a fast and accurate algorithm for solving bernstein vandermonde linear systems abstract a fast and accurate algorithm for solving a bernstein vandermonde linear system is presented the algorithm is derived by using results related to the bidiagonal decomposition of the inverse of a totally positive matrix by means of neville elimination the use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate

performance limitations of non minimum phase systems in the servomechanism problem abstract this paper studies the cheap regulator problem and the cheap servomechanism problem for systems which may be non minimum phase the study extends some well known properties of perfect regulation and the perfect tracking and disturbance rejection of minimum phase systems to non minimum phase systems it is shown that perfect rejection to disturbances applied to the plant input can be achieved no matter whether the system is minimum phase or non minimum phase whereas a fundamental limitation exists in the achievable transient performance of tracking and rejection to disturbances applied to the plant output for a non minimum phase system and that this limitation can be simply and completely characterized by the number and locations of those zeros of the system which lie in the right half of the complex plane furthermore this limitation provides a quantitative measure of the degree of difficulty which is inherent in the control of such non minimum phase systems

test cost sensitive multigranulation rough set model and minimal cost selection abstract multigranulation rough set is an expansion of the classical rough set by using multiple granular structures presently three important multigranulation rough sets have been proposed they are optimistic pessimistic and multigranulation approaches however such three multigranulation rough sets do not take the test cost into consideration which is an important issue in both data mining and machine learning to solve such problem we propose a test cost sensitive multigranulation rough set model in this paper we show that test cost sensitive multigranulation rough set is a generalization of optimistic pessimistic and multigranulation rough sets furthermore it is found that the traditional heuristic algorithm is not suitable for granular structure selection with lower test cost we then propose a backtracking algorithm for granular structure selection with minimal test cost the algorithms are tested on ten uci university of california irvine data sets experimental results show the effectiveness of backtracking algorithm by comparing with heuristic algorithm this study suggests potential application areas and new research trends concerning multigranulation rough set theory

a measurement theory view on the granularity of partitions measurement of granularity is one of the foundational issues in granular computing this paper investigates a class of measures of granularity of partitions the granularity of a set is defined by a strictly monotonic increasing transformation of the cardinality of the set the granularity of a partition is defined as the expected granularity of all blocks of the partition with respect to the probability distribution defined by the partition many existing measures of granularity are instances of the proposed class new measures of granularity of partitions are also introduced

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

on performance limitation in tracking a sinusoid this note studies the performance limitation of a feedback system with a given linear time invariant lti plant in tracking a sinusoidal signal it continues and goes beyond some recent studies in the same topic in which it is assumed that the controller can access all the past and future values of the reference signal in this note we consider the more realistic and more difficult situation where the controller only accesses the current and past values of the reference an explicit formula of the best attainable performance is obtained for a single input single output siso system which depends on the nonminimum phase zeros of the plant and the frequency of the reference sinusoid compared to the previously studied case when the future of the reference is available this formula clearly shows the extra effort one has to pay due to the lack of the reference information a partial result for a multiple input multiple output mimo system is also given

prediction of company acquisition in greece by means of the rough set approach abstract this paper presents a new approach to forecast the acquisition of a firm in greece based on the rough set theory a sample of acquired firms and a sample of equivalent non acquired firms are considered and the objective is to create patterns which would be able to distinguish between the two classes of firms based upon differences in their financial characteristics financial ratios for this purpose the rough set approach is used the information about the firms is organized in a financial information table in this table financial characteristics of the firms correspond to condition attributes and the classification is defined by a decision attribute telling if a firm has been acquired or not the rough set approach enables one to discover minimal subsets of condition attributes financial ratios ensuring an acceptable approximation of the classification of the firms analyzed and to derive decision rules from the financial information table which can be used to best distinguish in the future between acquired and non acquired firms a comparison of the rough set approach with the discriminant analysis on the same set of data shows an advantage of the new approach

automatic determination about precision parameter value based on inclusion degree with variable precision rough set model the rough set theory provides a powerful approach for attributes reduction and data analysis the variable precision rough set vprs model an extension of the original rough set approach tolerates misclassifications of the training data to some degree which promotes the applications of rough set theory in inconsistent information systems however in most existing algorithms of feature reduction based on vprs the precision parameter is introduced as prior knowledge which restricts their applications because it is not clear how to set the value by studying consistency in the measurement of a decision table and the threshold value of the consistent decision table this paper presents an algorithm for automatic determination of the precision parameter value from a decision table based on vprs at the same time the precision parameter value from our proposed method is compared with the thresholds from the decision theoretic rough set dtrs the influences of the precision parameter are also discussed on attribute reduction which shows the necessity of the estimated precision parameter from a decision table the simulation results including vprs and other classification methods in real data further indicate that different precision parameter values make a great difference on rules and setting a precise parameter near the threshold value of the consistent decision table can precisely reflect the decision distribution of the decision table

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

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

feature selection for unsupervised learning in this paper we identify two issues involved in developing an automated feature subset selection algorithm for unlabeled data the need for finding the number of clusters in conjunction with feature selection and the need for normalizing the bias of feature selection criteria with respect to dimension we explore the feature selection problem and these issues through fssem feature subset selection using expectation maximization em clustering and through two different performance criteria for evaluating candidate feature subsets scatter separability and maximum likelihood we present proofs on the dimensionality biases of these feature criteria and present a cross projection normalization scheme that can be applied to any criterion to ameliorate these biases our experiments show the need for feature selection the need for addressing these two issues and the effectiveness of our proposed solutions

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

quick attribute reduction in inconsistent decision tables this paper focuses on three types of attribute reducts in inconsistent decision tables assignment reduct distribution reduct and maximum distribution reduct it is quite inconvenient to judge these three types of reduct directly according to their definitions this paper proposes judgment theorems for the assignment reduct the distribution reduct and the maximum distribution reduct which are expected to greatly simplify the judging of these three types of reducts on this basis we derive three new types of attribute significance measures and construct the q ara quick assignment reduction algorithm the q dra quick distribution reduction algorithm and the q mdra quick maximum distribution reduction algorithm these three algorithms correspond to the three types of reducts we conduct a series of comparative experiments with twelve uci machine learning data repository university of california at irvine data sets including consistent and inconsistent decision tables to evaluate the performance of the three reduction algorithms proposed with the relevant algorithm quickreduct 9 34 the experimental results show that quickreduct possesses weak robustness because it cannot find the reduct even for consistent data sets whereas our proposed three algorithms show strong robustness because they can find the reduct for each data set in addition we compare the q dra quick distribution reduction algorithm with the cebarknc conditional entropy based algorithm for reduction of knowledge without a computing core 43 because both find the distribution reduct by using a heuristic search the experimental results demonstrate that q dra runs faster than cebarknc does because the distribution function of q dra has a lower calculation cost instructive conclusions for these reduction algorithms are drawn from the perspective of classification performance for the c4 5 and rbf svm classifiers last we make a comparison between discernibility matrix based methods and our algorithms the experimental results indicate that our algorithms are efficient and feasible

fundamental performance limitations in tracking sinusoidal signals this paper attempts to give a thorough treatment of the performance limitation of a linear time invariant multivariable system in tracking a reference signal which is a linear combination of a step signal and several sinusoids with different frequencies the tracking performance is measured by an integral square error between the output of the plant and the reference signal our purpose is to find the fundamental limit for the attainable tracking performance under any control structure and parameters in terms of the characteristics and structural parameters of the given plant as well as those of the reference signal under consideration it is shown that this fundamental limit depends on the interaction between the reference signal and the nonminimum phase zeros of the plant and their frequency dependent directional information

on the extension of some total positivity inequalities abstract some inequalities recently obtained for totally positive matrices are extended to more general classes of matrices among these larger classes of matrices sign consistent sign regular and matrices with signed bidiagonal decomposition are considered the class of matrices with a signed bidiagonal decomposition contains the nonsingular totally positive matrices and their inverses

interval ordered information systems interval information systems are generalized models of single valued information systems by introducing a dominance relation to interval information systems we propose a ranking approach for all objects based on dominance classes and establish a dominance based rough set approach which is mainly based on substitution of the indiscernibility relation by the dominance relation furthermore we discuss interval ordered decision tables and dominance rules to simplify knowledge representation and extract much simpler dominance rules we propose attribute reductions of interval ordered information systems and decision tables that eliminate only the information that are not essential from the viewpoint of the ordering of objects or dominance rules the approaches show how to simplify an interval ordered information system and find dominance rules directly from an interval ordered decision table these results will be helpful for decision making analysis in interval information systems

rough set approach to multi attribute decision analysis abstract we review the methodology of the rough set analysis of multi attribute decision problems rough set theory proved to be a useful tool for analysis of a vague description of decision situations it answers two basic questions related to multi attribute decision problems one about explanation of a decision situation and another about prescription of some decisions basing on analysis of a decision situation we define four classes of multi attribute decision problems depending on the structure of their representation its interpretation and the kind of questions related then we characterize the rough set methodology for each particular class of decision problems we use simple practical examples to illustrate this presentation a review of related literature is made throughout the paper

reduction target structure based hierarchical attribute reduction for two category decision theoretic rough sets abstract attribute reduction is an essential subject in rough set theory but because of quantitative extension it becomes a problem when considering probabilistic rough set prs approaches the decision theoretic rough set dtrs has a threshold semantics and decision feature and thus becomes a typical and fundamental prs based on reduction target structures this paper investigates hierarchical attribute reduction for a two category dtrs and is divided into five parts 1 the knowledge preservation property and reduct are explored by knowledge coarsening 2 the consistency preservation principle and reduct are constructed by a consistency mechanism 3 region preservation is analyzed and the separability between consistency preservation and region preservation is concluded thus the double preservation principle and reduct are studied 4 structure targets are proposed by knowledge structures and an attribute reduction is further described and simulated thus general reducts are defined to preserve the structure targets or optimal measures 5 the hierarchical relationships of the relevant four targets and reducts are analyzed and a decision table example is provided for illustration this study offers promotion rationality structure hierarchy and generalization and it establishes a fundamental reduction framework for two category dtrs the relevant results also provide some new insights into the attribute reduction problem for prs

characterization and reduction of concept lattices through matroid theory concept lattice theory is an efficient tool for data analysis reduction of attributes and objects for a context is an important issue in concept lattice theory we explore the relationships between matroidal spaces and concepts for a given context with the assistance of matroid theory we conduct the attribute reduction and object reduction for a context and formulate the concepts for a given context as a generalization of the linear independence in vector spaces matroids provide well established platforms for some algorithms such as greedy algorithm in this paper we mainly propose a notion of matroidal space by the family of circuits of a matroid these relationships characterize the constructions of not only concepts but also concept lattices meanwhile we demonstrate that reduction of attributes and objects proposed in this paper preserve the original hierarchy order in the concept lattice for a given context

axiomatic characterization of a general utility function and its particular cases in terms of conjoint measurement and rough set decision rules abstract utility or value functions play an important role of preference models in multiple criteria decision making we investigate the relationships between these models and the decision rule preference model obtained from the dominance based rough set approach the relationships are established by means of special cancellation properties used in conjoint measurement as axioms for representation of aggregation procedures we are considering a general utility function and three of its important special cases associative operator sugeno integral and ordered weighted maximum for each of these aggregation functions we give a representation theorem establishing equivalence between a very weak cancellation property the specific utility function and a set of rough set decision rules each result is illustrated by a simple example of multiple criteria decision making the results show that the decision rule model we propose has clear advantages over a general utility function and its particular cases

performance limitation of tracking control problem for a class of references this technical note deals with analysis of fundamental limitations for siso systems in tracking control problems in the existing results the fundamental limitations are analyzed based on detailed assumptions of reference signals such as the step the trigonometrical signals and so on on the other hand we define a class of reference signals in an abstract but general manner for the general class of reference inputs we give the analysis of the tracking performance limitations based on the achievable set of the outputs the analysis results can separate the contributions of the plant and the reference signal clearly

general solution of robust tracking problem in two degree of freedom control systems in this note we study linear time invariant multivariable systems with two degree of freedom compensators first we give a new parametrization of all stabilizing compensators from the viewpoint of independent design of transfer function and robustness property second utilizing this parametrization we solve a general robust tracking problem including the case of nonsquare plants and show that tracking and robustness can be independently specified in two degree of freedom control systems

degrees of conditional in dependence a framework for approximate bayesian networks and examples related to the rough set based feature selection bayesian networks provide the means for representing probabilistic conditional independence conditional independence is widely considered also beyond the theory of probability with linkages to e g the database multi valued dependencies and at a higher abstraction level of semi graphoid models the rough set framework for data analysis is related to the topics of conditional independence via the notion of a decision reduct to be considered within a wider domain of the feature selection given probabilistic version of decision reducts equivalent to the data based markov boundaries the studies were also conducted for other criteria of the rough set based feature selection e g those corresponding to the multi valued dependencies in this paper we investigate the degrees of approximate conditional dependence which could be a topic corresponding to the well known notions such as conditional mutual information and polymatroid functions however with many practically useful approximate conditional independence models unmanageable within the information theoretic framework the major paper s contribution lays in extending the means for understanding the degrees of approximate conditional dependence with appropriately generalized semi graphoid properties formulated and with the mathematical soundness of the bayesian network like representation of the approximate conditional independence statements thoroughly proved as an additional contribution we provide a case study of the approximate conditional independence model which would not be manageable without the above mentioned extensions

attribute reduction in decision theoretic rough set models rough set theory can be applied to rule induction there are two different types of classification rules positive and boundary rules leading to different decisions and consequences they can be distinguished not only from the syntax measures such as confidence coverage and generality but also the semantic measures such as decision monotocity cost and risk the classification rules can be evaluated locally for each individual rule or globally for a set of rules both the two types of classification rules can be generated from and interpreted by a decision theoretic model which is a probabilistic extension of the pawlak rough set model as an important concept of rough set theory an attribute reduct is a subset of attributes that are jointly sufficient and individually necessary for preserving a particular property of the given information table this paper addresses attribute reduction in decision theoretic rough set models regarding different classification properties such as decision monotocity confidence coverage generality and cost it is important to note that many of these properties can be truthfully reflected by a single measure c in the pawlak rough set model on the other hand they need to be considered separately in probabilistic models a straightforward extension of the c measure is unable to evaluate these properties this study provides a new insight into the problem of attribute reduction

an expanded double quantitative model regarding probabilities and grades and its hierarchical double quantitative attribute reduction a double quantitative model on probabilities and grades is proposed by integration the model with quantitative semantics expands dtrs model grs model pawlak model model regions are hierarchically and optimally calculated via two algorithms three reduct types srp reduct crp reduct app reduct are hierarchically studied the double quantitative reducts expand basic qualitative and quantitative reducts probabilities and grades serve as relative and absolute measures respectively they are used to establish the decision theoretic rough set dtrs and graded rough set grs two basic quantitative models the double quantification of probabilities and grades exhibits systematicness and completeness in view of the two dimensional feature of the approximate space however double quantitative construction becomes a problem and double quantitative reduction is rarely reported thus this paper mainly constructs an expanded double quantitative model by logically integrating probabilities and grades it further studies relevant double quantitative reduction by hierarchically preserving specific regions 1 first a novel model is established via the logic integration and expansion requirement and its regional system and granular hierarchy are studied via granular computing thus regional semantics is extracted via basic semantics granules regional calculation is realized by two algorithms and the algorithm regarding calculation granules exhibits optimization according to algorithm analyses 2 second three types of model regional preservation reducts and their hierarchy are discussed in the two category case thus srp reduct crp reduct and app reduct are studied by exploring four region preservation properties constructing two region classification regions and preserving four original approximations respectively furthermore a relevant reduction hierarchy is thoroughly achieved 3 moreover the model and its reduction are illustrated by two examples of decision tables the constructional model conducts double quantification regarding probabilities and grades thus it exhibits double quantitative semantics and benignly expands dtrs model grs model and pawlak model furthermore its hierarchical reduction reflects some double quantitative reduction essence thus its reduction expands qualitative pawlak reduction while guides quantitative dtrs reduction and grs reduction

on the schur and singular value decompositions of oscillatory matrices abstract a square matrix a is said to be oscillatory if it has nonnegative minors and some power ak of a is strictly totally positive i e ak has strictly positive minors we study the schur and singular value decompositions of oscillatory matrices some applications are provided

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 based on evidence theory in incomplete decision systems attribute reduction is a basic issue in knowledge representation and data mining this paper deals with attribute reduction in incomplete information systems and incomplete decision systems based on dempster shafer theory of evidence the concepts of plausibility reduct and belief reduct in incomplete information systems as well as relative plausibility reduct and relative belief reduct in incomplete decision systems are introduced it is shown that in an incomplete information system an attribute set is a belief reduct if and only if it is a classical reduct and a plausibility consistent set must be a classical consistent set in a consistent incomplete decision system the concepts of relative reduct relative plausibility reduct and relative belief reduct are all equivalent in an inconsistent incomplete decision system an attribute set is a relative plausibility reduct if and only if it is a relative reduct a plausibility consistent set must be a belief consistent set and a belief consistent set is not a plausibility consistent set in general

a rough set approach to attribute generalization in data mining abstract this paper presents a method for updating approximations of a concept incrementally the results can be used to implement a quasi incremental algorithm for learning classification rules from very large data bases generalized by dynamic conceptual hierarchies provided by users in general the process of attribute generalization may introduce inconsistency into a generalized relation this issue is resolved by using the inductive learning algorithm lers based on rough set theory

toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic there are three basic concepts that underlie human cognition granulation organization and causation informally granulation involves decomposition of whole into parts organization involves integration of parts into whole and causation involves association of causes with effects granulation of an object a leads to a collection of granules of a with a granule being a clump of points objects drawn together by indistinguishability similarity proximity or functionality for example the granules of a human head are the forehead nose cheeks ears eyes etc in general granulation is hierarchical in nature a familiar example is the granulation of time into years months days hours minutes etc modes of information granulation ig in which the granules are crisp c granular play important roles in a wide variety of methods approaches and techniques crisp ig however does not reflect the fact that in almost all of human reasoning and concept formation the granules are fuzzy f granular the granules of a human head for example are fuzzy in the sense that the boundaries between cheeks nose forehead ears etc are not sharply defined furthermore the attributes of fuzzy granules e g length of nose are fuzzy as are their values long short very long etc the fuzziness of granules their attributes and their values is characteristic of ways in which humans granulate and manipulate information

data analysis based on discernibility and indiscernibility rough set theory models similarities and difierences of objects based on the notions of indiscernibility and discernibility with respect to any subset of attributes one can deflne two pairs of dual relations the strong indiscernibility and weak discernibility relations and the weak indiscernibility and strong discernibility relations the similarities of objects are examined by the indiscernibility relations and the difierence by the discernibility relations respectively alternatively one can construct an indiscernibility matrix to represent the family of strong indiscernibility or weak discernibility relations one also can construct a discernibility matrix to represent the family of strong discernibility or weak indiscernibility relations the consideration of the matrix counterpart of relations and the relation counterpart of matrices brings more insights into rough set theory based on indiscernibility and discernibility three difierent types of reducts can be constructed keeping the indiscernibility discernibility and indiscernibility anddiscernibility relations respectively although the indiscernibility reducts have been intensively studied in the literature the other two types of reducts are relatively new and require more attention the existing methods for constructing the indiscernibility reducts also can be applied to construct the other two types of reducts an empirical experiment for letter recognition is reported for demonstrating the usefulness of the discussed relations and reducts

a fuzzy rough set approach for incremental feature selection on hybrid information systems in real applications there may exist many kinds of data e g boolean categorical real valued and set valued data and missing data in an information system which is called as a hybrid information system his a new hybrid distance hd in his is developed based on the value difference metric and a novel fuzzy rough set is constructed by combining the hd distance and the gaussian kernel considering the information systems often vary with time the updating mechanisms for attribute reduction feature selection are analyzed with the variation of the attribute set fuzzy rough set approaches for incremental feature selection on his are presented then two corresponding incremental algorithms are proposed respectively finally extensive experiments on eight datasets from uci and an artificial dataset show that the incremental approaches significantly outperform non incremental approaches with feature selection in the computational time

a set covering based approach to find the reduct of variable precision rough set abstract attribute reduction is one of the core problems in rough set rs theory in the variable precision rough set vprs model attribute reduction faces two difficulties firstly in the vprs model a reduct anomaly problem may arise and it may cause an inconsistency of positive regions and decision rules after attribute reduction secondly the attribution reduction problem has been proved an np hard problem accordingly we would need to find a tradeoff between calculating the minimal reduct and reducing computing complexity to avoid the combinatorial explosion problem we propose a new approach to calculate the reduct in vprs model this new method focuses on calculating a distribution reduct while avoiding the anomaly problem in the vprs model the basic idea of the proposed approach is to convert the reduct problem into a set covering problem scp according to the positive regions in the vprs model and consequently a set covering heuristic function schf algorithm is applied to calculate the reduct after this conversion this approach keeps the positive regions consistent after the attribute reduction and moreover based on the scp the performance ratio of the proposed method to calculate the minimal reduct ranges between ln u ln ln u o 1 and 1 o 1 ln u with a computational complexity having an upper bound as o mn m n 2 finally we demonstrate the practical application of the vprs model using real case scenario from china s electricity power yield to verify the validity of our proposed approach we then apply statistical evaluation to explain the economic significance of the attributes

similarity to totally positive matrices and accurate computations abstract matrices similar to nonsingular totally positive matrices are analyzed in the particular case of sbd matrices new characterizations are presented for more general classes of matrices methods to assure accurate computations are analyzed

incremental updating of rough approximations in interval valued information systems under attribute generalization interval valued information system ivis is a generalized model of single valued information system in which the attribute values of objects are all interval values instead of single values the attribute set in ivis is not static but rather dynamically changing over time with the collection of new information which results in the continuous updating of rough approximations for rough set based data analysis in this paper on the basis of the similarity based rough set model in ivis we develop incremental approaches for updating rough approximations in ivis under attribute generalization which refers to the dynamic changing of attributes firstly increment relationships between the original rough approximations and the updated ones when adding or deleting an attribute set are analyzed respectively and the incremental mechanisms for updating rough approximations in ivis are introduced which carry out the computation using the previous results from the original data set along with new results then the corresponding incremental algorithms are designed based on the proposed mechanisms finally comparative experiments on data sets from uci as well as artificial data sets are conducted respectively experimental results show that the proposed incremental algorithms can effectively reduce the running time for the computation of rough approximations in comparison with the static algorithm

consistency measure inclusion degree and fuzzy measure in decision tables classical consistency degree has some limitations for measuring the consistency of a decision table in which the lower approximation of a target decision is only taken into consideration in this paper we focus on how to measure the consistencies of a target concept and a decision table and the fuzziness of a rough set and a rough decision in rough set theory for three types of decision tables complete incomplete and maximal consistent blocks the membership functions of an object are defined through using the equivalence class tolerance class and maximal consistent blocks including itself respectively based on these membership functions we introduce consistency measures to assess the consistencies of a target set and a decision table and define fuzziness measures to compute the fuzziness of a rough set and a rough decision in these three types of decision tables in addition the relationships among the consistency inclusion degree and fuzzy measure are established as well these results will be helpful for understanding the essence of the uncertainty in decision tables and can be applied for rule extraction and rough classification in practical decision issues

robust neural modeling for the cross sectional analysis of accounting information abstract the performance of robust artificial neural network models in learning bivariate relationships between accounting magnitudes is assessed in this paper predictive performances of a number of modeling paradigms namely linear models log linear structures classical ratios and artificial neural networks are compared with regard to the problem of modeling a number of the most outstanding accounting ratio relations we conduct a large scale analysis carried out on a representative spanish data base several model fitting criteria are used for each model class namely least squares weighted least squares least absolute deviations lad and weighted lad regressions hence besides the standard least squares based version of each model we test a robust lad counterpart in principle more adequate for distributions strongly affected by outliers as typically appear in accounting ratio modeling our results strongly suggest that classical ratio models although much used in practical applications appear to be largely inadequate for predictive purposes with linear models both in their least squares and lad variants providing much more adequate specifications in a number of cases the linear specification is improved by considering flexible non linear structures neural networks because of their model free regression capabilities let us capture generic non linearities of unknown form in the modeled relations as well as providing when properly trained robust tools for modeling and prediction of general relationships

a note on dynamic relational systems abstract it is known that weak and strong definabilities are defined for multiple source approximation systems by khan and banerjee this paper presents in a more general setting a discussion on definabilities of sets in dynamic relational systems we prove that the inverse image of a weak or strong definable set with respect to relation preserving function is also weak or strong definable set respectively on the way we show that the inverse image of a reduct of attribute set is also a reduct under the object function of an information system homomorphism further we give the connections between definable sets and the topology determined by the intersections of the topologies of reflexive relations a quasi uniformity is a filter on the cartesian product of a given universe satisfying certain conditions in fact every quasi uniformity is a dynamic relational system where the relations are reflexive in this respect we discuss on the connections between approximation systems and quasi uniformities

maximal consistent block technique for rule acquisition in incomplete information systems in this paper the concept of a maximal consistent block is applied to formulate a new approximation to an object set in incomplete information systems with higher level of accuracy similar to the method in inform sci 112 1998 39 113 1999 271 the proposed rough set based rule acquisition method does not require change in the size of the original incomplete system it however has the additional advantage of using a set of simpler discernibility functions of an incomplete system this means that it can provide a more efficient computation for knowledge acquisition especially in large incomplete systems

catoptrical rough set model on two universes using granule based definition and its variable precision extensions rough set models on two universes are valuable generalizations of classic rough set model these models can be applied into formal concept analysis and decision making system this paper focuses on catoptrical approach for rough set models over two universes using granule based definition in this paper we propose a new catoptrical rough set model over dual universes the properties of the proposed model are discussed consequently a variable precision extension of the model is presented some properties of the extended model are obtained finally we investigate a further extension of the proposed model i e variable precision catoptrical rough set model with two parameters

decision region distribution preservation reduction in decision theoretic rough set model abstract in the pawlak rough set model the positive region the boundary region and the non negative region are monotonic with respect to the set inclusion of attributes however the monotonicity property of the decision regions positive region boundary region or non negative region with respect to the set inclusion of attributes does not hold in the decision theoretic rough set model therefore the decision regions may be changed after attribute reduction based on quantitative preservation or qualitative preservation of decision regions this effect is observed partly because three decision regions are defined by introducing the probabilistic threshold values in addition heuristic reduction algorithms based on decision regions may find super reducts because of the non monotonicity of decision regions to address the above issues this paper proposes solutions to the attribute reduction problem based on decision region preservation in the decision theoretic rough set model first the positive region distribution preservation reduct the boundary region distribution preservation reduct and the negative region distribution preservation reduct are introduced into the decision theoretic rough set model second three new monotonic measures are constructed by considering variants of the conditional information entropy from which we can obtain the heuristic reduction algorithms the results of the experimental analysis validate the monotonicity of new measures and verify the effectiveness of decision region distribution preservation reducts

set valued ordered information systems set valued ordered information systems can be classified into two categories disjunctive and conjunctive systems through introducing two new dominance relations to set valued information systems we first introduce the conjunctive disjunctive set valued ordered information systems and develop an approach to queuing problems for objects in presence of multiple attributes and criteria then we present a dominance based rough set approach for these two types of set valued ordered information systems which is mainly based on substitution of the indiscernibility relation by a dominance relation through the lower upper approximation of a decision some certain possible decision rules from a so called set valued ordered decision table can be extracted finally we present attribute reduction also called criteria reduction in ordered information systems approaches to these two types of ordered information systems and ordered decision tables which can be used to simplify a set valued ordered information system and find decision rules directly from a set valued ordered decision table these criteria reduction approaches can eliminate those criteria that are not essential from the viewpoint of the ordering of objects or decision rules

dynamic maintenance of approximations in set valued ordered decision systems under the attribute generalization set valued information systems are important type of data tables and generalized models of single valued information systems approximations are the focal point of approaches to knowledge discovery based on rough set theory which can be used to extract and represent the hidden knowledge in the form of decision rules attribute generalization refers to dynamic change of the attribute set in an information system with respect to the requirements of real life applications in this paper we focus on maintaining approximations dynamically in set valued ordered decision systems under the attribute generalization firstly a matrix based approach for computing approximations of upward and downward unions of decision classes is constructed by introducing the dominant and dominated matrices with respect to the dominance relation then incremental approaches for updating approximations are proposed which involves several modifications to relevant matrices without having to retrain from the start on all accumulated training data finally comparative experiments on data sets from uci as well as synthetic data sets show the proposed incremental updating methods are efficient and effective for dynamic attribute generalization

set valued information systems set valued information systems are generalized models of single valued information systems incomplete information systems can be viewed as disjunctively interpreted set valued information systems since some objects in set valued information systems may have more than one value for an attribute so we define tolerance relation and use the maximal tolerance classes to classify the universe of discourse in order to derive optimal decision rules from set valued decision information systems we propose the concept of relative reduct of maximal tolerance classes and define a kind of discernibility function to compute the relative reduct by boolean reasoning techniques finally we define three kinds of relative reducts for set valued information systems and used them to evaluate the significance of attributes

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

a novel approach to building a robust fuzzy rough classifier currently most robust classifiers with parameters focus on the determination of the optimal or suboptimal parameters there are no research studies or even discussions about robust classifiers on all of the possible parameters this paper considers the robust rough classifier and finds that the robust rough classifier satisfies a nested topological structure then the nested classifier which reflects the classifier on all of the possible parameters is proposed first some notions such as the robust discernibility vector the robust value reduct and the robust covering vector are proposed these notions can reflect the classical corresponding notions on all of the possible parameters it is more important that these notions share a common characteristic the nested structure the nested structure of these notions makes nested classifier theoretically possible furthermore some novel algorithms are designed to compute the robust value reduct the robust covering degree and the robust classifier these algorithms make the nested classifier technologically possible finally numerical experiments demonstrate that the nested classifier is effective and efficient for classification and predication

positive approximation and converse approximation in interval valued fuzzy rough sets methods of fuzzy rule extraction based on rough set theory are rarely reported in incomplete interval valued fuzzy information systems thus this paper deals with such systems instead of obtaining rules by attribute reduction which may have a negative effect on inducting good rules the objective of this paper is to extract rules without computing attribute reducts the data completeness of missing attribute values is first presented positive and converse approximations in interval valued fuzzy rough sets are then defined and their important properties are discussed two algorithms based on positive and converse approximations namely mine rules based on the positive approximation mrbpa and mine rules based on the converse approximation mrbca are proposed for rule extraction the two algorithms are evaluated by several data sets from the uc irvine machine learning repository the experimental results show that mrbpa and mrbca achieve better classification performances than the method based on attribute reduction

a novel method for attribute reduction of covering decision systems attribute reduction has become an important step in pattern recognition and machine learning tasks covering rough sets as a generalization of classical rough sets have attracted wide attention in both theory and application this paper provides a novel method for attribute reduction based on covering rough sets we review the concepts of consistent and inconsistent covering decision systems and their reducts and we develop a judgment theorem and a discernibility matrix for each type of covering decision system furthermore we present some basic structural properties of attribute reduction with covering rough sets based on a discernibility matrix we develop a heuristic algorithm to find a subset of attributes that approximate a minimal reduct finally the experimental results for uci data sets show that the proposed reduction approach is an effective technique for addressing numerical and categorical data and is more efficient than the method presented in the paper d g chen c z wang q h hu a new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets information sciences 177 17 2007 3500 3518

a test for strict sign regularity a characterization of strict sign regular matrices is obtained it is given in an algorithmic form which allows the easy construction of a test to check if a matrix is strictly sign regular the test is based on the neville elimination of several submatrices of a and requires less computational work than those which can be derived from other previously known characterizations of these matrices

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

dominance based rough set approach and knowledge reductions in incomplete ordered information system many methods based on the rough set to deal with incomplete information systems have been proposed in recent years however they are only suitable for the incomplete systems with regular attributes whose domains are not preference ordered this paper thus attempts to present research focusing on a complex incomplete information system the incomplete ordered information system in such incomplete information systems all attributes are considered as criterions a criterion indicates an attribute with preference ordered domain to conduct classification analysis in the incomplete ordered information system the concept of similarity dominance relation is first proposed two types of knowledge reductions are then formed for preserving two different notions of similarity dominance relations with introduction of the approximate distribution reduct into the incomplete ordered decision system the judgment theorems and discernibility matrixes associated with four novel approximate distribution reducts are obtained a numerical example is employed to substantiate the conceptual arguments

a new approach of optimal scale selection to multi scale decision tables as a special case of information table multi scale decision table can usually be observed in real life world in such table objects may take different values under the same attribute measured at different scales based on inclusion relation of subsets of attributes and coarse relation of scales of attributes multi layered granulations and stratified rough set approximations in multi scale decision tables are shown from the perspective of granular computing compared with a special case studied by wu and leung the multi scale decision tables of diverse attributes with different numbers of levels of scales are studied in this paper furthermore complement model and lattice model are proposed to analyze the optimal scale selection for multi scale decision tables correspondingly algorithms of the two models are designed and some experiments are performed to testify feasibilities of these proposed algorithms and to make comparisons of the models

knowledge reduction of dynamic covering decision information systems when varying covering cardinalities in covering based rough set theory non incremental approaches are time consuming for performing knowledge reduction of dynamic covering decision information systems when the cardinalities of coverings change as a result of object immigration and emigration because computing approximations of sets is an important step for knowledge reduction of dynamic covering decision information systems efficient approaches to calculating the second and sixth lower and upper approximations of sets using the type 1 and type 2 characteristic matrices respectively are essential in this paper we provide incremental approaches to computing the type 1 and type 2 characteristic matrices of dynamic coverings whose cardinalities vary with the immigration and emigration of objects we also design incremental algorithms to compute the second and sixth lower and upper set approximations experimental results demonstrate that the incremental approaches effectively improve the efficiency of set approximation computation finally we employ several examples to illustrate the feasibility of the incremental approaches for knowledge reduction of dynamic covering decision information systems when increasing the cardinalities of coverings

fuzzy preference based rough sets preference analysis is an important task in multi criteria decision making the rough set theory has been successfully extended to deal with preference analysis by replacing equivalence relations with dominance relations the existing studies involving preference relations cannot capture the uncertainty presented in numerical and fuzzy criteria in this paper we introduce a method to extract fuzzy preference relations from samples characterized by numerical criteria fuzzy preference relations are incorporated into a fuzzy rough set model which leads to a fuzzy preference based rough set model the measure of attribute dependency of the pawlak s rough set model is generalized to compute the relevance between criteria and decisions the definitions of upward dependency downward dependency and global dependency are introduced algorithms for computing attribute dependency and reducts are proposed and experimentally evaluated by using two publicly available data sets

right half plane poles and zeros and design tradeoffs in feedback systems this paper expresses limitations imposed by right half plane poles and zeros of the open loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed loop system the limitations are determined by integral relationships which must be satisfied by these functions the integral relationships are interpreted in the context of feedback design

quantitative information architecture granular computing and rough set models in the double quantitative approximation space of precision and grade because precision and grade act as fundamental quantitative information in approximation space they are used in relative and absolute quantifications respectively at present the double quantification regarding precision and grade is a novel and valuable subject but quantitative information fusion has become a key problem thus this paper constructs the double quantitative approximation space of precision and grade pg approx space and tackles the fusion problem using normal logical operations it further conducts double quantification studies on granular computing and rough set models 1 first for quantitative information organization and storage we construct space and plane forms of pg approx space using the cartesian product and for quantitative information extraction and fusion we establish semantics construction and semantics granules of pg approx space 2 second by granular computing we investigate three primary granular issues quantitative semantics complete system and optimal calculation accordingly six types of fundamental granules are proposed based on the semantic microscopic and macroscopic descriptions their semantics forms structures calculations and relationships are studied and the granular hierarchical structure is achieved 3 finally we investigate rough set models in pg approx space accordingly model regions are proposed by developing the classical regions model expansion is systematically analyzed some models are constructed as their structures are obtained and a concrete model is provided based on the quantitative information architecture this paper systematically conducts and investigates double quantification and establishes a fundamental and general exploration framework

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

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

incremental update of approximations in dominance based rough sets approach under the variation of attribute values dominance based rough sets approach drsa has received much attention since it is able to acquire knowledge from information with preference ordered attribute domains and decision classes in many real life applications the information systems may evolve over time dynamically in a dynamic information system the obtained knowledge e g approximations in drsa need to be updated for decision making and other related tasks as a useful technique the incremental update can be applied to process dynamic information with revising the obtained knowledge in this paper we propose an incremental approach for maintaining approximations of drsa when attribute values vary over time some numerical examples illustrate that the incremental approach can renew approximations of drsa without beginning from scratch experimental evaluations show that the incremental algorithm can effectively reduce the computational time in comparison with the non incremental one when the ratio of the attribute values varied is less than a threshold

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 sets theory for multicriteria decision analysis the original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation it operates on a data table composed of a set u of objects actions described by a set q of attributes its basic notions are indiscernibility relation on u lower and upper approximation of either a subset or a partition of u dependence and reduction of attributes from q and decision rules derived from lower approximations and boundaries of subsets identified with decision classes the original rough set idea is failing however when preference orders of attribute domains criteria are to be taken into account precisely it cannot handle inconsistencies following from violation of the dominance principle this inconsistency is characteristic for preferential information used in multicriteria decision analysis mcda problems like sorting choice or ranking in order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary the main change is the substitution of the indiscernibility relation by a dominance relation which permits approximation of ordered sets in multicriteria sorting to approximate preference relations in multicriteria choice and ranking problems another change is necessary substitution of the data table by a pairwise comparison table where each row corresponds to a pair of objects described by binary relations on particular criteria in all those mcda problems the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model it is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax in order to workout a recommendation in one of the mcda problems we propose exploitation procedures of the set of decision rules finally some other recently obtained results are given rough approximations by means of similarity relations rough set handling of missing data comparison of the rough set model with sugeno and choquet integrals and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive

induction of multiple fuzzy decision trees based on rough set technique the integration of fuzzy sets and rough sets can lead to a hybrid soft computing technique which has been applied successfully to many fields such as machine learning pattern recognition and image processing the key to this soft computing technique is how to set up and make use of the fuzzy attribute reduct in fuzzy rough set theory given a fuzzy information system we may find many fuzzy attribute reducts and each of them can have different contributions to decision making if only one of the fuzzy attribute reducts which may be the most important one is selected to induce decision rules some useful information hidden in the other reducts for the decision making will be losing unavoidably to sufficiently make use of the information provided by every individual fuzzy attribute reduct in a fuzzy information system this paper presents a novel induction of multiple fuzzy decision trees based on rough set technique the induction consists of three stages first several fuzzy attribute reducts are found by a similarity based approach and then a fuzzy decision tree for each fuzzy attribute reduct is generated according to the fuzzy id3 algorithm the fuzzy integral is finally considered as a fusion tool to integrate the generated decision trees which combines together all outputs of the multiple fuzzy decision trees and forms the final decision result an illustration is given to show the proposed fusion scheme a numerical experiment on real data indicates that the proposed multiple tree induction is superior to the single tree induction based on the individual reduct or on the entire feature set for learning problems with many attributes

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

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

mgrs a multi granulation rough set the original rough set model was developed by pawlak which is mainly concerned with the approximation of sets described by a single binary relation on the universe in the view of granular computing the classical rough set theory is established through a single granulation this paper extends pawlak s rough set model to a multi granulation rough set model mgrs where the set approximations are defined by using multi equivalence relations on the universe a number of important properties of mgrs are obtained it is shown that some of the properties of pawlak s rough set theory are special instances of those of mgrs moreover several important measures such as accuracy email protected quality of email protected and precision of email protected are presented which are re interpreted in terms of a classic measure based on sets the marczewski steinhaus metric and the inclusion degree measure a concept of approximation reduct is introduced to describe the smallest attribute subset that preserves the lower approximation and upper approximation of all decision classes in mgrs as well finally we discuss how to extract decision rules using mgrs unlike the decision rules and rules from pawlak s rough set model the form of decision rules in mgrs is or several pivotal algorithms are also designed which are helpful for applying this theory to practical issues the multi granulation rough set model provides an effective approach for problem solving in the context of multi granulations

approximate distribution reducts in inconsistent interval valued ordered decision tables abstract many methods based on the rough set theory to deal with information systems have been proposed in recent decades in practice some information systems are based on dominance relations and may be inconsistent because of various factors moreover taking the imprecise evaluations and assignments in the description of objects into account single valued information systems have been generalized to interval valued information systems in this paper by introducing a dominance relation to interval valued ordered information systems we establish a dominance based rough set approach which is mainly based on substitution of the indiscernibility relation by the dominance relation to extract the minimal decision rules approximate distribution reducts are proposed in inconsistent interval valued ordered decision tables this paper presents a theoretical method based on the discernibility matrix to enumerate all reducts and a practical approach on the basis of significance to find one reduct and two equivalent definitions of approximate distribution reducts are also introduced in addition numerical examples are employed to examine the validity of the approaches proposed in this paper

a weighted rough set based method developed for class imbalance learning in this paper we introduce weights into pawlak rough set model to balance the class distribution of a data set and develop a weighted rough set based method to deal with the class imbalance problem in order to develop the weighted rough set based method we design first a weighted attribute reduction algorithm by introducing and extending guiasu weighted entropy to measure the significance of an attribute then a weighted rule extraction algorithm by introducing a weighted heuristic strategy into lem2 algorithm and finally a weighted decision algorithm by introducing several weighted factors to evaluate extracted rules furthermore in order to estimate the performance of the developed method we compare the weighted rough set based method with several popular methods used for class imbalance learning by conducting experiments with twenty uci data sets comparative studies indicate that in terms of auc and minority class accuracy the weighted rough set based method is better than the re sampling and filtering based methods and is comparable to the decision tree and svm based methods it is therefore concluded that the weighted rough set based method is effective for class imbalance learning

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

incremental perspective for feature selection based on fuzzy rough sets feature selection based on fuzzy rough sets is an effective approach to select a compact feature subset that optimally predicts a given decision label despite being studied extensively most existing methods of fuzzy rough set based feature selection are restricted to computing the whole dataset in batch which is often costly or even intractable for large datasets to improve the time efficiency we investigate the incremental perspective for fuzzy rough set based feature selection assuming data can be presented in sample subsets one after another the key challenge for the incremental perspective is how to add and delete features with the subsequent arrival of sample subsets we tackle this challenge with strategies of adding and deleting features based on the relative discernibility relations that are updated as subsets arrive sequentially two incremental algorithms for fuzzy rough set based feature selection are designed based on the strategies one updates the selected features as each sample subset arrives and outputs the final feature subset where no sample subset is left the other updates the relative discernibility relations but only performs feature selection where there is no further subset arriving experimental comparisons suggest our incremental algorithms expedite fuzzy rough set based feature selection without compromising performance

relationships between generalized rough sets based on covering and reflexive neighborhood system rough set theory is an important tool to deal with uncertainty granularity and incompleteness of knowledge the covering based rough sets is one of the most important extensions of the classical pawlak rough sets in covering based rough set theory many types of rough sets were already defined in the literature to find out the relationships among different types of covering based rough sets and give an axiomatic definition is a very important issue the generalized rough sets in neighborhood system is another important extension of the classical pawlak rough sets in this paper we investigate relationships between covering based rough sets and generalized rough sets in neighborhood system then twenty three types of the element based the granule based and the subsystem based definitions of covering approximation operators are unified under the framework of generalized approximation operators in neighborhood systems moreover properties of the generalized approximation operators in neighborhood systems are presented

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