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

a new class of rational cubic spline fractal interpolation function and its constrained aspects abstract this paper pertains to the area of shape preservation and sets a theoretical foundation for the applications of preserving constrained nature of a given constraining data in fractal interpolation functions fifs techniques we construct a new class of rational cubic spline fifs rcsfifs with a preassigned quadratic denominator with two shape parameters which includes classical rational cubic interpolant appl math comp 216 2010 pp 2036 2049 as special case and improves the sufficient conditions for positivity convergence analysis of rcsfif to the original function in c 1 is studied in order to meet the needs of practical design or overcome the drawback of the tension effect in the proposed rcsfifs we improve our method by introducing a new tension parameter wi and construct a new class of rational cubic spline fifs with three shape parameters the scaling factors and shape parameters have a predictable adjusting role on the shape of curves the elements of the rational iterated function system in each subinterval are identified befittingly so that the graph of the resulting c 1 rational cubic spline fif constrained i within a prescribed rectangle ii above a prescribed straight line iii between two piecewise straight lines these parameters include in particular conditions on the positivity of the c 1 rational cubic spline fif several numerical examples are presented to ascertain the correctness and usability of developed scheme and to suggest how these schemes outperform their classical counterparts

type 2 fuzzy rough sets based on extended t norms a type 2 fuzzy rough set based on extended t norms is investigated from constructive and axiomatic approaches an approximation algorithm of type 2 fuzzy rough sets is listed based on cut sets weak type 2 fuzzy topologies are proposed to investigate the topological properties of type 2 fuzzy rough sets an interval valued type 2 fuzzy rough set is proposed and investigated this paper is devoted to investigate type 2 fuzzy rough sets based on extended t norms with respect to type 2 fuzzy relations with convex normal fuzzy truth values type 2 fuzzy rough sets are proposed on two finite universes of discourse and characterized from both constructive and axiomatic approaches in the constructive approach different classes of type 2 fuzzy rough sets are investigated an approximate algorithm of type 2 fuzzy rough sets based on cut sets is given with numerical examples in the axiomatic approach the axiomatic characterizations of type 2 fuzzy rough approximation operators are studied moreover the topological properties of type 2 fuzzy rough sets are discussed in the end this paper introduces interval valued type 2 fuzzy rough sets and discusses their properties

hybrid learning for interval type 2 fuzzy logic systems based on orthogonal least squares and back propagation methods this paper presents a novel learning methodology based on a hybrid algorithm for interval type 2 fuzzy logic systems since only the back propagation method has been proposed in the literature for the tuning of both the antecedent and the consequent parameters of type 2 fuzzy logic systems a hybrid learning algorithm has been developed the hybrid method uses a recursive orthogonal least squares method for tuning the consequent parameters and the back propagation method for tuning the antecedent parameters systems were tested for three types of inputs a interval singleton b interval type 1 non singleton and c interval type 2 non singleton experiments were carried out on the application of hybrid interval type 2 fuzzy logic systems for prediction of the scale breaker entry temperature in a real hot strip mill for three different types of coil the results proved the feasibility of the systems developed here for scale breaker entry temperature prediction comparison with type 1 fuzzy logic systems shows that hybrid learning interval type 2 fuzzy logic systems provide improved performance under the conditions tested

integrable nonlinear equations and liouville s theorem i a symplectic structure for stationary lax equations of the type l p 0 is constructed wherel is a matrix differential operator of the first order it is shown that the equation has a sufficient for the complete integrability amount of first integrals in involution the well known linearization of the equation by the abelian mapping is obtained in a natural manner in consequent exercising of liouville s procedure

relational interpretations of neighborhood operators and rough set approximation operators this paper presents a framework for the formulation interpretation and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations a special class of neighborhood systems called 1 neighborhood systems is introduced three extensions of pawlak approximation operators are analyzed properties of neighborhood and approximation operators are studied and their connections are examined

on characterization of generalized interval type 2 fuzzy rough sets this paper presents a systematic study of interval type 2 fuzzy rough sets integrating rough set theory with interval type 2 fuzzy set theory using constructive and axiomatic approaches from the perspective of a constructive approach a pair of lower and upper interval type 2 fuzzy rough approximation operators with respect to an interval type 2 fuzzy relation is defined the basic properties of the interval type 2 fuzzy rough approximation operators are studied using cut sets of interval type 2 fuzzy sets classical representations of interval type 2 fuzzy rough approximation operators are then presented and the connections between special interval type 2 fuzzy relations and interval type 2 fuzzy rough approximation operators are investigated adopting an axiomatic approach an operator oriented characterization of interval type 2 fuzzy rough sets is proposed in other words interval type 2 fuzzy rough approximation operators are characterized by axioms different axiom sets of interval type 2 fuzzy set theoretic operators guarantee the existence of different types of interval type 2 fuzzy relations that produce the same operators finally the relationship between interval type 2 fuzzy rough sets and interval type 2 fuzzy topological spaces is examined we obtain sufficient and necessary conditions for the conjecture that an interval type 2 fuzzy interior closure operator derived from an interval type 2 fuzzy topological space can associate with an interval type 2 fuzzy reflexive and transitive relation such that the corresponding lower upper interval type 2 fuzzy rough approximation operator is the interval type 2 fuzzy interior closure operator

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

rough set theory for the interval valued fuzzy information systems the notion of a rough set was originally proposed by pawlak z pawlak rough sets international journal of computer and information sciences 11 5 1982 341 356 later on dubois and prade d dubois h prade rough fuzzy sets and fuzzy rough sets international journal of general system 17 2 3 1990 191 209 introduced rough fuzzy sets and fuzzy rough sets as a generalization of rough sets this paper deals with an interval valued fuzzy information system by means of integrating the classical pawlak rough set theory with the interval valued fuzzy set theory and discusses the basic rough set theory for the interval valued fuzzy information systems in this paper we firstly define the rough approximation of an interval valued fuzzy set on the universe u in the classical pawlak approximation space and the generalized approximation space respectively i e the space on which the interval valued rough fuzzy set model is built secondly several interesting properties of the approximation operators are examined and the interrelationships of the interval valued rough fuzzy set models in the classical pawlak approximation space and the generalized approximation space are investigated thirdly we discuss the attribute reduction of the interval valued fuzzy information systems finally the methods of the knowledge discovery for the interval valued fuzzy information systems are presented with an example

an integrable su 3 spin chain abstract an integrable su 3 soin chain equation is constructed an explicit form of the lax pair representation is given and it is shown to be gauge equivalent to a set of nonlinear schrodinger type equations for three complex variables

positivity preserving interpolation of positive data by rational cubics this work is a contribution towards the graphical display of data when it is positive the data are required to be represented in such a way that its visual display looks smooth and pleasant its positive shape is preserved everywhere and the computation cost is economical a c 1 piecewise rational cubic function in its most general form has been utilized for this objective the method is implemented for the 1d data initially and then it is extended to an interpolating rational bicubic form for the data arranged over a rectangular grid simple sufficient conditions are developed on the free parameters in the description of the rational function to visualize the positive data in the form of positive curves and surfaces

c 2 positivity preserving rational interpolation splines in one and two dimensions abstract a class of rational quartic cubic interpolation spline with two local control parameters is presented which can be c 2 continuous without solving a linear system of consistency equations for the derivative values at the knots the effects of the local control parameters on generating interpolation curves are illustrated for c 2 interpolation the given interpolant can locally reproduce quadratic polynomials and has o h 2 or o h 3 convergence simple schemes for the c 2 interpolant to preserve the shape of 2d positive data are developed moreover based on the boolean sum of quintic interpolating operators a class of bi quintic partially blended rational quartic cubic interpolation surfaces is also constructed the given interpolation surface provides four local control parameters and can be c 2 continuous without using the second or higher mixed partial derivatives on a rectangular grid simple sufficient data dependent constraints are also derived on the local control parameters to preserve the shape of a 3d positive data set arranged over a rectangular grid

shape preserving piecewise rational interpolation with higher order continuity abstract a united form of the classical hermite interpolation and shape preserving interpolation is presented in this paper the presented interpolation method provides higher order continuous shape preserving interpolation splines the given interpolants are explicit piecewise rational expressions without solving a linear or nonlinear system of consistency equations by setting parameter values the interpolation curve can be generated by choosing the classical piecewise hermite interpolation polynomials or the presented piecewise rational expressions for monotonicity preserving and convexity preserving interpolation the appropriate values of a parameter are given on each subinterval numerical examples indicate that the given method produces visually pleasing curves

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

linear integral equations and multicomponent nonlinear integrable systems i abstract a systematic method for obtaining multicomponent generalizations of integrable nonlinear partial differential equations pde s is developed the method starts from a general type of linear integral equations containing integrations over an arbitrary contour with an arbitrary measure in the complex plane of k the spectral parameter special k dependent factors in the integrand are shown to induce an extra coupling between solutions of these integral equations with a different source term in this way the direct linearization is obtained of multicomponent generalizations of various nonlinear pde s such as the nonlinear schrodinger equation the derivative nonlinear schrodinger equations the isotropic heisenberg spin chain equation the complex sine gordon equation and the massive thirring model equations in the present paper i we present the general framework to derive finite matrix pde s and we also discuss backlund transformations and multicomponent lattice versions in a subsequent paper ii we treat a variety of examples of multicomponent pde s and discuss miura transformations and gauge equivalences

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

type 2 fuzzy hidden markov models and their application to speech recognition this paper presents an extension of hidden markov models hmms based on the type 2 t2 fuzzy set fs referred to as type 2 fuzzy hmms t2 fhmms membership functions mfs of t2 fss are three dimensional and this new third dimension offers additional degrees of freedom to evaluate the hmms fuzziness therefore t2 fhmms are able to handle both random and fuzzy uncertainties existing universally in the sequential data we derive the t2 fuzzy forward backward algorithm and viterbi algorithm using t2 fs operations in order to investigate the effectiveness of t2 fhmms we apply them to phoneme classification and recognition on the timit speech database experimental results show that t2 fhmms can effectively handle noise and dialect uncertainties in speech signals besides a better classification performance than the classical hmms

landau lifshitz and higher order nonlinear systems gauge generated from nonlinear schrodinger type equations new landau lifshitz ll and higher order nonlinear systems gauge generated from nonlinear schrodinger ns type equations are presented the consequences of gauge equivalence between different dynamical systems are discussed the gauge connections among various ll and ns equations are found and depicted through a schematic representation

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

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

application of fuzzy sets of type 2 to the solution of fuzzy equation systems abstract the present paper continues the authors investigation on fuzzy equation systems with tolerances we consider the situation that the outputs of an object are modelled by fuzzy sets of type 2 for special membership functions this problem can be attributed to a problem with tolerances moreover this concept enables us to define solutions for fuzzy outputs of type 1 when classical solution concepts may fail

aggregation using the linguistic weighted average and interval type 2 fuzzy sets the focus of this paper is the linguistic weighted average lwa where the weights are always words modeled as interval type 2 fuzzy sets it2 fss and the attributes may also but do not have to be words modeled as it2 fss consequently the output of the lwa is an it2 fs the lwa can be viewed as a generalization of the fuzzy weighted average fwa where the type 1 fuzzy inputs are replaced by it2 fss this paper presents the theory algorithms and an application of the lwa it is shown that finding the lwa can be decomposed into finding two fwas since the lwa can model more uncertainties it should have wide applications in distributed and hierarchical decision making

separation of variables in the classical integrable sl 3 magnetic chain there are two fundamental problems studied by the theory of hamiltonian integrable systems integration of equations of motion and construction of action angle variables a third problem however should be added to the list separation of variables though much simpler thant the two others it has important relations to quantum integrability separation of variables is constructed for thesl 3 magnetic chain an example of an integrable model associated to a nonhyperelliptic algebraic curve

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

generalized heisenberg ferromagnet models via hermitian symmetric spaces we study the generalized heisenberg ferromagnet model using a duality with the hermitian symmetric space nonlinear schrodinger equation we explain the mathematical structure of the hermiticity of the space in connection with the soliton surface theory and obtain various spin configurations in n dimension using this correspondence

positivity preserving rational bi cubic spline interpolation for 3d positive data this paper deals with the shape preserving interpolation problem for visualization of 3d positive data a required display of 3d data looks smooth and pleasant a rational bi cubic function involving six shape parameters is presented for this objective which is an extension of piecewise rational function in the form of cubic quadratic involving three shape parameters simple data dependent constraints for shape parameters are derived to conserve the inherited shape feature positivity of 3d data remaining shape parameters are left free for designer to modify the shape of positive surface as per industrial needs the interpolant is not only local c 1 but also it is a computationally economical in comparison with existing schemes several numerical examples are supplied to support the worth of proposed interpolant

a bivariate rational cubic interpolating spline with biquadratic denominator a bivariate rational bicubic interpolating spline bris with biquadratic denominator and six shape parameters is constructed in a rectangle domain the c1 continuous condition of bris discussed bris is proved to be bounded and its error is estimated in the case of the equally spaced knots the matrix expression and symmetry of bris are presented some properties of the basis of bris are given in the end a numerical example is given to illustrat the effect of the shape parameters on the shape of bris surface

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

gauge equivalence of sigma models with non compact grassmannian manifolds the gauge equivalence ge of sigma models associated with non compact grassmannian manifolds is investigated with emphasis on the necessary restrictions for the choice of gauge elements in such cases the importance of ge in solving a non linear system with the help of inverse scattering data of its gauge related counterpart is demonstrated the gauge relations between generalised landau lifshitz ll and non linear schrodinger nls type equations and also between non linear sigma models and generalised sine sinh gordon equations for non compact su p q s u u v u s t manifolds are established using h gauge invariance of ll the ge is extended to some higher order specific non linear systems the gauge connection among various ll and nls equations are schematically represented along with the recovery of earlier results important new results some with significant non compact structures are discovered

a model for type 2 fuzzy rough sets rough set theory is an important approach to granular computing type 1 fuzzy set theory permits the gradual assessment of the memberships of elements in a set hybridization of these assessments results in a fuzzy rough set theory type 2 fuzzy sets possess many advantages over type 1 fuzzy sets because their membership functions are themselves fuzzy which makes it possible to model and minimize the effects of uncertainty in type 1 fuzzy logic systems existing definitions of type 2 fuzzy rough sets are based on vertical slice or plane representations of type 2 fuzzy sets and the granular structure of type 2 fuzzy rough sets has not been discussed in this paper a definition of type 2 fuzzy rough sets based on a wavy slice representation of type 2 fuzzy sets is given then the concepts of granular type 2 fuzzy sets are proposed and their properties are investigated finally granular type 2 fuzzy sets are used to describe the granular structures of the lower and upper approximations of a type 2 fuzzy set and an example of attribute reduction is given

complete integrability for a discrete heisenberg chain haldane 1982 conjectured that a simple generalisation of the heisenberg spin chain leads to a completely integrable classical system without invoking the usual continuum approximation complete integrability is established for the generalised discrete chain by exhibiting a lax pair and an infinite set of non local conservation laws the associated inverse scattering problem is also formulated providing explicit soliton solutions

positive data modeling using spline function a rational cubic function with two parameters has been constructed to visualize the positive data the main focus of the work is the representation of the data in such a way that its view looks smooth and attractive in the first step simple data dependent constraints are derived on the parameters in the description of the rational cubic function to visualize the shape of positive data then it is extended to a rational bi cubic partially blended functions coons patches and derived constraints on parameters to visualize the shape of positive surface data the developed scheme is locally positive and economical the approximation order of rational cubic spline function is oh i 3

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

operations on type 2 fuzzy sets in this paper we discuss set operations on type 2 fuzzy sets including join and meet under minimum product t norm algebraic operations properties of membership grades of type 2 sets and type 2 relations and their compositions all this is needed to implement a type 2 fuzzy logic system fls

shape preserving piecewise rational interpolant with quartic numerator and quadratic denominator an explicit representation of a piecewise rational interpolant with quartic numerator and quadratic denominator is presented for positivity data monotone data and convex data the shape preserving properties of the interpolant are given the interpolant is c 2 continuous spline with a shape parameter w i on each subinterval the values of w i to guarantee shape preservation are estimated a convergence analysis establishes an error bound in terms of w i and shows that the interpolant is o h 2 or o h 3 accurate several examples are supplied to support the practical value of the given interpolation method

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

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

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

an interval type 2 fuzzy rough set model for attribute reduction rough set theory is a very useful tool for describing and modeling vagueness in ill defined environments traditional rough set theory is restricted to crisp environments however nowadays it has been extended to fuzzy environments resulting in the development of the so called fuzzy rough sets type 2 fuzzy sets possess many advantages over type 1 fuzzy sets but for the general type 2 fuzzy sets the computational complexity is severe on the other hand set theoretic and arithmetic computations for the interval type 2 fuzzy sets are very simple motivated by the aforementioned accomplishments in this paper the concept of fuzzy rough sets is generalized to interval type 2 fuzzy environments subsequently a method of attribute reduction within the interval type 2 fuzzy rough set framework is proposed lastly the properties of the interval type 2 fuzzy rough sets are presented

fuzzy interpolative reasoning based on the ratio of fuzziness of rough fuzzy sets in this paper we propose a method to construct a polygonal rough fuzzy set from a set of polygonal fuzzy sets representing the aggregation of multiple experts opinions and propose a new fuzzy interpolative reasoning method for sparse fuzzy rule based systems based on the ratio of fuzziness of the constructed polygonal rough fuzzy sets where the values of the antecedent variables and the consequence variable appearing in the fuzzy rules are represented by the constructed polygonal rough fuzzy sets the proposed fuzzy interpolative reasoning method can overcome the drawbacks of the existing method due to the fact that it can deal with fuzzy interpolative reasoning using polygonal rough fuzzy sets and it gets more reasonable fuzzy interpolative reasoning results than the existing method

applications of type 2 fuzzy logic systems to forecasting of time series abstract in this paper we begin with a type 1 fuzzy logic system fls trained with noisy data we then demonstrate how information about the noise in the training data can be incorporated into a type 2 fls which can be used to obtain bounds within which the true noisefree output is likely to lie we do this with the example of a one step predictor for the mackey glass chaotic time series m c mackey l glass oscillation and chaos in physiological control systems science 197 1977 287 280 we also demonstrate how a type 2 fls can be used to obtain better predictions than those obtained with a type 1 fls

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

a hybrid learning algorithm for a class of interval type 2 fuzzy neural networks in real life information about the world is uncertain and imprecise the cause of this uncertainty is due to deficiencies on given information the fuzzy nature of our perception of events and objects and on the limitations of the models we use to explain the world the development of new methods for dealing with information with uncertainty is crucial for solving real life problems in this paper three interval type 2 fuzzy neural network it2fnn architectures are proposed with hybrid learning algorithm techniques gradient descent backpropagation and gradient descent with adaptive learning rate backpropagation at the antecedents layer a interval type 2 fuzzy neuron it2fn model is used and in case of the consequents layer an interval type 1 fuzzy neuron model it1fn in order to fuzzify the rule s antecedents and consequents of an interval type 2 takagi sugeno kang fuzzy inference system it2 tsk fis it2 tsk fis is integrated in an adaptive neural network in order to take advantage the best of both models this provides a high order intuitive mechanism for representing imperfect information by means of use of fuzzy if then rules in addition to handling uncertainty and imprecision on the other hand neural networks are highly adaptable with learning and generalization capabilities experimental results are divided in two kinds in the first one a non linear identification problem for control systems is simulated here a comparative analysis of learning architectures it2fnn and anfis is done for the second kind a non linear mackey glass chaotic time series prediction problem with uncertainty sources is studied finally it2fnn proved to be more efficient mechanism for modeling real world problems