discernibility matrix simplification for constructing attribute reducts this paper proposes a reduct construction method based on discernibility matrix simplification the method works in a similar way to the classical gaussian elimination method for solving a system of linear equations elementary matrix simplification operations are introduced each operation transforms a matrix into a simpler form by applying these operations a finite number of times one can transform a discernibility matrix into one of its minimum i e the simplest forms elements of a minimum discernibility matrix are either the empty set or singleton subsets in which the union derives a reduct with respect to an ordering of attributes which is either computed based on a certain measure of attributes or directly given by a user two heuristic reduct construction algorithms are presented one algorithm attempts to exclude unimportant attributes from a reduct and the other attempts to include important attributes in a reduct

orthogonal polynomials and some q beta integrals of ramanujan abstract two integrals of ramanujan are used to define a q analogue of the euler beta integral on the real line and of the cauchy beta integral on the complex unit circle such integrals are connected to orthogonal biorthogonal and laurent polynomials explicit examples of laurent orthogonal polynomials are given on the real line and on the circle

on the combination and normalization of interval valued belief structures this paper investigates the issues of combination and normalization of interval valued belief structures within the framework of dempster shafer theory dst of evidence existing approaches are reviewed examined and critically analysed they either ignore the normalization or separate it from the combination process leading to irrational or suboptimal interval valued belief structures a new logically correct optimality approach is developed where the combination and the normalization are optimised together rather than separately numerical examples are provided throughout the paper

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

basic hypergeometric polynomials with zeros on the unit circle the objective in this manuscript is to present two specific sequences of polynomials with elements given in terms of basic hypergeometric polynomials or q hypergeometric polynomials it is shown that the zeros of these polynomials are simple and that they lie exactly on the unit circle

an efficient accelerator for attribute reduction from incomplete data in rough set framework feature selection attribute reduction from large scale incomplete data is a challenging problem in areas such as pattern recognition machine learning and data mining in rough set theory feature selection from incomplete data aims to retain the discriminatory power of original features to address this issue many feature selection algorithms have been proposed however these algorithms are often computationally time consuming to overcome this shortcoming we introduce in this paper a theoretic framework based on rough set theory which is called positive approximation and can be used to accelerate a heuristic process for feature selection from incomplete data as an application of the proposed accelerator a general feature selection algorithm is designed by integrating the accelerator into a heuristic algorithm we obtain several modified representative heuristic feature selection algorithms in rough set theory 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

space structure and clustering of categorical data learning from categorical data plays a fundamental role in such areas as pattern recognition machine learning data mining and knowledge discovery to effectively discover the group structure inherent in a set of categorical objects many categorical clustering algorithms have been developed in the literature among which k modes type algorithms are very representative because of their good performance nevertheless there is still much room for improving their clustering performance in comparison with the clustering algorithms for the numeric data this may arise from the fact that the categorical data lack a clear space structure as that of the numeric data to address this issue we propose in this paper a novel data representation scheme for the categorical data which maps a set of categorical objects into a euclidean space based on the data representation scheme a general framework for space structure based categorical clustering algorithms sbc is designed this framework together with the applications of two kinds of dissimilarities leads two versions of the sbc type algorithms to verify the performance of the sbc type algorithms we employ as references four representative algorithms of the k modes type algorithms experiments show that the proposed sbc type algorithms significantly outperform the k modes type algorithms

a novel attribute reduction approach for multi label data based on rough set theory multi label classification is an active research field in machine learning because of the high dimensionality of multi label data attribute reduction also known as feature selection is often necessary to improve multi label classification performance rough set theory has been widely used for attribute reduction with much success however little work has been done on applying rough set theory to attribute reduction in multi label classification in this paper a novel attribute reduction method based on rough set theory is proposed for multi label data first the uncertainties conveyed by labels are analyzed and a new type of attribute reduct is introduced called complementary decision reduct the relationships between complementary decision reduct and two representative types of attribute reducts are also investigated showing significant advantages of complementary decision reduct in revealing the uncertainties implied in multi label data second a discernibility matrix based approach is introduced for computing all complementary decision reducts and a heuristic algorithm is proposed for effectively computing a single complementary decision reduct experiments on real life data demonstrate that the proposed approach can effectively reduce unnecessary attributes and improve multi label classification accuracy

nested structure in parameterized rough reduction abstract in this paper by strict mathematical reasoning we discover the relationship between the parameters and the reducts in parameterized rough reduction this relationship named the nested reduction shows that the reducts act as a nested structure with the monotonically increasing parameter we present a systematic theoretical framework that provides some basic principles for constructing the nested structure in parameterized rough reduction some specific parameterized rough set models in which the nested reduction can be constructed are pointed out by strict mathematical reasoning based on the nested reduction we design several quick algorithms to find a different reduct when one reduct is already given here different refers to the reducts obtained on the different parameters all these algorithms are helpful for quickly finding a proper reduct in the parameterized rough set models the numerical experiments demonstrate the feasibility and the effectiveness of the nested reduction approach

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

maximal confidence intervals of the interval valued belief structure and applications an unsolved problem in the interval valued belief structure is the existence of maximal confidence intervals referring to the credible and maximal interval valued beliefs assigned to focal elements in which every point can be reached by a combination rule in this study the existence of maximal confidence intervals of the interval valued belief structure was investigated and validated in addition some applications of the maximal confidence interval in constructing valid and normalized interval valued belief structures are discussed a numerical experiment was conducted to illustrate the existence of maximal confidence intervals several propositions and a theorem were further proposed to demonstrate the validity of the maximal confidence interval by using the concept of the maximal confidence interval a procedure was proposed to construct valid and normalized interval valued belief structures and some meaningful guidance for assigning interval valued beliefs was also suggested to the experts finally a series of well designed examples was presented as applications of the maximal confidence interval the proposed procedure and the suggestions they indicated that valid and normalized interval valued belief structures can be achieved accurately and efficiently in practical applications with the proposed procedure and guidance on the basis of the concept of the maximal confidence interval

attribute reduction based on generalized fuzzy evidence theory in fuzzy decision systems attribute reduction is viewed as an important issue in data mining and knowledge representation this paper studies attribute reduction in fuzzy decision systems based on generalized fuzzy evidence theory the definitions of several kinds of attribute reducts are introduced the relationships among these reducts are then investigated in a fuzzy decision system it is proved that the concepts of fuzzy positive region reduct lower approximation reduct and generalized fuzzy belief reduct are all equivalent the concepts of fuzzy upper approximation reduct and generalized fuzzy plausibility reduct are equivalent and a generalized fuzzy plausibility consistent set must be a generalized fuzzy belief consistent set in a consistent fuzzy decision system an attribute set is a generalized fuzzy belief reduct if and only if it is a generalized fuzzy plausibility reduct but in an inconsistent fuzzy decision system a generalized fuzzy belief reduct is not a generalized fuzzy plausibility reduct in general

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

a unified framework for characterizing rough sets with evidence theory in various approximation spaces abstract different rough set models may be numerically characterized with evidence theory by adopting different methodologies that is to say the evidence theory based characteristics of rough sets vary with different approximation spaces and different rough set approximations to address this issue this paper proposes a theoretic framework within which rough set theory can be unifiedly characterized by evidence theory no matter what types of rough set approximations and approximation spaces they are the main works are presented as follows first we declare a principle that in any given approximation space a belief structure can be constructed based on which the lower and upper approximation operators can be measured by the belief and plausibility functions if and only if a basic condition is satisfied second in a given decision approximation space a belief structure is also derived based on which the lower and upper approximation operators can be always measured by the belief and plausibility functions without any additional condition as theoretical applications we employ the new principles to examine the evidence theory based characteristics of four types of substantial rough sets including the covering rough sets in covering approximation spaces the decision theoretic rough sets in pawlak approximation spaces the pawlak rough sets in pawlak decision approximation spaces and the multigranulation rough sets in pawlak decision approximation spaces respectively the proposed framework is applicable to various approximation spaces and various rough set approximations and hence can make us clear the basic principles for relating rough set theory with evidence theory

converse approximation and rule extraction from decision tables in rough set theory in this paper the concept of a granulation order is proposed in an information system the converse approximation of a target concept under a granulation order is defined and some of its important properties are obtained which can be used to characterize the structure of a set approximation for a subset of the universe in an information system its converge degree is monotonously increasing under a granulation order this means that a proper family of granulations can be chosen for a target concept approximation according to user requirements as an application of the converse approximation an algorithm based on the converse approximation called rebca is designed for decision rule extraction from a decision table which has a time complexity of o m2 c 2 u log 2 u and its practical applications are illustrated by two examples

pencils with prescribed constant subpencils abstract the results in this paper are within the scope of matrix pencil completion problems given an arbitrary matrix pencil we obtain necessary and sufficient conditions for the existence of a strictly equivalent pencil with a prescribed constant subpencil for algebraically closed fields

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

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

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

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

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