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Identification of a multi-input/single-output system enables the creation of a mathematical model that as accurately as possible represents the input-output relation induced by a physical system. When there is little information about the physical laws related to the system or when the system is too complex, methods such as parametric identification are used to define the system model. In that case, preliminary assumptions can be made about the system leading to a parametric aggregation function based model. This parametric model may be learned by estimating the parameters of this aggregation function from a representative set of inputs/outputs. A major difficulty is to design a model that is relatively simple yet precise enough to meet the user's needs. Linear models are commonly used because they meet these two contradictory constraints. However, use of a linear model is often at the expense of the accuracy of input-output relationship description. In a recent paper, a new modeling approach was proposed, under the name of macsum modeling [31] , which aims at replacing the linear model concept by a set of linear models. The obtained aggregation function leads to an interval-valued output. This represents the lack of accuracy of the model to predict the system output when the inputs are known. An interesting feature of this model is that it is ruled by a single precise parametric vector whereas the output is imprecise. Moreover, the vector dimension is equal to that of the input space. In this paper, we propose a method to learn such a model so that it best reflects the input-output relationship of the considered system. This method is based, as in the case of linear modeling, on a regression method. This is particularly new approach because the macsum aggregation is based on a Choquet integral and very few authors have proposed learning of an aggregation function based on the Choquet integral. [ABSTRACT FROM AUTHOR]

As powerful tools for modeling nondeterministic problems, Choquet integrals have attracted much attention in recent years. This study generalizes the existing set-valued Choquet integrals and formulates the theory of Choquet integrals with respect to set-valued fuzzy measures. First, a set-valued Choquet integral of a real-valued function is defined using the set of Choquet integrals of real-valued functions with respect to fuzzy measure selections of a set-valued fuzzy measure. The integrand is then extended to set-valued functions, after which various properties, convergence theorems, and set-valued Jensen's inequalities are obtained. Second, following how single-valued Choquet integrals are defined, an alternative set-valued Choquet integral of real-valued function is formulated using Aumann integrals. Through these steps, the existing set-valued Choquet integrals are covered. [ABSTRACT FROM AUTHOR]

A constructively k -additive set function μ defined on a measurable space (X , B) is represented by using some σ -additive measure μ [ k ]. In this paper, we define the base measurable space for μ [ k ] on which μ [ k ] is uniquely defined. Using this measure, we define the covering measure μ ¯ to be useful in the analysis of the measurable space. We elucidate that a measurable set A is a strong μ -null set (μ (B) = μ (B ∪ C) , ∀ B , ∀ C ⊂ A) if and only if μ ¯ (A) = 0. We prove the subadditivity of μ ¯ and this follows the linearity and completeness of L p -spaces with respect to the Choquet and Pan integrals over μ ¯. The equivalent classes of these function spaces are given by the relation "to take same value on the complement of a strong μ -null set." [ABSTRACT FROM AUTHOR]

Extending the concept of level measure μ ({ f ⩾ a }) we introduce a generalized level measure based on a family of conditional aggregations operators. As a recent research direction, conditional aggregation operators extend aggregation operators in order to model the dependence of function on a conditional set. Generalized level measures then provide a new insight into behaviour of level-dependent (monotone) measures and the related processes. We investigate in detail several basic properties of the generalized level measure, including connections with the family of level measures, the extension of the survival function μ ({ f > a }) and the transformation of monotone measures to hyperset. Applications in scientometrics and information science are described. [ABSTRACT FROM AUTHOR]

Recently, Paiva et al. introduced the concept of quasi-overlap functions on bounded lattices and investigated some vital properties of them. In this paper, we continue consider this research topic and focus on the constructions of quasi-overlap functions along with their generalized forms on bounded partially ordered sets. To be specific, firstly, we generalize the truth values set of quasi-overlap functions from bounded lattices to bounded partially ordered sets and introduce the notions of 0 P -quasi-overlap functions, 1 P -quasi-overlap functions and 0 P , 1 P -quasi-overlap functions on any bounded partially ordered set P by considering the weaker boundary conditions than the quasi-overlap functions on P. Secondly, we give the constructions of quasi-overlap functions, 0 P -quasi-overlap functions, 1 P -quasi-overlap functions and 0 P , 1 P -quasi-overlap functions on any bounded partially ordered set P via the so-called Galois s -connections and 0,1-homomorphisms, 1-homomorphisms, 0-homomorphisms and ord-homomorphisms, respectively. In particular, we prove that those constructions contain the methods of extending the known quasi-overlap functions, 0 P -quasi-overlap functions, 1 P -quasi-overlap functions and 0 P , 1 P -quasi-overlap functions from any bounded partially ordered set P to any other bounded partially ordered sets. Finally, we show that those extensions maintain some basic properties of the known quasi-overlap functions, 0 P -quasi-overlap functions, 1 P -quasi-overlap functions and 0 P , 1 P -quasi-overlap functions on P , such as, idempotent, Archimedean property and cancellation law. [ABSTRACT FROM AUTHOR]

In this note, we discuss the validity of formula (4.54) in Grabisch's book from 2016. This equation expresses the Choquet integral of nonnegative functions w.r.t. a game in the form of linear invertible transform of set functions. However, the formula (erroneously) excludes the empty set, which has consequences for deriving Choquet integral representations in the literature using this approach. We point out that results for several transforms (e.g. Möbius, or co-Möbius transform) are not affected, but it is not the case of others (e.g. Fourier transform). In this spirit, we also provide a new formula for the Choquet integral representation via the Shapley interaction transform. [ABSTRACT FROM AUTHOR]

After giving a calculation formula for the g-derivative for a fuzzy-valued function with unilateral derivatives of the endpoint functions of level sets and characterizing the partial order relations between two fuzzy intervals by their generalized difference, we present the optimality conditions of weak LU-minimum solutions for fuzzy optimization problems by using the partial generalized differentiability and directional generalized differentiability. The optimization conditions obtained in this article extend the results in the literature. Examples are also given to illustrate the main theorems. [ABSTRACT FROM AUTHOR]

A consensus control scheme is developed for a class of leader-follower multiagent systems with a fuzzy dead zone constraint. Differently from previous work, the method developed implements a predefined convergence of tracking errors for multiagent systems; especially, the dynamic model for each follower has a fuzzy dead zone input. To solve the problem, not only a set of smooth functions are used within the control design to increase the stability of the systems but also the unfuzziness and center-of-gravity method are used to analyze and process the actuator constraint model, where each slope of the dead zone is uncertain and fuzzy. It is verified that the method developed effectively guarantees that the tracking errors of multiagent systems can converge to a predefined interval; that is, the problem of better asymptotic consensus performance for nonlinear multiagent systems is solved. Simulation illustrates the results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study monotone set functions defined as the composition of an additive measure with a strictly increasing function. This function is a unary operator in continuous-valued logic, called the tau function, and it is a generator function-based parametric mapping. We provide a necessary and sufficient condition for the equality of two tau functions that are induced by different generator functions. Using the tau function and its properties, we introduce a new monotone measure that we call the tau-additive measure. This measure is computationally simple and it can be viewed as an upper or lower probability depending on the parameter settings of the tau function. We present the parameter-dependent submodularity and supermodularity of the tau-additive measure and show how this measure can be constructed from a set function on a finite set. This procedure is analogous to how the well-known lambda-additive measure can be constructed, but our method is computationally simpler. We demonstrate that the tau-additive measure can be used to approximate the lambda-additive measure. Lastly, exploiting these theoretical results, we present an application in the area of human resource management. [ABSTRACT FROM AUTHOR]

In this study, we generalize the concept of the k -order additivity of a set function. First, we discuss the Möbius transform for a non-discrete set function. Next, we generalize the definition of the k -order additivity of a set function using the Möbius transform and provide the equivalent conditions for the k -order additivity. Furthermore, we consider the k -order additivity of the distorted monotone measure. We prove that under certain conditions, a distorted measure is k -order additive if and only if the distortion function is a polynomial of k -th order. [ABSTRACT FROM AUTHOR]

We examine sufficient and necessary conditions for the Choquet integral on the support of its integrand to be equal to the integral on a set which includes this support (we consider the problem in the case of a locally compact Hausdorff space). [ABSTRACT FROM AUTHOR]

In this article, an Interval Type-3 Fuzzy Logic System (IT3FLS) for enhancing the performance in Bee Colony Optimization (BCO) is outlined. The efficiency of the IT3FLS approach is verified with results on a set of benchmark mathematical functions. The IT3FLS provides an approach that helps to identify the optimal values in α and β parameters that allows to improve the performance in the original BCO. The IT3FLS approach exhibits advantages in the optimization of the benchmark functions. It can be noted that a IT3FLS exhibits better results in the minimal values of the set of mathematical functions. The experimentation demostrates that the implementation of the IT3FLS approach enhances the performance of BCO when compared with respect to the variants utilizing Generalized Type-2 FLS (GT2FLS), Interval Type-2 FLS (IT2FLS) and Type-1 FLS (T1FLS). [ABSTRACT FROM AUTHOR]

Statistical depth functions are a standard tool in nonparametric statistics to extend order-based univariate methods to the multivariate setting. Since there is no universally accepted total order for fuzzy data (even in the univariate case) and there is a lack of parametric models, a fuzzy extension of depth-based methods is very interesting. In this paper, we adapt the multivariate depths projection depth and L r -type depth functions to the fuzzy setting, proposing different generalizations for the L r -type depths. We prove that the proposed fuzzy depth functions have very good properties, obtaining that the fuzzy projection depth is the second example in the literature to satisfy simultaneously the notion of semilinear and of geometric depth. This implies that the fuzzy projection depth is extremely well behave, to order fuzzy sets with respect to fuzzy random variables. Furthermore, we illustrate the good empirical behavior of the proposed fuzzy depth functions with a real data example of trapezoidal fuzzy sets and the used of fuzzy depths in depth-based classification procedures. Finally, as trapezoidal fuzzy sets can be represented by elements of R 4 , we justify our proposals by also showing empirically the superiority of the fuzzy depths over the multivariate projection depth applied to fuzzy sets. [ABSTRACT FROM AUTHOR]

We investigate the addition operation on fuzzy numbers with a fuzzy set of summands. This is a natural generalization of the corresponding operation which involves a crisp set of summands. It is shown that the resulting set is a type-2 fuzzy set. We construct a type-2 membership function of this set. The sum type-2 fuzzy set can be decomposed according to secondary membership grades into the corresponding collection of fuzzy numbers. It helps to represent the addition operation on fuzzy numbers with a fuzzy set of summands in a form which is convenient for a proper understanding and applications. Illustrative examples are given. [ABSTRACT FROM AUTHOR]

As a preceding result, it has been shown that the dominance relation is not transitive on the set of strict triangular norms. This result has been achieved thanks to new results on Mulholland inequality. Recently, Saminger-Platz, De Baets, and De Meyer have introduced the generalized Mulholland inequality which characterizes the dominance on all continuous Archimedean triangular norms in an analogous way as does Mulholland inequality on the strict triangular norms. Based on these new results, the present paper shows that the dominance relation is not transitive on the set of nilpotent triangular norms and, consequently, on the set of continuous Archimedean triangular norms. This result is achieved by introducing a new sufficient condition under which a given function solves the generalized Mulholland inequality and by showing that the set of the functions that solve the inequality is not closed with respect to compositions. [ABSTRACT FROM AUTHOR]

We characterize all pseudo-uninorms with continuous Archimedean underlying functions. By discussing all possible combinations of strict and nilpotent underlying functions we show that a pseudo-uninorm with continuous Archimedean underlying functions is not a uninorm only in the case when both underlying functions are strict. Moreover, we show that in the case of pseudo-uninorms with continuous Archimedean underlying functions the set of points where the commutativity could be violated reduces to { (0 , 1) , (1 , 0) }. [ABSTRACT FROM AUTHOR]

The recently defined concept of statistical depth function for fuzzy sets provides a theoretical framework for ordering fuzzy sets with respect to the distribution of a fuzzy random variable. One of the most used and studied statistical depth functions for multivariate data is simplicial depth, based on multivariate simplices. We introduce a notion of pseudosimplices generated by fuzzy sets and propose three generalizations of simplicial depth to fuzzy sets. Their theoretical properties are analyzed and the behavior of the proposals is illustrated through a study of both synthetic and real data. [ABSTRACT FROM AUTHOR]

We provide several improvements and corrections of results dealing with the smallest universal integral I S based on a semicopula S. We deal with a transformation of the integral into another semicopula-based universal integral for continuous and commutative semicopulas, monotone convergence theorems with new proofs and Fréchet-type mappings' properties corrected. During this way we discover several fine properties of the integral functional I S. We also discuss further convergences of measurable functions and the corresponding integrals in the light of the newest results from the literature answering several open questions stated in our previous articles of the series. [ABSTRACT FROM AUTHOR]

Inclusion is one of the most basic relations between sets. In this paper, we show how to represent the degree of inclusion between two L -fuzzy sets via a function. Specifically, such a function determines the minimal modifications needed in an L -fuzzy set to be included (in Zadeh's sense) into another. To reach such a goal, firstly we present the notion of f -inclusion, which defines a family of crisp binary relations between L -fuzzy sets that are used as indexes of inclusion and, subsequently, we define the φ -degree of inclusion as the most suitable f -inclusion under certain criterion. In addition, we also present three φ -degrees of similarity definable from the φ -degree of inclusion. We show that the φ -degree of inclusion and the φ -degrees of similarities satisfy versions of many common axioms usually required for measures of inclusion and similarity in the literature. [ABSTRACT FROM AUTHOR]

A fuzzy set approach to uncertain functions is in the focus of this paper. The uncertainty in functions entering a model, a differential equation, for instance, is represented by α -dependent sets of crisp functions, that is, α -cuts determined by an integral membership functional that is defined by means of a continuous auxiliary function whose design depends on the information the analyst has about the uncertainty. A scalar quantity of interest evaluating the model output is assumed and its membership function is inferred from the Zadeh extension principle by solving α -dependent worst- and best-case scenario problems. Two types of auxiliary functions are considered and their algorithmic advantages and disadvantages are discussed. [ABSTRACT FROM AUTHOR]

A new approach to non-additive set functions is presented. The notion of non-additive measure is extended using non-commutative and non-associative operations with two parameters and the cumulative distribution function of a random variable. One new non-additive measure, generalized pseudo-probability measure, is defined and some of its basic properties are proven. [ABSTRACT FROM AUTHOR]

In this work, we define a distance function on the set of bivariate subcopulas to generate a compact metric space. Moreover, the copula space equipped with the uniform distance is essentially a metric subspace of this subcopula space. We also characterize the convergence in this space, and provide the interrelationship with the convergence of the distribution functions. [ABSTRACT FROM AUTHOR]

The paper deals with an algebraic extension of MV-algebras based on the definition of generalized Boolean algebras. We introduce a new class of structures, not necessarily with a top element, which are called EMV-algebras, in a way that every EMV-algebra contains an MV-algebra. First, we present basic properties of EMV-algebras. We give some examples, introduce and investigate congruence relations, ideals and filters on these algebras. We establish a basic representation result saying that each EMV-algebra can be embedded into an EMV-algebra with top element and we characterize EMV-algebras either as structures which are termwise equivalent to MV-algebras or as maximal ideals of EMV-algebras with top element. We study the lattice of ideals of an EMV-algebra and prove that every EMV-algebra has at least one maximal ideal. We define an EMV-clan of fuzzy sets as a special EMV-algebra where all operations are defined by points. We show that any semisimple EMV-algebra is isomorphic to an EMV-clan of fuzzy functions on a set. The set of EMV-algebras is neither a variety nor a quasivariety, but rather a special class of EMV-algebras which we call a q-variety of EMV-algebras. We present an equational base for each proper q-subvariety of the q-variety of EMV-algebras. We establish categorical equivalencies among the category of proper EMV-algebras, the category of MV-algebras with a fixed special maximal ideal, and a special category of Abelian unital ℓ -groups. [ABSTRACT FROM AUTHOR]

We present a subset of the set of idempotent functions that is both a bisemilattice and a Birkhoff system under the join- and meet-convolution operations, and show this subset is a bounded distributive lattice under the convolution operations whether or not the domain lattice of the lattice functions is distributive. This answers the first open problem raised by De Miguel et al. (2018) [3]. [ABSTRACT FROM AUTHOR]

The Shilkret integral with respect to a completely maxitive capacity is fully determined by a possibility distribution. In this paper, we introduce a weaker topological form of maxitivity and show that under this assumption the Shilkret integral is still determined by its possibility distribution for functions that are sufficiently regular. Motivated by large deviations theory, we provide a Laplace principle for maxitive integrals and characterize the possibility distribution under certain separation and convexity assumptions. Moreover, we show a maxitive integral representation result for weakly maxitive non-linear expectations. The theoretical results are illustrated by providing large deviations bounds for sequences of capacities, and by deriving a monotone analogue of Cramér's theorem. [ABSTRACT FROM AUTHOR]

Ordered Weighted Averaging (OWA) functions are a popular tool for the aggregation of real values and have been used successfully in several fields of application. The extension of these OWA functions to the multivariate setting is not unique and has been addressed separately by different disciplines. In this paper, we introduce a unifying perspective by presenting under a common framework different classes of multivariate OWA functions and discuss the main fulfilled properties by each of these classes. • A unifying framework for multivariate OWA functions is presented. • Eight different classes of multivariate OWA functions studied in the literature are reviewed. • The properties fulfilled by each of the different classes are disclosed. [ABSTRACT FROM AUTHOR]

Abstract We show that a new integral, recently introduced by Lehrer and Teper (2008) [11] , provides all the tools to determine the necessary and sufficient conditions for the existence of an additive set function, m , sandwiched between two arbitrary set functions γ and ζ : γ ≤ m ≤ ζ. [ABSTRACT FROM AUTHOR]

Abstract: Set-valued functions are an important mathematical notion and play a crucial role in several practical areas. At the same time, pseudo-analysis as a background allows extension of some classical mathematical notions to the forms that are highly applicable in some nonstandard situations. This paper focuses on pseudo-integration of set-valued functions, which is generalization of Aumann''s research, and corresponding extensions of the Jensen and Chebyshev integral inequalities to the set-valued case. Since the integral inequalities in question are widely used in various aspects of mathematics, the main motivation for the presented research lies in the possibility of expanding the applicability of these inequalities by combining the properties of set-valued functions with pseudo-analysis. [Copyright &y& Elsevier]

Abstract: We introduce a generated Choquet integral with respect to absolutely monotone and sign stable set functions. Another type of integrals defined with respect to such a set function is obtained as a limit of an appropriate sequence of generated Choquet integrals. [Copyright &y& Elsevier]

Abstract: In this paper, we extend the concept of set-valued fuzzy measures (introduced and studied by Guo and Zhang [2004. On set-valued fuzzy measures. Information Sciences 160 (2004) 13–25]) to the class of multivalued set functions taking values in the family of non-empty closed subsets of a real normed space. Also, we introduce the notions of atom and non-atomicity of a multivalued set function and present some of their properties. We also establish an extension result by preserving non-atomicity for a fuzzy multisubmeasure/multimeasure. [Copyright &y& Elsevier]

Abstract: In this paper, we establish results concerning non-atomicity and the Darboux property for multisubmeasures/fuzzy multivalued set functions defined on the Baire (respectively, Borel) -ring (respectively, ) of a locally compact, Hausdorff space. [Copyright &y& Elsevier]

In this paper, we introduce the concept of pseudo-atoms of set functions, and establish some fundamental conclusions about pseudo-atoms on the assumption of null–null-additivity. We give the necessary and sufficient conditions which insure that the set of non-null-measure contains no pseudo-atoms. Further, we obtain several decomposition theorems about pseudo-atoms and atoms of set functions. [Copyright &y& Elsevier]

Spaces in which open sets take values in the L-interval I(L) are investigated. We prove a theorem concerning the insertion of a continuous function with values in I(L) with L a complete lattice. We establish certain factorizations of the functors ωI(L) and ιI(L) with L a hypercontinuous lattice. The latter result and the insertion theorem provide a partial answer to a recent open question related to insertion of lattice-valued functions. As another application of the insertion theorem we establish a relation between complete regularity in the categories of L-topological and I(L)-topological spaces with L a frame. [Copyright &y& Elsevier]

Recently, nonadditive set functions have been used in information fusion and data mining to describe the interaction among the predictive attributes. In this case, replacing the traditional weighted average or, more generally, the linear mapping from a higher dimensional space to a one-dimensional space, relevant nonlinear integrals should be used as the aggregation tool to express how the objective attribute depends on predictive attributes. Several different types of nonlinear integral, such as the Choquet integral and the Wang integral, have been discussed in literature. In these models, the objective attribute is the integral, while the predictive attributes are the integrand. In this paper, a new concept of indeterminate integral is introduced when the universe of discourse is finite. It is a family of integrals including nonlinear integrals mentioned above and, therefore, possesses a large adaptability as a new aggregation tool. Each specified type of indeterminate integral can be obtained from the family of indeterminate integrals by specifying a decomposition of the integrand. Such a new aggregation tool can be used in information fusion and data mining as well as in expert systems. [Copyright &y& Elsevier]