fuzzy reasoning model under quotient space structure in this paper we present a theoretical framework of fuzzy reasoning model under quotient space structure it consists of 1 introducing quotient space structure into fuzzy sets i e constructing fuzzy set representations of different grain size spaces and their relationships 2 introducing the concept of fuzzy sets into quotient space theory i e introducing fuzzy equivalence relation and discussing its corresponding reasoning in different grain size spaces and 3 discussing the relationship and transformation among different granular computing methodologies the framework proposed is aimed to combine two powerful abilities in order to enhance the efficiency of fuzzy reasoning one is the ability of computing with words based on fuzzy set methodology the other is the ability of hierarchical problem solving based on quotient space approach

matrices related to the bell polynomials abstract in this paper we study the matrices related to the partial exponential bell polynomials and those related to the bell polynomials with respect to as a result the factorizations of these matrices are obtained which give unified approaches to the factorizations of many lower triangular matrices moreover some combinatorial identities are also derived from the corresponding matrix representations

on several q special matrices including the q bernoulli and q euler matrices abstract in the spirit of our earlier articles 12 14 11 and our work 13 we define two dual q bernoulli polynomials with corresponding vector and matrix forms following aceto trigiante 1 the q l matrix the indefinite q integral of the q pascal matrix is the link between the q cauchy and the q bernoulli matrix the q analogue of the bernoulli complementary argument theorem can be expressed in matrix form through the diagonal a n matrix for the q euler polynomials corresponding results are obtained the umbral calculus for generating functions of q appell polynomials is shown to be equivalent to a transform method which maps polynomials to matrices a true q analogue of arponen 6 this is manifested by the vein 21 matrix which occurs as the transform of the q difference operator the aceto trigiante shifted q bernoulli matrix has a simple connection to the q bernoulli arponen matrix through the q pascal matrix we reintroduce certain q stirling numbers z q from 12 which will be needed for the polynomial matrix definitions

a condition for a subspace of b h to be an ideal abstract let h be a real or complex hilbert space with dim h 1 the subspace a b h is an ideal if and only if ta at a for every t b h a a every element of b h is the finite sum of ta at type operators

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

notes on particular symmetric polynomials with applications this paper presents some applications using several properties of three important symmetric polynomials elementary symmetric polynomials complete symmetric polynomials and the power sum symmetric polynomials the applications includes a simple proof of el mikkawy conjecture in m e a el mikkawy appl math comput 146 2003 759 769 and a very easy proof of the newton girard formula in addition a generalization of stirling numbers is obtained

measurement of general granules abstract the measurement of granules is often considered the most foundational problem in granular computing and it has been studied in many fields but in relative isolation this paper attempts to explore the measurement for general granules whether they are granules based on equivalence or general relations whether they are macrocosmic or microcosmic granules and whether they are granules on a finite or an infinite set for a finite set we discuss the measurement of granules corresponding to equivalence relations and introduce definitions of granularity fineness and granular entropy in order to distinguish those granules with the same granularity fineness and granular entropy we present conditional granularity conditional fineness and conditional granular entropy and obtain the necessary and sufficient condition which can judge one granule as coarser or finer than another while the conditional granularity is a subsethood measure which satisfies monotonic properties what s more we introduce relation matrix to measure the granules and it not only can be easily generalized to measure the granules on general relations but also connects granular computing to graph theory for an infinite set we can define different granularities in different questions but we suggest the basic idea of granularity that is the granularity of a granule is the sum of the relative information contents of its atomic granules

stirling matrix via pascal matrix abstract the pascal type matrices obtained from the stirling numbers of the first kind s n k and of the second kind s n k are studied respectively it is shown that these matrices can be factorized by the pascal matrices also the ldu factorization of a vandermonde matrix of the form v n x x 1 x n 1 for any real number x is obtained furthermore some well known combinatorial identities are obtained from the matrix representation of the stirling numbers and these matrices are generalized in one or two variables

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

on rough set and fuzzy sublattice let l be a lattice with the least element 0 and the greatest element 1 and let be a full congruence relation on l in this paper the notion of upper and lower approximations of a fuzzy subset of l is introduced and some important properties will be studied

generalized leibniz functional matrices and divided difference form of the lagrange burmann formula abstract in this paper we derive two versions of divided difference form of the lagrange burmann expansion formula the first version expresses all coefficients by a form of a determinant by means of the factorizations of the generalized leibniz functional matrices with respect to a sequence we obtain the second version with coefficients represented by a finite sum some new matrix inversions are also investigated

knowledge distance measure in multigranulation spaces of fuzzy equivalence relations abstract hierarchical quotient space structure hqss which is a typical representation of multigranulation spaces serves as the essential description tool of fuzzy equivalence relations however current studies on hqss have three main limitations i the inability to reflect the relationship between any two quotient spaces in an hqss ii classification isomorphism cannot characterize the degree of subdivision that exists in an hqss with changing granularities and iii the difficulty in characterizing the difference between two hqsss that are classification isomorphic in this paper we address these issues from the perspective of knowledge distance first we propose a partition based knowledge distance based on the earth mover s distance emd and we prove that our knowledge distance is equivalent to the binary granule based knowledge distance bgkd but is more intuitive than the bgkd then by studying the hierarchy of hqss we conclude that the granularity difference between any two quotient spaces in an hqss is equal to the knowledge distance between them in the view of knowledge distance the concepts of classification isomorphism and subdivision isomorphism are defined and discussed finally we define the concept of the knowledge difference sequence pair which not only discriminates whether two fuzzy equivalence relations are isomorphic but also characterizes the difference between any two hqsss with respect to them theoretical analysis shows that our work is valuable for advancing the application of the hierarchy granular computing theory

initialization of k modes clustering using outlier detection techniques we considered the initialization of k modes clustering from the view of outlier detection we proposed an initialization algorithm for k modes clustering via the distance based outlier detection technique we presented a partition entropy based outlier detection technique and designed an initialization algorithm via it we proposed a new distance metric weighted matching distance metric the effectiveness of our initialization algorithms was shown on several uci data sets the k modes clustering has received much attention since it works well for categorical data sets however the performance of k modes clustering is especially sensitive to the selection of initial cluster centers therefore choosing the proper initial cluster centers is a key step for k modes clustering in this paper we consider the initialization of k modes clustering from the view of outlier detection we present two different initialization algorithms for k modes clustering where the first is based on the traditional distance based outlier detection technique and the second is based on the partition entropy based outlier detection technique by using the above two outlier detection techniques to calculate the degree of outlierness of each object our algorithms can guarantee that the chosen initial cluster centers are not outliers moreover during the process of initialization we adopt a new distance metric weighted matching distance metric to calculate the distance between two objects described by categorical attributes experimental results on several uci data sets demonstrate the effectiveness of our initialization algorithms for k modes clustering

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

the structure of nonlinear lie derivation on von neumann algebras national natural science foundation of china 11071201 youth national natural science foundation of china 11001230 youth talented natural science foundation of fujian 2008f3103 fundamental research funds for the central universities 2010121001

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

information theoretic measures of uncertainty for rough sets and rough relational databases rough set theory has become well established as a mechanism for uncertainty management in a wide variety of applications including databases this paper addresses the measurement of uncertainty in rough sets and rough relational databases by introducing a measurement based on information theory this rough entropy is discussed as it applies to rough sets in general and in particular to aspects of the rough relational database model

quantitative qualitative region change uncertainty certainty in attribute reduction knowledge coarsening is investigated to describe attribute deletion granule merging and its region distribution are used to gain region change functions region change with certainty monotonicity is analyzed in qualitative pawlak model region change with uncertainty drifting is analyzed in quantitative dtrs model attribute reduction is a fundamental research theme in rough sets and granular computing grc its scientific construction originally depends on the region change law at present only region change non monotonicity monotonicity is mined in the quantitative qualitative model the in depth region change truth and its grc mechanism have significance especially for follow up attribute reduction this paper commences probing region change essence mainly from a novel uncertainty certainty viewpoint concretely we make comparative region change analyses based on grc by resorting to the qualitative pawlak model and quantitative dtrs model the decision theoretic rough set model 1 knowledge coarsening is investigated to describe attribute deletion 2 granule merging and its region distribution are studied to probe region change functions 3 region change is analyzed in pawlak model to mine qualitative region change certainty and its relevant properties 4 region change is analyzed in dtrs model to mine quantitative region change uncertainty and its relevant properties 5 comparative region change analyses are summarized and further experiment verification is provided knowledge coarsening and granule merging establish grc mechanisms for extensive region change analyses quantitative qualitative region change uncertainty certainty and relevant principles are discovered via dtrs model pawlak model by virtue of the grc technology and comparative strategy this study reveals region change uncertainty certainty to deepen region change non monotonicity monotonicity furthermore it underlies attribute reduction especially with regard to quantitative models

three layer granular structures and three way informational measures of a decision table abstract attribute reduction in rough set theory serves as a fundamental topic for information processing and its basis is usually a decision table d table d table attribute reduction concerns three hierarchical types and only classification based reduction is related to information theoretic representation aiming at inducing comprehensive d table attribute reduction with hierarchies and information this paper concretely constructs a d table s three layer granular structures and three way informational measures via granular computing and bayes theorem with regard to the d table the micro bottom meso middle and macro top are hierarchically organized according to the formal structure and systematic granularity then different layers produce different three way informational measures by developing bayes theorem thus three way weighted entropies originate from three way probabilities at the micro bottom and further evolve from the meso middle to the macro top and their granulation monotonicity and evolution systematicness are acquired furthermore three way informational measures are analyzed by three layer granular structures to achieve their hierarchical evolution superiority and algorithms finally structural and informational results are effectively illustrated by a d table example this study establishes d table s hierarchical structures to reveal constructional mechanisms and systematic relationships of informational measures the obtained results underlie the d table s hierarchical systematic and informational attribute reduction and they also enrich the three way decisions theory

extended bernoulli and stirling matrices and related combinatorial identities abstract in this paper we establish plenty of number theoretic and combinatoric identities involving generalized bernoulli polynomials and stirling numbers of both kinds which generalize various known identities these formulas are deduced from pascal type matrix representations of bernoulli and stirling numbers for this we define and factorize a modified pascal matrix corresponding to bernoulli and stirling cases

a novel attribute weighting algorithm for clustering high dimensional categorical data due to data sparseness and attribute redundancy in high dimensional data clusters of objects often exist in subspaces rather than in the entire space to effectively address this issue this paper presents a new optimization algorithm for clustering high dimensional categorical data which is an extension of the k modes clustering algorithm in the proposed algorithm a novel weighting technique for categorical data is developed to calculate two weights for each attribute or dimension in each cluster and use the weight values to identify the subsets of important attributes that categorize different clusters the convergence of the algorithm under an optimization framework is proved the performance and scalability of the algorithm is evaluated experimentally on both synthetic and real data sets the experimental studies show that the proposed algorithm is effective in clustering categorical data sets and also scalable to large data sets owning to its linear time complexity with respect to the number of data objects attributes or clusters

an entropy based uncertainty measurement approach in neighborhood systems abstract uncertainty measures can provide us with principled methodologies to analyze uncertain data and unveil the substantive characteristics of the data sets accuracy and roughness proposed by pawlak are two main tools to deal with uncertainty measurement issue in rough set theory many uncertainty measure methodologies for discrete valued information systems or discrete valued decision systems have been developed however there are only limited on the uncertainty measurement for neighborhood systems in this paper we address the issues of uncertainty of a neighborhood system and extend the traditional accuracy and roughness measures to deal with neighborhood systems in particular a concept called neighborhood entropy is first introduced to evaluate the uncertainty of a neighborhood information system consequently the entropy based roughness and approximation roughness measures of neighborhood system are presented theoretical analysis indicates that entropy based measures can be used to evaluate the uncertainty in neighborhood systems experiments are conducted on artificial data sets and standard uci data sets to test our proposed methodologies results demonstrate that the entropy based measures are effective and valid for evaluating the uncertainty of neighborhood 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

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

can fuzzy entropies be effective measures for evaluating the roughness of a rough set the roughness of a rough set arises from the existence of its boundary region in such a boundary region each object has a non zero rough membership degree when an object s rough membership degree is regarded as its fuzzy membership degree a rough set can induce a fuzzy set this relationship motivates us to assert that there may exist some inherent relations between the roughness of a rough set and the fuzziness of the fuzzy set induced from the rough set this assertion leads us to the question can the existing fuzzy entropies be used to evaluate the roughness of a rough set to answer this question we first analyze how the boundary region varies when the partition of the universe becomes coarser and then exploit this analysis in the introduction of a more appropriate definition on the roughness of a rough set to determine whether a fuzzy entropy can be used to evaluate the roughness of a rough set or not we develop three methods for estimating the ability of a fuzzy entropy to measure the roughness the experiments show that these methods are very effective and can be applied to select a fuzzy entropy as a measure of the roughness of a rough set

factorial stirling matrix and related combinatorial sequences abstract in this paper some relationships between the stirling matrix the vandermonde matrix the benoulli numbers and the eulerian numbers are studied from a matrix representation of k s n k which will be called the factorial stirling matrix where s n k are the stirling numbers of the second kind as a consequence a number of interesting and useful identities are obtained

ring derivations on standard operator algebras abstract let a be a standard operator algebra on an infinite dimensional banach space x we prove that every ring derivation d a b x is of the form d a ta at for some t b x

a comparative study of rough sets for hybrid data to discover knowledge from hybrid data using rough sets researchers have developed several fuzzy rough set models and a neighborhood rough set model these models have been applied to many hybrid data processing applications for a particular purpose thus neglecting the issue of selecting an appropriate model to address this issue this paper mainly concerns the relationships among these rough set models investigating fuzzy and neighborhood hybrid granules reveals an important relationship between these two granules analyzing the relationships among rough approximations of these models shows that hu s fuzzy rough approximations are special cases of neighborhood and wang s fuzzy rough approximations respectively furthermore one to one correspondence relationships exist between wang s fuzzy and neighborhood rough approximations this study also finds that wang s fuzzy and neighborhood rough approximations are cut sets of dubois fuzzy rough approximations and radzikowska and kerre s fuzzy rough approximations respectively

a new measure of uncertainty based on knowledge granulation for rough sets in rough set theory accuracy and roughness are used to characterize uncertainty of a set and approximation accuracy is employed to depict accuracy of a rough classification although these measures are effective they have some limitations when the lower upper approximation of a set under one knowledge is equal to that under another knowledge to overcome these limitations we address in this paper the issues of uncertainty of a set in an information system and approximation accuracy of a rough classification in a decision table an axiomatic definition of knowledge granulation for an information system is given under which these three measures are modified theoretical studies and experimental results show that the modified measures are effective and suitable for evaluating the roughness and accuracy of a set in an information system and the approximation accuracy of a rough classification in a decision table respectively and have a much simpler and more comprehensive form than the existing ones