several inequalities for the pan integral several inequalities for the pan integral are investigated it is shown that the chebyshev inequality holds for an arbitrary subadditive measure if and only if the integrands f g are comonotone thus we provide a new characterization for nonnegative comonotone functions it is also shown that the holder and minkowski inequalities for the pan integral hold if the monotone measure is subadditive since the pan integral coincides with the concave integral when is subadditive our results can also be applied to the concave integral

on the equality of integrals considering a finite space x several necessary conditions and one rather general sufficient condition describing when the choquet integral coincides with the pan integral with respect to the standard arithmetic operations are shown these conditions are characterized by using the minimal atoms of monotone measure under the introduced constraints the calculation of these coinciding two integrals is also given

an ergodic theorem for filtering with applications to stability abstract ergodic properties of the signal filtering pair are studied for continuous time finite markov chains observed in white noise the obtained law of large numbers is applied to the stability problem of the nonlinear filter with respect to initial conditions the furstenberg khasminskii formula is derived for the top lyapunov exponent of the zakai equation and is used to estimate the stability index of the filter

a new nonlinear classifier with a penalized signed fuzzy measure using effective genetic algorithm this paper proposes a new nonlinear classifier based on a generalized choquet integral with signed fuzzy measures to enhance the classification accuracy and power by capturing all possible interactions among two or more attributes this generalized approach was developed to address unsolved choquet integral classification issues such as allowing for flexible location of projection lines in n dimensional space automatic search for the least misclassification rate based on choquet distance and penalty on misclassified points a special genetic algorithm is designed to implement this classification optimization with fast convergence both the numerical experiment and empirical case studies show that this generalized approach improves and extends the functionality of this choquet nonlinear classification in more real world multi class multi dimensional situations

a sufficient condition of equivalence of the choquet and the pan integral abstract in this note we continue to investigate the relationship between the choquet integral and the pan integral on infinite space we will show that the m property of monotone measures is a sufficient condition that the choquet integral coincides with the pan integral in this discussion the spaces are not restricted to be finite thus the previous results obtained in finite space are further generalized and developed a characteristic of the m property of monotone measures is also presented

on the comonotonic property for sugeno integral the so called comonotonic property for sugeno integral is introduced and investigated where 0 2 0 is a binary operation under the assumption that lim b a email protected a and lim a a email protected b we prove that sugeno integral possesses comonotonic property if and only if equals to one of the four operators minimum maximum pf the first projection and pl the last projection we also discuss comonotonic property when this assumption does not hold

constructing fuzzy measures in expert systems abstract this paper is an overview of results regarding various representations of fuzzy measures and methods for constructing fuzzy measures in the context of expert systems which were obtained by the authors and their associates during the last three years included are methods for constructing fuzzy measures by various transformations by extension by statistical inference and by various data driven methods based either on the sugeno integral or the choquet integral and using neural networks genetic algorithms or fuzzy relation equations

a new type of nonlinear integrals and the computational algorithm in information fusion aggregations with various backgrounds require a variety of integrals to handle these integrals are generally nonlinear since the set functions used are nonadditive in many real problems in this study the set functions considered are nonnegative and vanishing at the empty set they are a class of set functions including fuzzy measures and even imprecise probabilities a new type of nonlinear integrals with respect to such a set function for nonnegative functions is introduced and its primary properties are detailed these type of integrals has a natural explanation and therefore has wide applicability we also show a comparison between the newly introduced nonlinear integral and other nonlinear integrals such as the choquet integral the natural extension in the theory of imprecise probabilities and the common pan integral with a flowchart the algorithm for calculating the integral is given in this paper when the universe of discourse the set of all information sources is finite

fuzzy numbers and fuzzification of the choquet integral to deal with linguistic attributes in databases the integrand of the choquet integral should be allowed to assume fuzzy numbers this paper provides a detailed discussion on one fuzzification of choquet integral in which the integrand as well as the integration result is fuzzy numbers based on the extension principle it is a generalized choquet integral for fuzzy valued integrand interval valued integrand as well as the crisp valued integrand two numerical methods are provided to calculate the membership function of the integration value of this generalized choquet integral with respect to the fuzzy measure and the signed fuzzy measure respectively the presented generalized choquet integral with respect to signed fuzzy measure can act as an aggregation tool which is especially useful in many information fusing and data mining problems such as regression and classification where not only crisp data but also heterogeneous fuzzy data are involved

a note on convergence properties of interval valued capacity functionals and choquet integrals based on the classical theory of random sets feng and nguyen 2007 5 studied the convergence of capacity functionals of random sets in terms of choquet integrals in this paper we consider an interval valued capacity functional which is motivated by the goal to generalize a capacity functional and the choquet integral with respect to an interval valued capacity in particular we discuss some convergence theorems for interval valued capacity functionals and interval valued probability measures in the hausdorff metric and choquet weak sense

some quantities represented by the choquet integral this paper shows that the choquet integral can represent some useful quantities such as supremum infimum essential supremum essential infimum mean median quantile 100 percentile and l estimators including trimmed mean

necessary and sufficient conditions of stability a multiloop generalization of the circle criterion a linear time invariant system with a vector output and a vector input is described this system is closed by an uncertain nonlinear time varying feedback the only information on this feedback is given by several integral quadratic inequalities i e the uncertainties under consideration generalize the so called conic nonlinearities necessary and sufficient frequency domain conditions of stability are obtained an advanced version of the s procedure losslessness theorem and some other tools of the absolute stability theory are used

robust performance unknown disturbances and known fixed inputs this paper addresses the robust performance problem when the system under consideration is subjected to norm bounded time varying uncertainty different performance objectives are considered and computable bounds are obtained for the worst case performance in various norms in some important cases it is shown that these bounds are tight giving simple exact formulas for computing robust performance measures one of the important features of these formulas is that they enable the direct and exact calculation of a number of worst cast performance measures while avoiding entirely the iterations needed for computing these measures by existing methods that rely on repeated scalings and spectral radius bound evaluations

the autocontinuity of set function and the fuzzy integral abstract the autocontinuity and some other concepts of a set function are introduced and sugeno s fuzzy measure with some annexed conditions is studied on a fuzzy measure space egoroffs theorem is proved it is also proved that f mean convergence is equivalent to convergence in measure for a sequence of measurable functions finally some convergence theorems of sequence of fuzzy integrals are proved especially using the concept of autocontinuity a necessary and sufficient condition for the convergence in measure theorem is given

a note on the coincidence of decomposition integrals and superdecomposition integrals abstract it is known that the decomposition integral i h and the superdecomposition integral i h related to the system h of all finite chains from a where a is a algebra of the subsets of a nonvoid set x coincide with each other for each monotone measure and for each nonnegative measurable function and they are equal to the choquet integral in this note we show that the converse is also true that is for a complete system h of finite set systems from a if i h i h holds for each monotone measure and for each nonnegative measurable function then the system h and the system of all finite chains from a are equivalent in some sense and the integrals involved are the choquet integral

on the equivalence of the choquet pan and concave integrals on finite spaces abstract in this paper we introduce the concept of maximal cluster of minimal atoms on monotone measure spaces and by means of this new concept we continue to investigate the relation between the choquet integral and the pan integral on finite spaces it is proved that the m property of a monotone measure is a sufficient condition that the choquet integral coincides with the pan integral based on the usual addition and multiplication thus combining our recent results we provide a necessary and sufficient condition that the choquet integral is equivalent to the pan integral on finite spaces meanwhile we also use the characteristics of minimal atoms of monotone measure to present another necessary and sufficient condition that these two kinds of integrals are equivalent on finite spaces the relationships among the choquet integral the pan integral and the concave integral are summarized

robust steady state tracking this paper addresses the robust performance problem when the performance measure is the steady state value of an error signal necessary and sufficient conditions are derived for robust steady state tracking of fixed inputs in the presence of structured time varying uncertainty are derived these conditions are easily computable and fit well with existing conditions on stability robustness and performance robustness when the performance measure is the level of disturbance rejection using these conditions it is shown that time varying perturbations of a nominal linear time invariant plant can result in large steady state tracking errors to fixed inputs even if the nominal plant and controller give zero steady state tracking errors the derived expressions for the worst case steady state tracking error give insight into how the time variation in the plant affects tracking errors and suggest that certain transfer function norms should be minimized to reduce the effect of these perturbations on the steady state value of error signals

set valued choquet integrals revisited this paper performs a further investigation on the set valued choquet integral given by jang and kwon fuzzy sets and systems 112 2000 233 some incorrect properties are revised it is pointed out that the definition for convergence of sequence of sets or set valued functions is unreasonable by using the usual kuratowski s convergence some new convergence theorems for the set valued choquet integral are shown meanwhile some other properties are obtained

technical communique optimization based design of fixed order controllers for command following for discrete time scalar systems we propose an approach for designing feedback controllers of fixed order to minimize an upper bound on the peak magnitude of the tracking error to a given command input the work makes use of linear programming to design over a class of closed loop systems recently proposed for the rejection of non zero initial conditions and bounded disturbances we incorporate performance robustness in the form of a guaranteed upper bound on the peak magnitude of the tracking error under plant coprime factor uncertainty

determining nonnegative monotone set functions based on sugeno s integral an application of genetic algorithms abstract regarding the set of all information sources as the universe of discourse we used a nonnegative monotone set function defined on its power set to describe the importance of each individual information source and their varied combinations such a set function is called an importance measure or a fuzzy measure the sugeno integral with respect to the nonnegative monotone set function possesses many desired properties such as the fuzzy linearity when the set function is fuzzy additive and can be adopted as an aggregation means in information fusion in real problems viewing the sugeno integral as a multi input single output system we use a genetic algorithm to determine the values of the importance measure from the input output data of the system

fuzzy measures defined by fuzzy integral and their absolute continuity abstract given a measurable space x f a fuzzy measure on x f and a nonnegative functionfonxthat is measurable with respect to f we can define a new set function on x f by the fuzzy integral it is known that is a lower semicontinuous fuzzy measure on x f and moreover if is finite then is a finite fuzzy measure as well in this paper we generalize in several different ways the concept of absolute continuity of set functions as defined in classical measure theory in addition we investigate the relationship among these generalizations by using the structural characteristics of set functions such as null additivity and autocontinuity and determine which types of absolute continuity of fuzzy measures are possessed by the fuzzy measure or the lower semicontinuous fuzzy measure obtained by the fuzzy integral

l1 robust performance of discrete time systems with structured uncertainty abstract this paper addresses the robust performance problem when a linear time invariant system is subjected to both norm bounded time varying uncertainty and worst case external bounded input the tightest upper bounds for steady state robust performance measures over classes of finite memory and fading memory perturbations are computed in the regulation and tracking problems since the classes of finite memory and fading memory perturbations are inappropriate to the purposes of model validation and adaptive control approximating subclasses of bounded memory perturbations and perturbations with exponentially decreasing impulse responses are considered it is shown that the robust performance bounds obtained are nonconservative in the case of bounded memory perturbations the worst case steady state robust performance measures for siso system under coprime factor perturbations are computed for illustration

on the null additivity and the autocontinuity of a fuzzy measure abstract in this paper further research on the null additivity and autocontinuity of fuzzy measures is made we prove that autocontinuity from below and autocontinuity from above of a fuzzy measure are equivalent when this fuzzy measure is finite furthermore if the space x is countable then the null additivity is equivalent to the autocontinuity

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

asymptotic structural characteristics of fuzzy measure and their applications abstract in fuzzy measure theory as sugeno s fuzzy measures lose additivity in general the concept almost which is well known in classical measure theory splits into two different concepts almost and pseudo almost in order to replace the additivity it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called waxing and waning and to introduce some new concepts such as autocontinuity converse autocontinuity and pseudo autocontinuity these concepts describe some asymptotic structural characteristics of a fuzzy measure in this paper by means of the asymptotic structural characteristics of fuzzy measure we also give four forms of generalization for both egoroff s theorem riesz s theorem and lebesgue s theorem respectively and prove the almost everywhere pseudo almost everywhere convergence theorem the convergence in measure pseudo in measure theorem of the sequence of fuzzy integrals in the last two theorems the employed conditions are not only sufficient but also necessary

suboptimal robust synthesis for mimo plant under coprime factor perturbations abstract suboptimal robust synthesis for mimo nominal system under coprime factor perturbations is considered in classical and non classical statements in the classical statement weights of perturbations and upper bound on magnitude bounded exogenous disturbance are assumed to be known to controller designer suboptimal synthesis within e tolerance is reduced to the solution of log 2 1 e standard mixed sensitivity problems of l 1 optimization in the non classical statement the upper bounds on perturbations and exogenous disturbance are to be estimated from measurement data and suboptimal synthesis is reduced to the solution of 1 e mixed sensitivity problems

on filtering of markov chains in strong noise the filtering problem for finite state markov chains is revisited in the low signal to noise regime we give a description of conditional measure concentration around the invariant distribution of the signal and derive asymptotic expressions for the performance indices of the minimum mean square error mmse and minimum a posteriori probability map filtering estimates

performance robustness of discrete time systems with structured uncertainty given an interconnection of a nominal discrete time plant and a stabilizing controller together with structured norm bounded nonlinear time varying perturbations necessary and sufficient conditions for robust stability and performance of the system are provided it is shown that performance robustness is equivalent to stability robustness in the sense that both problems can be dealt with in the framework of a general stability robustness problem the resulting stability robustness problem is shown to be equivalent to a simple algebraic one the solution of which provides the desired necessary and sufficient conditions for performance stability robustness these conditions provide an effective tool for robustness analysis and can be applied to a large class of problems in particular it is shown that some known results can be obtained immediately as special cases of these conditions

relationship between two types of superdecomposition integrals on finite spaces abstract this paper investigates the relationship between two types of superdecomposition integrals namely the convex integral and the pan integral from above on finite spaces to this end we introduce two new concepts related to monotone measures superadditivity with respect to singletons and minimal strictly subadditive set and discuss some of their properties in the case that the monotone measure is superadditive with respect to singletons we show that these two types of integrals are equivalent in other cases by means of the characteristics of minimal strictly subadditive sets we provide a set of necessary and sufficient conditions for which these two types of integrals coincide with each other

general chebyshev type inequalities for sugeno integrals chebyshev type inequalities for the sugeno integral on abstract spaces are studied in a rather general form thus closing the series of papers on the topic dealing with special cases restricted to the real line and product operation

on the convergence of the fuzzy valued functional defined by μ integrable fuzzy valued functions in this paper in the sense of sugeno s fuzzy measure we first introduce the integrable fuzzy valued function define the fuzzy valued functional a kind of new fuzzy integral and then we discuss some important elementary properties with respect to this kind of functional finally we give its convergence theorem

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

superdecomposition integrals this study introduces and discusses a new class of integrals based on superdecompositions of integrated functions including an analysis of their relationship with decomposition integrals which were introduced recently by even and lehrer the proposed superdecomposition integrals have several properties that are similar or dual with respect to decomposition integrals but they also have some significant differences the convex integral is obtained by considering all possible superdecompositions with no constraints on the applied sets which can be treated as the greatest convex homogeneous functional that is bounded from above by the measure we consider the relationship with the universal integral of klement et al is also discussed finally some possible generalizations are outlined

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

robust stability with time varying structured uncertainty considers the problem of assessing robust stability in the presence of block diagonally structured time varying dynamic uncertainty it is shown that robust stability holds only if there exist constant scalings which lead to a small gain condition the notion of stability here is finite gain stability over finite energy signals in sharp contrast to the case of time invariant dynamic uncertainty this result is not limited by the number uncertainty blocks these results parallel previous results regarding finite gain stability over persistent bounded signals

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

characterization and comparison of sugeno and choquet integrals abstract starting from the concept of equiordered functions we characterize two types of fuzzy integrals that can be defined on any kind of fuzzy measure sugeno and choquet integrals the characterization theorems that we obtain allow us to carry out a comparative study of these two fuzzy integrals pointing out their formal similarities and their conceptual differences

monotone set functions defined by choquet integral abstract given a nonnegative monotone set function and a nonnegative measurable function on a measurable space the choquet integral determines a new nonnegative monotone set function that is absolutely continuous with respect to the original one in a generalized sense for monotone set functions this new set function preserves almost all desirable structural characteristics of the original monotone set function such as continuity subadditivity superadditivity null additivity converse null additivity autocontinuity converse autocontinuity uniform autocontinuity uniform converse autocontinuity and fuzzy multiplicativity such a construction is a useful method to define sound fuzzy measures in various applications

robustness with integrators abstract two important practical criteria for most real control systems are that the controller has an integrator and that the closed loop system has robust stability in this paper we identify and characterize a grouping or classification of linear scalar discrete time plants comprising three classes according to the best possible stability robustness margins against coprime factor uncertainty that can be obtained with integral controllers

on fuzzy measures defined by fuzzy integrals on definit une mesure floue comme une fonction ensemble sur 7b f qui est une algebre de sous ensembles de l ensemble x

from a gaussian mixture model to nonadditive fuzzy systems this work explores how a kind of probabilistic system namely the gaussian mixture model gmm can be translated to an additive fuzzy system we will prove the mathematical equivalence between the conditional mean of a gmm and the defuzzified output of a generalized fuzzy model gfm the relationship between a gmm and a gfm and the conditions for gmm to gfm translation will be made explicit in the form of theorems the work will then extend to special cases of the gfm specifically the mamdani larsen and takagi sugeno fuzzy models the possibility of reverse translation that is from a gfm to a gmm will also be discussed finally we will consider the generality of a gmm specifically how it can approximate other distribution functions

modeling attitude to risk in human decision processes an application of fuzzy measures abstract several models of the human decision process have been proposed classical examples of which are utility theory and prospect theory in recent times the theory of fuzzy measures and integrals has emerged as an alternative meriting further investigation specifically one is interested in the degrees of disjunction and conjunction and the veto and favor indices that represent the tolerance measure of the decision maker though several theoretical expositions have appeared in contemporary literature empirical studies applying these concepts to the real world are scarce this paper reports two studies based on a model of strategic telecommunication investment decisions from a research work involving a survey of executives the first study involves building fuzzy models corresponding to each individual decision maker with the results grouped based on the decision makers propensity to risk as determined by their degrees of disjunction the shapley indices and the interaction effects are determined for each pooled dataset corresponding to each group to contrast this approach with those of conventional nomothetic comparisons of decision policies the decision makers are grouped based on a clustering analysis of the individual linear regression models the data for each cluster are pooled and the fuzzy measures learned from the dataset are analyzed for comparison purposes the results not only serve as a demonstration of fuzzy measure analysis as a viable approach to studying qualitative decision making but also provide useful methodological insights into applying fuzzy measures to strategic investment decisions under risk

robust e p stability and performance in this paper we study robust stability in the lp sense against structured perturbations of bounded lp norm the perturbations may be either linear time varying nonlinear time varying or nonlinear time invariant we develop exact tests for robust stability for each case in terms of convex optimization problems these results are extended to solve the corresponding robust performance problems with performance measured via the induced lp norm

analysis and design for robust performance with structured uncertainty abstract necessary and sufficient conditions for stability and performance robustness of discrete time systems are provided in terms of the spectral radius of a certain nonnegative matrix the conditions are easily computable and provide a simple and efficient method for computation of the robustness conditions for siso as well mimo perturbations the problem of robust controller synthesis is explored and an iteration scheme for controller synthesis is introduced

asymptotic stability of the wonham filter ergodic and nonergodic signals the stability problem of the wonham filter with respect to initial conditions is addressed the case of ergodic signals is revisited in view of a gap in the classic work of h kunita 1971 we give new bounds for the exponential stability rates which do not depend on the observations in the nonergodic case the stability is implied by identifiability conditions formulated explicitly in terms of the transition intensities matrix and the observation structure

aggregating membership values by a choquet fuzzy integral based operator a new type of aggregator the choquet fuzzy integral based operator is introduced and applied in aggregation of membership values in this paper we generalized the definition of the fuzzy integral as a tool for membership aggregation the generalization involves replacing the max or min operator in information integration with a choquet fuzzy integral based aggregator resulting in increased flexibility the characteristics of the choquet fuzzy integral are studied and a two phase aggregating process is proposed also we demonstrate a good classification result using the proposed aggregator to integrate the memberships of several clusters in a complex unconstrained handwritten digit recognition domain

two families of fuzzy integrals abstract in this paper we present two definitions of fuzzy integrals arising from the t seminorm and the t semiconorm concepts we show that sugeno s fuzzy integral is a particular case of them and we study their principal properties finally we suggest some applications of these results

relationship between the concave integrals and the pan integrals on finite spaces a r t i c l e i n f o a b s t r a c t this study discusses the relationship between the concave integrals and the pan integrals on finite spaces the minimal atom of a monotone measure is introduced and some properties are investigated by means of two important structure characteristics related to minimal atoms minimal atoms disjointness property and subadditivity for minimal atoms a necessary and sufficient condition is given that the concave integral coincides with the pan integral with respect to the standard arithmetic operations a nd on finite spaces following this result we have shown that these two integrals coincide if the underlying monotone measure is sub additive

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

choquet like integrals abstract integrals based on a pseudo addition and a pseudo multiplication with properties similar to those of the choquet and the sugeno integral respectively are introduced the structure of convenient pseudo arithmetical operations leading to a choquet like integral is studied

indeterminate integrals with respect to nonadditive measures recently nonadditive set functions have been used in information fusion and data mining to describe the interaction among the predictive attributes in this case replacing the traditional weighted average or more generally the linear mapping from a higher dimensional space to a one dimensional space relevant nonlinear integrals should be used as the aggregation tool to express how the objective attribute depends on predictive attributes several different types of nonlinear integral such as the choquet integral and the wang integral have been discussed in literature in these models the objective attribute is the integral while the predictive attributes are the integrand in this paper a new concept of indeterminate integral is introduced when the universe of discourse is finite it is a family of integrals including nonlinear integrals mentioned above and therefore possesses a large adaptability as a new aggregation tool each specified type of indeterminate integral can be obtained from the family of indeterminate integrals by specifying a decomposition of the integrand such a new aggregation tool can be used in information fusion and data mining as well as in expert systems

an interpretation of fuzzy measures and the choquet integral as an integral with respect to a fuzzy measure abstract in this paper the non additivity of sugeno s fuzzy measures is interpreted in terms of addition and the rationality of the choquet integral is discussed it is pointed out that a fuzzy measure on a set x expresses the interaction between the subsets of x and can be represented by an additive measure it is shown through concrete examples that the choquet integral is reasonable as an integral with respect to a fuzzy measure it is also found that the choquet integral is closely related with the representation of a fuzzy measure

the concave integral over large spaces this paper investigates the concave integral for capacities defined over large spaces we characterize when the integral with respect to capacity v can be represented as the infimum over all integrals with respect to additive measures that are greater than or equal to v we introduce the notion of loose extendability and study its relation to the concave integral a non additive version for the levi theorem and the fatou lemma are proven finally we provide several convergence theorems for capacities with large cores

a note on the monotone set functions defined by choquet integral abstract in this paper we answer the open question suggested by wang et al in monotone set functions defined by choquet integral fuzzy sets and systems 81 1996 241

on the equivalence of the choquet integral and the pan integrals from above abstract a monotone measure is said to have dual m property if for any a b there exists c with a c b such that c a and b c b c by using this concept we study the relationship of the choquet integral and the pan integral from above we prove that the choquet integral coincides with the pan integral from above if has dual m property when the underlying space is finite we prove that the dual m property is also necessary for the coincidence of these two integrals thus we provide a necessary and sufficient condition for the equivalence of the choquet integral and the pan integral from above on a finite space

a necessary and sufficient condition for robust bibo stability abstract this paper is concerned with obtaining necessary and sufficient conditions for robustly stabilizing a class of plants characterized by a known linear shift invariant plant with additive perturbations of the form of possibly nonlinear time varying l stable operators we will show that in some sense the small gain theorem is necessary and sufficient even when the perturbations are restiricted to a class of linear operators

h sub 2 control with time domain constraints theory and an application in this paper we study the problem of minimizing the h sub 2 norm of a given transfer function subject to time domain constraints on the time response of a different transfer function to a given test signal the main result of this paper shows that this problem admits a minimizing solution in r h sub 2 moreover rational solutions with performance arbitrarily close to optimal can be found by constructing families of approximating problems each one of these problems entails solving a finite dimensional quadratic programming problem whose dimension can be determined before hand these results are illustrated and experimentally validated by designing a controller for an active vision application

a theory of fuzzy measures representations the choquet integral and null sets abstract some theorems of t murofushi and m sugeno fuzzy sets and systems 29 1989 201 227 concerning representation of fuzzy measures and the choquet integral are generalized it is shown that if there is a certain relation between two measurable functions then the choquet integral is additive for these two functions in addition this article discusses null sets with respect to fuzzy measures and also fuzzy measures defined on a class which is not a algebra

convergence properties of the monotone expectation and its application to the extension of fuzzy measures we define a functional on fuzzy measures called monotone expectation based on choquet s integral operator and we study its convergence properties the monotone expectation is used to extend fuzzy measures to fuzzy subsets also sugeno s bound for the difference between his integral and the mathematical expectation is generalized

set characterization selection towards classification based on interaction index in many real world datasets both the individual and coordinated action of features may be relevant for class identification in this paper a computational strategy for relevant feature selection based on the characterization of redundant or complementary features is proposed the characterization is achieved using fuzzy measures and an interaction index computed from fuzzy measure coefficients fuzzy measure identification requires raw data to be turned into confidence degrees this key step is carried out considering the distributions of feature values across all the classes fuzzy measure coefficients are then estimated with an improved version of the heuristic least mean squares algorithm that includes an efficient management of untouched coefficients then a generalization of the shapley index for an arbitrary number of features is used simulations experiments on synthetic datasets are performed to study the behavior of this generalized interaction index for extreme datasets containing either redundant or complementary features as well as noise the index value is defined by mathematical formula this result is used to motivate feature selection guidelines that take into account feature interactions experimental results on benchmark datasets show that the proposal allows for the design of compact interpretable and competitive classification models