uncertainty measures for general fuzzy relations abstract relation is one of the important concepts in rough set models some uncertainty measures for relations were proposed in recent years by generalizing information entropy however the existing theories considered only special relations and do not address mixed type data well they always transformed each attribute into a binary relation with the same structure regardless of attribute types as a result the structure information can be lost for other types of attributes in this paper a new theory about uncertainty measures for general fuzzy relations is proposed the proposed theory can overcome the weakness of the existing theories we first introduce a novel entropy to compute the uncertainty information of fuzzy binary relations and then present the concepts of joint fuzzy entropy conditional fuzzy entropy and fuzzy mutual information the basic properties of these uncertainty measures are studied finally we apply the proposed theory to perform attribute reduction of heterogeneous data sets the experimental results show that our proposed method is feasible and effective in some cases the proposed theory can provide a fundamental framework that integrates all kinds of methods for the uncertainty information of special relations

fuzzy information entropy based adaptive approach for hybrid feature outlier detection abstract fuzzy information entropy based on fuzzy relation in fuzzy rough set theory is an important metric of uncertainty however the research of fuzzy information entropy for hybrid feature outlier detection has not been reported on this basis this paper constructs a hybrid feature outlier detection method based on fuzzy information entropy by using fuzzy approximate space with fuzzy similarity relation firstly the adaptive fuzzy radius of standard deviation and hybrid fuzzy similarity are employed to construct the fuzzy approximate space and the relative fuzzy entropy is defined based on the fuzzy information entropy then two kinds of metrics are constructed to describe the outlier degree of object finally the fuzzy entropy based outlier factor is integrated to implement outlier detection and the relevant fuzzy information entropy based outlier detection algorithm fieod is designed the fieod algorithm is compared with the main outlier detection algorithms on public data the experimental results reveal that the proposed method has better effectiveness and adaptability

a wavefront based gaussian beam method for computing high frequency wave propagation problems we present a novel wavefront method based on gaussian beams for computing high frequency wave propagation problems unlike standard geometrical optics gaussian beams compute the correct solution of the wave field also at caustics the method tracks a front of two canonical beams with two particular initial values for width and curvature in a fast post processing step from the canonical solutions we recreate any other gaussian beam with arbitrary initial data on the initial front this provides a simple mechanism to include a variety of optimization processes including error minimization and beam width minimization for a posteriori selection of optimal beam initial parameters the performance of the method is illustrated with two numerical examples

the information content of fuzzy relations and fuzzy rules based on different emphases some general information content measures of fuzzy relations are given distinguished from the measures related to fuzzy relations proposed before these measures try to estimate the information conveyed by multiple domain fuzzy relations without preassigned probability distribution since fuzzy rules can be fully captured by fuzzy relations from an input universe to an output universe the information content of fuzzy rules can be easily measured by the information content of fuzzy relations proposed by us however there also exists a difference between fuzzy relations and fuzzy rules rules especially classification rules always have a direction from the antecedent and consequent while relations do not have direction based on this difference some general measures for the information content of fuzzy rules are proposed in practice these measures can do well in the evaluation and selection of fuzzy rules finally the measures of the information content of fuzzy rules are used to evaluate the stability and sensitivity of fuzzy implication operators in fuzzy control

on a family of connectives for fuzzy sets abstract in this paper we find the general solution of the functional equation s t x y t x n y x where s is a t conorm t is a t norm and n is a strong negation on the unit interval in particular the result yields a family of connectives for fuzzy sets

fuzzy partitions with the connectives t s abstract the notion of a fuzzy partition obtained when the set operations are defined using the operators t and s and an appropriate concept of disjointness is according to butnariu 1983 equivalent with ruspini s definition we prove that such an equivalence holds if and only if the connectives for the fuzzy set operations are t and s

on hudetz entropy localization abstract in rahimi and riazi 2014 22 a type of entropy of continuous maps on compact metric spaces is defined via integrating a local entropy map this average entropy is represented in terms of the entropy in the sense of dumitrescu but the equality of these two types of entropy is left as an open problem in 22 in this paper we modify the problem stated in 22 by replacing the dumitrescu entropy by the hudetz entropy since in our case the two previous types of entropy take an infinite value and the problem stated in 22 reduces to a trivial equality then we conclude the paper by proving the equality of the modified average entropy and the hudetz entropy the key tool is a version of jaccob s theorem for the hudetz entropy

principles for constructing three way approximations of fuzzy sets a comparative evaluation based on unsupervised learning abstract three way approximations of fuzzy sets are an important scheme of granular computing by abstracting a fuzzy set to its discrete three option alternatives which adhere to human cognitive behaviors and reduce the computational burden the key point of such three way approximations of fuzzy sets is how to choose a suitable design leading to their realization undesired three way approximations might be produced if the selected mechanism is unsuitable to data distribution in this study the principles for constructing three way approximations of fuzzy sets are summarized the following taxonomy of these principles is provided namely i uncertainty balance based principle ii prototype based principle and iii model based invoking the tradeoff between classification error and the number of data that have to be classified a number of detailed optimization models are discussed in detail to evaluate the performance of different construction principles a general unsupervised learning framework based on three way approximations of fuzzy sets is exhibited some synthetic data sets along with sixteen data sets from uci repository are involved for experiments friedman testing followed by holm bonferroni testing are exploited to test the performance significance of the proposed criteria which can provide insights and deliver guidance when choosing a principle for constructing three way approximations of fuzzy sets in the real world scenarios the research methods in this paper can also be extended to supervised and semi supervised learning areas

necessary and sufficient conditions for generalized uniform fuzzy partitions abstract the fundamental concept in the theory of fuzzy transform f transform is the fuzzy partition which is a generalization of the classical concept of partition the original definition assumes that every two normal fuzzy subsets in a partition overlap in such a way that the sum of the membership degrees at each point is equal to 1 this condition can be generalized by relaxing the assumption of normality for fuzzy sets the result is a denser fuzzy partition that may improve the approximation properties and the smoothness of the inverse f transform a fuzzy partition with this property will be referred to as general the problem is how a general fuzzy partition can be effectively constructed if we use a generating function with special properties then it is not immediately clear whether it defines a general fuzzy partition in this paper we find a necessary and sufficient condition that will enable the optimal generalized fuzzy partition to be designed more easily which is important in various practical applications of the f transform for example image processing time series analysis and solving differential equations with boundary conditions

characterization of fuzzy measures constructed by means of triangular norms abstract following the ideas presented by the author e p klement j math anal appl 85 1982 543 565 finite t fuzzy measures are introduced t being a measurable triangular norm we show that a t fuzzy measure is always a fuzzy measure as considered earlier e p klement j math anal appl 25 1980 330 339 then we study the relation to the integral with respect to some classical measure finally for some special triangular norms t we give precise characterizations of the corresponding classes of t fuzzy measures

error estimates for gaussian beam superpositions gaussian beams are asymptotically valid high frequency solutions to hyperbolic partial differential equations concentrated on a single curve through the physical domain they can also be extended to some dispersive wave equations such as the schrodinger equation superpositions of gaussian beams provide a powerful tool to generate more general high frequency solutions that are not necessarily concentrated on a single curve this work is concerned with the accuracy of gaussian beam superpositions in terms of the wavelength epsilon we present a systematic construction of gaussian beam superpositions for all strictly hyperbolic and schrodinger equations subject to highly oscillatory initial data of the form ae i phi epsilon through a careful estimate of an oscillatory integral operator we prove that the k th order gaussian beam superposition converges to the original wave field at a rate proportional to epsilon k 2 in the appropriate norm dictated by the well posedness estimate in particular we prove that the gaussian beam superposition converges at this rate for the acoustic wave equation in the standard epsilon scaled energy norm and for the schrodinger equation in the l 2 norm the obtained results are valid for any number of spatial dimensions and are unaffected by the presence of caustics we present a numerical study of convergence for the constant coefficient acoustic wave equation in r 2 to analyze the sharpness of the theoretical results

entropy of fuzzy partitions a general model abstract the entropy of fuzzy partitions with regard to a fuzzy measure is analyzed a general model based on pseudo arithmetical operations and g functions will be created the entropy of fuzzy partitions introduced by ruspini inform and control 15 1969 22 and dumitrescu j math anal appl 176 1993 359 is shown to be reasonable only up to an isomorphism

on local entropy of fuzzy partitions in this paper we will introduce a new kind of fuzzy version of the local entropy of a continuous dynamical system on a compact metric space and will prove some of its properties then we will define an average entropy of fuzzy partitions by integrating the local fuzzy entropy we will also state the relationship between this entropy and the fuzzy entropy of dynamical systems based on the entropy of fuzzy partitions in the sense of dumitrescu

fuzzy measures and the entropy of fuzzy partitions abstract using an additive fuzzy measure a notion of the entropy of a finite fuzzy partition has been defined d dumitrescu on fuzzy partitions in itinerant seminar on functional equations approximation and convexity cluj napoca 1983 pp 57 60 d dumitrescu and m barbu fuzzy entropy and processes in itinerant seminar on functional equations approximation and convexity cluj napoca 1985 pp 71 74 this notion contains as a particular case a well known concept of entropy m a gil et al fuzzy sets and systems15 1985 65 78 h tanaka et al kybernetes5 1976 25 30 i j taneja bivariate entropies of finite partitions of fuzzy sets in preprints of second ifsa congress tokyo 1987 vol 2 pp 696 699 the aim of this paper is to review some basic properties of the considered entropy

fundamental triangular norm based tribes and measures abstract we complete the results of butnariu and klement about t s tribes and t s measures where t s s 0 is a frank s fundamental triangular norm for a denumerable universum x we show that any t s tribe is a semigenerated one and hence any t s measure is a generated one

i fuzzy equivalence relations and i fuzzy partitions a t fuzzy equivalence relation is a fuzzy binary relation on a set x which is reflexive symmetric and t transitive for a t norm t recently mesiar et al r mesiar b reusch h thiele fuzzy equivalence relations and fuzzy partitions j multi valued logic soft comput 12 2006 167 181 have generalised the t norm t to any general conjunctor c and investigated the minimal assumptions required on such operations called duality fitting conjunctors such that the fuzzification of the equivalence relation admits any value from the unit interval and also the one one correspondence between the fuzzy equivalence relations and fuzzy partitions is preserved in this work we conduct a similar study by employing a related form of c transitivity viz i transitivity where i is an implicator we show that although every i fuzzy equivalence relation can be shown to be a c fuzzy equivalence relation there exist c fuzzy equivalence relations that are not i fuzzy equivalence relations and hence these concepts are not equivalent most importantly we show that the class of duality fitting implicators i is much richer than the residuals of the duality fitting conjunctors in the study of mesiar et al we also show that the i fuzzy partitions have a constant wise structure

prestack gaussian beam depth migration kirchhoff migration is the most popular method of three dimensional prestack depth migration because of its flexibility and efficiency its effectiveness can become limited however when complex velocity structure causes multipathing of seismic energy an alternative is gaussian beam migration which is an extension of kirchhoff migration that overcomes many of the problems caused by multipathing unlike first arrival and most energetic arrival methods which retain only one traveltime this alternative method retains most arrivals by the superposition of gaussian beams this paper presents a prestack gaussian beam migration method that operates on common offset gathers the method is efficient because the computation of beam superposition isolates summations that do not depend on the seismic data and evaluates these integrals by considering their saddle points gaussian beam migration of the two dimensional marmousi test data set demonstrates the method s effectiveness for structural imaging in a case w

fuzzy probabilistic approximation spaces and their information measures rough set theory has proven to be an efficient tool for modeling and reasoning with uncertainty information by introducing probability into fuzzy approximation space a theory about fuzzy probabilistic approximation spaces is proposed in this paper which combines three types of uncertainty probability fuzziness and roughness into a rough set model we introduce shannon s entropy to measure information quantity implied in a pawlak s approximation space and then present a novel representation of shannon s entropy with a relation matrix based on the modified formulas some generalizations of the entropy are proposed to calculate the information in a fuzzy approximation space and a fuzzy probabilistic approximation space respectively as a result uniform representations of approximation spaces and their information measures are formed with this work