Susana Montes

University of Oviedo | UNIOVI · Department of Statistics and Operational Research and Mathematics Didactics

The imprecision inherent in human opinions is not properly modeled by crisp numbers. Other more complex structures like intervals or tuples capture better the imprecision of human assessments. This makes them very useful in decision problems. However, they cannot be easily compared. Despite they grasp better decision-makers inaccuracy, the lack of...

Estimating the mean of a population is a recurrent topic in statistics because of its multiple applications. If previous data is available, or the distribution of the deviation between the measurements and the mean is known, it is possible to perform such estimation by using L-statistics, whose optimal linear coefficients, typically referred to as...

Aggregation operators are unvaluable tools when different pieces of information have to be taken into account with respect to the same object. They allow to obtain a unique outcome when different evaluations are available for the same element/object. In this contribution we assume that the opinions are not given in form of isolated values, but inte...

The aggregation of several predictors in time series forecasting has been used intensely in the last decade in order to construct a better resulting model. Some of the most used alternatives are the ones related to the Induced Ordered Weighted Averaging (IOWA), in which the prediction values are ordered using a secondary vector, often related to th...

Granular computing, an attractive branch of artificial intelligence, focuses on constructing, processing and communicating information granules. Although various useful structures involving fuzzy sets, rough sets and their extensions have been discussed in relation to this literature, there is still a research gap regarding the connections between...

Aggregation functions have been widely used as a method to fuse data in a large number of applications. In most of them, the data can be modeled as a simple random sample. Thus, it is reasonable to treat the aggregated values as random variables. In this paper, the concept of aggregation functions of random variables with respect to a stochastic or...

The study of Gaussian Markov Random Fields has attracted the attention of a large number of scientific areas due to its increasing usage in several fields of application. Here, we consider the construction of Gaussian Markov Random Fields from a graph and a positive-definite matrix, which is closely related to the problem of finding the Maximum Lik...

Convexity is a deeply studied concept since it is very useful in many fields of mathematics, like optimization. When we deal with imprecision, the convexity is required as well and some important applications can be found fuzzy optimization, in particular convexity of fuzzy sets. In this paper we have extended the notion of convexity for interval-v...

In the field of statistics, linear combinations of order statistics, also known as L-statistics, have been widely used for the estimation of the mean of a population, which is equivalent to considering Ordered Weighted Averaging (OWA) operators over simple random samples. If previous data are available or the distribution of the deviation from the...

These days it is common to work with interval-valued data in areas like fuzzy analysis, clustering, medicine or decision-making. In this paper, we present a new similarity measure between intervals and compare it with other measures found in the literature. We also study some properties such as aliasing, that is obtaining the same similarity value...

Choquet integral is a widely used aggregation operator on one-dimensional and interval-valued information, since it is able to take into account the possible interaction among data. However, there are many cases where the information taken into account is vectorial, such as Long Short-Term Memories (LSTM). LSTM units are a kind of Recurrent Neural...

Multiple definitions have been put forward in the literature to measure the differences between two interval-valued fuzzy sets. However, in most cases, the outcome is just a real value, although an interval could be more appropriate in this environment. This is the starting point of this contribution. Thus, we revisit the axioms that a measure of t...

When a decision maker is asked to compare a set of alternatives, it may happen that the information provided is incomplete because she has no time to compare all the options or is unable to compare some alternatives against others. This contribution departs from an incomplete fuzzy weak preference relation by completing it on a consistent way with...

Starting from the notion of a multidistance, we formalize, through a suitable system of axioms, the concept of an inequality measure defined on a nonempty set with no additional structure implemented a priori. Among inequality measures, apart from multidistances we pay special attention to dispersions, and study their main features. Classical conce...

The notion of an orness measure for aggregation functions has been a relevant study subject whose history can be traced back to the early works of Dujmović in 1973. Intuitively, an orness measure quantifies the similarity of an aggregation function to the “or” function and results in an essential tool for decision engineering, field in which the ch...

This book constitutes the refereed proceedings of the 19th Conference of the Spanish Association for Artificial Intelligence, CAEPIA 2020, which was cancelled due to the COVID-19 pandemic, amalgamated with CAEPIA 2021, and held in Malaga, Spain, during September 2021. The 25 full papers presented were carefully selected from 40 submissions. The Co...

Dissimilarities are a very usual way to compare two fuzzy sets and also two interval-valued fuzzy sets. In both cases, the dissimilarity between two sets is a number. In this work, we introduce a generalization of the notion of dissimilarity for interval-valued fuzzy sets such that it assumes values on the set of subintervals instead of the set of...

Convexity is a very important property in many areas and the studies of this property are frequent. In this paper, we have extended the notion of convexity for interval-valued fuzzy sets based on different order between intervals. The considered orders are related and their behavior analyzed. In particular, we study the preservation of the convexit...

Characterizing the degree of similarity or difference between two sets is a very important topic, since it has many applications in different areas, including image processing or decision making. Several studies have been done about the comparison of fuzzy sets and its extensions, in particular for interval-valued fuzzy sets. However, in most of th...

Stochastic orders are mathematical methods allowing the comparison of random quantities. Probably the most usual one is stochastic dominance, which is based on the comparison of univariate cumulative distribution functions. Although it has been commonly applied, it does not consider the dependence between the random variables. This paper introduces...

This work presents a new algorithm based on Atanassov’s intuitionistic fuzzy sets and fuzzy mathematical morphology to leukocytes segmentation in color images. The main idea is based on modeling a color image as an Atanassov’s intuitionistic fuzzy set using the hue component in the HSV color space. Then, a pixel labeled as leukocyte is selected and...

Convexity is one of the most important geometric properties of sets and a useful concept in many fields of mathematics, like optimization. As there are also important applications making use of fuzzy optimization, it is obvious that the studies of convexity are also frequent. In this paper we have extended the notion of convexity for hesitant fuzzy...

The problem of ranking different candidates or alternatives according to the preferences of different voters or experts is a common study subject in the fields of social choice theory and preference modelling. Whereas the former field normally restricts its attention to preferences given in the form of rankings (with ties), the latter field embrace...

We investigate two related problems of categorization of alternatives. We are particularly concerned about liberalism and dictatorship, two noteworthy and opposing principles, in the context of fuzzy expressions of the opinions. First we study the problem of self-distributing the individuals in a society among fixed classes or categories. We identi...

In this work we consider some classes of functions with relaxed monotonicity conditions generalizing some other given classes of fusion functions. In particular, directionally increasing aggregation functions (called also pre-aggregation functions), directionally increasing conjunctors, or directionally increasing implications, etc., generalize the...

Fuzzy multisets represent a particularly challenging generalization of the concept of fuzzy sets. The membership degrees of fuzzy multisets are given by multisets in 0,1 rather than single values. Mathematically, they can be also seen as a generalization of the hesitant fuzzy sets. But in this general setting, the information about repetition is no...

We focus on the relationships among some consistency axioms in the framework of fuzzy choice functions. In order to help disclose the role of the t-norm in existing analyses, we start to study the situation that arises when we replace the standard t-norm with other t-norms. Our results allow us to conclude that unless we impose further structure on...

Two fuzzy subgroups \(\mu ,\eta \) of a group G are said to be equivalent if they have the same family of level set subgroups. Although it is well known that given two fuzzy subgroups \(\mu ,\eta \) of a group G their maximum is not always a fuzzy subgroup, it is clear that the maximum of two equivalent fuzzy subgroups is a fuzzy subgroup. We prove...

Probabilistic and fuzzy choice functions are used to describe decision situations in which some degree of uncertainty or imprecision is involved. We propose a way to equate these two formalisms by means of residual implication operations. Furthermore, a set of new rationality conditions for probabilistic choice functions is proposed and proved to b...

Stochastic dominance and statistical preference are two important tools for the pairwise comparison of random variables. However, pairwise methods are not always appropriate in the case of more than two alternatives. In this work, we generalize the notion of winning probability to the notion of multivariate winning probability. The latter allows to...

In the literature, there are two different approaches to define entropy of Atanassov intuitionistic fuzzy sets (AIFS, for short). The first approach, given by Szmidt and Kacprzyk, measures how far is an AIFS from its closest crisp set, while the second approach, given by Burrillo and Bustince, measures how far is an AIFS from its closest fuzzy set....

In their original and ordinary formulation, fuzzy sets associate each element in a reference set with one number, the membership value, in the real unit interval [0,1]. Among the various existing generalisations of the concept, we find fuzzy multisets. In this case, membership values are multisets in [0,1] rather than single values. Mathematically,...

The purpose of this paper is to study fuzzy operators induced by fuzzy relations and fuzzy relations induced by fuzzy operators. Many results are obtained about the relationship between \(*\)-preorders and fuzzy consequences operators for a fixed t-norm \(*\). We analyse these properties by considering a semi-copula (generalization of t-norm concep...

A common problem in social choice theory concerns the aggregation of the rankings expressed by several voters. Two different settings are often discussed depending on whether the aggregate is assumed to be a latent true ranking that voters try to identify or a compromise ranking that (partially) satisfies most of the voters. In a previous work, we...

We show that a definition of convexity based on the convexity of the score function does not guarantee preservation of convexity under intersections and provide a concept of convexity for hesitant fuzzy sets without this backdraw. We study the relationship between convex hesitant fuzzy sets and convex rough sets as their cuts.

The link between information theory and fuzzy logic has been proven in several previous papers. From this starting point, we propose here a review about the concept of divergence measures, which was proposed as a tool for comparing two fuzzy sets. The initial definition comes from the ideas behind the classical concept of divergence between two pro...

The link between information theory and fuzzy logic has been proven in several previous papers. From this starting point, we propose here a review about the concept of divergence measures, which was proposed as a tool for comparing two fuzzy sets. The initial definition comes from the ideas behind the classical concept of divergence between two pro...

In the literature we find two different approaches to define entropies of AIFSs. On the one hand, Szmidt and Kacprzyk’s entropy measures how far is an AIFS from a crisp set; on the other hand, Burrillo and Bustince’s approach measures how far is an AIFS from a fuzzy set. In this work we use divergence measures to define both types of entropies. We...

An important problem in fuzzy theory is how to determine if two elements are similar or not. There are different definitions to measure how similar or close are elements. One of the most important concepts in this context is that one of similarity. This paper aims at studying fuzzy similarities defined by fuzzy implications, logical equivalences ex...

In the Alto Valle of Rı́o Negro and Neuquén (Argentina) the most important economical activity is the pears and apples crop. The key to improve logistic, packaging and marketing strategies is related to the fruit size prediction. With this purpose, growth patterns have been developed for many years, based on the diameter’s fruits, depending on the...

We extend the notion of stochastic order to the pairwise comparison of fuzzy random variables. We consider expected utility, stochastic dominance and statistical preference, which are related to the comparisons of the expectations, distribution functions and medians of the underlying variables, and discuss how to generalize these notions to the fuz...

Aggregating the preferences of several voters on a set of candidates is a classical problem in several fields of application. In previous work, we have addressed this problem in the case where each voter expresses his/her preferences in the form of a ranking on the set of candidates, by searching for monotonicity of three different types of represe...

The Successive Likelihood Index Method establishes the degree of liability, and therefore the corresponding compensation, of the various errors that have caused an accident. From an expert judgment, the successive likelihood index of each error is calculated by a weighted arithmetic mean of their opinions. In this work we have considered other aver...

We analyze the existence of fuzzy sets of a universe that are convex with respect to certain particular classes of fusion operators that merge two fuzzy sets. In addition, we study aggregation operators that preserve various classes of generalized convexity on fuzzy sets. We focus our study on fuzzy subsets of the real line, so that given a mapping...

Measures of difference are studied for hesitant fuzzy sets, i.e. mappings where the values are multisets in the unit interval. Locality (the change of the value of the difference is dependent only on changes in singletons) of such mappings is discussed and the class of all local divergences is characterized. Entropies for hesitant fuzzy sets are al...

The comparison of sets is an important topic with application in several fields. Divergence measures were introduced as an adequate measure of comparison of two fuzzy sets and an alternative of the dissimilarities. The particular study for local divergences is here generalized to any t-conorm instead just the sum. The concept of divergence is revis...

In this work we introduce the definition of interval-valued fuzzy implication function with respect to any total order between intervals. We also present different construction methods for such functions. We show that the advantage of our definitions and constructions lays on that we can adapt to the interval-valued case any inequality in the fuzzy...

The choice of the ranking that best captures the preferences of several voters on a set of candidates has been a matter of study for centuries. An interesting point of view on this problem is centred on the notion of monotonicity. In this paper, we deal with an aspect of monotonicity that has not been addressed before: if there is a true ranking on...

Fuzzy relations and reciprocal relations are two popular tools for representing degrees of preference. It is important to note that they carry a different semantics and cannot be equated directly. We propose a simple transformation based on implication operators that allows to establish a one-to-one correspondence between both formalisms. It sets t...

In this paper MN-convex and MN-concave functions are examined and compared for particular cases of M and N. Characterization of pairs for the family of MN-convex (MN-concave) functions consisting of all functions and partial characterization of pairs for the family of MN-convex functions consisting of all constant functions will be presented. The c...

The aggregation of rankings is a recurrent task in several fields of application. In a recent work by Rademaker and De Baets, a ranking rule based on a natural monotonicity property was proposed in the context of social choice theory. This rule is built on the premise that, for a ranking to represent a group's opinion, it would be natural that the...

Hesitant fuzzy sets represent a useful tool in many areas such as decision making or image processing. Finite interval-valued hesitant fuzzy sets are a particular kind of hesitant fuzzy sets that generalize fuzzy sets, interval-valued fuzzy sets or Atanassov’s intuitionistic fuzzy sets, among others. Partitioning is a long-standing open problem due...

Nowadays, the representation and the treatment of color images are still open problems. Mathematical Morphology is the natural area for a rigorous formulation of many problems in image analysis. Moreover, it comprises powerful non linear techniques for filtering, texture analysis, shape analysis, edge detection or segmentation. A large number of mo...

We propose a new point of view of the long-standing problem where several voters have expressed a (strict) linear order (or ranking) over a set of candidates. For a ranking to represent a group's opinion, it would be natural that the strength with which is supported should not be less than both the strength with which and the strength with which ar...

Gray scale edge detection can be modeled using Fuzzy Sets and, in particular, Interval-Valued Fuzzy Sets. This work is focused on studying the performance of several Interval-Valued Fuzzy Sets construction methods for detecting edges in a gray scale image. These construction methods are based on considering information related to the neighborhood o...

In decision processes some objects may not be comparable with respect to a preference relation, especially if several criteria are considered. To provide a model for such cases a poset valued preference relation is introduced as a fuzzy relation on a set of alternatives with membership values in a partially ordered set. We analyze its properties an...

Partitions are at the basis of many processes as classification. They are classically defined in the context of crisp sets. However, partitions based on fuzzy sets have been proven to be more useful in real-life problems. This has motivated several different proposals to extend the definition of partition to the fuzzy sets context. Nevertheless, fu...

Stochastic dominance and statistical preference are stochastic orders with different interpretations: the former is based on the comparison of the marginal distributions while the latter is based on the joint distribution. Sklar's Theorem allows expressing the joint distribution in terms of the marginals by means of a copula. This paper investigate...

Genetic algorithms can be used to construct knowledge bases. They are based on the idea of “survival of the fittest” in the same way as natural evolution. Nature chooses the fittest ones in real life. In artificial intelligence we need a method that carries out the comparison and choice. Traditionally, this choice is based on fitness functions. Eac...

Ordering sets is a long-standing open problem due to its remarkable importance in many areas such as decision making, image processing or human reliability. This work is focused on introducing methods for ordering finitely generated sets as a generalization of those methods previously defined for ordering intervals. In addition, these orders betwee...

Probabilistic and fuzzy choice theory are used to describe decision situations in which a certain degree of imprecision is involved. In this work we propose a correspondence between probabilistic and fuzzy choice functions, based on implication operators. Given a probabilistic choice function a fuzzy choice function can be constructed and, furtherm...

Image processing represents an important challenge in different fields, especially in biomedical field. Mathematical Morphology uses concepts from set theory, geometry, algebra and topology to analyze the geometrical structure of an image. In addition, it is possible to consider methods where the starting point to analyze an image is a fuzzy relati...

Characterization of dissimilarity/divergence between intuitionistic fuzzy sets (IFSs) is important as it has applications in different areas including image segmentation and decision making. This study deals with the problem of comparison of intuitionistic fuzzy sets. An axiomatic definition of divergence measures for IFSs is presented, which are p...

Interval-valued fuzzy sets are an extension of fuzzy sets and are helpful when there is not enough information to define a membership function. This paper studies the behavior of a construction method for an interval-valued fuzzy relation built from a fuzzy relation. The behavior of this construction method is analyzed depending on the used t-norms...

We study interval-valued fuzzy sets as a model for the imprecise knowledge of the membership function of a fuzzy set. We compare three models for the probabilistic information about this membership function: the set of distributions of the measurable selections, the upper and lower probabilities of the associated random interval, and its p-box. We...

The comparison of random variables can be made by means of stochastic orders such as expected utility or statistical preference. One possible model when the random variables are imprecisely observed is to consider fuzzy random variables, so that the images become fuzzy sets. This paper proposes two comparison methods for fuzzy random variables: one...

The comparison of Atanassov intuitionistic fuzzy sets (AIF-sets) is a topic that has been widely studied due to its several applications in image segmentation or decision making, among other fields. Divergences for AIF-sets (AIF-divergences) were introduced as an adequate measure of comparison for AIF-sets. This study investigates a family of AIF-d...

Relations are used in many branches of mathematics to model concepts like is lower than, is equal to, etc. Initially, only crisp relations were considered, but in the last years, fuzzy relations have been revealed as a very useful tool in psychology, engineering, medicine, economics or any mathematically based field. A first approach to the concept...

Probabilistic and fuzzy choice theory are used to describe decision situations in which a certain degree of imprecision is involved. In this work we propose a correspondence between probabilistic and fuzzy choice functions, based on implication operators. Given a probabilistic choice function a fuzzy choice function can be constructed and, furtherm...