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

dual fuzzy neighbourhood spaces i abstract a fuzzy neighbourhood space fns is determined by means of a system of fuzzy neighbourhoods of the ordinary points of that space we provide a general method for constructing bases for all fuzzy neighbourhood systems in a natural way from the fuzzy topologies of a special type of fuzzy topological spaces fts s which we call the fuzzy neighbourhood base spaces this construction is shown to be functorial of those new fts s the full subcategory of dual fuzzy neighbourhood spaces is shown to be isomorphic to the category of fns s and a fns will coincide with its dual fts iff it is topologically generated

on fuzzy approximation operators in attribute reduction with fuzzy rough sets generally speaking there are four fuzzy approximation operators defined on a general triangular norm t norm framework in fuzzy rough sets different types of t norms specify various approximation operators one issue whether and how the different fuzzy approximation operators affect the result of attribute reduction is then arisen this paper addresses this issue from the theoretical viewpoint by reviewing attribute reduction with fuzzy rough sets and then describing and proving some theorems which demonstrate the effects of the fuzzy approximation operators on the results of attribute reduction first we review some notions of attribute reduction with fuzzy rough sets such as positive region dependency degree and attribute reduction we then present and prove some theorems which describe how and to what degree fuzzy approximation operators impact the performance of attribute reduction finally we report some experimental simulation results which demonstrate the effectiveness and correctness of the theoretical contributions one main contribution in this paper is that we have described and proven that each attribute reduction obtained using one type of fuzzy lower approximation operator always contains one reduction obtained using the other type of fuzzy lower approximation operator

set approximations in fuzzy formal concept analysis formal concept analysis and rough set theory are two important tools in knowledge representation and knowledge discovery in relational information systems the purpose of this paper is to study rough set approximations within formal concept analysis in fuzzy environment properties of existent fuzzy concept lattices derived from an adjoint pair of operations are first reviewed and examined based on both lattice theoretic and fuzzy set theoretic operators two new pairs of rough fuzzy set approximations within fuzzy formal contexts are then defined finally properties of the rough fuzzy set approximation operators are presented in detail

moving breathers and breather to soliton conversions for the hirota equation we find that the hirota equation admits breather to soliton conversion at special values of the solution eigenvalues this occurs for the first order as well as higher orders of breather solutions an analytic expression for the condition of the transformation is given and several examples of transformations are presented the values of these special eigenvalues depend on two free parameters that are present in the hirota equation we also find that higher order breathers generally have complicated quasi periodic oscillations along the direction of propagation various breather solutions are considered including the particular case of second order breathers of the nonlinear schrodinger equation

efficient parallel boolean matrix based algorithms for computing composite rough set approximations in information systems there may exist multiple different types of attributes like categorical attributes numerical attributes set valued attributes interval valued attributes missing attributes etc such information systems are called as composite information systems to process such attributes with rough set theory composite rough set model and corresponding matrix methods were introduced in our previous study calculation of rough set approximations of a concept is the key step for rule acquisition and attribute reduction in rough set based methods to accelerate the computation process of rough set approximations this paper first presents the boolean matrix representation of the lower and upper approximations in the composite information system then designs a parallel method for computing approximations based on matrix and implements it on multi gpu the experiments on data sets from uci and user defined data sets show that the proposed method can accelerate the computation process efficiently the multi gpu implementation achieves up to a speedup of 334 9 over the cpu implementation

combining numerical and linguistic information in group decision making abstract people give information about their personal preferences in many different ways depending on their background this paper deals with group decision making problems in which the solution depends on information of a different nature i e assuming that the experts express their preferences with numerical or linguistic values the aim of this paper is to present a proposal for this problem we introduce a fusion operator for numerical and linguistic information this operator combines linguistic values assessed in the same label set with numerical ones assessed in the interval 0 1 it is based on two transformation methods between numerical and linguistic values which are defined using the concept of the characteristic values proposed in this paper its application to group decision making problems is illustrated by means of a particular fusion operator guided by fuzzy majority considering that the experts express their opinions by means of fuzzy or linguistic preference relations this operator is used to develop a choice process for the alternatives allowing solutions to be obtained in line with the majority of the experts opinions

composite rough sets for dynamic data mining as a soft computing tool rough set theory has become a popular mathematical framework for pattern recognition data mining and knowledge discovery it can only deal with attributes of a specific type in the information system by using a specific binary relation however there may be attributes of multiple different types in information systems in real life applications such information systems are called as composite information systems in this paper a composite relation is proposed to process attributes of multiple different types simultaneously in composite information systems then an extended rough set model called as composite rough sets is presented we also redefine lower and upper approximations positive boundary and negative regions in composite rough sets through introducing the concepts of the relation matrix the decision matrix and the basic matrix we propose matrix based methods for computing the approximations positive boundary and negative regions in composite information systems which is crucial for feature selection and knowledge discovery moreover combined with the incremental learning technique a novel matrix based method for fast updating approximations is proposed in dynamic composite information systems extensive experiments on different data sets from uci and user defined data sets show that the proposed incremental method can process large data sets efficiently

min max attribute object bireducts on unifying models of reducts in rough set theory abstract a decision table describes a finite set of objects ob by using a finite set of condition attributes c and a finite set of decision attributes d pawlak defines attribute reducts by considering the entire decision table as a generalization we introduce the notion of min max attribute object bireducts of a sub table restricted by a pair b x of a subset of condition attributes b and a subset of objects x a pair r z is a min max attribute object bireduct of b x if and only if r is a minimal subset of b such that r and b make the same correct decisions for objects in x and z is a maximal subset of x for which b can make the correct decisions we propose the notion of the decidability of objects and introduce the decidable region of a set of objects as a generalization of the positive region of the set we define and interpret a min max attribute object bireduct based on the decidable region min max attribute object bireducts offer a general model and existing models of attribute reducts are special cases the results lead to a unified framework for studying four types of attribute reducts

active sample selection based incremental algorithm for attribute reduction with rough sets attribute reduction with rough sets is an effective technique for obtaining a compact and informative attribute set from a given dataset however traditional algorithms have no explicit provision for handling dynamic datasets where data present themselves in successive samples incremental algorithms for attribute reduction with rough sets have been recently introduced to handle dynamic datasets with large samples though they have high complexity in time and space to address the time space complexity issue of the algorithms this paper presents a novel incremental algorithm for attribute reduction with rough sets based on the adoption of an active sample selection process and an insight into the attribute reduction process this algorithm first decides whether each incoming sample is useful with respect to the current dataset by the active sample selection process a useless sample is discarded while a useful sample is selected to update a reduct at the arrival of a useful sample the attribute reduction process is then employed to guide how to add and or delete attributes in the current reduct the two processes thus constitute the theoretical framework of our algorithm the proposed algorithm is finally experimentally shown to be efficient in time and space

feature selection with partition differentiation entropy for large scale data sets feature selection especially for large data sets is a challenging problem in areas such as pattern recognition machine learning and data mining with the development of data collection and storage technologies the data has become bigger than ever thus making it difficult for learning from large data sets with traditional methods in this paper we introduce the partition differentiation entropy from the viewpoint of partition in rough sets to measure the significance and uncertainty of attributes and present a feature selection method for large scale data sets based on the information theoretical measurement of attribute significance given a large scale decision information system the proposed method first divides it into small sub information systems according to the decision classes then by computing partition differentiation entropy in the sub systems the partition differentiation entropy of the attribute subset in the original decision information system is obtained accordingly the important features are selected based on the value of partition differentiation entropy the experimental results show that the idea of the proposed method is feasible and valid

rogue waves in a resonant erbium doped fiber system with higher order effects we mainly investigate a coupled system of the generalized nonlinear schrodinger equation and the maxwell bloch equations which describes the wave propagation in an erbium doped nonlinear fiber with higher order effects including the forth order dispersion and quintic non kerr nonlinearity we derive the one fold darboux transformation of this system and construct the determinant representation of the n fold darboux transformation then the determinant representation of the nth new solutions en pn n which were generated from the known seed solutions e p is established through the n fold darboux transformation the solutions en pn n provide the bright and dark breather solutions of this system furthermore we construct the determinant representation of the nth order bright and dark rogue waves by taylor expansions and also discuss the hybrid solutions which are the nonlinear superposition of the rogue wave and breather solutions

covering based variable precision mathcal i mathcal t fuzzy rough sets with applications to multiattribute decision making at present there is no unified method for solving multiattribute decision making problems in this paper we propose two methods that benefit from some novel fuzzy rough set models some theoretical preliminaries pave the way first by means of a fuzzy logical implicator mathcal i and a triangular norm mathcal t four types of coverings based variable precision mathcal i mathcal t fuzzy rough set models are proposed they can be used to deal with misclassification and perturbation here misclassification refers to error or missing values in classification while perturbation refers to small changes in digital data second the properties and the relationships among these models are investigated finally we rely on their remarkable features in order to establish two approaches to multiattribute decision making some numerical examples illustrate the application of these new approaches the sensitivity and comparative analyses show that the respective ranking results produced by these decision making methods have a high consensus for multiattribute decision making problems with fuzzy evaluation information

fuzzy t locality spaces abstract a lowen fuzzy topological space is said to be localizable at points if it is completely determined in some prescribed manner by a system of fuzzy neighbourhoods called here fuzzy localities of its ordinary points we introduce for each lower semicontinuous triangular norm t a new type of those spaces which we call fuzzy t locality spaces the particularity of a fuzzy t locality space is that its fuzzy interior operator is obtained from its t locality system through a definition of fuzzy containment of fuzzy sets that employs the t residuated implication operator the origins of this new theory lie in the theory of probabilistic metric spaces indeed fuzzy t locality spaces provide a fuzzy topological framework for a naturally defined notion of metric t continuity of functions between probabilistic t metric spaces in a manner described in the last two sections of this article before that we establish characterizations and aspects of good behaviour for fuzzy t locality spaces including that every t locality system has an open basis we briefly study their level topologies and we show that all topologically generated spaces are fuzzy t locality spaces we prove that the category t fls of fuzzy t locality spaces and continuous functions between them is a topological category we show that it is closed in fts under the formation of optimal lifts of sources we provide a condition on t under which t fls becomes closed in fts also under the formation of co optimal lifts of sinks

a fast feature weighting algorithm of data gravitation classification the data gravitation classification dgc model is a new classification method that has gained great research interest in recent years feature weights are the key parameters in dgc models because the classification performances of these models are very sensitive to such feature weights the available dgc models use wrapper like algorithms to obtain their optimised feature weights although such algorithms produce high classification accuracies but they contribute to the high computational complexities of the dgc models in this study we propose a fast feature weight algorithm for dgc models called ffw dgc we use the concepts of feature discrimination and redundancy to measure the importance of a feature after which two fuzzy subsets are constructed to respectively represent these concepts next we combine the two fuzzy subsets to compute the feature weights used in gravitational computing we conduct our experiments on 25 standard data sets and 22 imbalance data sets and compare ffw dgc with 11 kinds of classifiers including the swarm intelligence based dgc pso dgc model competitive results of ffw dgc demonstrate that it can obtain high classification accuracies but also hundreds of times of speedup ratios compared with pso dgc

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

the logic of tied implications part 1 properties applications and representation a conjunction t ties an implication operator a if the identity a a a b z a t a b z holds a a abdel hamid n n morsi associatively tied implications fuzzy sets and systems 136 2003 291 311 we study the class of tied adjointness algebras which are five connective algebras on two partially ordered sets in which the implications are tied by triangular norms this class contains besides residuated implications several other implications employed in fuzzy logic nevertheless we show that the algebraic inequalities of residuated algebras remain true for our tied implications but in forms that distribute roles over the five connectives of the algebra we apply the properties of tied implications to a generalized modus ponens inference scheme with two successive rules we prove its equivalence to a scheme with one compound rule when both schemata are interpreted by the compositional rule of inference and all connectives are taken from one tied adjointness algebra then we quote another application of this rich theory a notion of many valued rough sets which exhibit the basic mathematical behaviour of the rough sets of pawlak a comparator h is said to be prelinear if it satisfies h y z h z y 1 for all y z hajek we introduce prelinear tied adjointness algebras in which two comparators are prelinear we provide a representation of those algebras as subdirect products of tied adjointness chains on the lines of hajek s representation of bl algebras but our representations are more economical because we employ minimal prime filters on residuated lattices only rather than all prime filters

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

using fsbt technique with rough set theory for personal investment portfolio analysis this study proposes a novel forward search and backward trace fsbt technique based on rough set theory to improve data analysis and extend the scope of observations made from sample data to solve personal investment portfolio problems rough set theory mathematically classifies data into class sets the class set with the most objects may generate one decision rule the rules generated from rst are rough and fragmented that are very difficult to interpret the information an empirical case is used to generate more than 85 rules by the rst method in comparison with fsbt method which only generated 14 rules this result can show our proposed method is better than traditional rst method based on class sets that contain the most objects much of human knowledge is described in natural language it is a very important thing to convert information from computer databases into normal human language sample data taken from features with the same backgrounds are used to compile different portfolios that investment companies and investment advisors can employ to satisfy the investor needs the method not only can provide decision making rules but also can offer alternative strategies for better data analysis we believe that the fsbt technique can be fully applied in research on investment marketing

localized waves and interaction solutions to a 3 1 dimensional generalized kp equation abstract based on the hirota bilinear method and long wave limit four kinds of localized waves namely solitons lumps breathers and rogue waves are constructed for the 3 1 dimensional generalized kp equation n soliton solutions are obtained by employing bilinear method then breathers two breathers and interaction breather solutions are obtained by selecting appropriate parameters on two soliton solution and four soliton solution these breathers own different dynamic behaviors in the different planes taking a long wave limit on the two and four soliton solutions under special parameter constraints one order lumps and rogue waves two order lumps and rogue waves and interaction solutions between lumps and rogue waves are derived applying the same method on the three soliton solution interaction solutions between kink solitons with periodic solutions lumps and rogue waves are constructed respectively the influence of parameters on the solution is analyzed the propagation directions phase shifts energies and shapes for these solutions can be affected and controlled by the parameters moreover graphics are presented to demonstrate the properties of the explicit analytical localized wave solutions

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

on fuzzy rough sets based on tolerance relations dubois and prade 1990 1 introduced the notion of fuzzy rough sets as a fuzzy generalization of rough sets which was originally proposed by pawlak 1982 8 later radzikowska and kerre introduced the so called i t fuzzy rough sets where i is an implication and t is a triangular norm in the present paper by using a pair of implications i j we define the so called i j fuzzy rough sets which generalize the concept of fuzzy rough sets in the sense of radzikowska and kerre and that of mi and zhang basic properties of i j fuzzy rough sets are investigated in detail

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

simple determinant representation for rogue waves of the nonlinear schrodinger equation we present a simple representation for arbitrary order rogue wave solution and a study on the trajectories of them explicitly we find that the trajectories of two valleys on whole temporal spatial distribution all look x shaped for rogue waves additionally we present different types of high order rogue wave structures which could be helpful towards realizing the complex dynamics of rogue waves

some properties of l fuzzy approximation spaces based on bounded integral residuated lattices in this paper we present a general framework for the study of l fuzzy rough sets based on bounded integral residuated lattices we introduce and discuss the notions of l fuzzy topological spaces left right lower and left right upper l fuzzy approximation operators based on a complete bounded integral residuated lattice investigate union intersection and composition operations of l fuzzy approximation spaces and study some properties of the l fuzzy topologies generated by these l fuzzy rough approximation operators and the l relations induced by an l fuzzy topology

l approximation spaces and l fuzzy quasi uniform spaces we introduce the notion of generalized l upper and generalized l lower approximation spaces under a family of information systems we define l upper and l lower quasi approximation operators in fuzzy rough sets and l fuzzy quasi uniform spaces as a view point of the entourage approach in a strictly two sided commutative bi quantale we investigate the relations between l quasi approximation spaces and l fuzzy quasi uniform spaces

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

on the relationship of extended necessity measures to implication operators on the unit interval abstract by an extended necessity measure on a set x we mean a function from ix to the unit interval i such that the measure of an arbitrary meet of fuzzy subsets of x equals the infimum of their measures we address the question of whether a given extended necessity measure on x is induced through a given implication operator on i from some unknown normalized fuzzy subset of x in the following manner the measure of ix is the degree of containment of in according to the implication we provide a simple complete answer to this question in the case is in one of the following three infinite sets of implication operators the s type implications the residuated implications and the continuous implications that satisfy the exchange principle

the model of fuzzy variable precision rough sets the fuzzy rough set frs model has been introduced to handle databases with real values however frs was sensitive to misclassification and perturbation here misclassification means error or missing values in classification and perturbation means small changes of numerical data the variable precision rough sets vprss model was introduced to handle databases with misclassification however it could not effectively handle the real valued datasets now it is valuable from theoretical and practical aspects to combine frs and vprs so that a powerful tool which not only can handle numerical data but also is less sensitive to misclassification and perturbation can be developed in this paper we set up a model named fuzzy vprss fvprss by combining frs and vprs with the goal of making frs a special case first we study the knowledge representation ways of frs and vprs and then propose the set approximation operators of fvprs second we employ the discernibility matrix approach to investigate the structure of attribute reductions in fvprs and develop an algorithm to find all reductions third in order to overcome the np complete problem of finding all reductions we develop some fast heuristic algorithms to obtain one near optimal attribute reduction finally we compare fvprs with rs frs and several flexible rs based approaches with respect to misclassification and perturbation the experimental comparisons show the feasibility and effectiveness of fvprs

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

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

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

three types of fuzzy covering based rough set models abstract in this paper three new types of fuzzy covering based rough set models are proposed first the membership degree based fuzzy covering rough set is defined in the fuzzy covering approximation space by separately comparing the membership degree of each object belonging to the approximated fuzzy set with those of the component fuzzy sets in the fuzzy covering for the same object second the membership function based fuzzy covering rough set is obtained by globally comparing the membership function of the approximated fuzzy set with those of the component fuzzy sets in the fuzzy covering finally the membership function based fuzzy covering rough set model is generalized to the fuzzy covering approximation space the properties of these new rough set models are discussed furthermore the relationship between these new rough sets and the relationship between the new rough set and pawlak rough set are described

using one axiom to characterize l fuzzy rough approximation operators based on residuated lattices abstract axiomatic characterization of approximation operators plays an important role in the study of rough set theory different axiom sets of abstract operators can illustrate different classes of rough set systems in this paper we are devoted to searching for a single axiom to characterize l fuzzy rough approximation operators based on residuated lattices axioms of l fuzzy set theoretic operators make sure of the existence of certain types of l fuzzy relations which produce the same operators we demonstrate that the lower upper l fuzzy rough approximation operators generated by a generalized l fuzzy relation can be characterized by only one axiom furthermore we also use one axiom to characterize l fuzzy rough approximation operators produced by the l fuzzy serial reflexive symmetric and t transitive relations as well as any of their compositions

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

neighborhood operator systems and approximations this paper presents a framework for the study of generalizing the standard notion of equivalence relation in rough set approximation space with various categories of k step neighborhood systems based on a binary relation on a finite universe six families of binary relations are obtained and the corresponding six classes of k step neighborhood systems are derived extensions of pawlak s rough set approximation operators based on such neighborhood systems are proposed properties of neighborhood operator systems and rough set approximation operators are investigated and their connections are examined

fuzzy rough set theory for the interval valued fuzzy information systems the concept of the rough set was originally proposed by pawlak as a formal tool for modelling and processing incomplete information in information systems then in 1990 dubois and prade first introduced the rough fuzzy sets and fuzzy rough sets as a fuzzy extension of the rough sets the aim of this paper is to present a new extension of the rough set theory by means of integrating the classical pawlak rough set theory with the interval valued fuzzy set theory i e the interval valued fuzzy rough set model is presented based on the interval valued fuzzy information systems which is defined in this paper by a binary interval valued fuzzy relations r f i uxu on the universe u several properties of the rough set model are given and the relationships of this model and the others rough set models are also examined furthermore we also discuss the knowledge reduction of the classical pawlak information systems and the interval valued fuzzy information systems respectively finally the knowledge reduction theorems of the interval valued fuzzy information systems are built

a fuzzy covering based rough set model and its generalization over fuzzy lattice in this paper we study a new type of fuzzy covering based rough set model by introducing the notion of fuzzy minimal description we mainly address the following issues in this paper first we present the definition of fuzzy minimal description and study its properties then we define a novel type of fuzzy covering based rough set model and investigate the properties of this model furthermore the axiomizations and matrix representations of the fuzzy covering lower and fuzzy covering upper approximations are the vital problems we investigate in this paper moreover we explore the conditions under which two fuzzy coverings generate the same fuzzy covering lower or fuzzy covering upper approximations finally we generalize the model to l fuzzy covering based rough set which is defined over fuzzy lattices similar to the fuzzy covering based rough set model we also address the issues mentioned above to the l fuzzy covering based rough set

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

fuzzy rough set based incremental attribute reduction from dynamic data with sample arriving abstract attribute reduction with fuzzy rough set is an effective technique for selecting most informative attributes from a given real valued dataset however existing algorithms for attribute reduction with fuzzy rough set have to re compute a reduct from dynamic data with sample arriving where one sample or multiple samples arrive successively this is clearly uneconomical from a computational point of view in order to efficiently find a reduct from such datasets this paper studies incremental attribute reduction with fuzzy rough sets at the arrival of one sample or multiple samples the relative discernibility relation is updated for each attribute on the basis of the updated relation an insight into the incremental process of attribute reduction with fuzzy rough sets is gained to reveal how to add new attributes into the current reduct and delete existing attributes from the current reduct applying the incremental process two incremental algorithms for attribute reduction with fuzzy rough sets are presented for one incoming sample and multiple incoming samples respectively experimental comparisons with several non incremental algorithms and the proposed incremental algorithm for one incoming sample show that our proposed incremental algorithm for multiple incoming samples can efficiently find one reduct with a comparable classification accuracy

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

permutable fuzzy consequence and interior operators and their connection with fuzzy relations fuzzy operators are an essential tool in many fields and the operation of composition is often needed in general composition is not a commutative operation however it is very useful to have operators for which the order of composition does not affect the result in this paper we analyze when permutability appears that is when the order of application of the operators does not change the outcome we characterize permutability in the case of the composition of fuzzy consequence operators and the dual case of fuzzy interior operators we prove that for these cases permutability is completely connected to the preservation of the operator type we also study the particular case of fuzzy operators induced by fuzzy relations through zadeh s compositional rule and the inf composition for this cases we connect permutability of the fuzzy relations using the sup composition with permutability of the induced operators special attention is paid to the cases of operators induced by fuzzy preorders and similarities finally we use these results to relate the operator induced by the transitive closure of the composition of two reflexive fuzzy relations with the closure of the operator this composition induces

n solitons breathers lumps and rogue wave solutions to a 3 1 dimensional nonlinear evolution equation abstract based on hirota bilinear method n solitons breathers lumps and rogue waves as exact solutions of the 3 1 dimensional nonlinear evolution equation are obtained the impacts of the parameters on these solutions are analyzed the parameters can influence and control the phase shifts propagation directions shapes and energies for these solutions the single kink soliton solution and interactions of two and three kink soliton overtaking collisions of the hirota bilinear equation are investigated in different planes the breathers in three dimensions possess different dynamics in different planes via a long wave limit of breathers with indefinitely large periods rogue waves are obtained and localized in time it is shown that the rogue wave possess a growing and decaying line profile that arises from a nonconstant background and then retreat back to the same nonconstant background again the results can be used to illustrate the interactions of water waves in shallow water moreover figures are given out to show the properties of the explicit analytic solutions

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

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

a graph approach for fuzzy rough feature selection abstract rough sets especially fuzzy rough sets have proven to be a powerful tool for dealing with vagueness and uncertainty in data analysis fuzzy rough feature selection has been shown to be highly useful in data dimensionality reduction however many fuzzy rough feature selection algorithms are still time consuming when dealing with the large scale data sets in this paper the problem of feature selection in fuzzy rough sets is studied in the framework of graph theory we propose a new mechanism for fuzzy rough feature selection it is shown that finding the attribute reduction of a fuzzy decision system can be translated into finding the transversal of a derivative hypergraph based on the graph representation model a novel graph theoretic algorithm for fuzzy rough feature selection is proposed the performance of the proposed method is compared with those of the state of the art methods on various classification tasks experimental results show that the proposed technique outperforms all other known feature selection methods in terms of the computation time especially for the large scale data sets it demonstrates promising performance moreover our proposed method can achieve better classification accuracies with the usage of small number of features

fuzzy rough based regularization in generalized multiple kernel learning generalized multiple kernel learning gmkl has been proposed in the literature for feature selection gmkl learns linear product and exponential combinations of given base kernels which makes it more robust and efficient than traditional multiple kernel learning mkl gmkl has been shown to be a good tool for feature selection as well time taken for the convergence of gmkl depends upon the initialization of kernel weights optimization schemes in gmkl initialize kernel weights randomly this produces variability in convergence time in this paper we propose a fuzzy rough set based kernel weight initialization for gmkl fr gmkl we show that this results in faster convergence than that obtained by random initialization in gmkl while retaining same level of accuracy we also show that the computation time of our proposed method is lower than that obtained through quadratic programming feature selection qpfs based as well as maximal relevance maxrel based initialization of the regularization parameter the performance of our proposed method is evaluated using five benchmark binary classification datasets and three benchmark multi class classification datasets from the uci repository

basins of attraction in piecewise smooth hamiltonian systems piecewise smooth hamiltonian systems arise in physical and engineering applications for such a system typically an infinite number of quasiperiodic attractors coexist here we use the term attractors to indicate invariant sets to which typically initial conditions approach as a result of the piecewise smoothness of the underlying system these attractors are therefore characteristically different from the attractors in dissipative dynamical systems we find that the basins of attraction of different attractors exhibit a riddledlike feature in that they mix with each other on arbitrarily fine scales this practically prevents prediction of attractors from specific initial conditions and parameters the mechanism leading to the complicated basin structure is found to be characteristically different from those reported previously for similar basin structure in smooth dynamical systems we demonstrate the phenomenon using a class of electronic relaxation oscillators with voltage protection and provide a theoretical explanation

generation of higher order rogue waves from multibreathers by double degeneracy in an optical fiber in this paper we construct a special kind of breather solution of the nonlinear schrodinger nls equation the so called breather positon b positon for short which can be obtained by taking the limit j 1 of the lax pair eigenvalues in the order n periodic solution which is generated by the n fold darboux transformation from a special seed solution plane wave further an order n b positon gives an order n rogue wave under a limit 1 0 here 0 is a special eigenvalue in a breather of the nls equation such that its period goes to infinity several analytical plots of order 2 breather confirm visually this double degeneration the last limit in this double degeneration can be realized approximately in an optical fiber governed by the nls equation in which an injected initial ideal pulse is created by a frequency comb system and a programable optical filter wave shaper according to the profile of an analytical form of the b positon at a certain position z 0 we also suggest a new way to observe higher order rogue waves generation in an optical fiber namely measure the patterns at the central region of the higher order b positon generated by above ideal initial pulses when 1 is very close to the 0 the excellent agreement between the numerical solutions generated from initial ideal inputs with a low signal to noise ratio and analytical solutions of order 2 b positon supports strongly this way in a realistic optical fiber system our results also show the validity of the generating mechanism of a higher order rogue waves from a multibreathers through the double degeneration

the necessary and sufficient conditions for a fuzzy relation being τ euclidean the connections between special types of fuzzy relations and properties of fuzzy rough approximation operators have been established in recent years but euclidean fuzzy relation has not been considered yet in the present paper the necessary and sufficient conditions for a fuzzy relation being euclidean are investigated in three different fuzzy rough approximation spaces meanwhile it is proved that in i fuzzy rough approximation space where i is an r implication the properties the euclidean i fuzzy rough approximation operators possess are just the same as those in rough fuzzy approximation space

the minimization of axiom sets characterizing generalized approximation operators in the axiomatic approach of rough set theory rough approximation operators are characterized by a set of axioms that guarantees the existence of certain types of binary relations reproducing the operators thus axiomatic characterization of rough approximation operators is an important aspect in the study of rough set theory in this paper the independence of axioms of generalized crisp approximation operators is investigated and their minimal sets of axioms are presented

dynamics of a piecewise linear map with a gap in this paper we consider periodic solutions of discontinuous non smooth maps we show how the fixed points of a general piecewise linear map with a discontinuity a map with a gap behave under parameter variation we show in detail all the possible behaviours of period 1 and period 2 solutions for positive gaps we find that period 2 solutions can exist independently of period 1 solutions conversely for negative gaps period 1 and period 2 solutions can coexist higher periodic orbits can also exist and be stable and we give several examples of how these solutions behave under parameter variation finally we compare our results with those of jain banerjee jain banerjee 2003 int j bifurcat chaos 13 3341 3351 and banerjee et al banerjee et al 2004 ieee trans circ syst ii 51 649 654 and explain their numerical simulations

second order nonlinear schrodinger equation breather solutions in the degenerate and rogue wave limits we present an explicit analytic form for the two breather solution of the nonlinear schr odinger equation with imaginary eigenvalues it describes various nonlinear combinations of akhmediev breathers and kuznetsov ma solitons the degenerate case when the two eigenvalues coincide is quite involved the standard inverse scattering technique does not generally provide an answer to this scenario we show here that the solution can still be found as a special limit of the general second order expression and appears as a mixture of polynomials with trigonometric and hyperbolic functions a further restriction of this particular case where the two eigenvalues are equal to i produces the second order rogue wave with two free parameters considered as differential shifts the illustrations reveal a precarious dependence of wave profile on the degenerate eigenvalues and differential shifts thus we establish a hierarchy of second order solutions revealing the interrelated nature of the general case the rogue wave and the degenerate breathers

characteristics of a piecewise smooth area preserving map we are reporting a study carried out in a system concatenated by two area preserving maps the system can be viewed as a model of an electronic relaxation oscillator with over voltage protection we found that a border collision bifurcation may interrupt a period doubling bifurcation cascade and that some special features such as quasicoexisting periodic orbits crossing border as well as the transition between quasitransience and chaotic orbits accompany the process these features belong to the so called quasidissipative properties here quasitransience denotes the behavior of iterations outside elliptic islands they are attracted to the islands as soon as it reaches the islands the iteration follows the conservative regulations exactly this induces a kind of escaping from strange sets the scaling behavior of the escaping rate is obtained numerically

learning fuzzy rules from fuzzy samples based on rough set technique although the traditional rough set theory has been a powerful mathematical tool for modeling incompleteness and vagueness its performance in dealing with initial fuzzy data is usually poor this paper makes an attempt to improve its performance by extending the traditional rough set approach to the fuzzy environment the extension is twofold one is knowledge representation and the other is knowledge reduction first we provide new definitions of fuzzy lower and upper approximations by considering the similarity between the two objects second we extend a number of underlying concepts of knowledge reduction such as the reduct and core to the fuzzy environment and use these extensions to propose a heuristic algorithm to learn fuzzy rules from initial fuzzy data finally we provide some numerical experiments to demonstrate the feasibility of the proposed algorithm one of the main contributions of this paper is that the fundamental relationship between the reducts and core of rough sets is still pertinent after the proposed extension

feature selection in mixed data feature selection in the data with different types of feature values i e the heterogeneous or mixed data is especially of practical importance because such types of data sets widely exist in real world the key issue for feature selection in mixed data is how to properly deal with different types of the features or attributes in the data set motivated by the fuzzy rough set theory which allows different fuzzy relations to be defined for different types of attributes to measure the similarity between objects and in view of the effectiveness of entropy to measure information uncertainty we propose in this paper a fuzzy rough set based information entropy for feature selection in a mixed data set it is proved that the newly defined entropy meets the common requirement of monotonicity and can equivalently characterize the existing attribute reductions in the fuzzy rough set theory then a feature selection algorithm is formulated based on the proposed entropy and a filter wrapper method is suggested to select the best feature subset in terms of classification accuracy an extensive numerical experiment is further conducted to assess the performance of the feature selection method and the results are satisfactory highlightsa novel fuzzy rough set based information entropy is constructed for mixed data the proposed entropy can equivalently characterize the existing attribute reductions in the fuzzy rough set theory a feature selection algorithm is formulated based on the proposed entropy a filter wrapper method is suggested to select a best feature subset

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

on the t transitivity of kernels abstract in this paper we present a view of kernels from a fuzzy set theoretical perspective indeed it turns out that kernels which are positive definite functions have to fulfill a consistency property given by the so called t transitivity of a fuzzy t equivalence relation with respect to the triangular norm t as a result we introduce a triangular norm t cos which is characterized as being the greatest one for which all kernels mapping to the unit interval and having constant 1 in its diagonal are t cos equivalences finally some applied relevant aspects are discussed as a starting point for further research