axiomatics for fuzzy rough sets abstract a fuzzy t rough set consists of a set x and a t similarity relation r on x where t is a lower semi continuous triangular norm we generalize the farinas prade definition for the upper approximation operator a i x i x of a fuzzy t rough set x r given originally for the special case t min to the case of arbitrary t we propose a new definition for the lower approximation operator a i x i x of x r our definition satisfies the two important identities a a a and a a a as well as a number of other interesting properties we provide axiomatics to fully characterize those upper and lower approximations

fuzzy rough set based active learning determining the informativeness of unlabeled samples is a key issue in active learning one solution to this is using the sample s in consistency between conditional features and decision labels in this paper a fuzzy rough set based active learning model is proposed to tackle this problem first the consistence degree of a labeled sample is computed by the lower approximations in fuzzy rough set which reflects its minimum membership in the decision class then the concept of sample covering is proposed to measure the relationship between labeled samples and un labeled samples afterward the memberships of an unlabeled sample be longing to different decision classes are computed based on the covering degrees of labeled samples on it finally these memberships are used to form a sample selection criterion to measure the sample s inconsistency by applying gaussian kernel based similarity relation to the aforemen tioned processes a support vector machine svm based active learning scheme is developed experimental results demonstrate the effectiveness of the proposed model

alternative c means clustering algorithms abstract in this paper we propose a new metric to replace the euclidean norm in c means clustering procedures on the basis of the robust statistic and the influence function we claim that the proposed new metric is more robust than the euclidean norm we then create two new clustering methods called the alternative hard c means ahcm and alternative fuzzy c means afcm clustering algorithms these alternative types of c means clustering have more robustness than c means clustering numerical results show that ahcm has better performance than hcm and afcm is better than fcm we recommend afcm for use in cluster analysis recently this afcm algorithm has successfully been used in segmenting the magnetic resonance image of ophthalmology to differentiate the abnormal tissues from the normal tissues

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

subnormality of unbounded composition operators over one circuit directed graphs exotic examples abstract a recent example of a non hyponormal injective composition operator in an l 2 space generating stieltjes moment sequences invented by three of the present authors was built over a non locally finite directed tree the main goal of this paper is to solve the problem of whether there exists such an operator over a locally finite directed graph and in the affirmative case to find the simplest possible graph with these properties simplicity refers to local valency the problem is solved affirmatively for the locally finite directed graph g 2 0 which consists of two branches and one loop the only simpler directed graph for which the problem remains unsolved consists of one branch and one loop the consistency condition the only efficient tool for verifying subnormality of unbounded composition operators is intensively studied in the context of g 2 0 which leads to a constructive method of solving the problem the method itself is partly based on transforming the krein and the friedrichs measures coming either from shifted al salam carlitz q polynomials or from a quartic birth and death process

the q stirling numbers continued fractions and the q charlier and q laguerre polynomials we give a simple proof of the continued fraction expansions of the ordinary generating functions of the q stirling numbers of both kinds by generalizing the method of touchard 1956 and milne 1978 we obtain the explicit formulas and measure of one set of the polynomials whose moments are the q stirling numbers

evidence theory based numerical characterization of multigranulation rough sets in incomplete information systems multigranulation rough sets are desirable features in the field of rough set where this concept is approximated by multiple granular structures in this study we employ belief and plausibility functions from evidence theory to characterize the set approximations and attribute reductions in multigranulation rough set theory first we show that in an incomplete information system the pessimistic multigranulation approximations can be measured by belief and plausibility functions whereas the optimistic multigranulation approximations do not possess this characteristic in general we also give a sufficient and necessary condition for the numerical measurement of optimistic multigranulation approximations by belief and plausibility functions second in an incomplete decision system the pessimistic multigranulation approximations are also measured by belief and plausibility functions in the end an attribute reduction algorithm for multigranulation rough sets is proposed based on evidence theory and its efficiency is examined by an example thus belief and plausibility functions can be employed to numerically characterize the attribute reductions and to construct an attribute reduction algorithm for multigranulation rough sets

rough fuzzy approximations on two universes of discourse in rough set theory the lower and upper approximation operators can be constructed via a variety of approaches various fuzzy generalizations of rough approximation operators have been made over the years this paper presents a framework for the study of rough fuzzy sets on two universes of discourse by means of a binary relation between two universes of discourse a covering and three relations are induced to a single universe of discourse based on the induced notions four pairs of rough fuzzy approximation operators are proposed these models guarantee that the approximating sets and the approximated sets are on the same universes of discourse furthermore the relationship between the new approximation operators and the existing rough fuzzy approximation operators on two universes of discourse are scrutinized and some interesting properties are investigated finally the connections of these approximation operators are made and conditions under which some of these approximation operators are equivalent are obtained

on representing and generating kernels by fuzzy equivalence relations kernels are two placed functions that can be interpreted as inner products in some hilbert space it is this property which makes kernels predestinated to carry linear models of learning optimization or classification strategies over to non linear variants following this idea various kernel based methods like support vector machines or kernel principal component analysis have been conceived which prove to be successful for machine learning data mining and computer vision applications when applying a kernel based method a central question is the choice and the design of the kernel function this paper provides a novel view on kernels based on fuzzy logical concepts which allows to incorporate prior knowledge in the design process it is demonstrated that kernels mapping to the unit interval with constant one in its diagonal can be represented by a commonly used fuzzy logical formula for representing fuzzy rule bases this means that a great class of kernels can be represented by fuzzy logical concepts apart from this result which only guarantees the existence of such a representation constructive examples are presented and the relation to unlabeled learning is pointed out

geometrical interpretation and applications of membership functions with fuzzy rough sets fuzzy rough sets are a generalization of crisp rough sets for measuring inconsistency between conditional attributes and decision attributes for many decision systems in many classification problems a membership function for the training sample belonging to a certain class can be computed by methods in fuzzy rough sets in this paper we present a geometrical interpretation and its applications of this kind of membership functions first we prove that every fuzzy similarity relation in fuzzy rough sets is a reproducing kernel which is related to a krein space thus fuzzy similarity relations can be geometrically explained in a krein space second we will present the interpretation of several types of membership functions geometrically by using the lower approximations in fuzzy rough sets in terms of square distances in krein spaces as practical applications of these membership functions we develop a new algorithm to find reducts and reformulate soft margin support vector machines by taking the membership degree for every training sample into considerations experimental results also demonstrate the effectiveness of the work proposed in this paper

fuzzy rough set model for set valued data abstract in many practical situations some of the attribute values for an object may be set valued the existing crisp rough set model for set valued information systems is based on a tolerance relation that examines whether two set values have a non empty intersection some information in the data will be lost according to the tolerance relation here we define a fuzzy relation and construct a fuzzy rough set model for set valued information systems the concepts of reduct core discernibility matrix and discernibility function for set valued information systems and relative reduct core discernibility matrix and discernibility function for set valued decision systems are defined and examined attribute reduction in set valued information systems based on discernibility matrices and functions are investigated

the design methodology of radial basis function neural networks based on fuzzy k nearest neighbors approach various approaches to partitioning of high dimensional input space have been studied with the intent of developing homogeneous clusters formed over input and output spaces of variables encountered in system modeling in this study we propose a new design methodology of a fuzzy model where the input space is partitioned by making use of some classification algorithm especially fuzzy k nearest neighbors k nn classifier being guided by some auxiliary information granules formed in the output space this classifier being regarded in the context of this design as a supervision mechanism is used to capture the distribution of data over the output space this type of supervision is beneficial when developing the structure in the input space it is demonstrated that data involved in a partition constructed by the fuzzy k nn method realized in the input space show a high level of homogeneity with regard to the data present in the output space this enhances the performance of the fuzzy rule based model whose premises in the rules involve partitions formed by the fuzzy k nn the design is illustrated with the aid of numeric examples that also provide a detailed insight into the performance of the fuzzy models and quantify several crucial design issues

constructive and axiomatic approaches of fuzzy approximation operators this paper presents a general framework for the study of rough set approximation operators in fuzzy environment in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized fuzzy rough and rough fuzzy respectively approximation operators is first defined the representations of both fuzzy rough approximation operators and rough fuzzy approximation operators are then presented the connections between fuzzy and crisp respectively relations and fuzzy rough and rough fuzzy respectively approximation operators are further established in axiomatic approach various classes of fuzzy approximation operators are characterized by different sets of axioms the minimal axiom sets of fuzzy approximation operators guarantee the existence of certain types of fuzzy or crisp relations producing the same operators

mercer kernel based clustering in feature space the article presents a method for both the unsupervised partitioning of a sample of data and the estimation of the possible number of inherent clusters which generate the data this work exploits the notion that performing a nonlinear data transformation into some high dimensional feature space increases the probability of the linear separability of the patterns within the transformed space and therefore simplifies the associated data structure it is shown that the eigenvectors of a kernel matrix which defines the implicit mapping provides a means to estimate the number of clusters inherent within the data and a computationally simple iterative procedure is presented for the subsequent feature space partitioning of the data

parameterized attribute reduction with gaussian kernel based fuzzy rough sets fuzzy rough sets are considered as an effective tool to deal with uncertainty in data analysis and fuzzy similarity relations are used in fuzzy rough sets to calculate similarity between objects on the other hand in kernel tricks a kernel maps data into a higher dimensional feature space where the resulting structure of the learning task is linearly separable while the kernel is the inner product of this feature space and can also be viewed as a similarity function it has been reported there is an overlap between family of kernels and collection of fuzzy similarity relations this fact motivates the idea in this paper to use some kernels as fuzzy similarity relations and develop kernel based fuzzy rough sets first we consider gaussian kernel and propose gaussian kernel based fuzzy rough sets second we introduce parameterized attribute reduction with the derived model of fuzzy rough sets structures of attribute reduction are investigated and an algorithm with discernibility matrix to find all reducts is developed finally a heuristic algorithm is designed to compute reducts with gaussian kernel fuzzy rough sets several experiments are provided to demonstrate the effectiveness of the idea

a queueing model and a set of orthogonal polynomials a single serving queueing model is studied where potential customers are discouraged at the rate n qn 0 q 1 n is the queue length the serving rate is n 1 qn n 0 1 the spectral function is computed and the corresponding set of orthogonal polynomials is studied in detail the slightly more general model with n qn 1 bqn n 1 qn 1 bqn and the analogous orthogonal polynomials are also investigated in both cases a method developed by pollaczek is used which has been used very successfully to study new sets of orthogonal polynomials by askey and ismail

generalized fuzzy rough sets this paper presents a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized approximation operators is defined the connections between fuzzy relations and fuzzy rough approximation operators are examined in axiomatic approach various classes of fuzzy rough approximation operators are characterized by different sets of axioms axioms of fuzzy approximation operators guarantee the existence of certain types of fuzzy relations producing the same operators

meta clustering of possibilistically segmented retail datasets this paper proposes a possibilistic meta clustering algorithm the possibility theory is used for possibilistic segmentation of the input data as well as for determining the possibilistic membership of objects to multiple clusters the meta clustering uses connections between information granules to send clustering knowledge from one granule to another the approach is demonstrated with the help of the k modes clustering algorithm for a real world retail store where a customer is connected to the products bought and products are connected to customers who buy them the meta clustering approach uses the results from clustering of customers as an input to cluster the products and recursively uses clustering of products as input to the clustering customers the customer granule is represented using static information from the database and dynamic information from the clustering of the products bought similarly a product granule is represented by static information from the database and dynamic part from the clustering of the customers who buy the product the static information from the database is represented using possibilistic segments

contiguous relations basic hypergeometric functions and orthogonal polynomials i abstract the second order q difference equation of a 2 1 and the contiguous relations of 2 1 s are used to derive continued fraction expansions for quotients of 2 1 s we identify the cases when these continued fractions are positive definite j fractions i e the cases when the denominators of the approximants of the j fractions are orthogonal polynomials the measures with respect to which the denominator polynomials are orthogonal are found and the orthogonal polynomials are studied in some detail

on the generalization of fuzzy rough sets rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets the existing generalizations of fuzzy rough sets are all based on special fuzzy relations fuzzy similarity relations t similarity relations it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches in this paper from the viewpoint of constructive approach we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined in axiomatic approach we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms the lattice and topological structures of fuzzy rough sets are also proposed in order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets a special lower approximation operator is applied to a fuzzy reasoning system which coincides with the mamdani algorithm

new approaches to fuzzy rough feature selection there has been great interest in developing methodologies that are capable of dealing with imprecision and uncertainty the large amount of research currently being carried out in fuzzy and rough sets is representative of this many deep relationships have been established and recent studies have concluded as to the complementary nature of the two methodologies therefore it is desirable to extend and hybridize the underlying concepts to deal with additional aspects of data imperfection such developments offer a high degree of flexibility and provide robust solutions and advanced tools for data analysis fuzzy rough set based feature fs selection has been shown to be highly useful at reducing data dimensionality but possesses several problems that render it ineffective for large datasets this paper proposes three new approaches to fuzzy rough fs based on fuzzy similarity relations in particular a fuzzy extension to crisp discernibility matrices is proposed and utilized initial experimentation shows that the methods greatly reduce dimensionality while preserving classification accuracy

feature space interpretation of svms with indefinite kernels kernel methods are becoming increasingly popular for various kinds of machine learning tasks the most famous being the support vector machine svm for classification the svm is well understood when using conditionally positive definite cpd kernel functions however in practice non cpd kernels arise and demand application in svm the procedure of plugging these indefinite kernels in svm often yields good empirical classification results however they are hard to interpret due to missing geometrical and theoretical understanding in this paper we provide a step toward the comprehension of svm classifiers in these situations we give a geometric interpretation of svm with indefinite kernel functions we show that such svm are optimal hyperplane classifiers not by margin maximization but by minimization of distances between convex hulls in pseudo euclidean spaces by this we obtain a sound framework and motivation for indefinite svm this interpretation is the basis for further theoretical analysis e g investigating uniqueness and for the derivation of practical guidelines like characterizing the suitability of indefinite svm

an axiomatic characterization of a fuzzy generalization of rough sets in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties several authors have proposed various fuzzy generalizations of rough approximations in this paper we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication then we find the assumptions which permit a given fuzzy set theoretic operator to represent a upper or lower approximation derived from a special fuzzy relation different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations and different sets of axioms of fuzzy set theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator finally we study the composition of two approximation spaces it is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces

structure of feature spaces related to fuzzy similarity relations as kernels fuzzy similarity relations have been considered as kernels in the domain of machine learning in previous works as a supplementary of this consideration this paper focuses on construction of feature space related to fuzzy similarity relations as indefinite kernels first we generalize the mercer theorem from positive definite kernels to fuzzy similarity relations with this generalization we construct the feature space related to a fuzzy similarity relation as a particular krein space the proposed feature space is a naturally generalization of the existing pseudo euclidean space from finite to infinite universe of discourses at the end of this paper we develop an open problem to set up the connection between transitivity and the positive definiteness of a fuzzy similarity relation results in this paper complete the mathematical foundation for further applications of fuzzy similarity relations under the framework of kernel tricks

minimization of axiom sets on fuzzy approximation operators axiomatic characterization of approximation operators is an important aspect in the study of rough set theory in this paper we examine the independence of axioms and present the minimal axiom sets characterizing fuzzy rough approximation operators and rough fuzzy approximation operators

local reduction of decision system with fuzzy rough sets one important and valuable topic in fuzzy rough sets is attribute reduction of decision system the existing attribute reductions with fuzzy rough sets consider all decision classes together and cannot identify key conditional attributes explicitly for special decision class in this paper we introduce the concept of local reduction with fuzzy rough sets for decision system the local reduction can identify key conditional attribute and offer a minimal description for every single decision class approach of discernibility matrix is employed to investigate the structure of local reduction at last several experiments are performed to show that the idea of local reduction is feasible and valid

fuzzy rough approximations for set valued data rough set theory is one of important tools of soft computing and rough approximations are the essential elements in rough set models however the existing fuzzy rough set model for set valued data which is directly constructed based on a kind of similarity relation fail to explicitly define fuzzy rough approximations to solve this issue in this paper we propose two types of fuzzy rough approximations and define two corresponding relative positive region reducts furthermore two discernibility matrices and two discernibility functions are introduced to acquire these new proposed reducts and the relationships among the new reducts and the existing reducts are also be provided theoretical analyses demonstrate that the new types of reducts have less redundancy and are more diverse no lower number of reducts than those obtained by means of the existing matrices and experimental results illustrate the new reducts found by our methods outperform those obtained by existing method

rough set analysis of a general type of fuzzy data using transitive aggregations of fuzzy similarity relations abstract in this paper we initially present and compile the tools and theories that are necessary for a rough set analysis of a general type of fuzzy data from this study we have developed two original contributions that can enhance the application of fuzzy rough sets techniques to data that may be affected by several sources of uncertainty in its measurement process for this type of analysis we have first considered the construction of homogeneous granules in which reflexivity symmetry and a certain transitivity property are satisfied this has encouraged us to study the application of t similarity relations in this framework and to examine the available theorems that permit us to aggregate this type of relations as a part of this study we have also developed a new theorem that allows us to use the dual of the generalized means as an aggregation operator in the fuzzy rough sets context another theoretical development that tries to respond to a practical problem is the concept of precision aggregation we use this new notion to export ziarko s variable precision rough set model to the analysis of fuzzy data a generalization that becomes a necessity for the fuzzy rough sets analysis of large databases the utility of this approach is demonstrated with an application example in the vehicle database donated by the turing institute

fuzzy rough attribute reduction with application to web categorization abstract due to the explosive growth of electronically stored information automatic methods must be developed to aid users in maintaining and using this abundance of information effectively in particular the sheer volume of redundancy present must be dealt with leaving only the information rich data to be processed this paper presents a novel approach based on an integrated use of fuzzy and rough set theories to greatly reduce this data redundancy formal concepts of fuzzy rough attribute reduction are introduced and illustrated with a simple example the work is applied to the problem of web categorization considerably reducing dimensionality with minimal loss of information experimental results show that fuzzy rough reduction is more powerful than the conventional rough set based approach classifiers that use a lower dimensional set of attributes which are retained by fuzzy rough reduction outperform those that employ more attributes returned by the existing crisp rough reduction method

nevanlinna extremal measures for some orthogonal polynomials related to birth and death processes abstract we consider two classes of birth and death processes which lead to indeterminate stieltjes moment problems and we calculate the entire functions c and d from the corresponding nevanlinna matrix this permits a direct computation of a nevanlinna extremal orthogonality measure for each of the processes

attributes reduction using fuzzy rough sets fuzzy rough sets are the generalization of traditional rough sets to deal with both fuzziness and vagueness in data the existing researches on fuzzy rough sets are mainly concentrated on the construction of approximation operators less effort has been put on the attributes reduction of databases with fuzzy rough sets this paper mainly focuses on the attributes reduction with fuzzy rough sets after analyzing the previous works on attributes reduction with fuzzy rough sets we introduce formal concepts of attributes reduction with fuzzy rough sets and completely study the structure of attributes reduction an algorithm using discernibility matrix to compute all the attributes reductions is developed based on these lines of thought we set up a solid mathematical foundation for attributes reduction with fuzzy rough sets the experimental results show that the idea in this paper is feasible and valid

kernel discriminant analysis for positive definite and indefinite kernels kernel methods are a class of well established and successful algorithms for pattern analysis thanks to their mathematical elegance and good performance numerous nonlinear extensions of pattern recognition techniques have been proposed so far based on the so called kernel trick the objective of this paper is twofold first we derive an additional kernel tool that is still missing namely kernel quadratic discriminant kqd we discuss different formulations of kqd based on the regularized kernel mahalanobis distance in both complete and class related subspaces secondly we propose suitable extensions of kernel linear and quadratic discriminants to indefinite kernels we provide classifiers that are applicable to kernels defined by any symmetric similarity measure this is important in practice because problem suited proximity measures often violate the requirement of positive definiteness as in the traditional case kqd can be advantageous for data with unequal class spreads in the kernel induced spaces which cannot be well separated by a linear discriminant we illustrate this on artificial and real data for both positive definite and indefinite kernels

generalized rough sets over fuzzy lattices this paper studies generalized rough sets over fuzzy lattices through both the constructive and axiomatic approaches from the viewpoint of the constructive approach the basic properties of generalized rough sets over fuzzy lattices are obtained the matrix representation of the lower and upper approximations is given according to this matrix view a simple algorithm is obtained for computing the lower and upper approximations as for the axiomatic approach a set of axioms is constructed to characterize the upper approximation of generalized rough sets over fuzzy lattices

a new kernel based fuzzy clustering approach support vector clustering with cell growing in this paper the support vector clustering is extended to an adaptive cell growing model which maps data points to a high dimensional feature space through a desired kernel function this generalized model is called multiple spheres support vector clustering which essentially identifies dense regions in the original space by finding their corresponding spheres with minimal radius in the feature space a multisphere clustering algorithm based on adaptive cluster cell growing method is developed whereby it is possible to obtain the grade of memberships as well as cluster prototypes in partition the effectiveness of the proposed algorithm is demonstrated for the problem of arbitrary cluster shapes and for prototype identification in an actual application to a handwritten digit data set

frsvms fuzzy rough set based support vector machines this paper aims to improve hard margin support vector machines svms by considering the membership of every training sample in constraints the membership is computed by employing the technique of fuzzy rough sets so that hard margin svms can be combined with fuzzy rough sets and the inconsistence between conditional features and decision labels can be taken into account at the same time in this paper we first propose fuzzy transitive kernel based fuzzy rough sets for binary classification we use a lower approximation operator in fuzzy transitive kernel based fuzzy rough sets to compute the membership for every training input and then we reformulate hard margin support vector machines into fuzzy rough set based svms frsvms with new constraints in which the membership is taken into account finally comparisons with soft margin svms and fuzzy svms are made the experimental results show that the proposed approach is feasible and valid it significantly improved the performance of the hard margin svms

kernelized fuzzy attribute c means clustering algorithm a novel kernelized fuzzy attribute c means clustering algorithm is proposed in this paper since attribute means clustering algorithm is an extension of fuzzy c means algorithm with weighting exponent m 2 and fuzzy attribute c means clustering is a general type of attribute c means clustering with weighting exponent m 1 we modify the distance in fuzzy attribute c means clustering algorithm with kernel induced distance and obtain kernelized fuzzy attribute c means clustering algorithm kernelized fuzzy attribute c means clustering algorithm is a natural generalization of kernelized fuzzy c means algorithm with stable function experimental results on standard iris database and tumor normal gene chip expression data demonstrate that kernelized fuzzy attribute c means clustering algorithm with gaussian radial basis kernel function and cauchy stable function is more effective and robust than fuzzy c means fuzzy attribute c means clustering and kernelized fuzzy c means as well

fuzzy rough data reduction with ant colony optimization feature selection refers to the problem of selecting those input features that are most predictive of a given outcome a problem encountered in many areas such as machine learning pattern recognition and signal processing in particular solution to this has found successful application in tasks that involve datasets containing huge numbers of features in the order of tens of thousands which would be impossible to process further recent examples include text processing and web content classification rough set theory has been used as such a dataset pre processor with much success but current methods are inadequate at finding minimal reductions the smallest sets of features possible to alleviate this difficulty a feature selection technique that employs a hybrid variant of rough sets fuzzy rough sets has been developed recently and has been shown to be effective however this method is still not able to find the optimal subsets regularly this paper proposes a new feature selection mechanism based on ant colony optimization in an attempt to combat this the method is then applied to the problem of finding optimal feature subsets in the fuzzy rough data reduction process the present work is applied to complex systems monitoring and experimentally compared with the original fuzzy rough method an entropy based feature selector and a transformation based reduction method pca comparisons with the use of a support vector classifier are also included

attribute reduction for heterogeneous data based on the combination of classical and fuzzy rough set models attribute reduction with rough sets aims to delete superfluous condition attributes from a decision system by considering the inconsistency between condition attributes and the decision labels however heterogeneous condition attributes including symbolic and real valued ones always coexist for most decision systems and different types of attributes induce different kinds of granular structures the existing rough set models do not have explicit mechanisms to address different kinds of granular structures reasonably and effectively in this paper we aim to perform attribute reduction for decision systems with symbolic and real valued condition attributes by composing classical rough set and fuzzy rough set models we first define a discernibility relation for every symbolic and real valued condition attribute to characterize its discernible ability related to the decision labels with these discernibility relations we can develop a dependence function to measure the inconsistency between heterogeneous condition attributes and decision labels and attribute reduction aims to keep this dependence function with a small perturbation the proposed attribute reduction deals with heterogeneous condition attributes from the viewpoint of discernible ability and can consider the mutual effects between two types of attributes without preprocessing into single typed ones an algorithm to find reducts is developed and experiments are performed to demonstrate that the proposed idea is effective

attribute selection with fuzzy decision reducts rough set theory provides a methodology for data analysis based on the approximation of concepts in information systems it revolves around the notion of discernibility the ability to distinguish between objects based on their attribute values it allows to infer data dependencies that are useful in the fields of feature selection and decision model construction in many cases however it is more natural and more effective to consider a gradual notion of discernibility therefore within the context of fuzzy rough set theory we present a generalization of the classical rough set framework for data based attribute selection and reduction using fuzzy tolerance relations the paper unifies existing work in this direction and introduces the concept of fuzzy decision reducts dependent on an increasing attribute subset measure experimental results demonstrate the potential of fuzzy decision reducts to discover shorter attribute subsets leading to decision models with a better coverage and with comparable or even higher accuracy

fuzzy rough sets assisted attribute selection attribute selection as refers to the problem of selecting those input attributes or features that are most predictive of a given outcome a problem encountered in many areas such as machine learning pattern recognition and signal processing unlike other dimensionality reduction methods attribute selectors preserve the original meaning of the attributes after reduction this has found application in tasks that involve datasets containing huge numbers of attributes in the order of tens of thousands which for some learning algorithms might be impossible to process further recent examples include text processing and web content classification as techniques have also been applied to small and medium sized datasets in order to locate the most informative attributes for later use one of the many successful applications of rough set theory has been to this area the rough set ideology of using only the supplied data and no other information has many benefits in as where most other methods require supplementary knowledge however the main limitation of rough set based attribute selection in the literature is the restrictive requirement that all data is discrete in classical rough set theory it is not possible to consider real valued or noisy data this paper investigates a novel approach based on fuzzy rough sets fuzzy rough feature selection frfs that addresses these problems and retains dataset semantics frfs is applied to two challenging domains where a feature reducing step is important namely web content classification and complex systems monitoring the utility of this approach is demonstrated and is compared empirically with several dimensionality reducers in the experimental studies frfs is shown to equal or improve classification accuracy when compared to the results from unreduced data classifiers that use a lower dimensional set of attributes which are retained by fuzzy rough reduction outperform those that employ more attributes returned by the existing crisp rough reduction method in addition it is shown that frfs is more powerful than the other as techniques in the comparative study

input space versus feature space in kernel based methods this paper collects some ideas targeted at advancing our understanding of the feature spaces associated with support vector sv kernel functions we first discuss the geometry of feature space in particular we review what is known about the shape of the image of input space under the feature space map and how this influences the capacity of sv methods following this we describe how the metric governing the intrinsic geometry of the mapped surface can be computed in terms of the kernel using the example of the class of inhomogeneous polynomial kernels which are often used in sv pattern recognition we then discuss the connection between feature space and input space by dealing with the question of how one can given some vector in feature space find a preimage exact or approximate in input space we describe algorithms to tackle this issue and show their utility in two applications of kernel methods first we use it to reduce the computational complexity of sv decision functions second we combine it with the kernel pca algorithm thereby constructing a nonlinear statistical denoising technique which is shown to perform well on real world data

a note on moment integrals and some applications abstract in this paper two moment integrals are given by the method of transformation in addition generalizations of sears s transformation are obtained by moment integrals moreover certain q mehler formulas for rogers szego polynomials are gained by moment integrals besides an open problem of trilinear generating function is deduced by moment integrals at last generalizations of u n 1 type kalnins miller transformation are achieved by moment integrals