on the structure of generalized rough sets in this paper we consider some fundamental properties of generalized rough sets induced by binary relations on algebras and show that 1 any reflexive binary relation determines a topology 2 if q is a reflexive and symmetric relation on a set x then o email protected x q a a is a topology such that a is open if and only if it is closed 3 conversely for every topological space x o satisfying the condition that a is open if and only if it is closed there exists a reflexive and symmetric relation r such that o email protected x r a a 4 let q be an equivalence relation on x for any pseudo w closed subset a of x q a is an w closed set if and only if w x x x email protected a for any email protected x moreover we consider properties of generalized rough sets

extensions and intentions in the rough set theory abstract the approach to rough set theory proposed in this paper is based on the mutual correspondence of the concepts of extension and intension it is different from the well known approaches in the literature in that the upper approximations and the lower approximations of unknown sets are considered as certain families of known sets this approach makes it possible to formulate necessary and sufficient conditions for the existence of operations on rough sets which are analogous to classical operations on sets the basic results presented in this paper based on certain ideas of the second author were formulated by the first author in his doctoral dissertation prepared under the supervision of the third author

fuzzy fractal dimensions and fuzzy modeling fractal dimensions describe self similarity structural complexity of various phenomena such as e g temporal signals images their determination through box counting or a correlation method is inherently associated with the use of information granules sets the intent of this study is to generalize the idea to the domain of fuzzy sets and reveal associations between the mechanisms of fractal analysis and granular computing including fuzzy modeling first we introduce the concept itself and discuss the role of fuzzy sets as a vehicle for constructing fractal dimensions second we propose an algorithmic framework necessary to carry out all computing and experimentally quantify a performance of regression models used to determine fractal dimensions and contrast it with the performance of the fractal models existing in the set based environment it is shown that fuzzy set approach produces more consistent models in terms of their performance we also postulate a power law of granularity and discuss its direct implications in the form of a variable granularity in fuzzy modeling in particular we show how the power law of granularity helps construct mappings between system s variables in rule based models experimental studies involving several frequently used categories of fuzzy sets illustrate the main features of the approach

rules in incomplete information systems abstract a new method of computing all optimal certain rules from an incomplete information system is presented and proved the method does not require changing the size of the original incomplete system additionally several existing rough set methods of computing decision rules from incomplete information systems are analyzed and compared we show which of these methods are capable of generating all optimal certain rules or a class of optimal certain rules and which methods may lead to generation of false rules

relationships among generalized rough sets in six coverings and pure reflexive neighborhood system rough set is a useful tool to deal with partition related uncertainty granularity and incompleteness of knowledge although the classical rough set is constructed on the basis of an indiscernibility relation it can also be generalized by using some weaker binary relations in this paper a systematic approach is used to study the generalized rough sets in six coverings and pure reflexive neighborhood system after two steps relationships among generalized rough sets in six coverings and pure reflexive neighborhood system are obtained the first step is to study the generalized rough sets in six coverings and to get relationships between every two covering rough set models the second step is to study the relationships between generalized rough sets in each covering and in pure reflexive neighborhood system the inclusion relations or equivalence relations among the seven upper lower approximations could be acquired finally the accuracy measures of generalized rough sets in six coverings and that in pure reflexive neighborhood system are compared the relationships among seven accuracy measures are also obtained some illustrative examples are employed to demonstrate our arguments

a comparative study of fuzzy sets and rough sets this paper reviews and compares theories of fuzzy sets and rough sets two approaches for the formulation of fuzzy sets are reviewed one is based on many valued logic and the other is based on modal logic two views of rough sets are presented set oriented view and operator oriented view rough sets under set oriented view are closely related to fuzzy sets which leads to non truth functional fuzzy set operators both of them may be considered as deviations of classical set algebra in contrast rough sets under operator oriented view are different from fuzzy sets and may be regarded as an extension of classical set algebra

relationship between generalized rough sets based on binary relation and covering rough set theory is a powerful tool for dealing with uncertainty granularity and incompleteness of knowledge in information systems this paper systematically studies a type of generalized rough sets based on covering and the relationship between this type of covering based rough sets and the generalized rough sets based on binary relation firstly we present basic concepts and properties of this kind of rough sets then we investigate the relationships between this type of generalized rough sets and other five types of covering based rough sets the major contribution in this paper is that we establish the equivalency between this type of covering based rough sets and a type of binary relation based rough sets through existing results in binary relation based rough sets we present axiomatic systems for this type of covering based lower and upper approximation operations in addition we explore the relationships among several important concepts such as minimal description reduction representative covering exact covering and unary covering in covering based rough sets investigation of this type of covering based will benefit to our understanding of other types of rough sets based on covering and binary relation

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

a short note on algebraic t rough sets our aim in this paper is to introduce the concept of a set valued homomorphism for groups which is a generalization of ordinary homomorphism then using the definitions of lower inverse and upper inverse we introduce the definition of uniform set valued homomorphism and prove that every set valued homomorphism is uniform finally the concept of probabilistic t rough set is defined

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

attribute reduction of data with error ranges and test costs in data mining applications we have a number of measurement methods to obtain a data item with different test costs and different error ranges test costs refer to time money or other resources spent in obtaining data items related to some object observational errors correspond to differences in measured and true value of a data item in supervised learning we need to decide which data items to obtain and which measurement methods to employ so as to minimize the total test cost and help in constructing classifiers this paper studies this problem in four steps first data models are built to address error ranges and test costs second error range based covering rough set is constructed to define lower and upper approximations positive regions and relative reducts a closely related theory deals with neighborhood rough set which has been successfully applied to heterogeneous attribute reduction the major difference between the two theories is the definition of neighborhood third the minimal test cost attribute reduction problem is redefined in the new theory fourth both backtrack and heuristic algorithms are proposed to deal with the new problem the algorithms are tested on ten uci university of california irvine datasets experimental results show that the backtrack algorithm is efficient on rational sized datasets the weighting mechanism for the heuristic information is effective and the competition approach can improve the quality of the result significantly this study suggests new research trends concerning attribute reduction and covering rough set

on the topological properties of fuzzy rough sets this paper is devoted to the discussion of the relationship between fuzzy rough set models and fuzzy topologies on a finite universe the tc axiom for fuzzy topology is proposed it is proved that the set of all lower approximation sets based on a reflexive and transitive fuzzy relation consists of a fuzzy topology which satisfies tc axiom and conversely a fuzzy topology which obey tc axiom is just the set of all lower approximation sets under a reflexive and transitive fuzzy relation that is to say there exists a one to one correspondence between the set of all reflexive and transitive fuzzy relations and the set of all fuzzy topologies which satisfy tc axiom

a class of rough multiple objective programming and its application to solid transportation problem this paper establishes a rough multiple objective programming model for a solid transportation problem furthermore a general model for rough multiple objective programming problem is presented properties of the feasible and efficient solutions of rough multiple objective programming problems are investigated in addition compromise solutions are obtained by using the interactive fuzzy satisfying method to solve the rough multiple objective solid transportation problem the rough simulation based genetic algorithm is proposed in which the rough simulation is embedded into the genetic algorithm finally an application of the solid transportation problem at xiluodu hydropower station is provided as an illustration

roughness in mv algebras in this paper by considering the notion of an mv algebra we consider a relationship between rough sets and mv algebra theory we introduce the notion of rough ideal with respect to an ideal of an mv algebra which is an extended notion of ideal in an mv algebra and we give some properties of the lower and the upper approximations in an mv algebra

on the topological properties of generalized rough sets in this paper we consider some topological properties of generalized rough sets induced by binary relations and show that1 any serial binary relation can induce a topology 2 let r be a binary relation on a universe u t r and e r denote the transitive closure and the equivalence closure of r respectively if r is a reflexive relation on u then r and t r induce the same topology i e t r t t r the interior and closure operators of the topology t r induced by r are the lower and upper approximation operators t r and t r respectively moreover r t r t r where r t r is the relation induced by the topology t r 3 when r is a reflexive and symmetric relation r and e r induce the same topology i e t r t e r the interior and closure operators of the topology t r induced by r are the lower and upper approximation operators e r and e r respectively moreover r t r e r 4 based on the above conclusions the notion of topological reduction of incomplete information systems is proposed and characterizations of reduction of consistent incomplete decision tables are obtained

multigranulation rough sets from partition to covering the classical multigranulation rough set mgrs theory offers a formal theoretical framework for solving the complex problem under multigranulation environment however it is noticeable that mgrs theory cannot be applied in multi source information systems with a covering environment in the real world to address this issue we firstly present in this paper three types of covering based multigranulation rough sets in which set approximations are defined by different covering approximation operators then by using two different approximation strategies i e seeking common reserving difference and seeking common rejecting difference two kinds of covering based multigranulation rough set are presented namely a covering based optimistic multigranulation rough set and a covering based pessimistic multigranulation rough sets finally we develop some properties and several uncertainty measures of the covering based multigranulation rough sets these results will enrich the mgrs theory and enlarge its application scope

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

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

generalized rough sets based on relations rough set theory has been proposed by pawlak as a tool for dealing with the vagueness and granularity in information systems the core concepts of classical rough sets are lower and upper approximations based on equivalence relations this paper studies arbitrary binary relation based generalized rough sets in this setting a binary relation can generate a lower approximation operation and an upper approximation operation but some of common properties of classical lower and upper approximation operations are no longer satisfied we investigate conditions for a relation under which these properties hold for the relation based lower and upper approximation operations

formal concept analysis based on the topology for attributes of a formal context formal concept analysis fca concerns the hierarchical structures induced by a binary relation between a pair of sets it is widely applied in data analysis information retrieval and knowledge discovery within the framework of fca computing all formal concepts is the main challenge due to its exponential complexity it has be proved that all formal concepts is a closure system on objects hence the closure operator on objects are general used to generate all formal concepts of a formal context in the lexicographic tree approach a base on objects is used to generate extensions of all formal concepts and construct the formal concept lattice inspired from the approach we concentrate a base on attributes and generate intensions of all formal concepts in this paper to this end we firstly analyze an example the lexicographic tree approach is used to generate all formal concepts from the time complexity point of view we present that it is not trivial to generate the base on attributes by a simple symmetrical way from the methods based on objects then we deduce a set valued mapping from attributes to the power set of attributes in a formal context and define a binary relation on attributes by the set valued mapping using the binary relation on attributes we construct an approximation space and a topology for attributes respectively and obtain a base for the topology we prove that intensions of all formal concepts are included in the topology for attributes this means that the base can be used to generate intensions of all formal concepts of the formal context and construct the formal concept lattice more general our results represent relationships and the hierarchical structures among attributes of the formal context we present some typical applications in which the topology for attributes and the base for the topology are applied for association rules discovery from a formal context and linguistic concept analysis

a study of rough sets based on 1 neighborhood systems abstract a special class of neighborhood systems called 1 neighborhood systems are important in rough set theory by using a concept core originated in general topology we define two types of rough sets based on 1 neighborhood systems in this paper we discuss properties of these rough sets from the perspective of both common 1 neighborhood systems and several special classes of 1 neighborhood systems such as reflexive symmetric transitive or euclidean 1 neighborhood systems by using these properties we discuss the relationship among several classes of 1 neighborhood systems with various special properties we give a necessary and sufficient condition for a reflexive and symmetric 1 neighborhood system being a unary we also prove that a reflexive and transitive 1 neighborhood system is representative the proofs of these results show that the rough sets we defined in this paper not only have application background but also have theoretic importance

topological properties of generalized approximation spaces rough set theory is a powerful mathematical tool for dealing with inexact uncertain or vague information the core concepts of rough set theory are information systems and approximation operators of approximation spaces approximation operators draw close links between rough set theory and topology this paper concerns generalized approximation spaces via topological methods and studies topological properties of rough sets classical separation axioms compactness and connectedness for topological spaces are extended to generalized approximation spaces relationships among separation axioms for generalized approximation spaces and relationships between topological spaces and their induced generalized approximation spaces are investigated an example is given to illustrate a new approach to recover missing values for incomplete information systems by regularity of generalized approximation spaces

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

roughness in n ary hypergroups this paper is aimed at studying the rough sets within the context of the commutative n ary hypergroups for this approach the presentation of a detailed study of the various types of n ary subhypergroups is of the greatest importance this study was initiated in v leoreanu fotea several types of n ary subhypergroups italian journal of pure and applied mathematics in press subsequently we shall use the findings in this study to define the upper and lower approximation of a subset with respect to an invertible n ary subhypergroup of a commutative n ary hypergroup in the final part the concept of n ary rough subhypergroup is introduced

on characterizations of i t fuzzy rough approximation operators in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties various fuzzy generalizations of rough approximations have been made in the literature this paper proposes a general framework for the study of i t fuzzy rough approximation operators within which both constructive and axiomatic approaches are used in the constructive approach a pair of lower and upper generalized fuzzy rough approximation operators determined by an implicator i and a triangular norm t is first defined basic properties of i t fuzzy rough approximation operators are investigated the connections between fuzzy relations and fuzzy rough approximation operators are further established in the axiomatic approach an operator oriented characterization of rough sets is proposed that is i t fuzzy approximation operators are defined by axioms different axiom sets of t upper and i lower fuzzy set theoretic operators guarantee the existence of different types of fuzzy relations which produce the same operators finally an open problem proposed by radzikowska and kerre in fuzzy sets and systems 126 2002 137 is solved

efficient bayesian inference for dynamic mixture models abstract a bayesian approach is presented for estimating a mixture of linear gaussian state space models such models are used to model interventions in time series and nonparametric regression markov chain monte carlo sampling is usually necessary to obtain the posterior distributions of such mixture models because it is difficult to obtain them analytically the methodological contribution of the article is to derive a set of recursions for dynamic mixture models that efficiently implement a markov chain monte carlo sampling scheme that converges rapidly to the posterior distribution the methodology is illustrated by fitting an autoregressive model subject to interventions to zinc concentration in sludge

topological approaches to covering rough sets rough sets a tool for data mining deal with the vagueness and granularity in information systems this paper studies covering based rough sets from the topological view we explore the topological properties of this type of rough sets study the interdependency between the lower and the upper approximation operations and establish the conditions under which two coverings generate the same lower approximation operation and the same upper approximation operation lastly axiomatic systems for the lower approximation operation and the upper approximation operation are constructed

regranulation a granular algorithm enabling communication between granular worlds abstract in this paper we describe a granular algorithm for translating information between two granular worlds represented as fuzzy rulebases these granular worlds are defined on the same universe of discourse but employ different granulations of this universe in order to translate information from one granular world to the other we must regranulate the information so that it matches the information granularity of the target world this is accomplished through the use of a first order interpolation algorithm implemented using linguistic arithmetic a set of elementary granular computing operations we first demonstrate this algorithm by studying the common fuzzy pd rulebase at several different granularities and conclude that the 3 3 granulation may be too coarse for this objective we then examine the question of what the natural granularity of a system might be this is studied through a 10 fold cross validation experiment involving three different granulations of the same underlying mapping for the problem under consideration we find that a 7 7 granulation appears to be the minimum necessary precision

reduction and axiomization of covering generalized rough sets this paper investigates some basic properties of covering generalized rough sets and their comparison with the corresponding ones of pawlak s rough sets a tool for data mining the focus here is on the concepts and conditions for two coverings to generate the same covering lower approximation or the same covering upper approximation the concept of reducts of coverings is introduced and the procedure to find a reduct for a covering is given it has been proved that the reduct of a covering is the minimal covering that generates the same covering lower approximation or the same covering upper approximation so this concept is also a technique to get rid of redundancy in data mining furthermore it has been shown that covering lower and upper approximations determine each other finally a set of axioms is constructed to characterize the covering lower approximation operation

quantum groups and deformations of preprojective algebras abstract let be a quiver containing finite cyclic blocks and be the deformed preprojective algebra of it is proved that a suitable quotient uq of for an appropriate vector is a finite dimensional quantum group some properties of uq are also obtained an isomorphism as hopf algebras is given between uq and a restricted quantum group u q sl 2

rough set approach to incomplete information systems abstract in the paper we present rough set approach to reasoning in incomplete information systems we propose reduction of knowledge that eliminates only that information which is not essential from the point of view of classification or decision making in our approach we make only one assumption about unknown values the real value of a missing attribute is one from the attribute domain however we do not assume which one we show how to find decision rules directly from such an incomplete decision table which are as little non deterministic as possible and have minimal number of conditions

mining incomplete data with singleton subset and concept probabilistic approximations abstract rough set theory provides a very useful idea of lower and upper approximations for inconsistent data for incomplete data these approximations are not unique in this paper we investigate properties of three well known generalizations of approximations singleton subset and concept these approximations were recently further generalized as to include an additional parameter interpreted as a probability in this paper we report novel properties of singleton subset and concept probabilistic approximations additionally we validated such approximations experimentally our main objective was to test which of the singleton subset and concept probabilistic approximations are the most useful for data mining our conclusion is that for a given incomplete data set all three approaches should be applied and the best approach should be selected as a result of ten fold cross validation finally we conducted experiments on complexity of rule sets and the total number of singleton subset and concept approximations

rough sets and boolean reasoning abstract in this article we discuss methods based on the combination of rough sets and boolean reasoning with applications in pattern recognition machine learning data mining and conflict analysis

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

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

nullity based matroid of rough sets and its application to attribute reduction rough sets were proposed to deal with vagueness and incompleteness of knowledge in information systems in this field there are many optimization issues such as attribute reduction matroids generalized from matrices have been widely used in many fields particularly greedy algorithm design which plays an important role in attribute reduction therefore it is meaningful to combine matroids with rough sets to solve the optimization problems in this paper we construct a type of matroid of rough sets based on the concept of nullity and apply it to attribute reduction first we propose a nullity operator for rough sets to induce a matroid and then we present a specific type of matroid called a nullity based matroid second given the relationship between nullities and matrices we present two types of matrices to characterize this type of matroid and its nullity operator third the dual of this type of matroid is induced by the second type of matrix finally we apply the obtained matroids to attribute reduction issues in information systems in summary this paper provides a new approach to studying rough sets

communication between information systems using fuzzy rough sets communication between information systems is a basic problem in granular computing and the concept of homomorphism is a useful mathematical tool to study this problem in this paper some properties of communication between information systems based on fuzzy rough sets are investigated the concepts of fuzzy relation mappings between universes are first proposed in order to construct a fuzzy relation of one universe according to the given fuzzy relation on the other universe the main properties of the mappings are studied the notions of homomorphism of information systems based on fuzzy rough sets are then proposed and it is proved that properties of relation operations in the original information system and structural features of the system such as approximations of arbitrary fuzzy sets and attribute reductions are guaranteed in its image system under the condition of homomorphism

the necessary and sufficient conditions for a fuzzy relation being τ euclidean the connections between special types of fuzzy relations and properties of fuzzy rough approximation operators have been established in recent years but euclidean fuzzy relation has not been considered yet in the present paper the necessary and sufficient conditions for a fuzzy relation being euclidean are investigated in three different fuzzy rough approximation spaces meanwhile it is proved that in i fuzzy rough approximation space where i is an r implication the properties the euclidean i fuzzy rough approximation operators possess are just the same as those in rough fuzzy approximation space

the minimization of axiom sets characterizing generalized approximation operators in the axiomatic approach of rough set theory rough approximation operators are characterized by a set of axioms that guarantees the existence of certain types of binary relations reproducing the operators thus axiomatic characterization of rough approximation operators is an important aspect in the study of rough set theory in this paper the independence of axioms of generalized crisp approximation operators is investigated and their minimal sets of axioms are presented

cost sensitive feature selection based on adaptive neighborhood granularity with multi level confidence neighborhood rough set model is considered as one of the effective granular computing models in dealing with numerical data this model is now widely discussed in feature selection and rule learning however there is no theoretical analysis on the issue of neighborhood granularity selection the influence of sampling resolution test and misclassification costs on modeling in this paper we design an adaptive neighborhood rough set model according to data precision and develop a fast backtracking algorithm for neighborhood rough sets based cost sensitive feature selection by considering the trade off between test costs and misclassification costs in the proposed model the neighborhood granularity based on the 3 rule of statistics is adaptive to data precision that is described by the multi level confidence of the feature subsets our experiments thoroughly performed on 12 datasets demonstrate the effectiveness of the model and the efficiency of the backtracking algorithm

knowledge acquisition in incomplete information systems a rough set approach abstract this paper deals with knowledge acquisition in incomplete information systems using rough set theory the concept of similarity classes in incomplete information systems is first proposed two kinds of partitions lower and upper approximations are then formed for the mining of certain and association rules in incomplete decision tables one type of optimal certain and two types of optimal association decision rules are generated two new quantitative measures random certainty factor and random coverage factor associated with each decision rule are further proposed to explain relationships between the condition and decision parts of a rule in incomplete decision tables the reduction of descriptors and induction of optimal rules in such tables are also examined