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

multi granulation fuzzy preference relation rough set for ordinal decision system preference analysis is a class of important issues in multi criteria ordinal decision making rough set is an effective approach to handle preference analysis in order to solve the multi criteria preference analysis problem this work improves the fuzzy preference relation rough set model with additive consistent fuzzy preference relation and expands it to multi granulation case cost is also an important issue in decision analysis taking the cost into consideration we also expand the model to cost sensitive multi granulation fuzzy preference relation rough set some theorems are represented and the classification and sample condensation algorithms based on our model are investigated some experiments are complete and the experimental results show that our model and algorithms are effective for preference decision making of ordinal decision system

adaptive neighborhood granularity selection and combination based on margin distribution optimization abstract granular computing aims to develop a granular view for interpreting and solving problems the model of neighborhood rough sets is one of effective tools for granular computing this model can deal with complex tasks of classification learning despite the success of the neighborhood model in attribute reduction and rule learning it still suffers from the issue of granularity selection namely it is an open problem to select a proper granularity of neighborhood for a specific task in this work we explore ensemble learning techniques for adaptively evaluating and combine the models derived from multiple granularity in the proposed framework base classifiers are trained in different granular spaces the importance of base classifiers is then learned by optimizing the margin distribution of the combined system experimental analysis shows that the proposed method can adaptively select a proper granularity and combining the models trained in multi granularity spaces leads to competent performance

on eigenvalues eigenvectors and singular values in robust stability analysis abstract recent papers have examined the problem of robustness of the stability of multi variable feedback systems to perturbations g in matrix form attention has been primarily focused on the use of the maximal singular value g this paper considers how structured information on the uncertainty in each element 9 con be used in a similar way based on eigenvalue and singular value analysis

on quick attribute reduction in decision theoretic rough set models compared with the pawlak rough set model decision theoretic rough set dtrs models have relatively powerful data processing capability for complex and low quality data however the non monotonicity of the criteria used in dtrs models causes many problems particularly the inefficiency of attribute reduction algorithms which greatly narrows their applications few systematic studies have been reported on how to construct efficient reduction methods for dtrs models this paper discusses this problem and constructs an efficient reduction algorithm for dtrs models incorporating three aspects taking advantage of common characteristics of data sets and employing division techniques a fast division approach to computing equivalence classes is proposed which is the most frequently used and the most time consuming basic operation in the reduction algorithm extracting a monotonic ingredient from decision systems an effective heuristic function is constructed for the reduction algorithm which can guide searches for the algorithm and has better ability to find shorter super reducts a novel algorithm is proposed to search the power set space of a given super reduct which has a shorter average search length than the existing backtracking methods finally the reduction algorithm is proposed that combines these improvements experimental results demonstrate that these improvements are effective and the proposed reduction algorithm is relatively efficient

adaptive robust control of uncertain systems with measurement noise abstract a class of dynamical systems with time varying uncertainty is considered the nominal portion i e the portion without uncertainty of the system is stabilizable the uncertainty is either assumed to satisfy the matching condition or is close to matching moreover the uncertainty is to be cone bounded the bound is however unknown the control design is only based on the deterministic properties related to the bound of uncertainty the feedback information is the state with measurement noise

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

feature analysis through information granulation and fuzzy sets feature analysis and feature selection are fundamental pursuits in pattern recognition we revisit and generalize an issue of feature selection by introducing a mechanism of soft fuzzy feature selection the underlying idea is to consider features to be granular rather than numeric by varying the level of granularity we modify the level of contribution of the specific feature to the overall feature space we admit an interval model of the features meaning that their values assume a form of numeric intervals the intervalization of the features exhibits a clear cut interpretation moreover a contribution of the features to the formation of the feature space can be easily controlled the broader the interval the less essential contribution of the feature to the entire feature space in limit when the intervals get broad enough one may view the feature to be completely eliminated dropped from the feature space the quantification of the features in terms of their importance is realized in the setting of the clustering fcm model namely a process of the binary or fuzzy feature selection is carried out and numerically quantified in the space of membership values generated by fuzzy clusters as the focal point of this study concerns an interval like form of information granules we reveal how such feature intervalization helps approximate fuzzy sets described by any type of membership function detailed computations give rise to a detailed quantification of such granular features numerical experiments provide a comprehensive numerical illustration of the problem

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

general c means clustering model partitional clustering is an important part of cluster analysis based on various theories numerous clustering algorithms have been developed and new clustering algorithms continue to appear in the literature it is known that occam s razor plays a pivotal role in data based models and partitional clustering is categorized as a data based model however no relation had previously been discovered between occam s razor and partitional clustering as we discuss in this paper the three main contributions of this paper can be summarized as follows 1 according to a novel definition of the mean a unifying generative framework for partitional clustering algorithms called a general c means clustering model gcm is presented and studied and 2 based on the local optimality test of the gcm the connection between occam s razor and partitional clustering is established for the first time as its application a comprehensive review of the existing objective function based clustering algorithms is presented based on gcm 3 under a common assumption about partitional clustering a theoretical guide for devising and implementing clustering algorithm is discovered these conclusions are verified by numerical experimental results

the numerical range a tool for robust stability studies the use of numerical range concepts for assessing robust stability of multivariable feedback systems is investigated the characterization of perturbations by their numerical range allows a more detailed description of their gains and phases and allows a robust stability theorem similar in structure to that of postlethwaite et al with the possibility of arbitrary large perturbations

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

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

incremental rough set approach for hierarchical multicriteria classification abstract multicriteria classification refers to classify objects evaluated by a set of criteria to preference ordered decision classes dominance based rough set approach has been successfully introduced to express and reason inconsistencies with a dominance principle in multicriteria classification problems hierarchical attribute values exist extensively within many real world applications which provide a hierarchical form to organize view and analyze data from different perspectives for accommodating the preference variability in this study we consider an extension of dominance based rough set approach by applying an incremental learning technique for hierarchical multicriteria classification while attribute values dynamically vary across different levels of granulations we formalize the dynamic characteristics of knowledge granules with the cut refinement and coarsening through attribute value taxonomies in the hierarchical multicriteria decision systems in consequence incremental algorithms for computing dominance based rough approximations of preference ordered decision classes are developed by applying the resulted prior knowledge as the input and only recomputing those outputs which depend on the changed attribute values this paper presents the theoretical foundation of the proposed approach example analysis and experimental evaluation are also provided for illustration of the feasibility and efficiency

fuzzy equivalence relation and its multigranulation spaces multigranulation rough sets focus on describing uncertain sets in different granularity spaces different approximation accuracy of an uncertain set can be obtained through changing knowledge granularities according to user s need however it is a bottleneck for establishing a multigranulation space to better describe an uncertain set in a proper granulation space and the typical way of a multigranulation space is a hierarchical quotient space determined by a fuzzy equivalence relation in this paper a new concept granularity of binary relation gb is put forward firstly and knowledge granulation gk is proved to be a special case of gb the changing rules of granularity series for describing a hierarchical quotient space structure are proposed and approximation accuracy series of an uncertain set is presented for automatically searching optimal knowledge space according to user s requirement a utility function with both knowledge space granularity and approximation accuracy is proposed then classification isomorphism and granularity isomorphism between two hierarchical quotient space structures are defined and discussed through the hierarchical quotient space structure of a fuzzy equivalence relation the essence of classification ability of a fuzzy equivalence relation is also revealed finally a new algorithm for establishing isomorphic fuzzy equivalence relations based on a hierarchical quotient space structure is presented and this algorithm can further explain the nature of those isomorphic fuzzy equivalence relations from a new perspective

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

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

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

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

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

minimum cost attribute reduction in decision theoretic rough set models in classical rough set models attribute reduction generally keeps the positive or non negative regions unchanged as these regions do not decrease with the addition of attributes however the monotonicity property in decision theoretic rough set models does not hold this is partly due to the fact that all regions are determined according to the bayesian decision procedure consequently it is difficult to evaluate and interpret region preservation attribute reduction in decision theoretic rough set models this paper provides a new definition of attribute reduct for decision theoretic rough set models the new attribute reduction is formulated as an optimization problem the objective is to minimize the cost of decisions theoretical analysis shows the meaning of the optimization problem both the problem definition and the objective function have good interpretation a heuristic approach a genetic approach and a simulated annealing approach to the new problem are proposed experimental results on several data sets indicate the efficiency of these approaches

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

dominance based rough set approach to incomplete ordered information systems dominance based rough set approach has attracted much attention in practical applications ever since its inception this theory has greatly promoted the research of multi criteria decision making problems involving preferential information this paper mainly deals with approaches to attribute reduction in incomplete ordered information systems in which some attribute values may be lost or absent by introducing a new kind of dominance relation named the characteristic based dominance relation to incomplete ordered information systems we expand the potential applications of dominance based rough set approach to eliminate information that is not essential attribute reduction in the sense of reducing attributes is needed an approach on the basis of the discernibility matrix and the discernibility function to computing all relative reducts is investigated in incomplete ordered information systems consistent incomplete ordered decision tables to reduce the computational burden a heuristic algorithm with polynomial time complexity for finding a unique relative reduct is designed by using the inner and outer significance measures of each criterion candidate moreover some numerical experiments are employed to verify the feasibility and effectiveness of the proposed algorithms

control parameterization and robustness of feedback systems an asymptotic analysis the problem of robust design is regarded as a parameterization problem and the choice of parameters to satisfy with respect to a class of bounded perturbations is considered application areas for the general results derived include multivariate root locus robustness design based on simplified or reduced plant models and singular perturbation analysis the results are based on asymptotic analysis of feedback structures and a proof of the existence of parameter ranges for robust stability

fuzzy granular structure distance a fuzzy granular structure refers to a mathematical structure of the collection of fuzzy information granules granulated from a dataset while a fuzzy information granularity is used to measure its uncertainty however the existing forms of fuzzy information granularity have two limitations one is that when the fuzzy information granularity of one fuzzy granular structure equals that of the other one can say that these two fuzzy granular structures possess the same uncertainty but these two fuzzy granular structures may be not equivalent to each other the other limitation is that existing axiomatic approaches to fuzzy information granularity are still not complete under which when the partial order relation among fuzzy granular structures cannot be found their coarseness fineness relationships will not be revealed to address these issues a so called fuzzy granular structure distance is proposed in this study which can well discriminate the difference between any two fuzzy granular structures besides this advantage the fuzzy granular structure distance has another important benefit it can be used to establish a generalized axiomatic constraint for fuzzy information granularity by using the axiomatic constraint the coarseness fineness of any two fuzzy granular structures can be distinguished in addition through taking the fuzzy granular structure distances of a fuzzy granular structure to the finest one and the coarsest one into account we also can build a bridge between fuzzy information granularity and fuzzy information entropy the applicable analysis on 12 real world datasets shows that the fuzzy granular structure distance and the generalized fuzzy information granularity have much better performance than existing methods

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

principal gains and principal phases in the analysis of linear multivariable feedback systems the concepts of principal gain and principal phase are introduced for linear multivariable systems and their use in the analysis of feedback behavior is demonstrated a sufficient nyquist type stability criterion is presented in terms of these quantities and is used to characterize the robustness of the closed loop stability property when the system model is subjected to a linear perturbation either multiplicative or additive at any point in the feedback configuration the results presented are less conservative than those obtained via the small gain theorem

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

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

probabilistic model criteria with decision theoretic rough sets in dealing with risk in real decision problems decision theoretic rough sets with loss functions aim to obtain optimization decisions by minimizing the overall risk with bayesian decision procedures two parameters generated by loss functions divide the universe into three regions as the decision of acceptance deferment and rejection in this paper we discuss the semantics of loss functions and utilize the differences of losses replace actual losses to construct a new four level approach of probabilistic rules choosing criteria ten types of probabilistic rough set models can be generated by the four level approach and form two groups of models two way probabilistic decision models and three way probabilistic decision models a reasonable decision with these criteria is demonstrated by an illustration of oil investment

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

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

the lattice and matroid representations of definable sets in generalized rough sets based on relations abstract the definable set is a core concept in rough set theory it plays an important role in the characterizations of rough sets in this paper we study the lattice and matroid representations of definable sets in generalized rough sets based on relations first we propose the lower definable lattice consisting of all lower definable sets with set inclusion order then we give some conditions under which the lower definable lattice is distributive or geometric or boolean furthermore we discuss the relationship between distributive lattices and the lower definable lattices in generalized approximation spaces based on reflexive and transitive relations on the one hand we show that the lower definable lattice in a generalized approximation space based on reflexive and transitive relation is distributive on the other hand we obtain the result that a distributive lattice can induce a lower definable lattice in a generalized approximation space based on a reflexive and transitive relation finally we investigate the combination of generalized rough sets and matroids in terms of the lower definable lattice we show that if a lower definable lattice is a lattice of closed sets of a matroid then it must be an open closed set lattice of a matroid in addition we prove that some lower definable lattices are not the lattices of closed sets of matroids these results of this paper will benefit to our understanding of the relationship between matroids and generalized rough sets based on relations

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

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

relations of reduction between covering generalized rough sets and concept lattices the reduction theory plays an important role in data analysis this paper studies the relation between the reduction of a covering and the attribute reduction of a concept lattice the reduction of a covering from the perspective of concept lattices is investigated conversely the attribute reduction of a formal context is studied in the framework of covering generalized rough sets the results in this paper show that the reduction of a covering can be viewed as the attribute reduction of a derivative formal context moreover every reduct of a given formal context can be seen as the reduct of an induced covering as an application of the theoretical results an approach to the attribute reduction of concept lattices based on covering generalized rough sets is proposed furthermore experiments are given to show the effectiveness of the proposed method

characterization of allowable perturbations for robust stability the problem of determining stability for a closed loop linear multivariable process in the presence of ill defined process perturbations is considered it is shown that m matrices may be used to characterize a class of allowable perturbations

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

rough set models in multigranulation spaces there exist several approaches to rough set approximations in a multigranulation space namely a family of equivalence relations in this paper we propose a unified framework to classify and compare existing studies an underlying principle is to explain rough sets in a multigranulation space through rough sets derived by using individual equivalence relations two basic models are suggested one model is based on a combination of a family of equivalence relations into an equivalence relation and the construction of approximations with respect to the combined relation by combining equivalence relations through set intersection and union respectively we construct two sub models the other model is based on the construction of a family of approximations from a set of equivalence relations and a combination of the family of approximations by using set intersection and union to combine a family of approximations respectively we again build two sub models as a result we have a total of four models we examine these models and give conditions under which some of them become the same

formal reasoning in preference based multiple source rough set model we propose a generalization of pawlak s rough set model for the multi agent situation where information from an agent can be preferred over that of another agent of the system while deciding membership of objects notions of lower upper approximations are given which depend on the knowledge base of the sources as well as on the position of the sources in the hierarchy giving the preference of sources a quantified modal logic is proposed to reason about the properties of the proposed approximations a sound and complete deductive system for the logic is also presented moreover it is shown how the properties of the proposed approximations can be deduced as theorems of this deductive system

cost sensitive rough set approach we consider test cost and decision cost simultaneously cost sensitive rough set was proposed and explored attribute reductions based on three different criteria were investigated cost sensitivity is an important problem which has been addressed by many researchers around the world as far as cost sensitivity in the rough set theory is concerned two types of important costs have been seriously considered on the one hand the decision cost has been introduced into the modeling of decision theoretic rough set on the other hand the test cost has been taken into account in attribute reduction however few researchers pay attention to the construction of rough set model with test cost and decision cost simultaneously to fill such a gap a new cost sensitive rough set approach is proposed in which the information granules are sensitive to test costs and approximations are sensitive to decision costs respectively furthermore with respect to different criteria of positive region preservation decision monotonicity and cost decrease three heuristic algorithms are designed to compute reducts respectively the comparisons among these three algorithms show us 1 positive region preservation based algorithm can keep the decision rules supported by lower approximation region unchanged 2 decision monotonicity based heuristic algorithm can obtain a reduct with more positive decision rules and higher classification accuracy 3 cost minimum based algorithm can generate a reduct with minor cost this study suggests potential application areas and new research trends concerning rough set theory

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

grouping granular structures in human granulation intelligence abstract human granulation intelligence means that people can observe and analyze the same problem from various granulation points of view which generally acknowledge an essential feature of human intelligence each granulation view can generate a granular structure through dividing a cognitive target into some meaningful information granules this means that a large number of granular structures can be generated from the cognitive target however people can group these granular structures and select some representative ones for problem solving this leads to an interesting research topic how to efficiently and effectively group a family of granular structures to address this issue we first introduce a granular structure distance to measure the difference between two granular structures within a unified knowledge representation then we propose a framework for grouping granular structures called ggs algorithm which is used to efficiently partition them moreover two indices denoted as dis and apd are also designed for evaluating the performance of a grouping result of granular structures finally experiments carried out for nine data sets show that the ggs algorithm comes as a sound solution from perspectives of its convergence effectiveness and scalability in this way we have proposed and experimented with the general framework for discovering the structure inherent in granular structures which can be afterwards used to simulate intelligent behavior of human s abilities of granular structure selection

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

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 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

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

a more efficient attribute self adaptive co evolutionary reduction algorithm by combining quantum elitist frogs and cloud model operators abstract in order to further improve the adaptability of attribute reduction and enhance its application performance in large scale attribute reduction a more efficient attribute self adaptive co evolutionary reduction algorithm by combining quantum elitist frogs and cloud model operators qecmascr is proposed in this paper firstly quantum chromosome is used to encode the evolutionary population and a multilevel elitist pool of quantum frogs is constructed in which quantum elitist frogs can fast guide the evolutionary population into the optimal area secondly a reversible cloud mode based on attribute entropy weight is designed to adjust the quantum cloud revolving angle so that the scope of search space can be adaptively controlled under the guidance of qualitative knowledge in addition both the quantum cloud mutation operator and quantum cloud entanglement operator are used to make quantum frogs be adaptive to get the optimal set of attribute reduction fast thirdly an improved decomposition framework of attribute self adaptive co evolution is adopted to capture interdependencies of decision variables it can decompose the large scale attribute set into reasonable scale subsets according to two kinds of the best performance fitness and assignment credit thus some optimal elitists in different memeplexes of multilevel elitist pool are selected out to evolve their representing attribute subsets which can increase the cooperation and efficiency of attribute reduction so the global minimum attribute reduction can be achieved steadily and efficiently experimental results indicate the proposed qecmascr algorithm achieves the better superior performance than existing representative algorithms moreover it is applied into mri segmentation and the effective and robust segmentation results further demonstrate its stronger applicability

estimation of transfer functions asymptotic theory and a bound of model uncertainty abstract the problem of transfer function estimation is considered we will discuss the problem in two related aspects firstly how to obtain a good black box transfer function model secondly how to derive a suitable description of the model uncertainty modelling errors we will emphasize the second point we shall discuss the asymptotic theory on the properties of transfer function estimates developed recently by ljung and yuan this new theory shows that the transfer function estimate is consistent the error of the estimate is asymptotically normal with a very simple expression for its variance their results will be extended here to cases where spectral analysis is used based on this theory we will propose the method for obtaining a low order nominal model of linear processes and define and estimate an upper bound of the model uncertainty an upper bound of the model uncertainty is a suitable description of the modelling errors for the robust controller 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

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

information theoretic measures associated with rough set approximations in rough set theory some information theoretic measures of uncertainty and granularity have been proposed a common feature of these measures is that they are only dependent on the partitions and the cardinality of a universe which means that they are independent of the lower and upper approximations of rough sets this seems somewhat unreasonable since the basic idea of rough set theory is to describe incomplete or inexact concepts by the lower and upper approximations in light of this we develop a new pair of information theoretic entropy and co entropy functions associated to partitions and approximations in this paper such functions are used to measure the uncertainty and granularity of an approximation space after introducing the novel notions of entropy and co entropy we then examine their properties in particular we disclose the relationship of co entropies between different universes the theoretical development is accompanied by illustrative numerical examples

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

fourier analysis on starlike lipschitz surfaces a theory of a class of singular integrals on starlike lipschitz surfaces in rn is established the class of singular integrals forms an operator algebra identical to the class of bounded holomorphic fourier multipliers as well as to the cauchy dunford bounded holomorphic functional calculus of the spherical dirac operator the study proposes a new method inducing clifford holomorphic functions from holomorphic functions of one complex variable by means of which problems on the sphere are reduced to those on the unit circle

α dominance relation and rough sets in interval valued information systems though rough set has been widely used to study systems characterized by insufficient and incomplete information its performance in dealing with initial interval valued data needs to be seriously considered for improving the suitability and scalability the aim of this paper is to present a parameterized dominance based rough set approach to interval valued information systems first by considering the degree that an interval valued data is dominating another one we propose the concept of dominance relation second we present the dominance based rough set model in interval valued decision systems finally we introduce lower and upper approximate reducts into dominance based rough set for simplifying decision rules we also present the judgement theorems and discernibility functions which describe how lower and upper approximate reducts can be calculated this study suggests potential application areas and new research trends concerning rough set approach to interval valued information systems

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

entropy measures and granularity measures for set valued information systems set valued information systems are generalized models of single valued information systems in this paper we propose two new relations for set valued information systems based on these two relations the concepts of knowledge information entropy knowledge rough entropy knowledge granulation and knowledge granularity measure are defined in set valued information systems and some properties are investigated moreover relationship between knowledge information entropy and knowledge granulation knowledge rough entropy and knowledge granularity measure are studied it is also shown that knowledge information entropy and knowledge granularity measure can be used to evaluate the certainty degree of knowledge in set valued information systems and knowledge rough entropy and knowledge granulation can be used to evaluate the uncertainty degree of knowledge in set valued information systems these results may supply a further understanding the essence of uncertainty and granularity in set valued information systems

constructive granular systems with universal approximation and fast knowledge discovery conventional gradient descent learning algorithms for soft computing systems have the learning speed bottleneck problem and the local minima problem to effectively solve the two problems the n variable constructive granular system with high speed granular constructive learning is proposed based on granular computing and soft computing and proved to be a universal approximator the fast granular constructive learning algorithm can highly speed up granular knowledge discovery by directly calculating all parameters of the n variable constructive granular system using training data and then construct the n variable constructive granular system with any required accuracy using a small number of granular rules predictive granular knowledge discovery simulation results indicate that the direct calculation based granular constructive algorithm is better than the conventional gradient descent learning algorithm in terms of learning speed learning error and prediction error

feature selection for monotonic classification monotonic classification is a kind of special task in machine learning and pattern recognition monotonicity constraints between features and decision should be taken into account in these tasks however most existing techniques are not able to discover and represent the ordinal structures in monotonic datasets thus they are inapplicable to monotonic classification feature selection has been proven effective in improving classification performance and avoiding overfitting to the best of our knowledge no technique has been specially designed to select features in monotonic classification until now in this paper we introduce a function which is called rank mutual information to evaluate monotonic consistency between features and decision in monotonic tasks this function combines the advantages of dominance rough sets in reflecting ordinal structures and mutual information in terms of robustness then rank mutual information is integrated with the search strategy of min redundancy and max relevance to compute optimal subsets of features a collection of numerical experiments are given to show the effectiveness of the proposed technique

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

three way decisions with probabilistic rough sets the rough set theory approximates a concept by three regions namely the positive boundary and negative regions rules constructed from the three regions are associated with different actions and decisions which immediately leads to the notion of three way decision rules a positive rule makes a decision of acceptance a negative rule makes a decision of rejection and a boundary rule makes a decision of abstaining this paper provides an analysis of three way decision rules in the classical rough set model and the decision theoretic rough set model the results enrich the rough set theory by ideas from bayesian decision theory and hypothesis testing in statistics the connections established between the levels of tolerance for errors and costs of incorrect decisions make the rough set theory practical in applications

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

an m matrix and majorant approach to robust stability and performance analysis for systems with structured uncertainty uncertain multiinput multioutput systems described in the frequency domain are discussed the theory of nonnegative matrices and m matrices is used along with majorant bounding techniques to develop robust stability and performance results for two types of uncertainty the first type is uncertainty with norm bounded subblocks the second type is uncertainty that has subblocks with known patterns but unknown gains this latter uncertainty representation allows the elements of a given uncertainty block to be correlated for uncertainties of this type a recursive analysis methodology is developed which yields increasingly nonconservative results throughout this paper the performance analysis determines norm bounds on the deviations of the outputs from their nominal values the results are illustrated by a numerical example

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

approximations and uncertainty measures in incomplete information systems there are mainly two methodologies dealing with uncertainty measurement issue in rough set theory pure rough set approach and information theory approach pure rough set approach is based on the concepts of accuracy roughness and approximation accuracy proposed by pawlak information theory approach is based on shannon s entropy or its variants several authors have extended the information theory approach into incomplete information systems however there are few studies on extending the pure rough set approach to incomplete information systems this paper focuses on constructing uncertainty measures in incomplete information systems by pure rough set approach three types of definitions of lower and upper approximations and corresponding uncertainty measurement concepts including accuracy roughness and approximation accuracy are investigated theoretical analysis indicates that two of the three types can be used to evaluate the uncertainty in incomplete information systems experiments on incomplete real life data sets have been conducted to test the two selected types the first type and the third type of uncertainty measures results show that the two types of uncertainty measures are effective

granular computing and dual galois connection a covering model for granular computing in a set theoretic setting is studied in this paper under this model a zooming in operator is redefined combinations of the zooming in and zooming out operators form two pairs of approximation operators of the original and the granulated universe of discourse their properties are examined in detail for the two pairs of lower and upper approximation operators it is proved that the duality is always true for a generalized approximation space the approximation representations are just the combination operators formed on the basis of the zooming out and zooming in operators relationships between these combination operators and the dual galois connection are also analyzed

semi supervised concept factorization for document clustering nonnegative matrix factorization nmf and concept factorization cf are two popular methods for finding the low rank approximation of nonnegative matrix different from nmf cf can be applied not only to the matrix containing negative values but also to the kernel space based on nmf and cf many methods such as graph regularized nonnegative matrix factorization gnmf and locally consistent clustering factorization lccf can significantly improve the performance of clustering unfortunately these are unsupervised learning methods in order to enhance the clustering performance with the supervisory information a semi supervised concept factorization sscf is proposed in this paper by incorporating the pairwise constraints into cf as the reward and penalty terms which can guarantee that the data points belonging to a cluster in the original space are still in the same cluster in the transformed space by comparing with the state of the arts algorithms km nmf cf lccf gnmf pccf experimental results on document clustering show that the proposed algorithm has better performance in terms of accuracy and mutual information

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

multigranulation rough set model based on granulation of attributes and granulation of attribute values abstract existing multigranulation rough set models often suppose the family of equivalence relations is existed and do not give the methods to construct the multigranulation space that is the multigranulation space is given so in multigranulation rough set models how to derive the family of equivalence relations and construct the multigranulation space is an important problem to be resolved in this paper considering there are different granulation of attributes and attribute values i propose multigranulation rough set model based on granulation of attributes and granulation of attribute values mraav which provides a method to construct the multigranulation space in multigranulation rough set models based on mraav multigranulation problem solving space is constructed the search strategy for problem solving in multigranulation problem solving space is presented which make the information processing process of rough set is structured and hierarchical finally the rule extraction algorithm based on mraav is given which can not only extract rules from multiple abstract concept levels but also with low costs the example and experimental results demonstrate the effectiveness of the proposed model and algorithms

the inverse fueter mapping theorem for axially monogenic functions of degree k abstract in this paper we first prove an important formula for the fractional laplacian and then we use it to invert the fueter mapping theorem for axially monogenic functions of degree k in fact we prove that for every axially monogenic function of degree k f x a x 0 x x x b x 0 x p k x x r n 1 there exists a holomorphic intrinsic function f k in c such that f x k f k x k n 1 2 f k x p k x where n can be any positive integer k can be any non negative integer f k is the slice monogenic function induced by f k and p k x is an inner spherical monogenic polynomial of degree k using the maps k k 0 1 2 we obtain a decomposition of a monogenic function for any value of the dimension n

an information fusion approach by combining multigranulation rough sets and evidence theory multigranulation rough set mgrs theory provides two kinds of qualitative combination rules that are generated by optimistic and pessimistic multigranulation fusion functions they are used to aggregate multiple granular structures from a set theoretic standpoint however the two combination rules seem to lack robustness because one is too relaxed and the other too restrictive to solve some practical problems dempster s combination rule in the evidence theory has been employed to aggregate information coming from multiple sources however it fails to deal with conflict evidence to overcome these limitations we focus on the combination of granular structures with both reliability and conflict from multiple sources which has been a challenging task in the field of granular computing we first address the connection between multigranulation rough set theory and the evidence theory then a two grade fusion approach involved in the evidence theory and multigranulation rough set theory is proposed which is based on a well defined distance function among granulation structures finally an illustrative example is given to show the effectiveness of the proposed fusion method the results of this study will be useful for pooling the uncertain data from different sources and significant for establishing a new direction of granular computing