optimization schemes for decomposition of fuzzy relations the paper proposes an optimization environment for decomposing fuzzy relations two basic categories of approaches are developed i based on direct gradient oriented optimization methods and ii exploiting the ideas of fuzzy neurocomputation these two approaches are carefully analyzed and contrasted making use of illustrative numerical material some immediate applications to pattern classification and fuzzy interpolation are studied in detail the study includes also some generalizations of the generic decomposition problem

quantum stochastic calculus associated with quadratic quantum noises we first study a class of fundamental quantum stochastic processes induced by the generators of a six dimensional non solvable lie algebra consisting of all linear combinations of the generalized gross laplacian and its adjoint annihilation operator creation operator conservation and time and then we study the quantum stochastic integrals associated with the class of fundamental quantum stochastic processes and the quantum ito formula is revisited the existence and uniqueness of solution of a quantum stochastic differential equation is proved the unitarity conditions of solutions of quantum stochastic differential equations associated with the fundamental processes are examined the quantum stochastic calculus extends the hudson parthasarathy quantum stochastic calculus

fuzzy relational equations and the inverse problem abstract the inverse problem concerned with fuzzy relations is investigated the conditions for the existence of a solution are shown and an analytical solution is given a method for the improvement of the solution is proposed

negative association ordering and convergence of resampling methods we study convergence and convergence rates for resampling schemes our first main result is a general consistency theorem based on the notion of negative association which is applied to establish the almost sure weak convergence of measures output from kitagawa s j comput graph statist 5 1996 1 25 stratified resampling method carpenter ckiffird and fearnhead s iee proc radar sonar navig 146 1999 2 7 systematic resampling method is similar in structure but can fail to converge depending on the order of the input samples we introduce a new resampling algorithm based on a stochastic rounding technique of in 42nd ieee symposium on foundations of computer science las vegas nv 2001 2001 588 597 ieee computer soc which shares some attractive properties of systematic resampling but which exhibits negative association and therefore converges irrespective of the order of the input samples we confirm a conjecture made by j comput graph statist 5 1996 1 25 that ordering input samples by their states in mathbb r yields a faster rate of convergence we establish that when particles are ordered using the hilbert curve in mathbb r d the variance of the resampling error is scriptstyle mathcal o n 1 1 d under mild conditions where n is the number of particles we use these results to establish asymptotic properties of particle algorithms based on resampling schemes that differ from multinomial resampling

fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max min composition in this paper we study the new linear objective function optimization with respect to the fuzzy relational inequalities defined by max min composition in which fuzzy inequality replaces ordinary inequality in the constraints fuzzy inequalities enable us to attain the optimal points that are better solutions than those resulting from the resolution of the similar problems with ordinary inequality constraints denoted by x o i in this paper we have presented an algorithm to generate such optimal solutions also two examples are given to illustrate the algorithm presented and some real application of this work

solutions of fuzzy relation equations based on continuous t norms this study is concerned with fuzzy relation equations with continuous t norms in the form atr b where a and b are the fuzzy subsets of x and y respectively email protected xxy is a fuzzy relation and t is a continuous t norm the problem is how to determine a from r and b first an if and only if condition of being solvable is presented novel algorithms are then presented for determining minimal solutions when x and y are finite the proposed algorithms generate all minimal solutions for the equations making them efficient solving procedures

a sequential particle filter method for static models summary particle filter methods are complex inference procedures which combine importance sampling and monte carlo schemes in order to explore consistently a sequence of multiple distributions of interest we show that such methods can also offer an efficient estimation

a similarity based bidirectional approximate reasoning method for decision making systems this paper presents a similarity based bidirectional approximate reasoning method which can express the decision maker s disposition for this we shall propose a new type of similarity measure based on the ordered weighted aggregation owa operator the proposed similarity measure has a structure to involve the disposition of a decision maker by choosing a suitable weighting vector of the owa operator from this property we derive a simple and flexible type of similarity based bidirectional approximate reasoning method which can support the making of a decision under more complex situations such as industrial inspection systems for example

a note on unsolvable systems of max min fuzzy equations abstract for unsolvable systems of linear equations of the form a x b over the max min fuzzy algebra r 0 1 max min we propose an efficient method for finding a chebychev best approximation of the matrix a r m n in the set s b d r m n d x b is solvable

multilevel splitting for estimating rare event probabilities we analyze the performance of a splitting technique for the estimation of rare event probabilities by simulation a straightforward estimator of the probability of an event evaluates the proportion of simulated paths on which the event occurs if the event is rare even a large number of paths may produce little information about its probability using this approach the method we study reinforces promising paths at intermediate thresholds by splitting them into subpaths which then evolve independently if implemented appropriately this has the effect of dedicating a greater fraction of the computational effort to informative runs we analyze the method for a class of models in which roughly speaking the number of states through which each threshold can be crossed is bounded under additional assumptions we identify the optimal degree of splitting at each threshold as the rarity of the event increases it should be set so that the expected number of subpaths reaching each threshold remains roughly constant thus implemented the method is provably effective in a sense appropriate to rare event simulations these results follow from a branching process analysis of the method we illustrate our theoretical results with some numerical examples for queueing models

genetic algorithms for learning in fuzzy relational structures abstract we propose an approach of integrated genetic learning in construction of fuzzy relational architectures described by fuzzy relational equations these equations have been widely utilized e g in fuzzy model identification fuzzy control and fuzzy controllers while the theoretical foundations of the equations are well developed they still call for more efficient and diversified schemes of learning the paper addresses this issue by synergistically combining some fundamental concepts of genetic algorithms ga and standard gradient based techniques into a unified scheme of a stratified learning especially we reveal how the ideas of gas can be effectively used at the level of the initialization of the gradient based learning schemes it is also elucidated how an optimal subpopulation of the strings can be helpful in forming feasibility regions guarding zones indispensable for supervising the subsequent phase of the parametric learning

multilevel sequential importance sampling for rare event estimation the estimation of the probability of rare events is an important task in reliability and risk assessment we consider failure events that are expressed in terms of a limit state function which dep

diagnosis problem solving using fuzzy relations this paper deals with the restoration and the identification of the causes diagnoses through the observed effects symptoms on the basis of fuzzy relations and zadeh s compositional rule of inference we propose an approach for building fuzzy systems of diagnosis which enables solving fuzzy relational equations together with design and tuning of fuzzy relations on the basis of expert and experimental information the essence of tuning consists of the selection such membership functions for fuzzy causes and effects and also fuzzy relations which minimize the difference between model and experimental results of diagnosis the genetic algorithm is used for solving the optimization problem the proposed approach is illustrated by the computer experiment and the real example of diagnosis

system of fuzzy relation equations with inf composition complete set of solutions the problem of solvability of a system of equations with inf composition is considered on finite universes equations are expressed using operations of a bl algebra we study complete set of solutions of the respective system in the particular case one equation and in the general case in both cases various conditions of solvability are found and proved we characterize all maximal solutions and prove that under certain conditions each solution of the system an equation is less than or equal to a respective maximal one as a result we are able to characterize the complete set of solutions examples of single equations and their systems are considered and complete sets of solutions are found for each

lower solutions of systems of fuzzy equations lower solutions of a system of fuzzy equations are determined by lower solutions of each equation the result is applied to fuzzy relation equations so that they are decomposed to simple fuzzy relation equations

finitary solvability conditions for systems of fuzzy relation equations in this paper we consider the problem of solving systems of fuzzy relation equations two types of these systems with different compositions are considered and processed simultaneously a new sufficient condition and new solvability criteria are proposed for systems of both types all these conditions are finitary and thus can be easily verified the sufficient condition and criteria of solvability characterize a relationship between a skeleton nxn matrix n is a number of equations and a vector of the right hand side

on the set of solutions of composite fuzzy relation equations in this paper we introduce some algorithms which have minimization properties about the fuzziness of solutions in the maxmin fuzzy relation equations

an inference network for bidirectional approximate reasoning based on an equality measure an inference network is proposed as a tool for bidirectional approximate reasoning the inference network can be designed directly from the given fuzzy data knowledge if a fuzzy input is given for the inference network then the network renders a reasonable fuzzy output after performing approximate reasoning based on an equality measure conversely due to the bidirectional structure the network can yield its corresponding reasonable fuzzy input for a given fuzzy output this property makes it possible to perform forward and backward reasoning in the knowledge base system

inverse problem in fuzzy relational equations abstract the paper deals with the problem of handling fuzzy relational equations of the form x r y in the case of an empty solution set with respect to x x r y discussing the essence of the structure of the fuzzy relation r with respect to provided y algorithms have been proposed which enable us to modify the original equation i e the fuzzy set y in a way leading to its genuine solution or at least an approximate one it is explained how these modifications relax a set of constraints needed to be satisfied for any solvable equation a grade of the modification of the original problem is expressed in terms of distortion of the fuzzy set y which forms a cornerstone of the proposed procedures numerical examples are provided some aspects related to the area of diagnostic models in which these equations are studied are also treated

on the existence of solutions of systems of fuzzy equations abstract a necessary and sufficient condition for the existence of solutions of a fuzzy equation given by e sanchez is extended to systems of fuzzy equations

decentralized fuzzy control of multivariable systems by active decomposition of control laws this paper considers the problem of decentralized fuzzy control of multivariable systems some definitions and theorems with regard to this problem are given decentralized fuzzy control algorithms based on active decomposition of control laws are presented and illustrated by numerical examples the algorithms use local sets of fuzzy relations decoupling the influence of state on control variables when added to the original fuzzy relations it is shown that the number of fuzzy relations is significantly reduced and thus real time control implementation is facilitated

approximate solutions of fuzzy relational equations in the previous chs we have discussed various problems concerning solutions of fuzzy relation equations obviously an underlying assumption is that there exists a nonempty set of solutions a situation may occurr and indeed it is quite common in which no solution exists nevertheless even in this case one might be interested to obtain an approximate solution and know to which extent it can be viewed as a solution this stream of investigations is particularly interesting for applicational purposes contrary to the topics already discussed in the previous chs this is a field of research which has not been developed enough so far it concerns studies on solvability properties of fuzzy relation equations in this ch we shall try to answer by using several techniques the following basic question how difficult is it to attain a situation in which the system of equations has solutions and then how to measure this property

extending the contraposition property of propositional logic for fuzzy abduction abduction deals with assumption based reasoning to explain an observation in the context of fuzzy reasoning abduction attempts to determine the membership function of the fuzzy propositions present in the antecedent of a rule when the membership functions for the propositions in the consequent of the rule are given currently available models of fuzzy abduction are capable of inferring the membership function of the antecedent clause accurately when the antecedent includes single fuzzy proposition however when the antecedent clause of a rule contains multiple fuzzy propositions these models fail to determine the independent membership function of the individual propositions present in the antecedent this paper presents a new formulation to handle the above problem by fuzzy extension of the well known contraposition property of propositional logic several interesting properties due to the fuzzy extension of the classical contraposition have been derived an algorithm for automated abduction using the extended contraposition property has been developed to demonstrate the principle of abduction with rules containing one or more fuzzy propositions in the antecedent consequent the time complexity of the proposed fuzzy abduction for a sequence of n chained rules where each rule has m fuzzy propositions is o mn considering a uniform cost for composition operation and t norm computation of the antecedent

processing in relational structures fuzzy relational equations abstract the paper provides an overview of methodology for processing fuzzy information in relational structures completed in terms of fuzzy relational equations the origin and the central role of relational calculus in general and fuzzy sets in particular is explained with special emphasis paid to its role and representation capabilities it is pointed out that fuzzy relational equations play a significant role as a platform for a uniform development of techniques in fuzzy sets many of the problems and frameworks developed in fuzzy sets so far can be easily translated into the language of fuzzy relational equations which immediately takes a significant advantage of their solid well developed formalisms techniques and clarity of exposition we will study the role of approximate solutions to fuzzy relational equations as a convenient tool to handle probabilistic type of uncertainty moreover those solutions can easily explain the origin and generate a way in which fuzzy sets of higher generality such as for instance type 2 sets interval valued sets can be algorithmically determined finally we study the role of equations in diverse fields of applications making use of an ample framework of general systems theory

fuzzy modelling fundamentals construction and evaluation abstract the paper is devoted to fundamental aspects of fuzzy modelling namely modelling articulated with the use of methodology of fuzzy sets open issues are addressed which summarize advantages of fuzzy models and point out some methodological deficiencies blurring existing approaches the problem of relevancy of fuzzy models and its direct impact on a suitable interpretation of derived results is studied in detail finally we analyze some particular fuzzy models developed in terms of relational equations

identification of fuzzy relational equations by fuzzy neural networks abstract although the identification of a fuzzy system by a feedforward neural network is obviously possible whenever the system was continuous however when we use fuzzy max min neural network some problems on the learning of the network arise in this paper we present a method to identify a fuzzy relation by a fuzzy max min neural network we adapt the learning algorithm backpropagation to learning of max min neural network by using one which we will denote as smooth derivative finally we present some examples and a comparison with a similar method

iteration algorithm for solving ax b in max min algebra abstract for solving the linear system a x b we introduce some iteration algorithms where a aij and b bi a given matrix and a column vector with elements from a brouwerian lattice and is a max min composite operation

solvability of fuzzy relational equations and manipulation of fuzzy data abstract the problem of characterization of fuzzy relational equations with respect to their solvability property is studied some aspects of manipulation of fuzzy data with the aid of fuzzy relational equations are considered two stages of manipulation process are indicated combining pieces of evidence and inferring their mutual correspondence both of them are formulated and solved by the use of fuzzy relational equations the role of a solvability index is extensively studied numerical considerations give an illustration of the approach we propose and deal with some real data used for fuzzy controllers

a multi resolution non parametric bayesian framework for identification of spatially varying model parameters this paper proposes a hierarchical multi resolution framework for the identification of model parameters and their spatially variability from noisy measurements of the response or output such parameters are frequently encountered in pde based models and correspond to quantities such as density or pressure fields elasto plastic moduli and internal variables in solid mechanics conductivity fields in heat diffusion problems permeability fields in fluid flow through porous media etc the proposed model has all the advantages of traditional bayesian formulations such as the ability to produce measures of confidence for the inferences made and providing not only predictive estimates but also quantitative measures of the predictive uncertainty in contrast to existing approaches it utilizes a parsimonious non parametric formulation that favors sparse representations and whose complexity can be determined from the data the proposed framework in non intrusive and makes use of a sequence of forward solvers operating at various resolutions as a result inexpensive coarse solvers are used to identify the most salient features of the unknown field s which are subsequently enriched by invoking solvers operating at finer resolutions this leads to significant computational savings particularly in problems involving computationally demanding forward models but also improvements in accuracy it is based on a novel adaptive scheme based on sequential monte carlo sampling which is embarrassingly parallelizable and circumvents issues with slow mixing encountered in markov chain monte carlo schemes the capabilities of the proposed methodology are illustrated in problems from nonlinear solid mechanics with special attention to cases where the data is contaminated with random noise and the scale of variability of the unknown field is smaller than the scale of the grid where observations are collected

quantum levy laplacian and associated heat equation abstract as a non commutative extension of the levy laplacian for entire functions on a nuclear space we define the quantum levy laplacian acting on white noise operators we solve a heat type equation associated with the quantum levy laplacian and study its relation to the classical levy heat equation the solution to the quantum levy heat equation is obtained also from a normal ordered white noise differential equation involving the quadratic quantum white noise

solvability criteria for systems of fuzzy relation equations abstract by solving systems of fuzzy relation equations qualitative process models can be obtained to give more information on the solving procedure and to help constructing models solvability criteria for systems of fuzzy relation equations are necessary in this article such criteria will be developed both sup and inf composition methods are considered in addition to some ideas on general t norms the minimum norm is evaluated in detail criteria of practical use will be developed these criteria will limit the variety of premise intersections to guarantee solvability nevertheless they will still allow to model the significant behaviour of the processes