an axiomatic approach to the measurement of the amount of interaction among criteria or players in the framework of multicriteria decision making based on fuzzy measures or cooperative game theory an axiomatization of the concept of amount of interaction index is given it is based on four axioms a natural property called absence of interaction a boundary condition for which the index reduces to the shapley or banzhaf interaction index a pseudo additivity axiom and a recursivity property the resulting index which can be interpreted as a measure of the average degree of independence among criteria or players takes a particularly simple and natural form for supermodular or submodular measures the approach to the notion of interaction adopted in this paper can be seen as complementary to that recently adopted by grabisch and roubens

an extension of fuzzy measures to multisets and its relation to distorted probabilities fuzzy measures are monotonic set functions on a reference set they generalize probabilities replacing the additivity condition by monotonicity the typical application of these measures is with fuzzy integrals fuzzy integrals integrate a function with respect to a fuzzy measure and they can be used to aggregate information from a set of sources opinions from experts or criteria in a multicriteria decision making problem in this context background knowledge on the sources is represented by means of the fuzzy measures for example interactions between criteria are represented by means of nonadditive measures in this paper we introduce fuzzy measures on multisets we propose a general definition and we then introduce a family of fuzzy measures for multisets which we show to be equivalent to distorted probabilities when the multisets are restricted to proper sets

an axiomatic approach to the definition of the entropy of a discrete choquet capacity to extend the classical shannon entropy to nonadditive measures marichal recently introduced the concept of generalized entropy for discrete choquet capacities we provide a first axiomatization of this new concept on the basis of three axioms the symmetry property a boundary condition for which the entropy reduces to the shannon entropy and a generalized version of the well known recursivity property we also show that this generalized entropy fulfills several properties considered as requisites for defining an entropy like measure lastly we provide an interpretation of it in the framework of aggregation by the discrete choquet integral

symmetric hilbert spaces arising from species of structures symmetric hilbert spaces such as the bosonic and the fermionic fock spaces over some lsquo one particle space mathcal k are formed by certain symmetrization procedures performed on the full fock space we investigate alternative ways of symmetrization by building on joyal s notion of a combinatorial species any such species f gives rise to an endofunctor gamma f of the category of hilbert spaces with contractions mapping a hilbert space mathcal k to a symmetric hilbert space gamma f mathcal k with the same symmetry as the species f a general framework for annihilation and creation operators on these spaces is developed and compared to the generalised brownian motions of r speicher and m bozejko as a corollary we find that the commutation relation a ia j a j a i f n delta ij with na i a i n a i admits a realization on a symmetric hilbert space whenever f has a power series with infinite radius of convergence and positive coefficients

a graphical interpretation of the choquet integral the choquet integral 1953 is a fuzzy measure used as a powerful aggregation operator over a finite set of elements we present a graphical interpretation of the choquet integral viewed as an aggregation operator in the case of two elements the interpretation relies on the interaction representation introduced by the author

on the dispersion measure of owa operators we describe some basic features of the owa operator we turn to the problem of determining the weights associated with this operator and particularly the maximal dispersion entropy approach we consider the possibility of using minimization of dispersion after discussing concerns with both maximization and minimization of dispersion we investigate the possibility of finding an optimal solution intermediate to these extremes we next consider alternative measures of dispersion we introduce a fundamental requirement for a measure of dispersion called the preference for equal division a number of general classes of dispersion measures are provided notable among these are those based on t norm and t conorm operators

a multivariate bayesian model for embryonic growth most longitudinal growth curve models evaluate the evolution of each of the anthropometric measurements separately when applied to a reference population this exercise leads to univariate reference curves against which new individuals can be evaluated however growth should be evaluated in totality that is by evaluating all body characteristics jointly recently cole et al suggested the superimposition by translation and rotation sitar model which expresses individual growth curves by three subject specific parameters indicating their deviation from a flexible overall growth curve this model allows the characterization of normal growth in a flexible though compact manner in this paper we generalize the sitar model in a bayesian way to multiple dimensions the multivariate sitar model allows us to create multivariate reference regions which is advantageous for prediction the usefulness of the model is illustrated on longitudinal measurements of embryonic growth obtained in the first semester of pregnancy collected in the ongoing rotterdam predict study further we demonstrate how the model can be used to find determinants of embryonic growth copyright 2015 john wiley sons ltd

determination of weights of interacting criteria from a reference set abstract in this paper we present a model allowing to determine the weights related to interacting criteria this is done on the basis of the knowledge of a partial ranking over a reference set of alternatives prototypes a partial ranking over the set of criteria and a partial ranking over the set of interactions between pairs of criteria

derivative estimation for longitudinal data analysis examining features of blood pressure measured repeatedly during pregnancy estimating velocity and acceleration trajectories allows novel inferences in the field of longitudinal data analysis such as estimating change regions rather than change points and testing group effects on nonlinear change in an outcome ie a nonlinear interaction in this article we develop derivative estimation for 2 standard approaches polynomial mixed models and spline mixed models we compare their performance with an established method principal component analysis through conditional expectation through a simulation study we then apply the methods to repeated blood pressure bp measurements in a uk cohort of pregnant women where the goals of analysis are to i identify and estimate regions of bp change for each individual and ii investigate the association between parity and bp change at the population level the penalized spline mixed model had the lowest bias in our simulation study and we identified evidence for bp change regions in over 75 of pregnant women using mean velocity difference revealed differences in bp change between women in their first pregnancy compared with those who had at least 1 previous pregnancy we recommend the use of penalized spline mixed models for derivative estimation in longitudinal data analysis

multi level repeated measures growth modelling using extended spline functions this paper explores the fitting of multi level models to growth data over a wide age range using a new class of extended spline models these extend conventional splines based on a function representation by allowing variable order functions and by including fractional polynomial terms the work focuses on modelling human growth in height and head circumference with example data sets the procedures can be used with covariates and for comparing population parameters

minimum classification error training for choquet integrals with applications to landmine detection a novel algorithm for discriminative training of choquet integral based fusion operators is described fusion is performed by choquet integration of classifier outputs with respect to fuzzy measures the fusion operators are determined by the parameters of fuzzy measures these parameters are found by minimizing a minimum classification error mce objective function the minimization is performed with respect to a special class of measures the sugeno lambda measures an analytic expression is derived for the gradient of the choquet integral with respect to the sugeno lambda measure the new algorithm is applied to a landmine detection problem and compared to previous techniques

an example of a generalized brownian motion we present an example of a generalized brownian motion it is given by creation and annihilation operators on a twisted fock space ofl2 these operators fulfill for a fixed 1 1 the relationsc f c g c g c f f g 1 f g l2 we show that the distribution of these operators with respect to the vacuum expectation is a generalized gaussian distribution in the sense that all moments can be calculated from the second moments with the help of a combinatorial formula we also indicate that our brownian motion is one component of ann dimensional brownian motion which is invariant under the quantum groups u n of woronowicz with v2

k intolerant capacities and choquet integrals we define an aggregation function to be at most k intolerant if it is bounded from above by its kth lowest input value applying this definition to the discrete choquet integral and its underlying capacity we introduce the concept of k intolerant capacities which when varying k from 1 to n cover all the possible capacities on n objects just as the concepts of k additive capacities and p symmetric capacities have been previously introduced essentially to overcome the problem of computational complexity of capacities k intolerant capacities are proposed here for the same purpose but also for dealing with intolerant or tolerant behaviors of aggregation we also introduce axiomatically indices to appraise the extent to which a given capacity is k intolerant and we apply them on a particular recruiting problem

assessing non additive utility for multicriteria decision aid abstract in the framework of multi attribute utility theory maut several methods have been proposed to build a decision maker s dm utility function representing his her preferences among such methods the uta method infers an additive utility function from a set of exemplary decisions using linear programming however the uta method does not guarantee to find a utility function which is coherent with the available information this drawback is due to the underlying utility model of uta viz the additive one which does not allow to include additional information such as an interaction among criteria in this paper we present a methodology for building a non additive utility function in the framework of the so called fuzzy integrals which permits to model preference structures with interaction between criteria like in the uta method we aim at searching a utility function representing the dm s preferences but unlike uta the functional form is a specific fuzzy integral choquet integral as a result we obtain weights which can be interpreted as the importance of coalitions of criteria exploiting the potential interaction between criteria as already proposed by other authors however within the same framework we obtain also the marginal utility functions relative to each one of the considered criteria that are evaluated on a common scale as a consequence of the implemented methodology finally we illustrate our approach with an example

on the extension of pseudo boolean functions for the aggregation of interacting criteria abstract the paper presents an analysis on the use of integrals defined for non additive measures or capacities as the choquet and the sipos integral and the multilinear model all seen as extensions of pseudo boolean functions and used as a means to model interaction between criteria in a multicriteria decision making problem the emphasis is put on the use besides classical comparative information of information about difference of attractiveness between acts and on the existence for each point of view of a neutral level allowing to introduce the absolute notion of attractive or repulsive act it is shown that in this case the sipos integral is a suitable solution although not unique properties of the sipos integral as a new way of aggregating criteria are shown with emphasis on the interaction among criteria

realizability of a model in infinite statistics following greenberg and others we study a space with a collection of operatorsa k satisfying the q mutator relations a l a k a l k l corresponding forq 1 to classical bose and fermi statistics we show that then n matrixa n q representing the scalar products ofn particle states is positive definite for alln ifq lies between 1 and 1 so that the commutator relations have a hilbert space representation in this case this has also been proved by fivel and by bozejko and speicher we also give an explicit factorization ofa n q as a product of matrices of the form 1 q jt 1 with 1 j n andt a permutation matrix in particular a n q is singular if and only ifq m 1 for some integerm of the formk 2 k 2 k n

probabilistically weighted owa aggregation in decision making under uncertainty we have a collection of alternatives from which we must choose one associated with each alternative is an uncertainty profile consisting of the set of possible outcomes that can occur if we choose this alternative along with some indication of the uncertainty associated with the outcomes since it is difficult to compare these uncertainty profiles one commonly used approach is to obtain a representative value an aggregation of the information in the uncertainty profile and use these to compare the alternatives the method that we use to aggregate the information in the uncertainty profile must reflect aspects of the decision attitude of the decision maker here we look at the role that the ordered weighted averaging owa can play in the aggregation process used to obtain these representative values since an uncertainty profile can involve a discrete set of possible outcomes or an interval of possible outcomes we provide the owa operator with the capability to perform aggregations in either of these environments in addition we provide the owa operator with the ability to perform aggregations in situations where the arguments being aggregated the potential outcomes have an associated uncertainty probability of occurrence particularly notable is the extension the owa operator to perform aggregation with arguments consisting of a continuum of values with an associated probability density function we also look at the cases where the uncertainty is modeled with a belief structure as well a possibility distribution

q gaussian processes non commutative and classical aspects we examine for 1 q 1 q gaussian processes i e families of operators non commutative random variables where the a t fulfill the q commutation relations for some covariance function equipped with the vacuum expectation state we show that there is a q analogue of the gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q gaussian processes into corresponding and much simpler questions in the underlying hilbert space in particular we use this idea to show that a large class of q gaussian processes possesses a non commutative kind of markov property which ensures that there exist classical versions of these non commutative processes this answers an old question of frisch and bourret fb

the h index and the number of citations two fuzzy integrals in this paper we review two of the most well known citation indexes and establish their connections with the choquet and sugeno integrals in particular we show that the recently established h index is a particular case of the sugeno integral and that the number of citations corresponds to the choquet integral in both cases they use the same fuzzy measure the results presented here permit one to envision new indexes defined in terms of fuzzy integrals using other types of fuzzy measures a few considerations in this respect are also included in this paper indexes for taking into account recent research and the publisher credibility are outlined

minimum variance capacity identification abstract in the framework of multi criteria decision making whose aggregation process is based on the choquet integral we present a maximum entropy like method enabling to determine if it exists the least specific capacity compatible with the initial preferences of the decision maker the proposed approach consists in solving a strictly convex quadratic program whose objective function is equivalently either the opposite of a generalized entropy measure or the variance of the capacity the application of the proposed approach is illustrated on two examples

the application of fuzzy integrals in multicriteria decision making abstract this paper presents a synthesis on the application of fuzzy integrals as an innovative tool for criteria aggregation in decision problems the main point is that fuzzy integrals are able to model interaction between criteria in a flexible way the methodology has been elaborated mainly in japan and has been applied there successfully in various fields such as design reliability evaluation of goods etc it seems however that this technique is still very little known in europe it is one of the aims of this review to disseminate this emerging technology in many industrial fields

tolerant or intolerant character of interacting criteria in aggregation by the choquet integral abstract in many multi criteria decision making problems the decision criteria present some interaction whose nature may vary from one situation to another for example some criteria may be statistically correlated thus making them somewhat redundant or opposed some others may be somewhat substitutive or complementary depending on the behavior of the decision maker some others may be decisive in the sense that the global score of any alternative obtained by aggregation is bounded by the partial score along one of them in this paper we analyze this latter form of interaction when a criterion bounds the global score from above it is called a blocker or a veto due to its rather intolerant character when it bounds the global score from below it is then called a pusher or a favor we thus investigate the tolerance of criteria or equivalently the tolerance of the weighted aggregation operator here the choquet integral that is used to aggregate criteria more specifically we propose axiomatically indices to appraise the extent to which each criterion behaves like a veto or a favor in the aggregation by the choquet integral previous to this we also propose global tolerance degrees measuring the extent to which the choquet integral is conjunctive or disjunctive

generalised brownian motion and second quantisation a new approach to the generalised brownian motion introduced by m bouzejko and r speicher is described based on symmetry rather than deformation the symmetrisation principle is provided by joyal s notions of tensorial and combinatorial species any such species v gives rise to an endofunctor fv of the category of hilbert spaces with contractions a generalised brownian motion is an algebra of creation and annihilation operators acting on fv h for arbitrary hilbert spaces h and having a prescription for the calculation of vacuum expectations in terms of a function t on pair partitions the positivity is encoded by a semigroup of broken pair partitions whose representation space with respect to t is v the existence of the second quantisation as functor t from hilbert spaces to noncommutative probability spaces is proved to be equivalent to the multiplicative property of the function t for a certain one parameter interpolation between the fermionic and the free brownian motion it is shown that the field algebras k are type ii1 factors when k is infinite dimensional

une theorie combinatoire des series formelles abstract this paper presents a combinatorial theory of formal power series the combinatorial interpretation of formal power series is based on the concept of species of structures a categorical approach is used to formulate it a new proof of cayley s formula for the number of labelled trees is given as well as a new combinatorial proof due to g labelle of lagrange s inversion formula polya s enumeration theory of isomorphism classes of structures is entirely renewed recursive methods for computing cycle index polynomials are described a combinatorial version of the implicit function theorem is stated and proved the paper ends with general considerations on the use of coalgebras in combinatorics

learning k maxitive fuzzy measures from data by mixed integer programming abstract fuzzy measures model interactions between the inputs in aggregation problems their complexity grows exponentially with the dimensionality of the problem and elicitation of fuzzy measure coefficients either from domain experts or from empirical data poses a significant challenge the notions of k additivity and k maxitivity simplify the fuzzy measures by limiting interactions to subsets of up to k elements learning fuzzy measures from data is an important elicitation technique which relies on solving an optimisation problem a heuristic learning algorithm to identify k maxitive fuzzy measures from the data on the basis of hlms heuristic least mean squares was recently presented in murillo et al 2017 11 we present an alternative formulation of the fitting problem which delivers a globally optimal solution through the solution of a mixed integer programming mip problem to deal with high computational cost of mip in moderate to large dimensions we also propose a simple mip relaxation technique which involves solving two related linear programming problems we also provide a linear programming formulation for fitting k tolerant fuzzy measures we discuss implementations of the fitting methods and present the results of numerical experiments

entropy of discrete choquet capacities we introduce a measure of entropy for any discrete choquet capacity and we interpret it in the setting of aggregation by the choquet integral

preference learning using the choquet integral the case of multipartite ranking we propose a novel method for preference learning or more specifically learning to rank where the task is to learn a ranking model that takes a subset of alternatives as input and produces a ranking of these alternatives as output just like in the case of conventional classifier learning training information is provided in the form of a set of labeled instances with labels or say preference degrees taken from an ordered categorical scale this setting is known as multipartite ranking in the literature our approach is based on the idea of using the discrete choquet integral as an underlying model for representing ranking functions being an established aggregation function in fields such as multiple criteria decision making and information fusion the choquet integral offers a number of interesting properties that make it attractive from a machine learning perspective too the learning problem itself comes down to properly specifying the fuzzy measure on which the choquet integral is defined this problem is formalized as a margin maximization problem and solved by means of a cutting plane algorithm the performance of our method is tested on a number of benchmark datasets

a q deformation of the gauss distribution the q deformed commutation relation aa qa a 1 for the harmonic oscillator is considered with q 1 1 an explicit representation generalizing the bargmann representation of analytic functions on the complex plane is constructed in this representation the distribution of a a in the vacuum state is explicitly calculated this distribution is to be regarded as the natural q deformation of the gaussian

fuzzy integral in multicriteria decision making abstract this paper presents a synthesis on the use of fuzzy integral as an aggregation operator in multicriteria decision making definitions essential properties are given and compared to those of usual aggregation operators a quick survey on applications is given

choquet fuzzy integral based hierarchical networks for decision analysis a choquet fuzzy integral based approach to hierarchical network implementation is investigated in this approach we generalized the fuzzy integral as an excellent component for decision analysis the generalization involves replacing the max or min operator in information aggregation with a fuzzy integral based neuron resulting in increased flexibility the characteristics of the choquet fuzzy integral are studied and a network based decision analysis framework is proposed the trainable hierarchical network can be implemented utilizing the fuzzy integral based neurons and connectives the training algorithms are derived and several examples given to illustrate the behaviors of the networks also we present a decision making experiment using the proposed network to learn appropriate functional relationships in the defective numeric fields detection domain

2 additive capacity identification methods from multicriteria correlation preference information the essential role of the particular families of capacities and the capacity identification methods is to help the decision maker to deal with the exponential complexity inherent in the construction process of the capacity the 2 additive capacities appear to be the most popular among the particular families of capacities since they permit to model interactions between criteria while preserving simplicity besides the preference with respect to the decision criteria most of the capacity identification methods also need to provide the desired overall evaluations of the decision alternatives in the learning set which is a time consuming task for the decision maker in this paper we propose some models to identify 2 additive capacities only from a kind of refined preference information with respect to the decision criteria called the multicriteria correlation preference information mccpi the mccpi is a group of 2 d preference information which can be described and obtained by the refined diamond diagram the common principle of the proposed models is to minimize the different kinds of deviations between the mccpi and the most desired 2 additive capacity ies a multicriteria decision making example is presented to show the feasibility of the proposed methods and a 2 d scale of the mccpi is also introduced in the further discussion of the illustrative example

applying fuzzy measures for considering interaction effects in root dispersal models this paper presents a successful application of fuzzy measures for describing interaction in fine root biomass dispersal models fine roots are roots with diameter smaller than 2mm which are responsible for the soil water reception of trees a root dispersal model based on point process theory is presented in this approach interaction effects are taken into account by modelling the intensity function using discrete choquet integrals it is shown that the combination of cluster point processes and the modelling of interaction effects by fuzzy measures lead to improved models for the root dispersion an example for the root dispersion in a stand with beeches and spruces in saxony germany is presented

fock space associated to coxeter groups of type b abstract in this article we construct a generalized gaussian process coming from coxeter groups of type b it is given by creation and annihilation operators on an q fock space which satisfy the commutation relation b q x b q y q b q y b q x x y i x y q 2 n where x y are elements of a complex hilbert space with a self adjoint involution x x and n is the number operator with respect to the grading on the q fock space we give an estimate of the norms of creation operators we show that the distribution of the operators b q x b q x with respect to the vacuum expectation becomes a generalized gaussian distribution in the sense that all mixed moments can be calculated from the second moments with the help of a combinatorial formula related with set partitions our generalized gaussian distribution is associated to the orthogonal polynomials called the q meixner pollaczek polynomials yielding the q hermite polynomials when 0 and free meixner polynomials when q 0

a universal integral as common frame for choquet and sugeno integral the choquet and the sugeno integral provide a useful tool in many problems in engineering and social choice where the aggregation of data is required however their applicability is restricted because of the special operations used in the construction of these integrals therefore we provide a concept of integrals generalizing both the choquet and the sugeno case for functions with values in the nonnegative real numbers universal integrals are introduced and investigated which can be defined on arbitrary measurable spaces and for arbitrary monotone measures for a fixed pseudo multiplication on the nonnegative real numbers the smallest and the greatest universal integrals are given finally another construction method for obtaining universal integrals is introduced and the restriction to the unit interval i e to fuzzy integrals is considered

a review of methods for capacity identification in choquet integral based multi attribute utility theory applications of the kappalab r package the application of multi attribute utility theory whose aggregation process is based on the choquet integral requires the prior identification of a capacity the main approaches to capacity identification proposed in the literature are reviewed and their advantages and inconveniences are discussed all the reviewed methods have been implemented within the kappalab r package their application is illustrated on a detailed example

k order additive discrete fuzzy measures and their representation abstract in order to face with the complexity of discrete fuzzy measures we propose the concept of k orderadditive fuzzy measure including usual additive measures and fuzzy measures every discrete fuzzy measure is a k order additive fuzzy measure for a unique k a related topic of the paper is to introduce an alternative representation of fuzzy measures called the interaction representation which sets and extends in a common framework the shapley value and the interaction index proposed by murofushi and soneda

multivariate multilevel spline models for parallel growth processes application to weight and mean arterial pressure in pregnancy growth models are commonly used in life course epidemiology to describe growth trajectories and their determinants or to relate particular patterns of change to later health outcomes however methods to analyse relationships between two or more change processes occurring in parallel in particular to assess evidence for causal influences of change in one variable on subsequent changes in another are less developed we discuss linear spline multilevel models with a multivariate response and show how these can be used to relate rates of change in a particular time period in one variable to later rates of change in another variable by using the variances and covariances of individual level random effects for each of the splines we describe how regression coefficients can be calculated for these associations and how these can be adjusted for other parameters such as random effect variables relating to baseline values or rates of change in earlier time periods and compare different methods for calculating the standard errors of these regression coefficients we also show that these models can equivalently be fitted in the structural equation modelling framework and apply each method to weight and mean arterial pressure changes during pregnancy obtaining similar results for multilevel and structural equation models this method improves on the multivariate linear growth models which have been used previously to model parallel processes because it enables nonlinear patterns of change to be modelled and the temporal sequence of multivariate changes to be determined with adjustment for change in earlier time periods

learning choquet integral based metrics for semisupervised clustering we consider an application of fuzzy measures to the problem of metric learning in semisupervised clustering we investigate the necessary and sufficient conditions on the underlying fuzzy measure that make the discrete choquet integral suitable for defining a metric as a byproduct we can obtain the analogous conditions for the ordered weighted averaging owa operators which constitute a special case we then generalize these results for power based choquet and owa operators we show that this metric learning problem can be formulated as a linear programming problem and specify the required sets of linear constraints we present the results of numerical experiments on artificial and real world datasets which illustrate the potential usefulness and limitations of this construction

non additive robust ordinal regression a multiple criteria decision model based on the choquet integral abstract within the multicriteria aggregation disaggregation framework ordinal regression aims at inducing the parameters of a decision model for example those of a utility function which have to represent some holistic preference comparisons of a decision maker dm usually among the many utility functions representing the dm s preference information only one utility function is selected since such a choice is arbitrary to some extent recently robust ordinal regression has been proposed with the purpose of taking into account all the sets of parameters compatible with the dm s preference information until now robust ordinal regression has been implemented to additive utility functions under the assumption of criteria independence in this paper we propose a non additive robust ordinal regression on a set of alternatives a whose utility is evaluated in terms of the choquet integral which permits to represent the interaction among criteria modelled by the fuzzy measures parameterizing our approach in our methodology besides holistic pairwise preference comparisons of alternatives from a subset of reference alternatives a a the dm is also requested to express the intensity of preference on pairs of alternatives from a and to supply pairwise comparisons on the importance of criteria and the sign and intensity of interaction among pairs of criteria the output of our approach defines a set of fuzzy measures capacities such that the corresponding choquet integral is compatible with the dm s preference information moreover using linear programming our decision model establishes two preference relations for any a b a the necessary weak preference relation if for all compatible fuzzy measures the utility of a is not smaller than the utility of b and the possible weak preference relation if for at least one compatible fuzzy measure the utility of a is not smaller than the utility of b

an axiomatic approach of the discrete choquet integral as a tool to aggregate interacting criteria the most often used operator to aggregate criteria in decision making problems is the classical weighted arithmetic mean however in many problems the criteria considered interact and a substitute to the weighted arithmetic mean has to be adopted we show that under rather natural conditions the discrete choquet integral is an adequate aggregation operator that extends the weighted arithmetic mean by taking into consideration of the interaction among criteria the axiomatic that supports the choquet integral is presented and an intuitive approach is proposed as well

combinatorial structure of the polytope of 2 additive measures in this article we study the polytope of 2 additive measures an important subpolytope of the polytope of fuzzy measures for this polytope we obtain its combinatorial structure namely the adjacency structure and the structure of two dimensional faces three dimensional faces and so on based on this information we build a triangulation of this polytope satisfying that all simplices in the triangulation have the same volume as a consequence this allows a very simple and appealing way to generate points in a random way in this polytope an interesting problem arising in the practical identification of 2 additive measures finally we also derive the volume and the centroid of this polytope

the q t gaussian process the q t fock space fq t h introduced in this paper is a deformation of the q fock space of bozejko and speicher the corresponding creation and annihilation operators now satisfy the commutation relation aq t f aq t g qaq t g aq t f f g htn turn mathjax off a defining relation of the chakrabarti jagannathan deformed quantum oscillator algebra the moments of the deformed gaussian element sq t h aq t h aq t h are encoded by the joint statistics of crossings and nestings in pair partitions the q 0 t specialization yields a natural single parameter deformation of the full boltzmann fock space of free probability with the corresponding semicircular measure variously encoded via the rogers ramanujan continued fraction the t airy function the t catalan numbers of carlitz riordan and the first order statistics of the reduced wigner process

classification by nonlinear integral projections a new method based on nonlinear integral projections for classification is presented the contribution rate of each combination of the feature attributes including each singleton toward the classification is represented by a fuzzy measure the nonadditivity of the fuzzy measure reflects the interactions among the feature attributes the weighted choquet integral with respect to the fuzzy measure serves as an aggregation tool to project the feature space onto a real axis optimally according to an error criterion and the classifying attribute is properly numerical analysed on the axis simultaneously making the classification simple to implement the classification we need to determine the unknown parameters the values of fuzzy measure and the weight function this can be done by running an adaptive genetic algorithm on the given training data the new classifier is tested by recovering the preset parameters from a set of artificial training data generated from these parameters it also performs well on several real world data sets beyond discriminating classes this method can also learn the scaling requirements and the respective importance indexes of the feature attributes as well as the relationships among them a comprehensive discussion on the semantic and geometric meanings of the parameters is given moreover we show how these parameters values can be used for short listing important feature attributes to reduce the complexity dimensions of the classification problem our new method also compares favorably with other methods on some well known real world benchmarks

the choquet integral for the aggregation of interval scales in multicriteria decision making this paper addresses the question of which models fit with information concerning the preferences of the decision maker over each attribute and his preferences about aggregation of criteria interacting criteria we show that the conditions induced by these information plus some intuitive conditions lead to a unique possible aggregation operator the choquet integral

classification of heterogeneous fuzzy data by choquet integral with fuzzy valued integrand as a fuzzification of the choquet integral the defuzzified choquet integral with fuzzy valued integrand dcifi takes a fuzzy valued integrand and gives a crisp valued integration result in this paper the dcifi acts as a projection to project high dimensional heterogeneous fuzzy data to one dimensional crisp data to handle the classification problems involving different data forms such as crisp data interval values fuzzy numbers and linguistic variables simultaneously the nonadditivity of the signed fuzzy measure applied in the dcifi can represent the interaction among the measurements of features towards the discrimination of classes values of the signed fuzzy measure in the dcifi are considered to be unknown parameters which should be learned before the classifier is used to classify new data we have implemented a genetic algorithm ga based adaptive classifier learning algorithm to optimally learn the signed fuzzy measure values and the classified boundaries simultaneously the performance of our algorithm has been tested both on synthetic and real data the experimental results are satisfactory and outperform those of existing methods such as the fuzzy decision trees and the fuzzy neuro networks

an empirical study of statistical properties of the choquet and sugeno integrals this paper investigates the statistical properties of the choquet and sugeno integrals used as multiattribute models the investigation is done on an empirical basis and focuses on two topics the distribution of the output of these integrals when the input is corrupted with noise and the robustness of these models when they are identified using some set of learning data through some learning procedure

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

compromise principle based methods of identifying capacities in the framework of multicriteria decision analysis abstract the basic aim of the compromise principle employed in this paper is to seek the capacity by which each alternative has a relatively equal chance to reach as close as possible to its highest reachable overall evaluation according to the compromise principle three types of capacity identification methods the simple arithmetic average based compromise method the least squares based compromise method and the linear programming based compromise method are proposed the input information required for the compromise principle based identification methods consists of the preference information with respect to the decision criteria and the partial evaluations of the alternatives provided by the decision maker an illustrative example is given to show the processes of the proposed methods and a comparison analysis with the maximum entropy principle based identification method is also presented

on the structure of non commutative white noises we consider the concepts of continuous bernoulli systems and non commutative white noises we address the question of isomorphism of continuous bernoulli systems and show that for large classes of quantum levy processes one can make quite precise statements about the time behaviour of their moments