a simplified expression of the shapley function for fuzzy game abstract in this paper a simplified expression of the shapley function for games with fuzzy coalition is proposed which can be regarded as the generalization of shapley functions defined in some particular games with fuzzy coalition the simplified expression of the shapley function is compared with two definitions established by butnariu tsurumi et al a conclusion is drawn that the simplified expression of the shapley function is equivalent to butnariu s definition when characteristic function is a game with proportional values and is equivalent to tsurumi s definition when characteristic function is a game with choquet integral forms furthermore from an angle of interaction between two participation levels the properties of the two games defined by butnariu and tsurumi are respectively studied

modeling interaction phenomena using fuzzy measures on the notions of interaction and independence interaction phenomena among elements of a finite set can be modeled by a discrete fuzzy measure the notion of marginal interaction between two elements is at the root of the definition of interaction we first extend this notion by defining that of marginal interaction among pairwise disjoint subsets and we study some of its properties then we generalize the notion of interaction index to pairwise disjoint subsets finally we formalize the intuitive notion of independence and we propose the concept of measure of marginal amount of interaction in order to fill the gap between the notions of interaction and independence

on the set of imputations induced by the k additive core an extension to the classical notion of core is the notion of k additive core that is the set of k additive games which dominate a given game where a k additive game has its mobius transform or harsanyi dividends vanishing for subsets of more than k elements therefore the 1 additive core coincides with the classical core the advantages of the k additive core is that it is never empty once k geq 2 and that it preserves the idea of coalitional rationality however it produces k imputations that is imputations on individuals and coalitions of at most k individuals instead of a classical imputation therefore one needs to derive a classical imputation from a k order imputation by a so called sharing rule the paper investigates what set of imputations the k additive core can produce from a given sharing rule

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

weighted banzhaf power and interaction indexes through weighted approximations of games the banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game the banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players it was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo boolean functions considering certain weighted versions of this approximation problem we define a class of weighted interaction indexes that generalize the banzhaf interaction index we show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties finally we give an interpretation of the banzhaf and shapley interaction indexes as centers of mass of this subclass of interaction indexes

on the structure of some families of fuzzy measures the generation of fuzzy measures is an important question arising in the practical use of these operators in this paper we deal with the problem of developing a random generator of fuzzy measures more concretely we study some of the properties that any random generator should satisfy these properties lead to some theoretical problems concerning the group of isometries that we tackle in this paper for some subfamilies of fuzzy measures

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

on the use of the choquet integral with fuzzy numbers in multiple criteria decision support this paper presents a multiple criteria decision support approach in order to build a ranking and suggest a best choice on a set of alternatives the partial evaluations of the alternatives on the points of view can be fuzzy numbers the aggregation is performed through the use of a fuzzy extension of the choquet integral we detail how to assess the coefficients of the aggregation operator by using alternatives which are well known to the decision maker and which originate from his domain of expertise

the symmetric sugeno integral we propose an extension of the sugeno integral for negative numbers in the spirit of the symmetric extension of choquet integral also called sipos integral our framework is purely ordinal since the sugeno integral has its interest when the underlying structure is ordinal we begin by defining negative numbers on a linearly ordered set and we endow this new structure with a suitable algebra very close to the ring of real numbers in a second step we introduce the mobius transform on this new structure lastly we define the symmetric sugeno integral and show its similarity with the symmetric choquet integral

sensor selection for estimation with correlated measurement noise in this paper we consider the problem of sensor selection for parameter estimation with correlated measurement noise we seek optimal sensor activations by formulating an optimization problem in which the estimation error given by the trace of the inverse of the bayesian fisher information matrix is minimized subject to energy constraints fisher information has been widely used as an effective sensor selection criterion however existing information based sensor selection methods are limited to the case of uncorrelated noise or weakly correlated noise due to the use of approximate metrics by contrast here we derive the closed form of the fisher information matrix with respect to sensor selection variables that is valid for any arbitrary noise correlation regime and develop both a convex relaxation approach and a greedy algorithm to find near optimal solutions we further extend our framework of sensor selection to solve the problem of sensor scheduling where a greedy algorithm is proposed to determine non myopic multi time step ahead sensor schedules lastly numerical results are provided to illustrate the effectiveness of our approach and to reveal the effect of noise correlation on estimation performance

the combination of evidence in the transferable belief model a description of the transferable belief model which is used to quantify degrees of belief based on belief functions is given the impact of open and closed world assumption on conditioning is discussed the nature of the frame of discernment on which a degree of belief will be established is discussed a set of axioms justifying dempster s rule for the combination of belief functions induced by two distinct evidences is presented

conditional belief functions as lower envelopes of conditional probabilities in a finite setting the aim is to provide a characterization of full conditional measures on a finite boolean algebra obtained as lower envelope of the extensions of a full conditional probability defined on another finite boolean algebra such conditional measures are conditional belief functions defined by means of a generalized bayesian conditioning rule relying on a linearly ordered class of belief functions this notion of bayesian conditioning for belief functions is compared with other well known conditioning rules by looking for those conditional measures that can be seen as lower conditional probabilities

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

citation based journal ranks the use of fuzzy measures this article considers a real world application of the choquet integral based classifier to the problem of ranking of scientific journals we attempted to predict rankings given by panels of experts by using available citation indices we compare the choquet integral fit by the least absolute deviation criterion as a classifier against other ordinal classification methods before interpreting the fuzzy measure weights and indices our findings show that the journal rankings data set is difficult to model accurately due to inconsistencies and lack of monotonicity but that the choquet integral still performs well as a classifier

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

the electrostatic limit for the 3d zakharov system abstract we consider the vectorial zakharov system describing langmuir waves in a weakly magnetized plasma in its original derivation zakharov 1972 the evolution for the electric field envelope is governed by a schrodinger type equation with a singular parameter which is usually large in physical applications motivated by this we study the rigorous limit as this parameter goes to infinity by using some strichartz type estimates to control separately the fast and slow dynamics in the problem we show that the evolution of the electric field envelope is asymptotically constrained onto the space of irrotational vector fields

information driven distributed maximum likelihood estimation based on gauss newton method in wireless sensor networks in this paper we develop an energy efficient distributed estimation method that can be used in applications such as the estimation of a diffusive source and the localization and tracking of an acoustic target in wireless sensor networks we first propose a statistical measurement model in which we separate the linear and nonlinear parameters this modeling strategy reduce the processing complexity we then study the distributed implementation of the gauss newton method in the maximum likelihood estimation after that we propose a fully distributed estimation approach based on an incremental realization of the gauss newton method we derive three modifications of the basic algorithm to improve the distributed processing performance while still considering the energy restriction we implement the idea of information driven collaborative signal processing and provide a sensor node scheduling scheme in which the cramer rao bound crb is used as the performance and information utility measure to select the next sensor node numerical examples are used to study the performance of the distributed estimation and we show that of the methods considered here the proposed multiple iteration kalman filtering method has the most advantages for wireless sensor networks

convergence theorems for monotone measures in classical measure theory there are a number of convergence theorems such as the egorov the riesz and the lusin theorem among others we consider monotone measures i e monotone set functions vanishing in the empty set and defined on a measurable space and discuss how and to which extent classical convergence theorems can be carried over to this more general case

interaction transform for bi set functions over a finite set set functions appear as a useful tool in many areas of decision making and operations research and several linear invertible transformations have been introduced for set functions such as the mobius transform and the interaction transform the present paper establish similar transforms and their relationships for bi set functions i e functions of two disjoint subsets bi set functions have been recently introduced in decision making bi capacities and game theory bi cooperative games and appear to open new areas in these fields

sensor placement by maximal projection on minimum eigenspace for linear inverse problems this paper presents two new greedy sensor placement algorithms named minimum nonzero eigenvalue pursuit mnep and maximal projection on minimum eigenspace mpme for linear inverse problems with greater emphasis on the mpme algorithm for performance comparison with existing approaches in both mnep and mpme we select the sensing locations one by one in this way the least number of required sensor nodes can be determined by checking whether the estimation accuracy is satisfied after each sensing location is determined for the mpme algorithm the minimum eigenspace is defined as the eigenspace associated with the minimum eigenvalue of the dual observation matrix for each sensing location the projection of its observation vector onto the minimum eigenspace is shown to be monotonically decreasing w r t the worst case error variance wcev of the estimated parameters we select the sensing location whose observation vector has the maximum projection onto the minimum eigenspace of the current dual observation matrix the proposed mpme is shown to be one of the most computationally efficient algorithms our monte carlo simulations showed that mpme outperforms the convex relaxation method the sparsense method and the framesense method in terms of wcev and the mean square error mse of the estimated parameters especially when the number of available sensor nodes is very limited

new axiomatizations of the shapley interaction index for bi capacities bi capacities are a natural generalization of capacities or fuzzy measures in a context of decision making where underlying scales are bipolar they are able to capture a wide variety of decision behaviours after a short presentation of the basis structure we introduce the shapley value and the interaction index for capacities afterwards the case of bi capacities is studied with new axiomatizations of the interaction index

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

on the zakharov and zakharov schulman systems abstract we consider the initial value problem for the zakharov system formula which models the long wave langmuir turbulence in a plasma using the standard iteration scheme in the original system it is shown that the system is locally well posed uniformly on the ionic speed of sound in appropriate sobolev spaces the method provides similar results for the ivp for the zakharov schulman system formula where u r d 0 c r d 0 r and formula with a i j k a k i j real constants which models the interactions of small amplitude high frequency waves with acoustic type waves

generalizations of k order additive discrete fuzzy measures abstract two types of generalizations of k order additive discrete fuzzy measures recently introduced by grabisch are shown first a special type of discrete fuzzy measures generalizing the pseudo additive measures with respect to a pan addition is introduced following the ideas of grabisch a mobius like transformation of discrete fuzzy measures is presented and an evaluation formula for the corresponding choquet like integral is given including the sugeno integral next k order additive fuzzy measures on general spaces are introduced combination of both generalizations is discussed and several open problems are addressed

random generation of k interactive capacities abstract the theory of capacities provides powerful formal methodology to account for criteria dependencies in multiple criteria decision problems the discrete choquet and sugeno integrals aggregate criteria valuations accounting for criteria synergies and redundancies we address an important problem of randomly generating capacities of special classes for simulation studies and for capacity learning through evolutionary algorithms we discuss two efficient methods suitable for k interactive capacities the results are supported by the extensive numerical evidence and provide a useful tool for large scale simulations

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

dynamic bit allocation for object tracking in wireless sensor networks in this paper we study the target tracking problem in wireless sensor networks wsns using quantized sensor measurements where the total number of bits that can be transmitted from sensors to the fusion center is limited at each time step of tracking a total of r available bits need to be distributed among the n sensors in the wsn for the next time step the optimal solution for the bit allocation problem can be obtained by using a combinatorial search which may become computationally prohibitive for large n and r therefore we develop two new suboptimal bit allocation algorithms which are based on convex optimization and approximate dynamic programming a dp we compare the mean squared error mse and computational complexity performances of convex optimization and a dp with other existing suboptimal bit allocation schemes based on generalized breiman friedman olshen and stone gbfos algorithm and greedy search simulation results show that a dp convex optimization and gbfos yield similar mse performance which is very close to that based on the optimal exhaustive search approach and they outperform greedy search and nearest neighbor based bit allocation approaches significantly computationally a dp is more efficient than the bit allocation schemes based on convex optimization and gbfos especially for a large sensor network

mobile sensor network control using mutual information methods and particle filters this paper develops a set of methods enabling an information theoretic distributed control architecture to facilitate search by a mobile sensor network given a particular configuration of sensors this technique exploits the structure of the probability distributions of the target state and of the sensor measurements to control the mobile sensors such that future observations minimize the expected future uncertainty of the target state the mutual information between the sensors and the target state is computed using a particle filter representation of the posterior probability distribution making it possible to directly use nonlinear and non gaussian target state and sensor models to make the approach scalable to increasing network sizes single node and pairwise node approximations to the mutual information are derived for general probability density models with analytically bounded error the pairwise node approximation is proven to be a more accurate objective function than the single node approximation the mobile sensors are cooperatively controlled using a distributed optimization yielding coordinated motion of the network these methods are explored for various sensing modalities including bearings only sensing range only sensing and magnetic field sensing all with potential for search and rescue applications for each sensing modality the behavior of this non parametric method is compared and contrasted with the results of linearized methods and simulations are performed of a target search using the dynamics of actual vehicles monte carlo results demonstrate that as network size increases the sensors more quickly localize the target and the pairwise node approximation provides superior performance to the single node approximation the proposed methods are shown to produce similar results to linearized methods in particular scenarios yet they capture effects in more general scenarios that are not possible with linearized methods

the measure of interaction among players in games with fuzzy coalitions in this paper the derivative among participation levels of players in games with fuzzy coalitions which measures the simultaneous interaction among players with certain participation levels is defined some properties of the derivative of fuzzy game with k monotonicity are studied furthermore the interaction index among participation levels is established based on the assumption that fuzzy characteristic functions are continuous the integral index of absolute interaction among levels is proposed and the integral index of absolute interaction among players is defined as well at last the shapley total integral index of interaction among players is constructed and a series of properties relative to the index are proved

analysis and design of optimum sparse array configurations for adaptive beamforming in this paper we analyze the effect of nonuniform array configurations on adaptive beamforming for enhanced signal to interference plus noise ratio sinr the array is configured using a given number of antennas or through a selection of subset of antennas from a larger available set leading to a sparse array in both cases the bounds on the highest achievable sinr for a given number of antennas are formulated and used to offer new insights into open loop adaptive beamforming the upper and lower bounds underscore the role of array configurations in optimizing performance for interference free and interference active environments respectively this paper considers the general case of multiple sources and interferences in the field of view we formulate three angles namely eigenspace angle conventional angle and minimum canonical angle for characterizing spatial separation between the source and interference subspaces which represent performance loss incurred by interference nulling three sparse array design methods incorporating these angles are proposed simulation examples confirm the role of subspace angles in optimum beamforming and validate the utility of antenna selection algorithms for sparse array design

k balanced games and capacities in this paper we present a generalization of the concept of balanced game for finite games balanced games are those having a nonempty core and this core is usually considered as the solution of the game based on the concept of k additivity we define the so called k balanced games and the corresponding generalization of core the k additive core whose elements are not directly imputations but k additive games we show that any game is k balanced for a suitable choice of k so that the corresponding k additive core is not empty for the games in the k additive core we propose a sharing procedure to get an imputation and a representative value for the expectations of the players based on the pessimistic criterion moreover we look for necessary and sufficient conditions for a game to be k balanced for the general case it is shown that any game is either balanced or 2 balanced finally we treat the special case of capacities

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

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

optimal sensor selection for discrete event systems with partial observation for discrete event systems under partial observation we study the problem of selection of an optimal set of sensors that can provide sufficient yet minimal events observation information the sufficiency of the observed information is captured as the fulfillment of a desired formal property selection of sensors can be viewed as a selection of an observation mask and also of an equivalence class of events a sensor set is called optimal if any coarser selection of the corresponding equivalence class of events results in some significant loss of the events observation information we study an optimal selection of sensors over the set of general nonprojection observation masks we show that this problem is np hard in general for mask monotonic properties we present a top down and a bottom up algorithm each of polynomial complexity we show that observerness is not mask monotonic we show that the computational complexity can be further improved if the property is preserved under the projection via an intermediary observation mask that is an observer our results are obtained in a general setting so that they can be adapted for an optimal selection of sensors for a variety of applications

existence of self similar blow up solutions for zakhrov equation in dimension two part i we consider the zakharov equation in space dimension two left begin array 20 c iu t delta u nu frac 1 c 0 2 n tt delta n delta left u right 2 end array right we prove the existence of blow up solutions stable self similar blow up solutions for this problem and we study various properties of these solutions

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

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

k additivity and c decomposability of bi capacities and its integral k additivity is a convenient way to have less complex bi capacities this paper gives a new characterization of k additivity introduced by grabisch and labreuche of bi capacities and contrasts between the existing characterization and the new one that differs from the one of saminger and mesiar besides in the same way for capacities a concept of c decomposability distinct from the proposal of saminger and mesiar but closely linked to k additivity is introduced for bi capacities moreover the concept of c decomposability applies to the choquet integral with respect to bi capacities

on policy capturing with fuzzy measures policy capturing methods generally apply linear regression analysis to model human judgment in this paper we examine the application of fuzzy set and fuzzy measure theories to obtain subjective descriptions of cue importance for policy capturing at the heart of the approach is a method of learning fuzzy measures the shapley values associated with the fuzzy measures provide a basis for comparison with the results of linear regression however the fuzzy measure theoretical approach provides additional insight into interaction effects corresponding to the nonlinear noncompensatory nature of the underlying decision model to illustrate the methodology we estimated the importance of factors and the interactions among them that influence decisions related to strategic investments in telecommunications infrastructure and compared the results from the fuzzy approach to those obtained from traditional statistical methods

interaction transform of set functions over a finite set the paper introduces a new transform of set functions over a finite set which is linear and invertible as the well known mobius transform in combinatorics this transform leads to the interaction index a central concept in multicriteria decision making the interaction index of a singleton happens to be the shapley value of the set function or in terms of cooperative game theory of the value function of the game properties of this new transform are studied in detail and some illustrative examples are given

blow up results of viriel type for zakharov equations we consider the zakharov equations in n for n 2 n 3 we first establish a viriel identity for such equations and then prove a blow up result for solutions with a negative energy

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

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

sensor selection based on generalized information gain for target tracking in large sensor networks in this paper sensor selection problems for target tracking in large sensor networks with linear equality or inequality constraints are considered first we derive an equivalent kalman filter for sensor selection i e generalized information filter then under a regularity condition we prove that the multistage look ahead policy that minimizes either the final or the average estimation error covariances of next multiple time steps is equivalent to a myopic sensor selection policy that maximizes the trace of the generalized information gain at each time step moreover when the measurement noises are uncorrelated between sensors the optimal solution can be obtained analytically for sensor selection when constraints are temporally separable when constraints are temporally inseparable sensor selections can be obtained by approximately solving a linear programming problem so that the sensor selection problem for a large sensor network can be dealt with quickly although there is no guarantee that the gap between the performance of the chosen subset and the performance bound is always small numerical examples suggest that the algorithm is near optimal in many cases finally when the measurement noises are correlated between sensors the sensor selection problem with temporally inseparable constraints can be relaxed to a boolean quadratic programming problem which can be efficiently solved by a gaussian randomization procedure along with solving a semi definite programming problem numerical examples show that the proposed method is much better than the method that ignores dependence of noises

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

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

super quantum measures on effect algebras with the riesz decomposition properties we give one basis of the space of super quantum measures on finite effect algebras with the riesz decomposition properties rdp for short then we prove that the super quantum measures and quantum interference functions on finite effect algebras with the rdp are determined each other at last we investigate the relationships between the super quantum measures and the diagonally positive signed measures on finite effect algebras with the rdp in detail

optimal periodic sensor scheduling in networks of dynamical systems we consider the problem of finding optimal time periodic sensor schedules for estimating the state of discrete time dynamical systems we assume that multiple sensors have been deployed and that the sensors are subject to resource constraints which limits the number of times each can be activated over one period of the periodic schedule we seek an algorithm that strikes a balance between estimation accuracy and total sensor activations over one period we make a correspondence between active sensors and the nonzero columns of the estimator gain we formulate an optimization problem in which we minimize the trace of the error covariance with respect to the estimator gain while simultaneously penalizing the number of nonzero columns of the estimator gain this optimization problem is combinatorial in nature and we employ the alternating direction method of multipliers admm to find its locally optimal solutions numerical results and comparisons with other sensor scheduling algorithms in the literature are provided to illustrate the effectiveness of our proposed method

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

on a stochastic sensor selection algorithm with applications in sensor scheduling and sensor coverage in this note we consider the following problem suppose a set of sensors is jointly trying to estimate a process one sensor takes a measurement at every time step and the measurements are then exchanged among all the sensors what is the sensor schedule that results in the minimum error covariance we describe a stochastic sensor selection strategy that is easy to implement and is computationally tractable the problem described above comes up in many domains out of which we discuss two in the sensor selection problem there are multiple sensors that cannot operate simultaneously e g sonars in the same frequency band thus measurements need to be scheduled in the sensor coverage problem a geographical area needs to be covered by mobile sensors each with limited range thus from every position the sensors obtain a different view point of the area and the sensors need to optimize their trajectories the algorithm is applied to these problems and illustrated through simple examples

rough sets theory for multicriteria decision analysis the original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation it operates on a data table composed of a set u of objects actions described by a set q of attributes its basic notions are indiscernibility relation on u lower and upper approximation of either a subset or a partition of u dependence and reduction of attributes from q and decision rules derived from lower approximations and boundaries of subsets identified with decision classes the original rough set idea is failing however when preference orders of attribute domains criteria are to be taken into account precisely it cannot handle inconsistencies following from violation of the dominance principle this inconsistency is characteristic for preferential information used in multicriteria decision analysis mcda problems like sorting choice or ranking in order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary the main change is the substitution of the indiscernibility relation by a dominance relation which permits approximation of ordered sets in multicriteria sorting to approximate preference relations in multicriteria choice and ranking problems another change is necessary substitution of the data table by a pairwise comparison table where each row corresponds to a pair of objects described by binary relations on particular criteria in all those mcda problems the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model it is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax in order to workout a recommendation in one of the mcda problems we propose exploitation procedures of the set of decision rules finally some other recently obtained results are given rough approximations by means of similarity relations rough set handling of missing data comparison of the rough set model with sugeno and choquet integrals and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive

on the polytope of non additive measures in this paper we deal with the problem of studying the structure of the polytope of non additive measures for finite referential sets we give a necessary and sufficient condition for two extreme points of this polytope to be adjacent we also show that it is possible to find out in polynomial time whether two vertices are adjacent these results can be extended to the polytope given by the convex hull of monotone boolean functions we also give some results about the facets and edges of the polytope of non additive measures we prove that the diameter of the polytope is 3 for referentials of three elements or more finally we show that the polytope is combinatorial and study the corresponding properties more concretely we show that the graph of non additive measures is hamilton connected if the cardinality of the referential set is not 2

interaction indices for multichoice games abstract models in multicriteria decision analysis mcda can be analyzed by means of an importance index and an interaction index for every group of criteria we consider first discrete models in mcda without further restriction which amounts to considering multichoice games that is cooperative games with several levels of participation we propose and axiomatize two interaction indices for multichoice games the signed interaction index and the absolute interaction index in a second part we consider the continuous case supposing that the continuous model is obtained from a discrete one by means of the choquet integral we show that as in the case of classical games the interaction index defined for continuous aggregation functions coincides with the signed interaction index up to a normalizing coefficient

robust optimization of the choquet integral we study the problem of the choquet integral maximization for the case when preferences of the decision maker do not define a unique capacity but rather some convex set we introduce a robust version of the problem based on the minimax regret criterion and construct a global optimization algorithm it is shown that the decision rule based on regret minimization over a set of capacities is equivalent to maximization of the choquet integral with respect to a certain capacity belonging to this set we compare our method to various capacity identification approaches developed in the literature and provide a semantic interpretation in terms of information value

on the structure of the k additive fuzzy measures in this paper we present some results concerning the vertices of the set of fuzzy measures being at most k additive we provide an algorithm to compute them we give some examples of the results obtained with this algorithm and give lower bounds on the number of vertices for the n 1 additive case proving that it grows much faster than the number of vertices of the general fuzzy measures the results in the paper suggest that the structure of k additive measures might be more complex than expected from their definition and in particular that they are more complex than general fuzzy measures

a weight based approach to the measurement of the interaction among criteria in the framework of aggregation by the bipolar choquet integral in the context of aggregation the choquet integral arises as a natural generalization of the weighted arithmetic mean in their seminal work on bi capacities grabisch and labreuche have recently proposed an extension of the choquet integral adapted to the situation where the values to be aggregated lie on a bipolar scale in such a framework they have derived generalizations of the shapley value and of the shapley interaction index that can be used to apprehend the main behavioral characteristics of the aggregation founding our approach on the intuitive notion of weight of a criterion we propose alternative definitions of importance and interaction indices that may be more easily interpretable and we study their properties

applying young diagrams to 2 symmetric fuzzy measures with an application to general fuzzy measures abstract in this paper we apply young diagrams a well known object appearing in combinatorics and group representation theory to study some properties of the polytope of 2 symmetric fuzzy measures with respect to a given partition a subpolytope of the set of fuzzy measures the main result in the paper allows to build a simple and fast algorithm for generating points on this polytope in a random fashion besides we also study some other properties of this polytope as for example its volume in the last section we give an application of this result to the problem of identification of general fuzzy measures

energy convergence for singular limits of zakharov type systems we prove existence and uniqueness of solutions to the klein gordon zakharov system in the energy space h1 l2 on some time interval which is uniform with respect to two large parameters c and these two parameters correspond to the plasma frequency and the sound speed in the simultaneous high frequency and subsonic limit we recover the nonlinear schrodinger system at the limit we are also able to say more when we take the limits separately

on the closure of families of fuzzy measures under eventwise aggregations the paper contains some results on eventwise aggregation of fuzzy measures from the various families of lower probabilities this investigation includes coherent lower probabilities k monotone measures belief measures etc we study sufficient and necessary conditions on aggregation operators satisfying the closure property it is necessary to guarantee that the aggregation belongs to the same family as generating measures we also propose an efficient way for constructing aggregation operators using the found unique properties of multilinear extension this representation method for fuzzy measures generalizes to the n dimensional case the well known one giving distorted probabilities

a practical approach for estimating weights of interacting criteria from profile sets aggregating multi criteria data is an important problem with many applications commonly used additive aggregation methods such as the weighted arithmetic mean cannot account for the criteria interactions encountered in many practical multi criteria decision problems the choquet integral is a suitable aggregation operator in the presence of interacting criteria it replaces the weight vector with a fuzzy measure that models the importance of each subset or coalition of criteria rather than just the importance of individual criteria however estimating the fuzzy measures in practice has been problematic conventional approaches are cognitively challenging for decision makers while more recent approaches suffer from prohibitive data requirements in this paper we present a formulation for the weights of the choquet integral that uses principal component analysis to account for criteria interaction this novel unsupervised approach to estimating the required fuzzy measures overcomes the limitations of other methods the approach is applied to two case studies in environmental and sustainability analysis and the results are compared with those of the weighted arithmetic mean the first case is a triple bottom line analysis of 135 australian industry sectors evaluated against 11 criteria while the second case is an environmental life cycle assessment of 8 alternative biosolids management options evaluated against 5 criteria these examples demonstrate the ability of the proposed approach to account for criteria interaction in these and other decision contexts

extreme points of some families of non additive measures non additive measures are a valuable tool to model many different problems arising in real situations however two important difficulties appear in their practical use the complexity of the measures and their identification from sample data for the first problem additional conditions are imposed leading to different subfamilies of non additive measures related to the second point in this paper we study the set of vertices of some families of non additive measures namely k additive measures and p symmetric measures these extreme points are necessary in order to properly apply a new method of identification of non additive measures based on genetic algorithms whose cross over operator is the convex combination we solve the problem through techniques of linear programming

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

convolutions of singular measures and applications to the zakharov system abstract uniform l 2 estimates for the convolution of singular measures with respect to transversal submanifolds are proved in arbitrary space dimension the results of bennett bez are used to extend previous work of bejenaru herr tataru as an application it is shown that the 3d zakharov system is locally well posed in the full subcritical regime

inconsistency and non additive capacities the analytic hierarchy process in the framework of choquet integration we examine the ahp in the framework of choquet integration and we propose an extension of the standard ahp aggregation scheme on the basis of the shapley values associated with the criteria in our model a measure of dominance inconsistency between criteria is defined in terms of the totally inconsistent matrix associated with the main pairwise comparison matrix of the ahp the measure of dominance inconsistency is then used to construct a non additive capacity whose associated shapley values reduce to the standard ahp priority weights in the consistency case in the general inconsistency case however the extended aggregation scheme based on the shapley weighted mean tends to attenuate resp emphasize the priority weights of the criteria with higher resp lower average dominance inconsistency with respect to the other criteria

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

k maxitive aggregation functions abstract inspired by the idea of k maxitive measures we introduce and study k maxitive aggregation functions in particular 1 maxitive aggregation functions are shown to be just maxitive aggregation functions i e maxima of distorted inputs we introduce among others a representation of symmetric k maxitive aggregation functions moreover we show that for any k maxitive capacity and the smallest universal integral based on an arbitrary fixed semicopula including the sugeno and shilkret integrals the resulting aggregation function is k maxitive

complex structured decision making model a hierarchical frame work for complex structured data we introduce a hierarchical framework we call complex structured decision making model for complexly structured knowledge representation in intelligent decision making we show that our model extends non hierarchical flat decision making models to hierarchical decision making models that are similar to comprehensible human decision making processes further we make an argument that hierarchial representation of knowledge in a complex structured decision making model simplifies the approximation of aggregation functions to easily adapt to the underline relation of the system additionally using a real world complex structured data set we show that hierarchically organized fuzzy integrals e g choquet integral and sugeno integral and fuzzy signatures outperform these non hierarchical fuzzy integrals