rough prime ideals and rough fuzzy prime ideals in semigroups in this paper we shall introduce the notions of rough prime ideals and rough fuzzy prime ideals in a semigroup and give some properties of such ideals also we will discuss the relations between the upper lower rough prime ideals and the upper lower approximations of their homomorphism images

axiomatics for fuzzy rough sets abstract a fuzzy t rough set consists of a set x and a t similarity relation r on x where t is a lower semi continuous triangular norm we generalize the farinas prade definition for the upper approximation operator a i x i x of a fuzzy t rough set x r given originally for the special case t min to the case of arbitrary t we propose a new definition for the lower approximation operator a i x i x of x r our definition satisfies the two important identities a a a and a a a as well as a number of other interesting properties we provide axiomatics to fully characterize those upper and lower approximations

neighborhood systems and approximate retrieval an approximate retrieval model is proposed based on the notion of neighborhood systems the knowledge used in the model consists of an information table in which each object is represented by its values on a finite set of attributes and neighborhood systems on attribute values which provide semantic similarity or closeness of different values an information table can be used for exact retrieval with the introduction of neighborhood systems to information tables one is able to perform approximate retrieval operations on neighborhood systems are introduced based on power algebras an ordering relation representing the information of a neighborhood system is suggested and examined approximate retrieval is carried out by the relaxation of the original query using neighborhood systems and the combination of intermediate results using neighborhood system operations the final retrieval results are presented according to the proposed ordering relation in contrast to many existing systems a main advantage of the proposed model is that the retrieval results are a non linear ordering of objects

efficient parallel boolean matrix based algorithms for computing composite rough set approximations in information systems there may exist multiple different types of attributes like categorical attributes numerical attributes set valued attributes interval valued attributes missing attributes etc such information systems are called as composite information systems to process such attributes with rough set theory composite rough set model and corresponding matrix methods were introduced in our previous study calculation of rough set approximations of a concept is the key step for rule acquisition and attribute reduction in rough set based methods to accelerate the computation process of rough set approximations this paper first presents the boolean matrix representation of the lower and upper approximations in the composite information system then designs a parallel method for computing approximations based on matrix and implements it on multi gpu the experiments on data sets from uci and user defined data sets show that the proposed method can accelerate the computation process efficiently the multi gpu implementation achieves up to a speedup of 334 9 over the cpu implementation

information granularity in fuzzy binary grc model zadeh s seminal work in theory of fuzzy information granulation in human reasoning is inspired by the ways in which humans granulate information and reason with it this has led to an interesting research topic granular computing grc although many excellent research contributions have been made there remains an important issue to be addressed what is the essence of measuring a fuzzy information granularity of a fuzzy granular structure what is needed to answer this question is an axiomatic constraint with a partial order relation that is defined in terms of the size of each fuzzy information granule from a fuzzy binary granular structure this viewpoint is demonstrated for fuzzy binary granular structure which is called the binary grc model by lin we study this viewpoint from from five aspects in this study which are fuzzy binary granular structure operators partial order relations measures for fuzzy information granularity an axiomatic approach to fuzzy information granularity and fuzzy information entropies

roughness of a fuzzy set an integration between the theories of fuzzy sets and rough sets has been attempted by providing a measure of roughness of a fuzzy set several properties of this new measure are established some of the possible applications for handling uncertainties in the field of pattern recognition are mentioned

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

relational interpretations of neighborhood operators and rough set approximation operators this paper presents a framework for the formulation interpretation and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations a special class of neighborhood systems called 1 neighborhood systems is introduced three extensions of pawlak approximation operators are analyzed properties of neighborhood and approximation operators are studied and their connections are examined

on robust fuzzy rough set models rough sets especially fuzzy rough sets are supposedly a powerful mathematical tool to deal with uncertainty in data analysis this theory has been applied to feature selection dimensionality reduction and rule learning however it is pointed out that the classical model of fuzzy rough sets is sensitive to noisy information which is considered as a main source of uncertainty in applications this disadvantage limits the applicability of fuzzy rough sets in this paper we reveal why the classical fuzzy rough set model is sensitive to noise and how noisy samples impose influence on fuzzy rough computation based on this discussion we study the properties of some current fuzzy rough models in dealing with noisy data and introduce several new robust models the properties of the proposed models are also discussed finally a robust classification algorithm is designed based on fuzzy lower approximations some numerical experiments are given to illustrate the effectiveness of the models the classifiers that are developed with the proposed models achieve good generalization performance

constructive and axiomatic approaches of fuzzy approximation operators this paper presents a general framework for the study of rough set approximation operators in fuzzy environment in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized fuzzy rough and rough fuzzy respectively approximation operators is first defined the representations of both fuzzy rough approximation operators and rough fuzzy approximation operators are then presented the connections between fuzzy and crisp respectively relations and fuzzy rough and rough fuzzy respectively approximation operators are further established in axiomatic approach various classes of fuzzy approximation operators are characterized by different sets of axioms the minimal axiom sets of fuzzy approximation operators guarantee the existence of certain types of fuzzy or crisp relations producing the same operators

soft fuzzy rough sets for robust feature evaluation and selection the fuzzy dependency function proposed in the fuzzy rough set model is widely employed in feature evaluation and attribute reduction it is shown that this function is not robust to noisy information in this paper as datasets in real world applications are usually contaminated by noise robustness of data analysis models is very important in practice in this work we develop a new model of fuzzy rough sets called soft fuzzy rough sets which can reduce the influence of noise we discuss the properties of the model and construct a new dependence function from the model then we use the function to evaluate and select features the presented experimental results show the effectiveness of the new model

rough set theory for the interval valued fuzzy information systems the notion of a rough set was originally proposed by pawlak z pawlak rough sets international journal of computer and information sciences 11 5 1982 341 356 later on dubois and prade d dubois h prade rough fuzzy sets and fuzzy rough sets international journal of general system 17 2 3 1990 191 209 introduced rough fuzzy sets and fuzzy rough sets as a generalization of rough sets this paper deals with an interval valued fuzzy information system by means of integrating the classical pawlak rough set theory with the interval valued fuzzy set theory and discusses the basic rough set theory for the interval valued fuzzy information systems in this paper we firstly define the rough approximation of an interval valued fuzzy set on the universe u in the classical pawlak approximation space and the generalized approximation space respectively i e the space on which the interval valued rough fuzzy set model is built secondly several interesting properties of the approximation operators are examined and the interrelationships of the interval valued rough fuzzy set models in the classical pawlak approximation space and the generalized approximation space are investigated thirdly we discuss the attribute reduction of the interval valued fuzzy information systems finally the methods of the knowledge discovery for the interval valued fuzzy information systems are presented with an example

learning optimization in simplifying fuzzy rules it is important that an optimal learning problem is proved to be np hard and the heuristic algorithm for solving the problem has to be given this paper deals with a learning problem appearing in the process of simplifying fuzzy rules proves that the solution optimization is np hard and gives its heuristic algorithm this heuristic regarded as a new fuzzy learning algorithm has many significant advantages

roughness based on fuzzy ideals the theory of rough set proposed by pawlak and the theory of fuzzy set proposed by zadeh are complementary generalizations of classical set theory many sets are naturally endowed with two binary operations addition and multiplication one concept which does this is a ring this paper concerns a relationship between rough sets fuzzy sets and ring theory it is a continuation of ideas presented by kuroki and wang n kuroki p p wang the lower and upper approximations in a fuzzy group inform sci 90 1996 203 220 we consider a ring as a universal set and we assume that the knowledge about objects is restricted by a fuzzy ideal in fact we apply the notion of fuzzy ideal of a ring for definitions of the lower and upper approximations in a ring some characterizations of the above approximations are made and some examples are presented

rough approximations on a complete completely distributive lattice with applications to generalized rough sets the generalizations of rough sets considered with respect to similarity relation covers and fuzzy relations are main research topics of rough set theory however these generalizations have shown less connection among each other and have not been brought into a unified framework which has limited the in depth research and application of rough set theory in this paper the complete completely distributive ccd lattice is selected as the mathematical foundation on which definitions of lower and upper approximations that form the basic concepts of rough set theory are proposed these definitions result from the concept of cover introduced on a ccd lattice and improve the approximations of the existing crisp generalizations of rough sets with respect to similarity relation and covers when t similarity relation is considered the existing fuzzy rough sets are the special cases of our proposed approximations on a ccd lattice thus these generalizations of rough sets are brought into a unified framework and a wider mathematical foundation for rough set theory is established

generalized fuzzy rough sets this paper presents a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized approximation operators is defined the connections between fuzzy relations and fuzzy rough approximation operators are examined in axiomatic approach various classes of fuzzy rough approximation operators are characterized by different sets of axioms axioms of fuzzy approximation operators guarantee the existence of certain types of fuzzy relations producing the same operators

performance measures independent of adjustment an explanation and extension of taguchi s signal to noise ratios parameter design is a method popularized by japanese quality expert g taguchi for designing products and manufacturing processes that are robust to uncontrollable variations in parameter design taguchi s stated objective is to find the settings of product or process design parameters that minimize average quadratic loss that is the average squared deviation of the response from its target value yet in practice to choose the settings of design parameters he maximizes a set of measures called signal to noise ratios in general he gives no connection between these two optimization problems in this article we show that for certain underlying models for the product or process response maximization of the signal to noise ratio leads to minimization of average quadratic loss the signal to noise ratios take advantage of the existence of special design parameters called adjustment parameters when these parameters exist use of the signal to noise ratio allows the parameter design optimization procedu

on the generalization of fuzzy rough sets rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets the existing generalizations of fuzzy rough sets are all based on special fuzzy relations fuzzy similarity relations t similarity relations it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches in this paper from the viewpoint of constructive approach we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined in axiomatic approach we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms the lattice and topological structures of fuzzy rough sets are also proposed in order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets a special lower approximation operator is applied to a fuzzy reasoning system which coincides with the mamdani algorithm

families of elliptic curves over quartic number fields with prescribed torsion subgroups in this paper we construct infinite families of elliptic curves with given torsion group structures over cubic number fields this result provides explicit examples of the theoretical result recently developed by the first two authors and a schweizer they determined all the group structures which occur infinitely often as the torsion of elliptic curves over cubic number fields in fact this paper presents an efficient way of constructing such families of elliptic curves with prescribed torsion group structures over cubic number fields

on fuzzy rough sets based on tolerance relations dubois and prade 1990 1 introduced the notion of fuzzy rough sets as a fuzzy generalization of rough sets which was originally proposed by pawlak 1982 8 later radzikowska and kerre introduced the so called i t fuzzy rough sets where i is an implication and t is a triangular norm in the present paper by using a pair of implications i j we define the so called i j fuzzy rough sets which generalize the concept of fuzzy rough sets in the sense of radzikowska and kerre and that of mi and zhang basic properties of i j fuzzy rough sets are investigated in detail

rough approximation of a fuzzy concept on a hybrid attribute information system and its uncertainty measure abstract one important and valuable topic in rough set theory is the concept of approximation over various generalized information systems although rough set models and approaches over hybrid attribute sets have been studied by many researchers recently the studies focus on the rough approximation of a crisp concept over a hybrid attribute information system this paper considers the rough approximation of a fuzzy concept over a hybrid attribute information system we define a hybrid indiscernibility relation by compounding the fuzzy indiscernibility similarity relation over numerical attributes with the equivalence relation over symbolic attributes then we present the lower and upper approximations of a fuzzy set based on the hybrid indiscernibility relation i e the fuzzy rough set over a hybrid attribute information system also some interesting properties of the hybrid fuzzy rough approximation operators are presented in detail and the relationship between the fuzzy rough set and the existing rough set model over a hybrid attribute information system is established meanwhile based on the definition of the cut set of a fuzzy set of the universe of discourse the representations of the lower and upper approximations of a fuzzy set are given with respect to hybrid attribute approximation space furthermore the concept of the cut set of the hybrid indiscernibility relation of the hybrid attribute approximation space is defined then the characterizations of the hybrid fuzzy rough approximation operators are established as well two different types of characterization theorems of the hybrid fuzzy rough approximation operators are presented based on the cut set of the hybrid indiscernibility relation over the hybrid attribute information system at the same time we study the uncertainty measure of the fuzzy rough set over a hybrid attribute information system by using knowledge granulation the main contribution of this paper is twofold one is to extend the existing rough set approach over hybrid attribute information systems to a fuzzy environment another is to present a new way to define the rough approximation of a fuzzy concept as well the uncertainty measure over hybrid attribute information systems

defining equations of modular curves we obtain defining equations of modular curves x0 n x1 n and x n by explicitly constructing modular functions using generalized dedekind eta functions as applications we describe a method of obtaining a basis for the space of cusp forms of weight 2 on a congruence subgroup we also use our model of x0 37 to find explicit modular parameterization of rational elliptic curves of conductor 37

reasoning within expressive fuzzy rough description logics it is generally accepted that the management of imprecision and vagueness will yield more intelligent and realistic knowledge based applications description logics dls are suitable well known logics for managing structured knowledge that have gained considerable attention the last decade the current research progress and the existing problems of uncertain or imprecise knowledge representation and reasoning in dls are analyzed in this paper an integration between the theories of fuzzy dls and rough dls has been attempted by providing fuzzy rough dls based on fuzzy rough set theory the syntax semantics and properties of fuzzy rough dls are given it is proved that the satisfiability subsumption entailment and abox consistency reasoning in fuzzy rough dls may be reduced to the abox consistency reasoning in the corresponding fuzzy dls

neighborhood rough set based heterogeneous feature subset selection feature subset selection is viewed as an important preprocessing step for pattern recognition machine learning and data mining most of researches are focused on dealing with homogeneous feature selection namely numerical or categorical features in this paper we introduce a neighborhood rough set model to deal with the problem of heterogeneous feature subset selection as the classical rough set model can just be used to evaluate categorical features we generalize this model with neighborhood relations and introduce a neighborhood rough set model the proposed model will degrade to the classical one if we specify the size of neighborhood zero the neighborhood model is used to reduce numerical and categorical features by assigning different thresholds for different kinds of attributes in this model the sizes of the neighborhood lower and upper approximations of decisions reflect the discriminating capability of feature subsets the size of lower approximation is computed as the dependency between decision and condition attributes we use the neighborhood dependency to evaluate the significance of a subset of heterogeneous features and construct forward feature subset selection algorithms the proposed algorithms are compared with some classical techniques experimental results show that the neighborhood model based method is more flexible to deal with heterogeneous data

joint estimation of location dispersion and random effects in robust design this article presents a methodology for simultaneously modeling three components of a general mixed model approach to robust design location fixed effects dispersion effects and random effects control noise and random factors can be accommodated with this approach parameters associated with all three are estimated jointly using residual maximum likelihood and assuming normality simulated and real datasets illustrate the key concepts and advantages over previously proposed approaches

gonality of the modular curve x1 n in this paper we compute the gonality over q of the modular curve x1 n for all n 40 and give upper bounds for each n 250 this allows us to determine all n for which x1 n has infinitely points of degree 8 we conjecture that the modular units of q x1 n are freely generated by f 2 f n 2 1 where f k is obtained from the equation for x1 k

an axiomatic characterization of a fuzzy generalization of rough sets in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties several authors have proposed various fuzzy generalizations of rough approximations in this paper we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication then we find the assumptions which permit a given fuzzy set theoretic operator to represent a upper or lower approximation derived from a special fuzzy relation different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations and different sets of axioms of fuzzy set theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator finally we study the composition of two approximation spaces it is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces

torsion subgroups of rational elliptic curves over the compositum of all d4 extensions of the rational numbers abstract let e q be an elliptic curve and let q d 4 be the compositum of all extensions of q whose galois closure has galois group isomorphic to a quotient of a subdirect product of a finite number of transitive subgroups of d 4 in this article we first show that q d 4 is in fact the compositum of all d 4 extensions of q and then we prove that the torsion subgroup of e q d 4 is finite and determine the 24 possibilities for its structure we also give a complete classification of the elliptic curves that have each possible torsion structure in terms of their j invariants

fuzzy rough sets assisted attribute selection attribute selection as refers to the problem of selecting those input attributes or features that are most predictive of a given outcome a problem encountered in many areas such as machine learning pattern recognition and signal processing unlike other dimensionality reduction methods attribute selectors preserve the original meaning of the attributes after reduction this has found application in tasks that involve datasets containing huge numbers of attributes in the order of tens of thousands which for some learning algorithms might be impossible to process further recent examples include text processing and web content classification as techniques have also been applied to small and medium sized datasets in order to locate the most informative attributes for later use one of the many successful applications of rough set theory has been to this area the rough set ideology of using only the supplied data and no other information has many benefits in as where most other methods require supplementary knowledge however the main limitation of rough set based attribute selection in the literature is the restrictive requirement that all data is discrete in classical rough set theory it is not possible to consider real valued or noisy data this paper investigates a novel approach based on fuzzy rough sets fuzzy rough feature selection frfs that addresses these problems and retains dataset semantics frfs is applied to two challenging domains where a feature reducing step is important namely web content classification and complex systems monitoring the utility of this approach is demonstrated and is compared empirically with several dimensionality reducers in the experimental studies frfs is shown to equal or improve classification accuracy when compared to the results from unreduced data classifiers that use a lower dimensional set of attributes which are retained by fuzzy rough reduction outperform those that employ more attributes returned by the existing crisp rough reduction method in addition it is shown that frfs is more powerful than the other as techniques in the comparative study

on generalized fuzzy belief functions in infinite spaces determined by a fuzzy implication operator a general type of fuzzy belief structure and its induced dual pair of fuzzy belief and plausibility functions in infinite universes of discourse are first defined relationship between the belief structure based and the belief space based fuzzy dempster shafer models is then established it is shown that the lower and upper fuzzy probabilities induced by the fuzzy belief space yield a dual pair of fuzzy belief and plausibility functions for any fuzzy belief structure there must exist a fuzzy belief space such that the fuzzy belief and plausibility functions defined by the given fuzzy belief structure are just the lower and upper fuzzy probabilities induced by the fuzzy belief space respectively essential properties of the fuzzy belief and plausibility functions are also examined the fuzzy belief and plausibility functions are respectively a fuzzy monotone choquet capacity and a fuzzy alternating choquet capacity of infinite order

the model of fuzzy variable precision rough sets the fuzzy rough set frs model has been introduced to handle databases with real values however frs was sensitive to misclassification and perturbation here misclassification means error or missing values in classification and perturbation means small changes of numerical data the variable precision rough sets vprss model was introduced to handle databases with misclassification however it could not effectively handle the real valued datasets now it is valuable from theoretical and practical aspects to combine frs and vprs so that a powerful tool which not only can handle numerical data but also is less sensitive to misclassification and perturbation can be developed in this paper we set up a model named fuzzy vprss fvprss by combining frs and vprs with the goal of making frs a special case first we study the knowledge representation ways of frs and vprs and then propose the set approximation operators of fvprs second we employ the discernibility matrix approach to investigate the structure of attribute reductions in fvprs and develop an algorithm to find all reductions third in order to overcome the np complete problem of finding all reductions we develop some fast heuristic algorithms to obtain one near optimal attribute reduction finally we compare fvprs with rs frs and several flexible rs based approaches with respect to misclassification and perturbation the experimental comparisons show the feasibility and effectiveness of fvprs

measures for evaluating the decision performance of a decision table in rough set theory as two classical measures approximation accuracy and consistency degree can be employed to evaluate the decision performance of a decision table however these two measures cannot give elaborate depictions of the certainty and consistency of a decision table when their values are equal to zero to overcome this shortcoming we first classify decision tables in rough set theory into three types according to their consistency and introduce three new measures for evaluating the decision performance of a decision rule set extracted from a decision table we then analyze how each of these three measures depends on the condition granulation and decision granulation of each of the three types of decision tables experimental analyses on three practical data sets show that the three new measures appear to be well suited for evaluating the decision performance of a decision rule set and are much better than the two classical measures

constructive and algebraic methods of the theory of rough sets this paper reviews and compares constructive and algebraic approaches in the study of rough sets in the constructive approach one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation different classes of rough set algebras are obtained from different types of binary relations in the algebraic approach one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators various classes of rough set algebras are characterized by different sets of axioms axioms of approximation operators guarantee the existence of certain types of binary relations producing the same operators

fuzzy rough granular neural networks fuzzy granules and classification we introduce a fuzzy rough granular neural network frgnn model based on the multilayer perceptron using a back propagation algorithm for the fuzzy classification of patterns we provide the development strategy of the network mainly based upon the input vector initial connection weights determined by fuzzy rough set theoretic concepts and the target vector while the input vector is described in terms of fuzzy granules the target vector is defined in terms of fuzzy class membership values and zeros crude domain knowledge about the initial data is represented in the form of a decision table which is divided into subtables corresponding to different classes the data in each decision table is converted into granular form the syntax of these decision tables automatically determines the appropriate number of hidden nodes while the dependency factors from all the decision tables are used as initial weights the dependency factor of each attribute and the average degree of the dependency factor of all the attributes with respect to decision classes are considered as initial connection weights between the nodes of the input layer and the hidden layer and the hidden layer and the output layer respectively the effectiveness of the proposed frgnn is demonstrated on several real life data sets

theory and applications of granular labelled partitions in multi scale decision tables abstract granular computing and acquisition of if then rules are two basic issues in knowledge representation and data mining a formal approach to granular computing with multi scale data measured at different levels of granulations is proposed in this paper the concept of labelled blocks determined by a surjective function is first introduced lower and upper label block approximations of sets are then defined multi scale granular labelled partitions and multi scale decision granular labelled partitions as well as their derived rough set approximations are further formulated to analyze hierarchically structured data finally the concept of multi scale information tables in the context of rough set is proposed and the unravelling of decision rules at different scales in multi scale decision tables is discussed

explicit determination of nontrivial torsion structures of elliptic curves over quadratic number fields we determine equations of the modular curves x1 n for n 11 13 14 15 16 17 and 18 except for n 17 these are the only existing elliptic or hyperelliptic x1 n applying these x1 n we calculate tables of elliptic curves e over quadratic fields k with torsion groups of one of the following isomorphism types etor k z mz m 11 13 14 15 16 and 18

hybrid attribute reduction based on a novel fuzzy rough model and information granulation feature subset selection has become an important challenge in areas of pattern recognition machine learning and data mining as different semantics are hidden in numerical and categorical features there are two strategies for selecting hybrid attributes discretizing numerical variables or numericalize categorical features in this paper we introduce a simple and efficient hybrid attribute reduction algorithm based on a generalized fuzzy rough model a theoretic framework of fuzzy rough model based on fuzzy relations is presented which underlies a foundation for algorithm construction we derive several attribute significance measures based on the proposed fuzzy rough model and construct a forward greedy algorithm for hybrid attribute reduction the experiments show that the technique of variable precision fuzzy inclusion in computing decision positive region can get the optimal classification performance number of the selected features is the least but accuracy is the best

a novel algorithm for finding reducts with fuzzy rough sets attribute reduction is one of the most meaningful research topics in the existing fuzzy rough sets and the approach of discernibility matrix is the mathematical foundation of computing reducts when computing reducts with discernibility matrix we find that only the minimal elements in a discernibility matrix are sufficient and necessary this fact motivates our idea in this paper to develop a novel algorithm to find reducts that are based on the minimal elements in the discernibility matrix relative discernibility relations of conditional attributes are defined and minimal elements in the fuzzy discernibility matrix are characterized by the relative discernibility relations then the algorithms to compute minimal elements and reducts are developed in the framework of fuzzy rough sets experimental comparison shows that the proposed algorithms are effective

fuzzy rough attribute reduction with application to web categorization abstract due to the explosive growth of electronically stored information automatic methods must be developed to aid users in maintaining and using this abundance of information effectively in particular the sheer volume of redundancy present must be dealt with leaving only the information rich data to be processed this paper presents a novel approach based on an integrated use of fuzzy and rough set theories to greatly reduce this data redundancy formal concepts of fuzzy rough attribute reduction are introduced and illustrated with a simple example the work is applied to the problem of web categorization considerably reducing dimensionality with minimal loss of information experimental results show that fuzzy rough reduction is more powerful than the conventional rough set based approach classifiers that use a lower dimensional set of attributes which are retained by fuzzy rough reduction outperform those that employ more attributes returned by the existing crisp rough reduction method

fuzzy rough set theory for the interval valued fuzzy information systems the concept of the rough set was originally proposed by pawlak as a formal tool for modelling and processing incomplete information in information systems then in 1990 dubois and prade first introduced the rough fuzzy sets and fuzzy rough sets as a fuzzy extension of the rough sets the aim of this paper is to present a new extension of the rough set theory by means of integrating the classical pawlak rough set theory with the interval valued fuzzy set theory i e the interval valued fuzzy rough set model is presented based on the interval valued fuzzy information systems which is defined in this paper by a binary interval valued fuzzy relations r f i uxu on the universe u several properties of the rough set model are given and the relationships of this model and the others rough set models are also examined furthermore we also discuss the knowledge reduction of the classical pawlak information systems and the interval valued fuzzy information systems respectively finally the knowledge reduction theorems of the interval valued fuzzy information systems are built

learning fuzzy rules from fuzzy samples based on rough set technique although the traditional rough set theory has been a powerful mathematical tool for modeling incompleteness and vagueness its performance in dealing with initial fuzzy data is usually poor this paper makes an attempt to improve its performance by extending the traditional rough set approach to the fuzzy environment the extension is twofold one is knowledge representation and the other is knowledge reduction first we provide new definitions of fuzzy lower and upper approximations by considering the similarity between the two objects second we extend a number of underlying concepts of knowledge reduction such as the reduct and core to the fuzzy environment and use these extensions to propose a heuristic algorithm to learn fuzzy rules from initial fuzzy data finally we provide some numerical experiments to demonstrate the feasibility of the proposed algorithm one of the main contributions of this paper is that the fundamental relationship between the reducts and core of rough sets is still pertinent after the proposed extension

attributes reduction using fuzzy rough sets fuzzy rough sets are the generalization of traditional rough sets to deal with both fuzziness and vagueness in data the existing researches on fuzzy rough sets are mainly concentrated on the construction of approximation operators less effort has been put on the attributes reduction of databases with fuzzy rough sets this paper mainly focuses on the attributes reduction with fuzzy rough sets after analyzing the previous works on attributes reduction with fuzzy rough sets we introduce formal concepts of attributes reduction with fuzzy rough sets and completely study the structure of attributes reduction an algorithm using discernibility matrix to compute all the attributes reductions is developed based on these lines of thought we set up a solid mathematical foundation for attributes reduction with fuzzy rough sets the experimental results show that the idea in this paper is feasible and valid

an interval type 2 fuzzy rough set model for attribute reduction rough set theory is a very useful tool for describing and modeling vagueness in ill defined environments traditional rough set theory is restricted to crisp environments however nowadays it has been extended to fuzzy environments resulting in the development of the so called fuzzy rough sets type 2 fuzzy sets possess many advantages over type 1 fuzzy sets but for the general type 2 fuzzy sets the computational complexity is severe on the other hand set theoretic and arithmetic computations for the interval type 2 fuzzy sets are very simple motivated by the aforementioned accomplishments in this paper the concept of fuzzy rough sets is generalized to interval type 2 fuzzy environments subsequently a method of attribute reduction within the interval type 2 fuzzy rough set framework is proposed lastly the properties of the interval type 2 fuzzy rough sets are presented

fuzzy interpolative reasoning based on the ratio of fuzziness of rough fuzzy sets in this paper we propose a method to construct a polygonal rough fuzzy set from a set of polygonal fuzzy sets representing the aggregation of multiple experts opinions and propose a new fuzzy interpolative reasoning method for sparse fuzzy rule based systems based on the ratio of fuzziness of the constructed polygonal rough fuzzy sets where the values of the antecedent variables and the consequence variable appearing in the fuzzy rules are represented by the constructed polygonal rough fuzzy sets the proposed fuzzy interpolative reasoning method can overcome the drawbacks of the existing method due to the fact that it can deal with fuzzy interpolative reasoning using polygonal rough fuzzy sets and it gets more reasonable fuzzy interpolative reasoning results than the existing method

generalized rough sets over fuzzy lattices this paper studies generalized rough sets over fuzzy lattices through both the constructive and axiomatic approaches from the viewpoint of the constructive approach the basic properties of generalized rough sets over fuzzy lattices are obtained the matrix representation of the lower and upper approximations is given according to this matrix view a simple algorithm is obtained for computing the lower and upper approximations as for the axiomatic approach a set of axioms is constructed to characterize the upper approximation of generalized rough sets over fuzzy lattices

the notion of roughness of a fuzzy set abstract the roughness of a set according to the notion introduced by pawlak in 1991 can be regarded as the mz distance between its upper and the lower approximations with this idea in mind we have generalized pawlak s definition by replacing the mz distance by a general distance measure we also generalize the notion of roughness of fuzzy sets introduced by huyhn and nakamori in 2005

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

can fuzzy entropies be effective measures for evaluating the roughness of a rough set the roughness of a rough set arises from the existence of its boundary region in such a boundary region each object has a non zero rough membership degree when an object s rough membership degree is regarded as its fuzzy membership degree a rough set can induce a fuzzy set this relationship motivates us to assert that there may exist some inherent relations between the roughness of a rough set and the fuzziness of the fuzzy set induced from the rough set this assertion leads us to the question can the existing fuzzy entropies be used to evaluate the roughness of a rough set to answer this question we first analyze how the boundary region varies when the partition of the universe becomes coarser and then exploit this analysis in the introduction of a more appropriate definition on the roughness of a rough set to determine whether a fuzzy entropy can be used to evaluate the roughness of a rough set or not we develop three methods for estimating the ability of a fuzzy entropy to measure the roughness the experiments show that these methods are very effective and can be applied to select a fuzzy entropy as a measure of the roughness of a rough set

cost driven parameter design parameter design methodology has focused primarily on the quadratic loss function which can often be solved by using a two step procedure involving the minimization of a dispersion measure and then adjusting the mean to target in some practical situations however the loss can be nonquadratic or highly skewed by building on a theory of leon and wu we develop a modeling and data analysis strategy for a general loss function in which the quality characteristic follows a location scale model the only difference from the two step procedure just mentioned is the adjustment step in which the mean is moved to that side of the target with lower cost the technique is illustrated on an experiment involving epitaxial layer growth in integrated circuit fabrication

positive approximation an accelerator for attribute reduction in rough set theory feature selection is a challenging problem in areas such as pattern recognition machine learning and data mining considering a consistency measure introduced in rough set theory the problem of feature selection also called attribute reduction aims to retain the discriminatory power of original features many heuristic attribute reduction algorithms have been proposed however quite often these methods are computationally time consuming to overcome this shortcoming we introduce a theoretic framework based on rough set theory called positive approximation which can be used to accelerate a heuristic process of attribute reduction based on the proposed accelerator a general attribute reduction algorithm is designed through the use of the accelerator several representative heuristic attribute reduction algorithms in rough set theory have been enhanced note that each of the modified algorithms can choose the same attribute reduct as its original version and hence possesses the same classification accuracy experiments show that these modified algorithms outperform their original counterparts it is worth noting that the performance of the modified algorithms becomes more visible when dealing with larger data sets

a roughness measure for fuzzy sets recently an attempt of integration between the theories of fuzzy sets and rough sets has resulted in providing a roughness measure for fuzzy sets m banerjee s k pal roughness of a fuzzy set information sciences 93 1996 235 246 essentially banerjee and pal s roughness measure depends on parameters that are designed as thresholds of definiteness and possibility in membership of the objects to a fuzzy set in this paper we first remark that this measure of roughness has several undesirable properties and then propose a parameter free roughness measure for fuzzy sets based on the notion of the mass assignment of a fuzzy set several interesting properties of this new measure are examined furthermore we also discuss how the proposed approach is used to describe the rough approximation quality of a fuzzy classification

a general model of decision theoretic three way approximations of fuzzy sets based on a heuristic algorithm abstract a model of decision theoretic three way approximations of fuzzy sets that can calculate a pair of interpretable thresholds by combining shadowed sets with three way decisions was established by deng in that model the significant contribution is that the value 0 5 which denotes a membership grade with the highest uncertainty is used to replace the uncertain region with the interval 0 1 in shadowed sets from the principle of the minimum decision cost although the lowest overall cost can be achieved precisely in some data distributions based on the value 0 5 there is a difference between the obtained overall cost and the least cost in some cases therefore in this paper based on deng s model the concept of a general three way approximation of a fuzzy set is proposed to replace 0 5 with a variable value c 0

torsion of elliptic curves over cyclic cubic fields we determine all the possible torsion groups of elliptic curves over cyclic cubic fields over non cyclic totally real cubic fields and over complex cubic fields

a comparative study of rough sets for hybrid data to discover knowledge from hybrid data using rough sets researchers have developed several fuzzy rough set models and a neighborhood rough set model these models have been applied to many hybrid data processing applications for a particular purpose thus neglecting the issue of selecting an appropriate model to address this issue this paper mainly concerns the relationships among these rough set models investigating fuzzy and neighborhood hybrid granules reveals an important relationship between these two granules analyzing the relationships among rough approximations of these models shows that hu s fuzzy rough approximations are special cases of neighborhood and wang s fuzzy rough approximations respectively furthermore one to one correspondence relationships exist between wang s fuzzy and neighborhood rough approximations this study also finds that wang s fuzzy and neighborhood rough approximations are cut sets of dubois fuzzy rough approximations and radzikowska and kerre s fuzzy rough approximations respectively

a new measure of uncertainty based on knowledge granulation for rough sets in rough set theory accuracy and roughness are used to characterize uncertainty of a set and approximation accuracy is employed to depict accuracy of a rough classification although these measures are effective they have some limitations when the lower upper approximation of a set under one knowledge is equal to that under another knowledge to overcome these limitations we address in this paper the issues of uncertainty of a set in an information system and approximation accuracy of a rough classification in a decision table an axiomatic definition of knowledge granulation for an information system is given under which these three measures are modified theoretical studies and experimental results show that the modified measures are effective and suitable for evaluating the roughness and accuracy of a set in an information system and the approximation accuracy of a rough classification in a decision table respectively and have a much simpler and more comprehensive form than the existing ones