reducing the overall uncertainty.

Marcello D’Orazio, Marco Di Zio, Mauro Scanu

### Beyond Fuzzy, Possibilistic and Rough: An Investigation of Belief Functions in Clustering

In evidential clustering, uncertaintyDenoeux, Thierry about the assignment of objectsKanjanatarakul, Orakanya to clusters is represented by Dempster-Shafer mass functions. The resulting clustering structure, called a credal partition, is shown to be more general than hard, fuzzy, possibility and rough partitions, which are recovered as special cases. Different algorithms to generate a credal partition are reviewed. We also describe different ways in which a credal partition, such as produced by the EVCLUS or ECM algorithms, can be summarized into any of the simpler clustering structures.

Thierry Denœux, Orakanya Kanjanatarakul

### Small Area Estimation in the Presence of Linkage Errors

In Official Statistics, interest for data integrationTuoto, Tiziana has been increasingly growing, though the effect of this procedure on the statistical analyses has been disregarded for a long time. In recent years, however, it is largely recognized that linkage is not an error-free procedure and linkage errors, as false links and missed links can invalidate standard estimates. More recently, growing attention is devoted to the effect of linkage errors on the subsequent analyses. For instance, Samart and Chambers (Samart in Aust N Z J Stat 56, 2014 [14]) consider the effect of linkage errors on mixed effect models. Their proposal finds a natural application in the context of longitudinal studies, where repeated measures are taken on the same individuals. In official statistics, the mixed models is largely exploited for small area estimation to increase detailed information at local level. In this work, an EBLUP estimator that takes account of the linkage errors is derived.

Loredana Di Consiglio, Tiziana Tuoto

### A Test for Truncation Invariant Dependence

A test is proposed to check whether a randomDi Lascio, F. Marta L.sampleJaworski, Piotr comes from a truncationDurante, Fabrizio invariant copula C, that is, if C is the copula of a pair (U, V) of random variables uniformly distributed on [0, 1], then C is also the copula of the conditional distribution function of $$(U,V\mid U\le \alpha )$$(U,V∣U≤α) for every $$\alpha \in (0,1]$$α∈(0,1]. The asymptotic normality of the test statistics is shown. Moreover, a procedure is described to simplify the approximation of the asymptotic variance of the test. Its performance is investigated in a simulation study.

F. Marta L. Di Lascio, Fabrizio Durante, Piotr Jaworski

### Finite Mixture of Linear Regression Models: An Adaptive Constrained Approach to Maximum Likelihood Estimation

In order to overcomeRocci, Roberto the problemsDi Mari, Roberto due to the unboundednessGattone, Stefano Antonio of the likelihood, constrained approaches to maximum likelihood estimation in the context of finite mixtures of univariate and multivariate normals have been presented in the literature. One main drawback is that they require a knowledge of the variance and covariance structure. We propose a fully data-driven constrained method for estimation of mixtures of linear regression models. The method does not require any prior knowledge of the variance structure, it is invariant under change of scale in the data and it is easy and ready to implement in standard routines.

Roberto Di Mari, Roberto Rocci, Stefano Antonio Gattone

### A Multivariate Analysis of Tourists’ Spending Behaviour

According to the micro-economicDisegna, Marta theories regardingDurante, Fabrizio consumption behaviour, the determinants affecting the joint propensity of purchasingFoscolo, Enrico different goods and services are investigated. For this purpose, a copula-based model is suggested to understand how different expenditure categories are dependent with each other. A real application drawn from the tourism field illustrates the proposed approach and shows its main advantages. The findings could guide local practitioners and managers in creating new promotional campaigns able to attract visitors willing to pay on a bundle of goods and services correlated with each other.

Marta Disegna, Fabrizio Durante, Enrico Foscolo

### Robust Fuzzy Clustering via Trimming and Constraints

A methodology for robust fuzzyDotto, FrancescoclusteringFarcomeni, Alessio is proposed. This methodology can be widely applied in very different statistical problems given that it is based on probability likelihoods. Robustness is achieved by trimming a fixed proportion of “most outlying” observations which are indeed self-determined by the data set at hand. Constraints on the clusters’ scatters are also needed to get mathematically well-defined problems and to avoid the detection of non-interesting spurious clusters. The main lines for computationally feasible algorithms are provided and some simple guidelines about how to choose tuning parameters are briefly outlined. The proposed methodology is illustrated through two applications. The first one is aimed at heterogeneously clustering under multivariate normal assumptions and the second one might be useful in fuzzy clusterwise linear regression problems.

Francesco Dotto, Alessio Farcomeni, Luis Angel García-Escudero, Agustín Mayo-Iscar

### One-Factor Lévy-Frailty Copulas with Inhomogeneous Trigger Rates

A new parametric familyEngel, Janina of high-dimensionalScherer, Matthias, non-exchangeable extreme-valueSpiegelberg, Leonhard copulas is presented. The construction is based on the Lévy-frailty construction and stems from a subfamily of the Marshall–Olkin distribution. In contrast to the classical Lévy-frailty construction, non-exchangeability is achieved by inhomogeneous trigger-rate parameters. This family is studied with respect to its distributional properties and a sampling algorithm is developed. Moreover, a new estimator for its parameters is given. The estimation strategy consists in minimizing the mean squared error of the underlying Bernstein function and certain strongly consistent estimates thereof.

Janina Engel, Matthias Scherer, Leonhard Spiegelberg

### A Perceptron Classifier and Corresponding Probabilities

In this paper a fault tolerant probabilisticFalkowski, Bernd-Jürgen kernel version with smoothing parameter of Minsky’s perceptron classifier for more than two classes is sketched. Moreover a probabilistic interpretation of the output is exhibited. The price one has to pay for this improvement appears in the non-determinism of the algorithm. Nevertheless an efficient implementation using for example Java concurrent programming and suitable hardware is shown to be possible. Encouraging preliminary experimental results are presented.

Bernd-Jürgen Falkowski

### Fuzzy Signals Fed to Gaussian Channels

We add fuzzinessFranzoi, Laura to the signals sent throughSgarro, Andrea a continuous Gaussian transmission channel: fuzzy signals are modeled by means of triangular fuzzy numbers. Our approach is mixed, fuzzy/stochastic: below we do not call into question the probabilistic nature of the channel, and fuzziness will concern only the channel inputs. We argue that fuzziness is an adequate choice when one cannot control crisply each signal fed to the channel. Using the fuzzy arithmetic of interactive fuzzy numbers, we explore the impact that a fuzziness constraint has on channel capacity; in our model we are ready to tolerate a given fuzziness error F. We take the chance to put forward a remarkable case of “irrelevance” in fuzzy arithmetic.

Laura Franzoi, Andrea Sgarro

### Fuzzy Clustering Through Robust Factor Analyzers

In fuzzy clustering, data elementsGarcía-Escudero, Luis Angel can belong to more than one clusterMayo-Iscar, Agustin, and membership levelsGreselin, Francesca are associated with each element, to indicate the strength of the association between that data element and a particular cluster. Unfortunately, fuzzy clustering is not robust, while in real applications the data is contaminated by outliers and noise, and the assumed underlying Gaussian distributions could be unrealistic. Here we propose a robust fuzzy estimator for clustering through Factor Analyzers, by introducing the joint usage of trimming and of constrained estimation of noise matrices in the classic Maximum Likelihood approach.

Luis Angel García-Escudero, Francesca Greselin, Agustin Mayo Iscar

### Consensus-Based Clustering in Numerical Decision-Making

In this paper, we considerPérez-Román, David that a set of agentsGarcía-Lapresta, José Luis assess a set of alternatives through numbers in the unit interval. In this setting, we introduce a measure that assigns a degree of consensus to each subset of agents with respect to every subset of alternatives. This consensus measure is defined as 1 minus the outcome generated by a symmetric aggregation function to the distances between the corresponding individual assessments. We establish some properties of the consensus measure, some of them depending on the used aggregation function. We also introduce an agglomerative hierarchical clustering procedure that is generated by similarity functions based on the previous consensus measures.

José Luis García-Lapresta, David Pérez-Román

### Spatial Outlier Detection Using GAMs and Geographical Information Systems

A spatial (local) outlierCabrero-Ortega, Yolanda is a value that differsGarcía-Pérez, Alfonso from its neighbors. The usual way in which these are detected is a complicated task, especially if the data refer to many locations. In this paper we propose a different approach to this problem that consists in considering outlying slopes in an interpolation map of the observations, as indicators of local outliers. To do this, we transfer geographical properties and tools to this task using a Geographical Information System (GIS) analysis. To start, we use two completely different techniques in the detection of possible spatial outliers: First, using the observations as heights in a map and, secondly, using the residuals of a robust Generalized Additive Model (GAM) fit. With this process we obtain areas of possible spatial outliers (called hotspots) reducing the set of all locations to a small and manageable set of points. Then we compute the probability of such a big slope at each of the hotspots after fitting a classical GAM to the observations. Observations with a very low probability of such slope will finally be labelled as spatial outliers.

Alfonso García-Pérez, Yolanda Cabrero-Ortega

### Centering and Compound Conditionals Under Coherence

There is wide supportGilio, Angelo in logicOver, David E., philosophyPfeifer, Niki, and psychologySanfilippo, Giuseppe for the hypothesis that the probability of the indicative conditional of natural language, $$P(\textit{if } A \textit{ then } B)$$P(ifAthenB), is the conditional probability of B given A, P(B|A). We identify a conditional which is such that $$P(\textit{if } A \textit{ then } B)= P(B|A)$$P(ifAthenB)=P(B|A) with de Finetti’s conditional event, B|A. An objection to making this identification in the past was that it appeared unclear how to form compounds and iterations of conditional events. In this paper, we illustrate how to overcome this objection with a probabilistic analysis, based on coherence, of these compounds and iterations. We interpret the compounds and iterations as conditional random quantities, which sometimes reduce to conditional events, given logical dependencies. We also show, for the first time, how to extend the inference of centering for conditional events, inferring B|A from the conjunction A and B, to compounds and iterations of both conditional events and biconditional events, B||A, and generalize it to n-conditional events.

Angelo Gilio, David E. Over, Niki Pfeifer, Giuseppe Sanfilippo

### Approximate Bayesian Methods for Multivariate and Conditional Copulae

We describe a simpleGrazian, Clara method forLiseo, Brunero making inference on a functional of a multivariate distribution. The method is based on a copula representation of the multivariate distribution, where copula is a flexible probabilistic tool that allows the researcher to model the joint distribution of a random vector in two separate steps: the marginal distributions and a copula function which captures the dependence structure among the vector components. The method is also based on the properties of an approximate Bayesian Monte Carlo algorithm, where the proposed values of the functional of interest are weighted in terms of their empirical likelihood. This method is particularly useful when the likelihood function associated with the working model is too costly to evaluate or when the working model is only partially specified.

Clara Grazian, Brunero Liseo

### The Sign Test for Interval-Valued Data

Two versionsGrzegorzewski, Przemysław of the generalizedSpiewak, Martyna sign test for interval-valued data are proposed. Each version correspond to a different view on the interval outcomes of the experiment—either the epistemic or the ontic one. As it is shown, each view yield different approaches to data analysis and statistical inference.

Przemysław Grzegorzewski, Martyna Śpiewak

### Probability Distributions Related to Fuzzy P-Values

In the paper weHryniewicz, Olgierd have considered different approaches for the calculation of the p-value for fuzzy statistical tests. For the particular problem of testing hypotheses about the mean in the normal distribution with known standard deviation, and a certain type of fuzziness (both in data and tested hypotheses) we have found probability distributions of the respective defuzzified p-values. These distributions let us evaluate the compatibility of the observed data with the assumed hypothetical model.

Olgierd Hryniewicz

### Probabilistic Semantics and Pragmatics for the Language of Uncertainty

The idea that the probabilityKaufmann, Stefan of a conditional is the corresponding conditional probability has led something of an embattled existence in philosophy and linguistics. Part of the reason for the reluctance to embrace it has to do with certain technical difficulties (especially triviality). Even though solutions to the triviality problem are known to exist, their widespread adoption is hindered by their narrow range of data coverage and unclear relationship to established frameworks for modeling the dynamics of belief and conversation. This paper considers the case of Bernoulli models and proposes steps towards broadening the coverage of their application.

Stefan Kaufmann

### Dynamic Analysis of the Development of Scientific Communities in the Field of Soft Computing

This paper is dedicatedKutynina, Katerina to the researchLepskiy, Alexander of the dynamics of development and interactions among several scientific communities in the field of fuzzy logic and soft computing. This analysis was performed with the help of the following characteristics: conferences participants’ renewal, the level of cooperation in scientific communities, participation of one community’s key players in activities of the other ones, comparative number of most active participants in each community, uniformity of key players’ participation in different conferences.

Ekaterina Kutynina, Alexander Lepskiy

### Talk to Your Neighbour: A Belief Propagation Approach to Data Fusion

Data fusion is a majorLaurenza, Eleonora task in data management. Frequently, different sources store data about the same real-world entities, however with conflicts in the values of their features. Data fusion aims at solving those conflicts in order to obtain a unique global view over those sources. Some solutions to the problem have been proposed in the database literature, yet they have a number of limitations for real cases: for example they leave too many alternatives to users or produce biased results. This paper proposes a novel algorithm for data fusion actually addressing conflict resolution in databases and overcoming some existing limitations.

Eleonora Laurenza

### The Qualitative Characteristics of Combining Evidence with Discounting

The qualitative characteristicsLepskiy, Alexander of the combining evidence with the help of Dempster’s rule with discounting is studied in this paper in the framework of Dempster-Shafer theory. The discount coefficient (discounting rate) characterizes the reliability of information source. The conflict between evidence and change of ignorance after applying combining rule are considered in this paper as important characteristics of quality of combining. The quantity of ignorance is estimated with the help of linear imprecision index. The set of crisp and fuzzy discounting rates for which the value of ignorance after combining does not increases is described.

Alexander Lepskiy

### Measuring the Dissimilarity Between the Distributions of Two Random Fuzzy Numbers

In a previous paper the fuzzyCasals, María Rosa characterizing functionGil, María Ángeles of a random fuzzyLópez, María Teresa number was introducedLubiano, María Asunción as an extensionSinova, Beatriz of the moment generating function of a real-valued random variable. Properties of the fuzzy characterizing function have been examined, among them, the crucial one proving that it unequivocally determines the distribution of a random fuzzy number in a neighborhood of 0. This property suggests to consider the empirical fuzzy characterizing function as a tool to measure the dissimilarity between the distributions of two random fuzzy numbers, and its expected descriptive potentiality is illustrated by means of a real-life example.

María Asunción Lubiano, María Ángeles Gil, Beatriz Sinova, María Rosa Casals, María Teresa López

### An Empirical Analysis of the Coherence Between Fuzzy Rating Scale- and Likert Scale-Based Responses to Questionnaires

In dealing with questionnairesDe La Rosa de Sáa, Sara concerning satisfactionGil, María Ángeles, quality perceptionLubiano, María Asunción, attitude, judgementMontenegro, Manuel, etc., the fuzzySalas, Antonia rating scale has been introduced as a flexible way to respond to questionnaires’ items. Designs for this type of questionnaires are often based on Likert scales. This paper aims to examine three different real-life examples in which respondents have been allowed to doubly answer: in accordance with either a fuzzy rating scale or a Likert one. By considering a minimum distance-based criterion, each of the fuzzy rating scale answers is associated with one of the Likert scale labels. The percentages of coincidences between the two responses in the double answer are computed by the criterion-based association. Some empirical conclusions are drawn from the computation of such percentages.

María Asunción Lubiano, Antonia Salas, Sara De la Rosa de Sáa, Manuel Montenegro, María Ángeles Gil

### Asymptotic Results for Sums of Independent Random Variables with Alternating Laws

Stochastic models governed by an alternating dynamicsMacci, Claudio arise in various applications. In several cases these models can be described by sums of independent random variables with alternating laws. The aim of this paper is to study the asymptotic behavior of these sums in the fashion of large deviations.

Claudio Macci

### Dispersion Measures and Multidistances on

After introducing a definitionMartín, Javier of dispersionMayor, Gaspar measure on the Euclidean space $$\mathbb {R}^k$$Rk, we deal with the connection between these measures and the so called multidistances. In this way, we show that thr standard deviation is a relevant example of multidistance and, on the other hand, several significant families of multidistances are, at the same time, dispersion measures. Sufficient conditions for a multidistance to be a dispersion measure are also established.

Javier Martín, Gaspar Mayor

### Full Conglomerability, Continuity and Marginal Extension

We investigate fully conglomerable coherentMiranda, Enrique lower previsionsZaffalon, Marco in the sense of Walley, and some particular cases of interest: envelopes of fully conglomerable linear previsions, envelopes of countably additive linear previsions and fully disintegrable linear previsions. We study the connections with continuity and countable super-additivity, and show that full conglomerability can be characterised in terms of a supremum of marginal extension models.

Enrique Miranda, Marco Zaffalon

### On Extreme Points of p-Boxes and Belief Functions

The extreme pointsDestercke, Sébastien of convex probabilityMontes, Ignacio sets play an important practical role, especially as a tool to obtain specific, easier to manipulate sets. Although this problem has been studied for many models (probability intervals, possibility distributions), it remains to be studied for imprecise cumulative distributions (a.k.a. p-boxes). This is what we do in this paper, where we characterize the maximal number of extreme points of a p-box, give a family of p-boxes that attains this number and show an algorithm that allows to compute the extreme points of a given p-box. To achieve all this, we also provide what we think to be a new characterization of extreme points of a belief function.

Ignacio Montes, Sebastien Destercke

### Modelling the Dependence in Multivariate Longitudinal Data by Pair Copula Decomposition

The aim of the workNai Ruscone, Marta is to propose a new flexibleOsmetti, Silvia Angela way of modeling the dependence between the components of non-normal multivariate longitudinal-data by using the copula approach. The presence of longitudinal data is increasing in the scientific areas where several variables are measured over a sample of statistical units at different times, showing two types of dependence: between variables and across time. We propose to model jointly the dependence structure between the responses and the temporal structure of each processes by pair copula contruction (PCC). The use of the copula allows the relaxation of the assumption of multinormality that is typical of the usual model for multivariate longitudinal data. The use of PCC allows us to overcome the problem of the multivariate copulae used in the literature which suffer from rather inflexible structures in high dimension. The result is a new extremly flexible model for multivariate longitudinal data, which overcomes the problem of modeling simultaneous dependence between two or more non-normal responses over time. The explanation of the methodology is accompanied by an example.

Marta Nai Ruscone, Silvia Angela Osmetti

### Predictability in Probabilistic Discrete Event Systems

Predictability is a key propertyDague, Philippe allowing one to expectNouioua, Farid in advance the occurrenceYe, Lina of a fault in a system based on its observed events. Existing works give a binary answer to the question of knowing whether a system is predictable or not. In this paper, we consider discrete event systems where probabilities of the transitions are available. We show how to take advantage of this information to perform a Markov chain-based analysis and extract probability values that give a finer appreciation of the degree of predictability. This analysis is particularly important in case of non predictable systems.

Farid Nouioua, Philippe Dague, Lina Ye

### A Sandwich Theorem for Natural Extensions

The recently introducedPelessoni, Renato weak consistency notionsVicig, Paolo of 2-coherence and 2-convexity are endowed with a concept of 2-coherent, respectively, 2-convex natural extension, whose properties parallel those of the natural extension for coherent lower previsions. We show that some of these extensions coincide in various common instances, thus producing the same inferences.

Renato Pelessoni, Paolo Vicig

### Envelopes of Joint Probabilities with Given Marginals Under Absolute Continuity or Equivalence Constraints

The aim is to determinePetturiti, Davide the envelopesVantaggi, Barbara of the class of joint probabilities (provided it is not empty) with assigned marginals, under the constraint of absolute continuity or equivalence with respect to a given reference measure.

Davide Petturiti, Barbara Vantaggi

### Square of Opposition Under Coherence

Various semanticsPfeifer, Niki for studying the squareSanfilippo, Giuseppe of opposition have been proposed recently. So far, only [14] studied a probabilistic version of the square where the sentences were interpreted by (negated) defaults. We extend this work by interpreting sentences by imprecise (set-valued) probability assessments on a sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square in terms of acceptability and show how to construct probabilistic versions of the square of opposition by forming suitable tripartitions. Finally, as an application, we present a new square involving generalized quantifiers.

Niki Pfeifer, Giuseppe Sanfilippo

### Testing of Coarsening Mechanisms: Coarsening at Random Versus Subgroup Independence

Since coarse(ned) data naturallyAugustin, Thomas induce set-valuedCattaneo, MarcoestimatorsPlass, Julia, analystsSchollmeyer, Georg often assume coarsening at random (CAR) to force them to be single-valued. Using the PASS data as an example, we re-illustrate the impossibility to test CAR and contrast it to another type of uninformative coarsening called subgroup independence (SI). It turns out that SI is testable here.

Julia Plass, Marco E. G. V. Cattaneo, Georg Schollmeyer, Thomas Augustin

### Two-Sample Similarity Test for the Expected Value of Random Intervals

The similarity degreeBlanco-Fernández, Angela between the expectationRamos-Guajardo, Ana B. of two random intervals is studied by means of a hypothesis testing procedure. For this purpose, a similarity measure for intervals is introduced based on the so-called Jaccard index for convex sets. The measure ranges from 0 (if both intervals are not similar at all, i.e., if they are not overlapped) to 1 (if both intervals are equal). A test statistic is proposed and its limit distribution is analyzed by considering asymptotic and bootstrap techniques. Some simulation studies are carried out to examine the behaviour of the approach.

Ana B. Ramos-Guajardo, Ángela Blanco-Fernández

### Handling Uncertainty in Structural Equation Modeling

This paper attempts to proposePalumbo, Francesco an overviewRomano, Rosaria of a recent method named partial possibilistic regression path modeling (PPRPM), which is a particular structural equation model that combines the principles of path modeling with those of possibilistic regression to model the net of relations among variables. PPRPM assumes that the randomness can be referred to the measurement error, that is the error in modeling the relations among the observed variables, and the vagueness to the structural error, that is the uncertainty in modeling the relations among the latent variables behind each block. PPRPM gives rise to possibilistic regressions that account for the imprecise nature or vagueness in our understanding phenomena, which is manifested by yielding interval path coefficients of the structural model. However, possibilistic regression is known to be a model sensitive to extreme values. That is way recent developments of PPRPM are focused on robust procedures for the detection of extreme values to omit or lessen their effect on the modeling. A case study on the motivational and emotional aspects of teaching is used to illustrate the procedure.

Rosaria Romano, Francesco Palumbo

### Detecting Inconsistencies in Revision Problems

When dealing with complexGebhardt, Jörg knowledge, inconsistenciesKruse, Rudolf become a bigSchmidt, Fabian problem. One important aspect of handling inconsistencies is their detection. In this paper we consider approaches to detect different types of inconsistencies that may occur in the formulation of revision problems. The general discussion focuses on the revision of probability distributions. In our practical analysis, we refer to probability distributions represented as Markov networks.

Fabian Schmidt, Jörg Gebhardt, Rudolf Kruse

### Tukey’s Biweight Loss Function for Fuzzy Set-Valued M-estimators of Location

The Aumann-type meanSinova, Beatriz is probablyVan Aelst, Stefan the best-known measure for the location of a random fuzzy set. Despite its numerous probabilistic and statistical properties, it inherits from the mean of a real-valued random variable the high sensitivity to outliers or data changes. Several alternatives extending the concept of median to the fuzzy setting have already been proposed in the literature. Recently, the adaptation of location M-estimators has also been tackled. The expression of fuzzy-valued location M-estimators as weighted means under mild conditions allows us to guarantee that these measures take values in the space of fuzzy sets. It has already been shown that these conditions hold for the Huber and Hampel families of loss functions. In this paper, the strong consistency and the maximum finite sample breakdown point when the Tukey biweight (or bisquare) loss function is chosen are analyzed. Finally, a real-life example will illustrate the influence of the choice of the loss function on the outputs.

Beatriz Sinova, Stefan Van Aelst

### Technical Gestures Recognition by Set-Valued Hidden Markov Models with Prior Knowledge

Hidden Markov modelsAntonucci, Alessandro are popular tools for gestureDestercke, Sébastien recognition. Once the generative processes of gesturesSoullard, Yann have been identified, an observation sequence is usually classified as the gesture having the highest likelihood, thus ignoring possible prior information. In this paper, we consider two potential improvements of such methods: the inclusion of prior information, and the possibility of considering convex sets of probabilities (in the likelihoods and the prior) to infer imprecise, but more reliable, predictions when information is insufficient. We apply the proposed approach to technical gestures, typically characterized by severe class imbalance. By modelling such imbalances as a prior information, we achieve more accurate results, while the imprecise quantification is shown to produce more reliable estimates.

Yann Soullard, Alessandro Antonucci, Sébastien Destercke

### Time Series Modeling Based on Fuzzy Transform

It is well known thatGuerra, Maria LetiziasmoothingSorini, Laerte is appliedStefanini, Luciano to better see patterns and underlying trends in time series. In fact, to smooth a data set means to create an approximating function that attempts to capture important features in the data, while leaving out noises. In this paper we choose, as an approximation function, the inverse fuzzy transform (introduced by Perfilieva in Fuzzy Sets Syst 157:993–1023, 2006 [3]) that is based on fuzzy partitioning of a closed real interval into fuzzy subsets. The empirical distribution we introduce can be characterized by its expectiles in a similar way as it is characterized by quantiles.

Luciano Stefanini, Laerte Sorini, Maria Letizia Guerra

### Back to “Reasoning”

Is rigor always strictlyTabacchi, Marco Elio related to precisionTermini, Settimo and accuracy? This is a fundamental question in the realm of Fuzzy Logic; the first instinct would be to answer in the positive, but the question is much more complex than it appears, as true rigor is obtained also by a careful examination of the context, and limiting to a mechanical transfer of techniques, procedures and conceptual attitudes from one domain to another, such as from the pure engineering feats or the ones of mathematical logic to the study of human reasoning, does not guarantee optimal results. Starting from this question, we discuss some implications of going back to the very concept of reasoning as it is used in natural language and in everyday life. Taking into account the presence—from the start—of uncertainty and approximation in one of its possible forms seems to indicate the need of a different approach from the simple extension of tools and concepts from mathematical logic.

Marco Elio Tabacchi, Settimo Termini

### Lexicographic Choice Functions Without Archimedeanicity

We investigate the connectionDe Cooman, Gert between choiceMiranda, EnriquefunctionsVan Camp, Arthur and lexicographic probabilities, by means of the convexity axiom considered by Seidenfeld et al. (Synthese 172:157–176, 2010 [7]) but without imposing any Archimedean condition. We show that lexicographic probabilities are related to a particular type of sets of desirable gambles, and investigate the properties of the coherent choice function this induces via maximality. Finally, we show that the convexity axiom is necessary but not sufficient for a coherent choice function to be the infimum of a class of lexicographic ones.

Arthur Van Camp, Enrique Miranda, Gert de Cooman

### Composition Operator for Credal Sets Reconsidered

This paper is the second attemptVejnarova, Jirina to introduce the composition operator, already known from probability, possibility, evidence and valuation-based systems theories, also for credal sets. We try to avoid the discontinuity which was present in the original definition, but simultaneously to keep all the properties enabling us to design compositional models in a way analogous to those in the above-mentioned theories. These compositional models are aimed to be an alternative to Graphical Markov Models. Theoretical results achieved in this paper are illustrated by an example.

Jiřina Vejnarová

### A Nonparametric Linearity Test for a Multiple Regression Model with Fuzzy Data

A linearity test for a multipleWang, Dabuxilatu regression model with LR-fuzzy responses and LR-fuzzy explanatory variables is considered. The regression model consists of several multiple regression models from response center or spreads to the explanatory centers and spreads. A multiple nonparametric regression model to be employed as a reference in the testing approach is estimated, and with which the linearity of the regression model is tested. Some simulation example is also presented.

Dabuxilatu Wang

### Treat a Weak Dependence Among Causes and Lives in Insurance by Means of Fuzzy Sets

In this paper, we apply the copulasWang, Dabuxilatu and fuzzyWang, Tengteng sets to approximate the dependencies in causes and lives, where under each cause of decrement the decrement times of the lives are assumed to be weak dependence. We propose utilizing a mixture of both randomness and fuzziness to describe the concept of weak dependence. An application is considered for a general symmetric status of multiple life under dependent causes of decrement.

Dabuxilatu Wang, Tengteng Wang

### A Portfolio Diversification Strategy via Tail Dependence Clustering

We provide a two-stage portfolioDurante, Fabrizio selection procedureFoscolo, Enrico in order to increasePappadà, Roberta the diversificationWang, Hao benefits in a bear market. By exploiting tail dependence-based risky measures, a cluster analysis is carried out for discerning between assets with the same performance in risky scenarios. Then, the portfolio composition is determined by fixing a number of assets and by selecting only one item from each cluster. Empirical calculations on the EURO STOXX 50 prove that investing on selected assets in trouble periods may improve the performance of risk-averse investors.

Hao Wang, Roberta Pappadà, Fabrizio Durante, Enrico Foscolo

### An Upper Bound Estimation About the Sample Average of Interval-Valued Random Sets

In this paper, we give an upper boundGuan, LiestimationWang, Xia about the probability of the event that the sample average of i.i.d. interval-valued random sets is included in a closed set. The main tool is Cramér theorem in the classic theory of large deviation principle about real-valued random variables.

Xia Wang, Li Guan

### On Asymptotic Properties of the Multiple Fuzzy Least Squares Estimator

The multiple fuzzy linearChoi, Seung HoeregressionGrzegorzewski, Przemysław model with fuzzyYoon, Jin Hee input–fuzzy output is considered. Assuming that fuzzy inputs and fuzzy outputs are modeled by triangular fuzzy numbers, we prove the consistency and asymptotic normality of the least squares estimators.

Jin Hee Yoon, Seung Hoe Choi, Przemyslaw Grzegorzewski

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