consistency based search in feature selection feature selection is an effective technique in dealing with dimensionality reduction for classification it is used to find an optimal subset of relevant features such that the overall accuracy of classification is increased while the data size is reduced and the comprehensibility is improved feature selection methods contain two important aspects evaluation of a candidate feature subset and search through the feature space existing algorithms adopt various measures to evaluate the goodness of feature subsets this work focuses on inconsistency measure according to which a feature subset is inconsistent if there exist at least two instances with same feature values but with different class labels we compare inconsistency measure with other measures and study different search strategies such as exhaustive complete heuristic and random search that can be applied to this measure we conduct an empirical study to examine the pros and cons of these search methods give some guidelines on choosing a search method and compare the classifier error rates before and after feature selection

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

a short note on some properties of rough groups it is a useful method in research of group theory to construct a new group by using known groups lower and upper approximation operators of rough sets are applied into group theory and so the notion of a rough group has been introduced in this paper we first point out that there are still some incomplete propositions in n kuroki p p wang the lower and upper approximations in a fuzzy group inform sci 90 1996 203 220 although some authors have showed several incorrect statements in the literature we then present improved versions of the incomplete propositions and continue to study the image and inverse image of rough approximations of a subgroup with respect to a homomorphism between two groups

rough set theory applied to fuzzy ideal theory we use covers of the universal set to define approximation operators on the power set of the given set in section 1 we determine basic properties of the upper approximation operator and show how it can be used to give algebraic structural properties of certain subsets we define a particular cover on the set of ideals of a commutative ring with identity in such a way that both the concepts of the fuzzy prime spectrum of a ring and rough set theory can simultaneously be brought to bear on the study of fuzzy ideals of a ring

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

reducts and constructs in classic and dominance based rough sets approach abstract the idea of the reduct as defined in the classic rough sets approach crsa has proven to be inspiring enough to get into closely related theories including the dominance based rough sets approach drsa the procedure of reduction is generally similar to that of feature selection but narrower as it is the descriptive rather than the predictive aspect of data exploration that constitutes its principal goal crsa reducts are thus defined as minimal subsets of attributes that retain sufficiently high quality of object description developed within crsa the crsa reducts have given rise to the generalized notion of crsa constructs which have turned out to be superior to reducts in numerous practical experiments with real life data sets the generalization process is continued in this paper in which a definition of constructs in the context of drsa is introduced the definition fully analogous to that of crsa constructs differs only in that it is context based in drsa while context free in crsa consequently the presented drsa constructs are expected to have analogous properties to that of crsa constructs including superiority to drsa reducts in experiments with real life data sets

a new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets traditional rough set theory is mainly used to extract rules from and reduce attributes in databases in which attributes are characterized by partitions while the covering rough set theory a generalization of traditional rough set theory does the same yet characterizes attributes by covers in this paper we propose a way to reduce the attributes of covering decision systems which are databases characterized by covers first we define consistent and inconsistent covering decision systems and their attribute reductions then we state the sufficient and the necessary conditions for reduction finally we use a discernibility matrix to design algorithms that compute all the reducts of consistent and inconsistent covering decision systems numerical tests on four public data sets show that the proposed attribute reductions of covering decision systems accomplish better classification performance than those of traditional rough sets

reducts within the variable precision rough sets model a further investigation abstract one fundamental aspect of the variable precision rough sets vprs model involves a search for subsets of condition attributes which provide the same information for classification purposes as the full set of available attributes such subsets are labelled approximate reducts or reducts being defined for a specified classification error denoted by this paper undertakes a further investigation of the criteria for a reduct within vprs certain anomalies and interesting implications are identified an additional condition is suggested for finding reducts which assures a more general level knowledge equivalent to that of the full set of attributes

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

applying electromagnetism like mechanism for feature selection in recent years methods of feature selection have been increasingly emphasized as venues for reducing cost and shortening the length of time required for computation in data mining this study utilizes electromagnetism like mechanism as a wrapper approach to feature selection birbil and fang proposed em in 2003 em uses the attraction repulsion mechanism of the electromagnetism theory to ascertain the optimal solution although em has been applied to the topic of optimization in continuous space and a small number of studies on discrete problems it has not been applied to the subject of feature selection in this study em combined with 1 nearest neighbor 1nn was applied for feature selection and classification this study utilized the total force exerted on a particle and evaluated this force to determine which features are to be selected the most crucial features were selected according to the proposed method based on the minimum miss classification rate which was attained through 1nn an unknown datum was classified by 1nn based on the chosen reduced model to estimate the effectiveness of the proposed method a numerical experiment was conducted using several data sets with diverse sizes features separability and classes experimental results indicated that the proposed method outperformed other well known algorithms in not only balanced classification accuracy but also efficiency of feature selection lastly this study used an actual case concerning gestational diabetes mellitus to demonstrate the workability of the proposed method

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

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

chain recurrence for multi valued dynamical systems on noncompact spaces abstract it is well known that the chain recurrence set of continuous map f is the complement of union of b a a where a is an attractor and b a is a basin of a we prove that this result also holds when f is a compact valued continuous relation

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

granular matrix a new approach for granular structure reduction and redundancy evaluation granular structure is a mathematical expression of knowledge in granular computing and a direct determinant of the data processing efficiency to improve the efficiency of data processing many scholars have studied the reduction of granular structure the attribute reduction and the granular reduction are two types of reduction on different layers of a granular structure with the latter being both an essential step for granular structure reduction and the foundation of the attribute reduction yet compared with the attribute reduction the granular reduction has received less attention from scholars therefore a fuzzy granular reduction theory and a granular matrix based on the fuzzy beta coverings is proposed in this article the insufficiency of the existing granular reduction theory for fuzzy beta coverings is pointed out and proper sufficient and necessary conditions for two fuzzy beta coverings generating the same upper and lower approximations are also given in this article in addition to reduce and evaluate a fuzzy beta covering a novel reduction algorithm based on a granular matrix is proposed for the first time also since fuzzy covering reduction is np hard a heuristic greedy algorithm is designed to obtain a reduct numerical experiments show that the redundancy rates of neighborhood granule sets induced by some big scale data sets exceed 99 which indicates that the existing neighborhood granulation methods need to be urgently improved based on this concise granular structures and much more efficient feature selection algorithms can be proposed in the future

attribute reduction and optimal decision rules acquisition for continuous valued information systems for continuous valued information systems the attribute values of objects for the same attribute represent not only their ordinal relationship but also their relative distances therefore the classical rough set model is not suitable for deducing attribute reductions and optimal decision rules for continuous valued information systems though some discretization methods are proposed to transform the continuous valued information systems into discrete ones but those methods are too categorical and may lead to loss of information in some cases to solve such information loss problem we propose a tolerance rough set model in this paper with a given level the proposed model can divide a universe into some maximal tolerance classes also two types of lower and upper approximations are defined accordingly then the reductions of the maximal tolerance class and optimal decision rules based on the proposed attribute descriptors are defined and the approximate discernibility function for the maximal tolerance class is constructed and used to compute all the corresponding optimal decision rules via using boolean reasoning techniques finally the general reductions and consistent reductions for continuous valued information systems are discussed

fuzzy information systems and their homomorphisms abstract with the arrival of the information age information acquisition and communication have become more and more important in the field of information technology this paper uses the concept of homomorphism as a basic tool to study the communication between fuzzy information systems the concepts of consistent and compatible mappings with respect to fuzzy sets are firstly defined and their basic properties are studied then a pair of lower and upper rough fuzzy approximation operators is constructed by means of the concept of fuzzy mappings basic invariant properties of the approximation operators are investigated finally the concepts of fuzzy information system and its homomorphism are introduced and some invariant properties of fuzzy information systems under homomorphisms are examined it is proved that the attribute reductions of an original information system and its image system are equivalent to each other in the context of fuzzy attributes these results may have potential applications in attribute reduction and classification issues

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

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

rough sets methodology for sorting problems in presence of multiple attributes and criteria abstract we consider a sorting classification problem in the presence of multiple attributes and criteria called the mac however the domains of attributes are not preference ordered while the domains of criteria scales are totally ordered by preference relations among the attributes we distinguish between qualitative attributes and quantitative attributes in order to construct a comprehensive preference model that could be used to support the sorting task we consider preferential information of the decision maker dm in the form of assignment examples i e exemplary assignments of some reference actions to the decision classes the preference model inferred from these examples is a set of if then decision rules the rules are derived from rough approximations of decision classes made up of reference actions they satisfy conditions of completeness and dominance and manage with possible ambiguity inconsistencies in the set of examples our idea of rough approximations involves three relations together indiscernibility similarity and dominance defined on qualitative and quantitative attributes and on criteria respectively the usefulness of this approach is illustrated by an example

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

extended results on the relationship between information systems the relationship between information systems is an important topic in rough set field and it is studied through mappings many kinds of consistent functions have been proposed and they can be used to describe the invariance of attribute reducts between two information systems efficiently however they cannot reflect relationships between neighborhoods generated from two information systems in this paper to achieve this aim firstly we introduce a neighborhood continuous function that is inspired by the concept of continuous function in topology then we address its properties and discuss relationships between neighborhood continuous functions and several kinds of existing consistent functions furthermore we investigate some important properties of neighborhood continuous function with respect to relation mappings finally based on neighborhood continuous functions the notion of a neighborhood homomorphism between different information systems is proposed it is noted that the reduct feature of a single information system can be described by neighborhood continuous functions while the reduct invariance of two different information systems can be described by neighborhood homomorphisms

a rst a generalization of rough set theory the paper presents a transition from the crisp rough set theory to a fuzzy one called alpha rough set theory or in short a rst all basic concepts or rough set theory are extended i e information system indiscernibility dependency reduction core definability approximations and boundary the resulted theory takes into account fuzzy data and allows the approximation of fuzzy concepts besides the control of knowledge granularity is natural in a rst which is based on a parameterized indiscernibility relation a rst is developed to recognize non deterministic relationships using notions as a dependency a reduct and so forth on the other hand we introduce a notion of relative dependency as an alternative of the absolute definibility presented in rough set theory the extension a rst leads naturally to the new concept of alpha rough sets which represents sets with fuzzy non empty boundaries 2000 elsevier science inc all rights reserved

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

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

related family a new method for attribute reduction of covering information systems in terms of attribute reduction of covering based rough sets the discernibility matrix is used as a conventional method to compute all attribute reducts however it is inapplicable to attribute reduction in certain circumstances in this article a new method referred to as the related family is introduced to compute all attribute reducts and relative attribute reducts for covering rough sets its core idea is to remove superfluous attributes while keeping the approximation space of covering information system unchanged the related family method is more powerful than the discernibility matrix method since the former can handle complicated cases that could not be handled by the latter in addition a simplified version of the related family and its corresponding heuristic algorithm are also presented

a systematic study on attribute reduction with rough sets based on general binary relations attribute reduction is considered as an important preprocessing step for pattern recognition machine learning and data mining this paper provides a systematic study on attribute reduction with rough sets based on general binary relations we define a relation information system a consistent relation decision system and a relation decision system and their attribute reductions furthermore we present a judgment theorem and a discernibility matrix associated with attribute reduction in each type of system based on the discernibility matrix we can compute all the reducts finally the experimental results with uci data sets show that the proposed reduction methods are an effective technique to deal with complex data sets

the homotopy principle and algorithms for linear programming linear programming techniques are formulated within the unifying framework of the homotopy principle and newton s method key strategies that determine the effectiveness of an implementation are considered in detail a complexity analysis is developed for an elevator predictor newton corrector algorithm started at an arbitrary interior primal dual feasible point this analysis is based on a fundamental theorem of smale algorithms for solving equations proc internat congress of mathematicians universityof california berkeley ca 1986 in the form given by renegar and shub report no 807 school of operations research and industrial engineering cornell university ithaca ny 1988

knowledge reduction in random information systems via dempster shafer theory of evidence knowledge reduction is one of the main problems in the study of rough set theory this paper deals with knowledge reduction in random information systems based on dempster shafer theory of evidence the concepts of belief and plausibility reducts in random information systems are first introduced it is proved that both of belief reduct and plausibility reduct are equivalent to classical reduct in random information systems the relative belief and plausibility reducts in consistent and inconsistent random decision systems are then proposed and compared to the relative reduct and relationships between the new reducts and some existing ones are examined

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

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

concept analysis via rough set and afs algebra formal concept analysis fca was originally proposed by wille 1982 which is an important theory for data analysis and knowledge discovery afs axiomatic fuzzy set algebra was proposed by liu x liu the fuzzy theory based on afs algebras and afs structure journal of mathematical analysis and applications 217 1998 459 478 x liu the topology on afs algebra and afs structure journal of mathematical analysis and applications 217 1998 479 489 which is a semantic methodology relating to the fuzzy theory combining above two theories we propose afs formal concept which can be viewed as the generalization and development of monotone concept proposed by deogun and saquer 2003 moreover we show that the set of all afs formal concepts forms a complete lattice afs formal concept can be applied to represent the logic operations of queries in information retrieval furthermore we give an approach to find the afs formal concepts whose intents extents approximate any element of afs algebra by virtue of rough set theory

approaches to knowledge reduction based on variable precision rough set model this paper deals with approaches to knowledge reduction based on variable precision rough set model the concepts of lower distribution reduct and upper distribution reduct based on variable precision rough sets vprs are first introduced their equivalent definitions are then given and the relationships among lower and upper distribution reducts and alternative types of knowledge reduction in inconsistent systems are investigated it is proved that for some special thresholds lower distribution reduct is equivalent to the maximum distribution reduct whereas upper distribution reduct is equivalent to the possible reduct the judgement theorems and discernibility matrices associated with the lower and upper distribution reducts are also established from which we can obtain the approaches to knowledge reduction in vprs