Non-additive Set Function Research Articles

Nonadditive integration and some solutions of cooperative games

In the paper, we propose three schemes of nonadditive integration based on several extensions of nonadditive set function and integrand to the appropriate symmetric power of the original measurable space. A survey on the integral representation of some classic objects of the cooperative game theory, derived by nonadditive integration, is given. A universal approach for investigation of both finite and infinite games is developed. We pay a particular attention to the Shapley value, Owen multilinear extension, and support function of the core of a convex cooperative game.

Volterra-Choquet nonlinear operators

In this paper we study to what extent properties of the classical linear Volterra operators can be transferred to the nonlinear Volterra-Choquet operators, obtained by replacing the classical linear integral with respect to the Lebesgue measure, by the nonlinear Choquet integral with respect to a nonadditive set function. Compactness, Lipschitz and cyclicity properties are studied.

Convergence results in Birkhoff weak integrability

This article presents some properties of real Birkhoff weak integrable functions with respect to a non-additive set function, such as H??lder inequality, Minkowski inequality, Lebesgue type convergence theorems and Fatou type theorem.

Solving nonlinear programming problems based on the Choquet integral by a genetic algorithm

Nonadditive set functions represent contribution rate of individual feature attributes and combinations of feature attributes toward the target. Their nonadditivity describes the interaction among contributions. The generalized weighted Choquet integral with respect to a nonadditive set function serves as an aggregation tool, which may be used in the nonlinear objective function of optimization problems, to project n-dimensional feature space onto a real axis with a corresponding projection value. In this paper, we propose a special genetic algorithm model to identify the values of feature attributes that maximize the value of the objective function involving a generalized weighted Choquet integral. Thus, we may find the values of the feature attributes giving the greatest contribution toward the target. This is a generalization of linear programming to nonlinear cases.

Multi-Attribution Classification Based on Lower Integral with Genetic Algorithm

Fuzzy integral has been widely used in multi-attribution classification when the interactions exist between the attributions. Because the fuzzy measure defined on the attributions represents the weights of all the attributions and the interactions between them. The lower integral is a type of fuzzy integral with respect to fuzzy measures, which represents the minimum potential of efficiency for a group of attributions with interaction. The value of lower integrals can be evaluated through solving a linear programming problem. Considering the lower integral as a classifier, this paper investigates its implementation and performance. The difficult step in the implementation is how to learn the non-additive set function used in lower integrals. And Genetic algorithm is used to solve the problem. Finally, numerical simulations on some benchmark data sets are given.

The Gould integral in Banach lattices

In this paper, we study a Gould type integral in a new frame of Banach lattices. We consider the Gould integral of real functions relative to a non-additive set function taking values in a Banach lattice. Some continuity properties of this integral and relationships between integrability and total measurability are presented.

Inclusion and Chain Variations of Non-Additive Set Function on T-Tribe

Inclusion and Chain Variations of Non-Additive Set Function on T-Tribe

Credibility measure of fuzzy sets and applications

In order to measure a fuzzy event, credibility measure is proposed as a non-additive set function. In this paper, this concept is extended to fuzzy sets. First, a mean measure is defined by Lebesgue integral and some properties are investigated such as the monotonicity theorem, self-duality theorem and subadditivity theorem. Furthermore, by using Sugeno integral, an equilibrium measure is proposed and studied. Finally, these concepts are applied to optimisation problems, and the mean measure maximisation model and equilibrium measure maximisation model are proposed.

Lower integrals and upper integrals with respect to nonadditive set functions

Nonadditive set functions have been used as a proper mathematical model for describing the contribution rates, as well as their interaction, from predictive attributes toward some target. In information fusion and data mining, a mathematical tool is needed to aggregate contributions from each predictive attribute to the target. Essentially, it is a mapping from a higher dimensional space (sample space) to a lower dimensional space (decision space). The mapping, of course, depends on the nonadditive set function. Such a tool is generally called an integral. Due to the nonadditivity of the set function, the integral is a nonlinear functional defined on the set of the so-called nonnegative measurable functions, where each function is a point in the sample space. The upper integral, the lower integral, and the Choquet integral are some special types of nonlinear integrals. When the set of attributes is finite, all nonlinear integrals are generally called indeterminate integrals. In this case, the upper integral and the lower integral are two extreme specified indeterminate integrals. A way for calculating the newly defined integrals is shown in this paper when the universal set (the set of attributes) is finite. Recalling the concept of indeterminate integral, the value of any specified indeterminate integral of a given function can be expressed as a convex combination of the values of its upper integral and lower integral. In such a point of view, the inverse problem of the indeterminate integral with respect to a given nonadditive set function is rather easy to be solved.

A characterization of α-maximin solutions of fair division problems

This paper investigates the problem of fair division of a measurable space among a finite number of individuals and characterizes some equity concepts when preferences of each individual are represented by a nonadditive set function on a σ-algebra. We show that if utility functions of individuals satisfy continuity from below and strict monotonicity, then positive Pareto optimality is equivalent to α-maximin optimality for some α in the unit simplex and Pareto-optimal α-equitability is equivalent to α-maximin optimality. These characterizations are novel in the literature.

Ordinal aggregation and quantiles

Consider the problem of aggregating a profile of interpersonally comparable utilities into a social utility. We require that the units of measurement of utility used for agents is the same as the units of measurement for society (ordinal covariance) and a mild Pareto condition (monotonicity). We provide several representations of such social aggregation operators: a canonical representation, a Choquet expectation representation, a minimax representation, and a quantile representation (with respect to a possibly non-additive set function on the agents). We also isolate an additional condition that gives us a quantile representation with respect to a probability measure, in both the finite and infinite agents case.

Identification of Fuzzy Measures with Distorted Probability Measures

We consider the identification of fuzzy measures using a class of distorted probabilities – a scale transformation of probabilities. A fuzzy measure, which is a nonadditive set function with a high degree of freedom, enables us to express complicated interactions among evaluative items. Because of the high degree of freedom, however, it is difficult to identify all of the values μ(A) for every event A from known data μ(B), B ∈ A, where A is generally a small subclass of events. In this paper, we classify fuzzy measures by introducing “type,” and propose an identifying fuzzy measures using a classified class of distorted probabilities.

Handling Interaction in Fuzzy Production Rule Reasoning

When fuzzy production rules are used to approximate reasoning, interaction exists among rules that have the same consequent. Due to this interaction, the weighted average model frequently used in approximate reasoning does not work well in many real-world problems. In order to model and handle this interaction, this paper proposes to use a nonadditive nonnegative set function to replace the weights assigned to rules having the same consequent, and to draw the reasoning conclusion based on an integral with respect to the nonadditive nonnegative set function, rather than on the weighted average model. Handling interaction in fuzzy production rule reasoning in this way can lead to a good understanding of the rules base and an improvement of reasoning accuracy. This paper also investigates how to determine from data the nonadditive set function that cannot be specified by a domain expert.

Darboux property of a non-additive set function

As is known, the range of a finite positive non-atomic measure on a σ-algebra is a closed interval. In the present paper it is proved that this property holds also in a broad class of non-additive set functions on an F-algebra if the non-atomicity is replaced by the Saks property.

Discover dependency pattern among attributes by using a new type of nonlinear multiregression

Multiregression is one of the most common approaches used to discover dependency pattern among attributes in a database. Nonadditive set functions have been applied to deal with the interactive predictive attributes involved, and some nonlinear integrals with respect to nonadditive set functions are employed to establish a nonlinear multiregression model describing the relation between the objective attribute and predictive attributes. The values of the nonadditive set function play a role of unknown regression coefficients in the model and are determined by an adaptive genetic algorithm from the data of predictive and objective attributes. Furthermore, such a model is now improved by a new numericalization technique such that the model can accommodate both categorical and continuous numerical attributes. The traditional dummy binary method dealing with the mixed type data can be regarded as a very special case of our model when there is no interaction among the predictive attributes and the Choquet integral is used. When running the algorithm, to avoid a premature during the evolutionary procedure, a technique of maintaining diversity in the population is adopted. A test example shows that the algorithm and the relevant program have a good reversibility for the data. © 2001 John Wiley & Sons, Inc.16: 949–962 (2001)

Nonlinear nonnegative multiregressions based on Choquet integrals

Using a nonadditive set function to describe the interaction among attributes, a new nonlinear nonnegative multiregression is established based on Choquet integrals with respect to the set function. Regarding the values of the set function as unknown regression parameters, an evolutionary computation can be used to determine them when necessary data are available. Such a model is a generalization of the traditional linear multiregression. It provides an effective regression tool in some real problems where the linear multiregression model and the second-order multiregression model fail. This new method has a wide applicability in the areas of information fusion and data mining, as well as in the areas of decision making, image processing, pattern recognition, medical and industrial diagnoses, and expert systems.

Using genetic algorithms to determine nonnegative monotone set functions for information fusion in environments with random perturbation

Due to some inherent interactions among diverse information sources, the classical weighted average method is not adequate for information fusion in many real problems. To describe the interactions, an intuitive and effective way is to use an appropriate nonadditive set function. Instead of the weighted average method, which is essentially the Lebesgue integral, we should thus use the Choquet integral or some other nonlinear integrals. To apply this alternative, more realistic approach to information fusion, we need to determine the nonadditive set function from given input-output data, viewing the nonlinear integral as a multi-input one-output system. In this paper, we employ an adaptive genetic algorithm to construct an approximate optimal nonnegative monotone set function from given input-output data in an environment with random perturbation. An example for diverse strengths of random perturbation is shown to demonstrate the efficiency of this algorithm.

A genetic algorithm for determining nonadditive set functions in information fusion

As a classical aggregation tool, the weighted average method is widely used in information fusion. It is the Lebesgue integral with respect to the weights essentially. Due to some inherent interaction among diverse information sources, the weighted average method does not work well in many real problems. To describe the interaction, an intuitive and effective way is to replace the additive weights with a nonadditive set function defined on the power set of the set of all information sources. Instead of the weighted average method, we should use the Choquet integral or some other nonlinear integrals, especially, the new nonlinear integral introduced by the authors recently. The crux of making such an improvement is how to determine the nonadditive set function from given input-output data when the nonlinear integral is viewed as a multi-input single-output system. In this paper, we employ a specially designed genetic algorithm to realize the optimization in determining the nonadditive set function.

Risk-adjusted credibility premiums using distorted probabilities

Denneberg (1990) and Wang (1996a) propose that one calculate risk-adjusted insurance premiums as the expectation with respect to a distorted probability measure, a non-additive set function. This premium principle is supported by the theories of decision making of Yaari (1987) and of Schmeidler (1989). Denneberg (1994a) presents three conditioning rules for updating non-additive set functions in light of available information. In this work, we show how to apply these three update rules to calculate a risk-adjusted credibility premium and, thereby, combine credibility theory with this relatively new premium principle. Our main result is that, for some pairs of distortion function and update rule, one gets the same risk-adjusted credibility premium by distorting the predictive probability distribution, as required by the theory of Yaari, or by updating the distorted probability, as required by the theory of Schmeidler.

Choquet integrals in potential theory

This is a survey of various applications of the notion of the Choquet integral to questions in Potential Theory, i.e. the integral of a function with respect to a non-additive set function on subsets of Euclidean $n$-space, capacity. The Choquet integral is, in a sense, a nonlinear extension of the standard Lebesgue integral with respect to the linear set function, measure. Applications include an integration principle for potentials, inequalities for maximal functions, stability for solutions to obstacle problems, and a refined notion of pointwise differentiation of Sobolev functions.