outlier detection with the kernelized spatial depth function statistical depth functions provide from the deepest point a center outward ordering of multidimensional data in this sense depth functions can measure the extremeness or outlyingness of a data point with respect to a given data set hence they can detect outliers observations that appear extreme relative to the rest of the observations of the various statistical depths the spatial depth is especially appealing because of its computational efficiency and mathematical tractability in this article we propose a novel statistical depth the kernelized spatial depth ksd which generalizes the spatial depth via positive definite kernels by choosing a proper kernel the ksd can capture the local structure of a data set while the spatial depth fails we demonstrate this by the half moon data and the ring shaped data based on the ksd we propose a novel outlier detection algorithm by which an observation with a depth value less than a threshold is declared as an outlier the proposed algorithm is simple in structure the threshold is the only one parameter for a given kernel it applies to a one class learning setting in which normal observations are given as the training data as well as to a missing label scenario where the training set consists of a mixture of normal observations and outliers with unknown labels we give upper bounds on the false alarm probability of a depth based detector these upper bounds can be used to determine the threshold we perform extensive experiments on synthetic data and data sets from real applications the proposed outlier detector is compared with existing methods the ksd outlier detector demonstrates a competitive performance

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

support vector machine soft margin classifiers error analysis the purpose of this paper is to provide a pac error analysis for the q norm soft margin classifier a support vector machine classification algorithm it consists of two parts regularization error and sample error while many techniques are available for treating the sample error much less is known for the regularization error and the corresponding approximation error for reproducing kernel hilbert spaces we are mainly concerned about the regularization error it is estimated for general distributions by a k functional in weighted lq spaces for weakly separable distributions i e the margin may be zero satisfactory convergence rates are provided by means of separating functions a projection operator is introduced which leads to better sample error estimates especially for small complexity kernels the misclassification error is bounded by the v risk associated with a general class of loss functions v the difficulty of bounding the offset is overcome polynomial kernels and gaussian kernels are used to demonstrate the main results the choice of the regularization parameter plays an important role in our analysis

soft data mining computational theory of perceptions and rough fuzzy approach data mining and knowledge discovery is described from pattern recognition point of view along with the relevance of soft computing key features of the computational theory of perceptions and its significance in pattern recognition and knowledge discovery problems are explained role of fuzzy granulation f granulation in machine and human intelligence and its modeling through rough fuzzy integration are discussed merits of fuzzy granular computation in terms of performance and computation time for the task of case generation in large scale case based reasoning systems are illustrated through an example

pseudopolar based estimation of large translations rotations and scalings in images one of the major challenges related to image registration is the estimation of large motions without prior knowledge this work presents a fourier based approach that estimates large translations scalings and rotations the algorithm uses the pseudopolar pp fourier transform to achieve substantial improved approximations of the polar and log polar fourier transforms of an image thus rotations and scalings are reduced to translations which are estimated using phase correlation by utilizing the pp grid we increase the performance accuracy speed and robustness of the registration algorithms scales up to 4 and arbitrary rotation angles can be robustly recovered compared to a maximum scaling of 2 recovered by state of the art algorithms the algorithm only utilizes one dimensional fast fourier transform computations whose overall complexity is significantly lower than prior works experimental results demonstrate the applicability of the proposed algorithms

robust fuzzy clustering of relational data popular relational data clustering algorithms relational dual of fuzzy c means rfcm non euclidean rfcm nerfcm both by hathaway et al and fanny by kaufman and rousseeuw are examined a new algorithm which is a generalization of fanny called the fuzzy relational data clustering frc algorithm is introduced having an identical objective functional as rfcm however the frc does not have the restriction of rfcm which is that the relational data is derived from euclidean distance as the measure of dissimilarity between the objects and it also does not have limitations of fanny including the use of a fixed membership exponent or a fuzzifier exponent m the frc algorithm is further improved by incorporating the concept of dave s object data noise clustering nc algorithm done by proposing a concept of noise dissimilarity next based on the constrained minimization which includes an inequality constraint for the memberships and corresponding kuhn tucker conditions a noise resistant frc algorithm is derived which works well for all types of non euclidean dissimilarity data thus it is shown that the extra computations for data expansion spl beta spread transformation required by the nerfcm algorithm are not necessary this new algorithm is called robust non euclidean fuzzy relational data clustering robust ne frc and its robustness is demonstrated through several numerical examples advantages of this new algorithm are faster convergence robustness against outliers and ability to handle all kinds of relational data including non euclidean the paper also presents a new and better interpretation of the noise class

menger s theorem for fuzzy graphs the concept of the strongest path plays a crucial role in fuzzy graph theory in classical graph theory all paths in a graph are strongest with a strength value of one in this article we introduce menger s theorem for fuzzy graphs and discuss the concepts of strength reducing sets and t connected fuzzy graphs we also characterize t connected and t arc connected fuzzy graphs

a comprehensive study of implicator conjunctor based and noise tolerant fuzzy rough sets definitions properties and robustness analysis abstract both rough and fuzzy set theories offer interesting tools for dealing with imperfect data while the former allows us to work with uncertain and incomplete information the latter provides a formal setting for vague concepts the two theories are highly compatible and since the late 1980s many researchers have studied their hybridization in this paper we critically evaluate most relevant fuzzy rough set models proposed in the literature to this end we establish a formally correct and unified mathematical framework for them both implicator conjunctor based definitions and noise tolerant models are studied we evaluate these models on two different fronts firstly we discuss which properties of the original rough set model can be maintained and secondly we examine how robust they are against both class and attribute noise by highlighting the benefits and drawbacks of the different fuzzy rough set models this study appears a necessary first step to propose and develop new models in future research

granular variable precision l fuzzy rough sets based on residuated lattices abstract firstly as for the beginning of this paper we introduce the granular variable precision l fuzzy rough sets based on residuated lattices with arbitrary l fuzzy relations and then the equivalent expressions for the lower and upper approximation operators are given with l fuzzy implications and l fuzzy coimplications on arbitrary l fuzzy relations which can calculate the lower and upper approximation operators efficiently finally some basic properties of the granular variable precision l fuzzy rough sets on residuated lattices are investigated and the conclusions on the granular variable precision fuzzy rough sets are generalized into the situations of the truth values set becoming residuated lattices

rfrr robust fuzzy rough reduction this paper proposes a robust method of dimension reduction using fuzzy rough sets in which the reduction results can reflect the reducts obtained on all of the possible parameters here the reducts being obtained on all of the possible parameters mean that all of the reducts are obtained on different degrees of robustness to handle noise this method is completely different from the existing methods of fuzzy rough reduction the differences are shown in three aspects the concept the tool and the algorithm first the key concept of attribute reduction is redefined in a new way that is the robust fuzzy rough reduct which is shortened to a robust reduct is proposed to reflect the classical reducts obtained on all of the possible parameters the new robust reduct is not a crisp subset of condition attributes rather it is a fuzzy subset whose most interesting property is that any cut set of the robust reduct is a classical reduct on a certain parameter second the tool used to measure the discernibility power is different from the existing discernibility measures in this paper the robustness of each attribute to handle misclassification and perturbation is considered by considering both the robustness and the discernibility a robust fuzzy discernibility matrix is designed finally the algorithms used to find the robust reducts are designed based upon the robust fuzzy discernibility matrix which is completely different from the existing algorithms used to find the classical reducts

roughness of a fuzzy set an integration between the theories of fuzzy sets and rough sets has been attempted by providing a measure of roughness of a fuzzy set several properties of this new measure are established some of the possible applications for handling uncertainties in the field of pattern recognition are mentioned

relationships among generalized rough sets in six coverings and pure reflexive neighborhood system rough set is a useful tool to deal with partition related uncertainty granularity and incompleteness of knowledge although the classical rough set is constructed on the basis of an indiscernibility relation it can also be generalized by using some weaker binary relations in this paper a systematic approach is used to study the generalized rough sets in six coverings and pure reflexive neighborhood system after two steps relationships among generalized rough sets in six coverings and pure reflexive neighborhood system are obtained the first step is to study the generalized rough sets in six coverings and to get relationships between every two covering rough set models the second step is to study the relationships between generalized rough sets in each covering and in pure reflexive neighborhood system the inclusion relations or equivalence relations among the seven upper lower approximations could be acquired finally the accuracy measures of generalized rough sets in six coverings and that in pure reflexive neighborhood system are compared the relationships among seven accuracy measures are also obtained some illustrative examples are employed to demonstrate our arguments

classical r matrices and compatible poisson structures for lax equations on poisson algebras given a classical r matrix on a poisson algebra we show how to construct a natural family of compatible poisson structures for the hamiltonian formulation of lax equations examples for which our formalism applies include the benny hierarchy the dispersionless toda lattice hierarchy the dispersionless kp and modified kp hierarchies the dispersionless dym hierarchy etc

the higher order hamiltonian structures for the modified classical yang baxter equation we consider constructing the higher order hamiltonian structures on the dual of the lie algebra from the first hamiltonian structure of the coadjoint orbit method for this purpose we show that the structure of the lie algebrag is inherited to the algebra of vector fields ong through the solution of the modified classical yang baxter equation classicalr matrix we study the algebra that generates the compatible poisson brackets

rational solutions of the master symmetries of the kdv equation in this article we characterize a certain class of rational solutions of the hierarchy of master symmetries for kdv the result is that the generic rational potentials that decay at infinity and remain rational by all the flows of the master symmetry kdv hierarchy are bispectral potentials for the schrodinger operator by bispectral potentials we mean that the corresponding schrodinger operators possess families of eigenfunctions that are also eigenfunctions of a differential operator in the spectral variable this complements certain results of airault mckean moser 4 duistermaat grunbaum 10 and magri zubelli 40 as a consequence of bispectrality the rational solutions of the master symmetries turn out to be solutions of a generalized string equation

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

iterative relief for feature weighting algorithms theories and applications relief is considered one of the most successful algorithms for assessing the quality of features in this paper we propose a set of new feature weighting algorithms that perform significantly better than relief without introducing a large increase in computational complexity our work starts from a mathematical interpretation of the seemingly heuristic relief algorithm as an online method solving a convex optimization problem with a margin based objective function this interpretation explains the success of relief in real application and enables us to identify and address its following weaknesses relief makes an implicit assumption that the nearest neighbors found in the original feature space are the ones in the weighted space and relief lacks a mechanism to deal with outlier data we propose an iterative relief i relief algorithm to alleviate the deficiencies of relief by exploring the framework of the expectation maximization algorithm we extend i relief to multiclass settings by using a new multiclass margin definition to reduce computational costs an online learning algorithm is also developed convergence analysis of the proposed algorithms is presented the results of large scale experiments on the uci and microarray data sets are reported which demonstrate the effectiveness of the proposed algorithms and verify the presented theoretical results

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 robust fuzzy rough set models rough sets especially fuzzy rough sets are supposedly a powerful mathematical tool to deal with uncertainty in data analysis this theory has been applied to feature selection dimensionality reduction and rule learning however it is pointed out that the classical model of fuzzy rough sets is sensitive to noisy information which is considered as a main source of uncertainty in applications this disadvantage limits the applicability of fuzzy rough sets in this paper we reveal why the classical fuzzy rough set model is sensitive to noise and how noisy samples impose influence on fuzzy rough computation based on this discussion we study the properties of some current fuzzy rough models in dealing with noisy data and introduce several new robust models the properties of the proposed models are also discussed finally a robust classification algorithm is designed based on fuzzy lower approximations some numerical experiments are given to illustrate the effectiveness of the models the classifiers that are developed with the proposed models achieve good generalization performance

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

soft fuzzy rough sets for robust feature evaluation and selection the fuzzy dependency function proposed in the fuzzy rough set model is widely employed in feature evaluation and attribute reduction it is shown that this function is not robust to noisy information in this paper as datasets in real world applications are usually contaminated by noise robustness of data analysis models is very important in practice in this work we develop a new model of fuzzy rough sets called soft fuzzy rough sets which can reduce the influence of noise we discuss the properties of the model and construct a new dependence function from the model then we use the function to evaluate and select features the presented experimental results show the effectiveness of the new model

symmetric phase only matched filtering of fourier mellin transforms for image registration and recognition presents a new method to match a 2d image to a translated rotated and scaled reference image the approach consists of two steps the calculation of a fourier mellin invariant fmi descriptor for each image to be matched and the matching of the fmi descriptors the fmi descriptor is translation invariant and represents rotation and scaling as translations in parameter space the matching of the fmi descriptors is achieved using symmetric phase only matched filtering spomf the performance of the fmi spomf algorithm is the same or similar to that of phase only matched filtering when dealing with image translations the significant advantage of the new technique is its capability to match rotated and scaled images accurately and efficiently the innovation is the application of spomf to the fmi descriptors which guarantees high discriminating power and excellent robustness in the presence of noise this paper describes the principle of the new method and its discrete implementation for either image detection problems or image registration problems practical results are presented for various applications in medical imaging remote sensing fingerprint recognition and multiobject identification

scale affine invariant interest point detectors in this paper we propose a novel approach for detecting interest points invariant to scale and affine transformations our scale and affine invariant detectors are based on the following recent results 1 interest points extracted with the harris detector can be adapted to affine transformations and give repeatable results geometrically stable 2 the characteristic scale of a local structure is indicated by a local extremum over scale of normalized derivatives the laplacian 3 the affine shape of a point neighborhood is estimated based on the second moment matrix our scale invariant detector computes a multi scale representation for the harris interest point detector and then selects points at which a local measure the laplacian is maximal over scales this provides a set of distinctive points which are invariant to scale rotation and translation as well as robust to illumination changes and limited changes of viewpoint the characteristic scale determines a scale invariant region for each point we extend the scale invariant detector to affine invariance by estimating the affine shape of a point neighborhood an iterative algorithm modifies location scale and neighborhood of each point and converges to affine invariant points this method can deal with significant affine transformations including large scale changes the characteristic scale and the affine shape of neighborhood determine an affine invariant region for each point we present a comparative evaluation of different detectors and show that our approach provides better results than existing methods the performance of our detector is also confirmed by excellent matching resultss the image is described by a set of scale affine invariant descriptors computed on the regions associated with our points

fast branch bound algorithms for optimal feature selection a novel search principle for optimal feature subset selection using the branch bound method is introduced thanks to a simple mechanism for predicting criterion values a considerable amount of time can be saved by avoiding many slow criterion evaluations we propose two implementations of the proposed prediction mechanism that are suitable for use with nonrecursive and recursive criterion forms respectively both algorithms find the optimum usually several times faster than any other known branch bound algorithm as the algorithm computational efficiency is crucial due to the exponential nature of the search problem we also investigate other factors that affect the search performance of all branch bound algorithms using a set of synthetic criteria we show that the speed of the branch bound algorithms strongly depends on the diversity among features feature stability with respect to different subsets and criterion function dependence on feature set size we identify the scenarios where the search is accelerated the most dramatically finish in linear time as well as the worst conditions we verify our conclusions experimentally on three real data sets using traditional probabilistic distance criteria

attribute reduction in decision theoretic rough set models rough set theory can be applied to rule induction there are two different types of classification rules positive and boundary rules leading to different decisions and consequences they can be distinguished not only from the syntax measures such as confidence coverage and generality but also the semantic measures such as decision monotocity cost and risk the classification rules can be evaluated locally for each individual rule or globally for a set of rules both the two types of classification rules can be generated from and interpreted by a decision theoretic model which is a probabilistic extension of the pawlak rough set model as an important concept of rough set theory an attribute reduct is a subset of attributes that are jointly sufficient and individually necessary for preserving a particular property of the given information table this paper addresses attribute reduction in decision theoretic rough set models regarding different classification properties such as decision monotocity confidence coverage generality and cost it is important to note that many of these properties can be truthfully reflected by a single measure c in the pawlak rough set model on the other hand they need to be considered separately in probabilistic models a straightforward extension of the c measure is unable to evaluate these properties this study provides a new insight into the problem of attribute reduction

a quantum group structure in integrable conformal field theories we discuss a quantization prescription of the conformal algebras of a class ofd 2 conformal field theories which are integrable we first give a geometrical construction of certain extensions of the classical virasoro algebra known as classicalw algebras in which these algebras arise as the lie algebra of the second hamiltonian structure of a generalized korteweg de vries hierarchy this fact implies that thew algebras obtained as a reduction with respect to the nilpotent subalgebras of the kac moody algebra describe the integrability of a toda field theory subsequently we determine the coadjoint operators of thew algebras and relate these to classical yang baxter matrices the quantization of these algebras can be carried out using the concept of a so called quantum group we derive the condition under which the representations of these quantum groups admit a hilbert space completion by exploring the relation with the braid group then we consider a modification of the miura transformation which we use to define a quantumw algebra this leads to an alternative interpretation of the coset construction for kac moody algebras in terms of nonlinear integrable hierarchies subsequently we use the connection between the induced braid group representations and the representations of the mapping class group of riemann surfaces to identify an action of thew algebras on the moduli space of stable curves and we give the invariants of this action this provides a generalization of the situation for the virasoro algebra where such as invariant is given by the so called mumford form which describes the partition function of the bosonic string

factor analysis latent subspace modeling and robust fuzzy clustering using t distributions factor analysis is a latent subspace model commonly used for local dimensionality reduction tasks fuzzy c means fcm type fuzzy clustering approaches are closely related to gaussian mixture models gmms and expectation maximization em like algorithms have been employed in fuzzy clustering with regularized objective functions student s t mixture models smms have been proposed recently as an alternative to gmms resolving their outlier vulnerability problems in this paper we propose a novel fcm type fuzzy clustering scheme providing two significant benefits when compared with the existing approaches first it provides a well established observation space dimensionality reduction framework for fuzzy clustering algorithms based on factor analysis allowing concurrent performance of fuzzy clustering and within each cluster local dimensionality reduction second it exploits the outlier tolerance advantages of smms to provide a novel soundly founded nonheuristic robust fuzzy clustering framework by introducing the effective means to incorporate the explicit assumption about student s t distributed data into the fuzzy clustering procedure this way the proposed model yields a significant performance increase for the fuzzy clustering algorithm as we experimentally demonstrate

granular variable precision fuzzy rough sets with general fuzzy relations the variable precision fuzzy rough sets were proposed to remedy the defects of preexisting fuzzy rough set models however the variable precision fuzzy rough sets were only defined and investigated on fuzzy similarity relations in this paper the granular variable precision fuzzy rough sets with general fuzzy relations are proposed on arbitrary fuzzy relations the equivalent expressions of the approximation operators are given with fuzzy co implications on arbitrary fuzzy relations which can calculate efficiently the approximation operators the granular variable precision fuzzy rough sets are characterized from the constructive approach which are investigated on different fuzzy relations the conclusions on the variable precision fuzzy rough sets are generalized into the granular variable precision fuzzy rough sets

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

nested structure in parameterized rough reduction abstract in this paper by strict mathematical reasoning we discover the relationship between the parameters and the reducts in parameterized rough reduction this relationship named the nested reduction shows that the reducts act as a nested structure with the monotonically increasing parameter we present a systematic theoretical framework that provides some basic principles for constructing the nested structure in parameterized rough reduction some specific parameterized rough set models in which the nested reduction can be constructed are pointed out by strict mathematical reasoning based on the nested reduction we design several quick algorithms to find a different reduct when one reduct is already given here different refers to the reducts obtained on the different parameters all these algorithms are helpful for quickly finding a proper reduct in the parameterized rough set models the numerical experiments demonstrate the feasibility and the effectiveness of the nested reduction approach

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

compatible poisson structures and the virasoro algebra this paper is dedicated to henry mckean whose master craftsmanship enriched such diverse areas of mathematics and provided his disciples with a lofty measure by which to gauge their efforts and so redouble them

feature subset selection using separability index matrix effective feature subset selection fss is an important step when designing engineering systems that classify complex data in real time the electromyographic emg signal based walking assistance system is a typical system that requires an efficient computational architecture for classification the performance of such a system depends largely on a criterion function that assesses the quality of selected feature subsets however many well known conventional criterion functions use less relevant features for classification or they have a high computational cost here we propose a new criterion function that provides more effective fss the proposed criterion function known as a separability index matrix sim provides features pertinent to the classification task and a very low computational cost this new function produces to a simple feature selection algorithm when combined with the forward search paradigm we performed extensive experimental comparisons in terms of classification accuracy and computational costs to confirm that the proposed algorithm outperformed other filter type feature selection methods that are based on various distance measures including inter intra euclidean mahalanobis and bhattacharyya distances we then applied the proposed method to a gait phase recognition problem in our emg signal based walking assistance system we demonstrated that the proposed method performed competitively when compared with other wrapper type feature selection methods in terms of class separability and recognition rate

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

some mathematical structures for computational information abstract this paper is about the basic underlying mathematical structures of various types of imprecise information set theory and probability measures are basic ingredients in extracting some specific type of information and in reasoning with uncertain knowledge but in this era of information technology it is desirable to consider also more complex types of information for example information coming from human perception see e g l a zadeh ieee trans circuits and systems 45 1 1999 105 119 as a first approximation the classical ingredients could be generalized to adequately model new types of information as such we will describe in this paper four types of non classical sets namely random sets rough sets conditional sets and fuzzy sets as well as their associated non additive measures

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

input feature selection by mutual information based on parzen window mutual information is a good indicator of relevance between variables and have been used as a measure in several feature selection algorithms however calculating the mutual information is difficult and the performance of a feature selection algorithm depends on the accuracy of the mutual information in this paper we propose a new method of calculating mutual information between input and class variables based on the parzen window and we apply this to a feature selection algorithm for classification problems

neighborhood rough set based heterogeneous feature subset selection feature subset selection is viewed as an important preprocessing step for pattern recognition machine learning and data mining most of researches are focused on dealing with homogeneous feature selection namely numerical or categorical features in this paper we introduce a neighborhood rough set model to deal with the problem of heterogeneous feature subset selection as the classical rough set model can just be used to evaluate categorical features we generalize this model with neighborhood relations and introduce a neighborhood rough set model the proposed model will degrade to the classical one if we specify the size of neighborhood zero the neighborhood model is used to reduce numerical and categorical features by assigning different thresholds for different kinds of attributes in this model the sizes of the neighborhood lower and upper approximations of decisions reflect the discriminating capability of feature subsets the size of lower approximation is computed as the dependency between decision and condition attributes we use the neighborhood dependency to evaluate the significance of a subset of heterogeneous features and construct forward feature subset selection algorithms the proposed algorithms are compared with some classical techniques experimental results show that the neighborhood model based method is more flexible to deal with heterogeneous data

local affine image matching and synthesis based on structural patterns a general purpose block to block affine transformation estimator is described the estimator is based on fourier slice analysis and fourier spectral alignment it shows encouraging performance in terms of both speed and accuracy compared to existing methods the key elements of its success are attributed to the ability to 1 locate an arbitrary number of affine invariant points in the spectrum that latch onto significant structural features 2 match the estimated invariant points with the target spectrum by the slicewise phase correlation and 3 use affine invariant points to directly compute all linear parameters of the full affine transform by spectral alignment experimental results using a wide range of textures are presented potential applications include affine invariant image segmentation registration affine symmetric image coding and motion analysis

catoptrical rough set model on two universes using granule based definition and its variable precision extensions rough set models on two universes are valuable generalizations of classic rough set model these models can be applied into formal concept analysis and decision making system this paper focuses on catoptrical approach for rough set models over two universes using granule based definition in this paper we propose a new catoptrical rough set model over dual universes the properties of the proposed model are discussed consequently a variable precision extension of the model is presented some properties of the extended model are obtained finally we investigate a further extension of the proposed model i e variable precision catoptrical rough set model with two parameters

on a ktb modal fuzzy logic abstract an extension of the classical ktb modal logic to the fuzzy case is considered a new notion of similarity is introduced and a fuzzy logic based on this relation is proposed also the validity of formulas is defined by introducing a notion of logical consequence then the decision procedure of this logic is described in the tableau method

a robust minimax approach to classification when constructing a classifier the probability of correct classification of future data points should be maximized we consider a binary classification problem where the mean and covariance matrix of each class are assumed to be known no further assumptions are made with respect to the class conditional distributions misclassification probabilities are then controlled in a worst case setting that is under all possible choices of class conditional densities with given mean and covariance matrix we minimize the worst case maximum probability of misclassification of future data points for a linear decision boundary this desideratum is translated in a very direct way into a convex second order cone optimization problem with complexity similar to a support vector machine problem the minimax problem can be interpreted geometrically as minimizing the maximum of the mahalanobis distances to the two classes we address the issue of robustness with respect to estimation errors in the means and covariances of the classes via a simple modification of the input data we also show how to exploit mercer kernels in this setting to obtain nonlinear decision boundaries yielding a classifier which proves to be competitive with current methods including support vector machines an important feature of this method is that a worst case bound on the probability of misclassification of future data is always obtained explicitly

template matching in rotated images a rotationally invariant template matching using normalized invariant moments is described it is shown that if normalized invariant moments in circular windows are used then template matching in rotated images becomes similar to template matching in translated images a speedup technique based on the idea of two stage template matching is also described in this technique the zeroth order moment is used in the first stage to determine the likely match positions and the second and third order moments are used in the second stage to determine the best match position among the likely ones

attribute reduction based on generalized fuzzy evidence theory in fuzzy decision systems attribute reduction is viewed as an important issue in data mining and knowledge representation this paper studies attribute reduction in fuzzy decision systems based on generalized fuzzy evidence theory the definitions of several kinds of attribute reducts are introduced the relationships among these reducts are then investigated in a fuzzy decision system it is proved that the concepts of fuzzy positive region reduct lower approximation reduct and generalized fuzzy belief reduct are all equivalent the concepts of fuzzy upper approximation reduct and generalized fuzzy plausibility reduct are equivalent and a generalized fuzzy plausibility consistent set must be a generalized fuzzy belief consistent set in a consistent fuzzy decision system an attribute set is a generalized fuzzy belief reduct if and only if it is a generalized fuzzy plausibility reduct but in an inconsistent fuzzy decision system a generalized fuzzy belief reduct is not a generalized fuzzy plausibility reduct in general

tau functions as highest weight vectors for algebra for each we construct a highest weight module of the lie algebra the highest weight vectors are specific tau functions of the nth gelfand dickey hierarchy we show that these modules are quasifinite and we give a complete description of the reducible ones together with a formula for the singular vectors

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

a novel approach to building a robust fuzzy rough classifier currently most robust classifiers with parameters focus on the determination of the optimal or suboptimal parameters there are no research studies or even discussions about robust classifiers on all of the possible parameters this paper considers the robust rough classifier and finds that the robust rough classifier satisfies a nested topological structure then the nested classifier which reflects the classifier on all of the possible parameters is proposed first some notions such as the robust discernibility vector the robust value reduct and the robust covering vector are proposed these notions can reflect the classical corresponding notions on all of the possible parameters it is more important that these notions share a common characteristic the nested structure the nested structure of these notions makes nested classifier theoretically possible furthermore some novel algorithms are designed to compute the robust value reduct the robust covering degree and the robust classifier these algorithms make the nested classifier technologically possible finally numerical experiments demonstrate that the nested classifier is effective and efficient for classification and predication

constructive and algebraic methods of the theory of rough sets this paper reviews and compares constructive and algebraic approaches in the study of rough sets in the constructive approach one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation different classes of rough set algebras are obtained from different types of binary relations in the algebraic approach one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators various classes of rough set algebras are characterized by different sets of axioms axioms of approximation operators guarantee the existence of certain types of binary relations producing the same operators

exact poisson pencils τ structures and topological hierarchies abstract we discuss in the framework of the dubrovin zhang perturbative approach to integrable evolutionary pdes in 1 1 dimensions the role of a special class of poisson pencils called exact poisson pencils in particular we show that in the semisimple case exactness of the pencil is equivalent to the constancy of the so called central invariants of the theory that were introduced by dubrovin liu and zhang

rough set theory applied to lattice theory in this paper we intend to study a connection between rough sets and lattice theory we introduce the concepts of upper and lower rough ideals filters in a lattice then we offer some of their properties with regard to prime ideals filters the set of all fixed points compact elements and homomorphisms

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

a rst a generalization of rough set theory the paper presents a transition from the crisp rough set theory to a fuzzy one called alpha rough set theory or in short a rst all basic concepts or rough set theory are extended i e information system indiscernibility dependency reduction core definability approximations and boundary the resulted theory takes into account fuzzy data and allows the approximation of fuzzy concepts besides the control of knowledge granularity is natural in a rst which is based on a parameterized indiscernibility relation a rst is developed to recognize non deterministic relationships using notions as a dependency a reduct and so forth on the other hand we introduce a notion of relative dependency as an alternative of the absolute definibility presented in rough set theory the extension a rst leads naturally to the new concept of alpha rough sets which represents sets with fuzzy non empty boundaries 2000 elsevier science inc all rights reserved

robust fuzzy rough classifiers fuzzy rough sets generalized from pawlak s rough sets were introduced for dealing with continuous or fuzzy data this model has been widely discussed and applied these years it is shown that the model of fuzzy rough sets is sensitive to noisy samples especially sensitive to mislabeled samples as data are usually contaminated with noise in practice a robust model is desirable we introduce a new model of fuzzy rough set model called soft fuzzy rough sets and design a robust classification algorithm based on the model experimental results show the effectiveness of the proposed algorithm

rough set approach to incomplete information systems abstract in the paper we present rough set approach to reasoning in incomplete information systems we propose reduction of knowledge that eliminates only that information which is not essential from the point of view of classification or decision making in our approach we make only one assumption about unknown values the real value of a missing attribute is one from the attribute domain however we do not assume which one we show how to find decision rules directly from such an incomplete decision table which are as little non deterministic as possible and have minimal number of conditions

the nearest subclass classifier a compromise between the nearest mean and nearest neighbor classifier we present the nearest subclass classifier nsc which is a classification algorithm that unifies the flexibility of the nearest neighbor classifier with the robustness of the nearest mean classifier the algorithm is based on the maximum variance cluster algorithm and as such it belongs to the class of prototype based classifiers the variance constraint parameter of the cluster algorithm serves to regularize the classifier that is to prevent overfitting with a low variance constraint value the classifier turns into the nearest neighbor classifier and with a high variance parameter it becomes the nearest mean classifier with the respective properties in other words the number of prototypes ranges from the whole training set to only one per class in the experiments we compared the nsc with regard to its performance and data set compression ratio to several other prototype based methods on several data sets the nsc performed similarly to the k nearest neighbor classifier which is a well established classifier in many domains also concerning storage requirements and classification speed the nsc has favorable properties so it gives a good compromise between classification performance and efficiency

extension of phase correlation to subpixel registration in this paper we have derived analytic expressions for the phase correlation of downsampled images we have shown that for downsampled images the signal power in the phase correlation is not concentrated in a single peak but rather in several coherent peaks mostly adjacent to each other these coherent peaks correspond to the polyphase transform of a filtered unit impulse centered at the point of registration the analytic results provide a closed form solution to subpixel translation estimation and are used for detailed error analysis excellent results have been obtained for subpixel translation estimation of images of different nature and across different spectral bands

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

a two component model of texture for analysis and synthesis a model of natural texture based on a structural component that uses affine coordinate transformations and a stochastic residual component is presented it is argued that the selection of an appropriate analysis scale can be formulated in terms of a tradeoff between the rate at which parameters are generated and the distortion resulting from the approximation by the structural component an efficient algorithm for identifying the parameters of the structural model is described and its utility demonstrated on a number of synthetic and natural textures

differential equations in the spectral parameter for matrix differential operators we use matrix darboux transformation to generate a class of matrix differential operators l that have the following property there exist a family x k of eigenfunctions of l also satisfying a differential equation in the spectral parameter k of the form b k k x whereb k k is a diffrential and is a non constant function of x we also obtain nth order scalar operators which have this property

maxi min margin machine learning large margin classifiers locally and globally in this paper we propose a novel large margin classifier called the maxi min margin machine m4 this model learns the decision boundary both locally and globally in comparison other large margin classifiers construct separating hyperplanes only either locally or globally for example a state of the art large margin classifier the support vector machine svm considers data only locally while another significant model the minimax probability machine mpm focuses on building the decision hyperplane exclusively based on the global information as a major contribution we show that svm yields the same solution as m4 when data satisfy certain conditions and mpm can be regarded as a relaxation model of m4 moreover based on our proposed local and global view of data another popular model the linear discriminant analysis can easily be interpreted and extended as well we describe the m4 model definition provide a geometrical interpretation present theoretical justifications and propose a practical sequential conic programming method to solve the optimization problem we also show how to exploit mercer kernels to extend m4 for nonlinear classifications furthermore we perform a series of evaluations on both synthetic data sets and real world benchmark data sets comparison with svm and mpm demonstrates the advantages of our new model

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

a matrix integral solution to two dimensional w p gravity thep th gel fand dickey equation and the string equation l p 1 have a common solution expressible in terms of an integral overn n hermitean matrices for largen the integrand being a perturbation of a gaussian generalizing kontsevich s integral beyond the kdv case it is equivalent to showing that is a vacuum vector for aw generated from the coefficients of the vertex operator this connection is established via a quadratic identity involving the wave function and the vertex operator which is a disguised differential version of the fay identity the latter is also the key to the spectral theory for the two compatible symplectic structures of kdv in terms of the stress energy tensor associated with the virasoro algebra given a differential operator l d p q 2 t d p 2 cdots q p t with d frac partial dx t t 1 t 2 t 3 x equiv t 1 consider the deformation equations1 0 1 begin gathered frac partial l partial t n l n p l n 1 2 n 0 mod p hfill p reduced kp equation hfill end gathered ofl for which there exists a differential operatorp possibly of infinite order such that 0 2 l p 1 string equation in this note we give a complete solution to this problem in section 1 we give a brief survey of useful facts about thei function t the wave function t z solution of t n l n p x andl 1 p z and the corresponding infinitedimensional planev 0 of formal power series inz for largez v 0 span psi t z for all t in mathbb c infty in sato s grassmannian the three theorems below form the core of the paper their proof will be given in subseuqent sections each of which lives on its own right

generalized fuzzy rough sets determined by a triangular norm the theory of rough sets has become well established as an approach for uncertainty management in a wide variety of applications various fuzzy generalizations of rough approximations have been made over the years this paper presents a general framework for the study of t fuzzy rough approximation operators in which both the constructive and axiomatic approaches are used by using a pair of dual triangular norms in the constructive approach some definitions of the upper and lower approximation operators of fuzzy sets are proposed and analyzed by means of arbitrary fuzzy relations the connections between special fuzzy relations and the t upper and t lower approximation operators of fuzzy sets are also examined in the axiomatic approach an operator oriented characterization of rough sets is proposed that is t fuzzy approximation operators are defined by axioms different axiom sets of t upper and t lower fuzzy set theoretic operators guarantee the existence of different types of fuzzy relations producing the same operators the independence of axioms characterizing the t fuzzy rough approximation operators is examined then the minimal sets of axioms for the characterization of the t fuzzy approximation operators are presented based on information theory the entropy of the generalized fuzzy approximation space which is similar to shannon s entropy is formulated to measure uncertainty in t generalized fuzzy rough sets a notion of fuzziness is introduced some basic properties of this measure are examined for a special triangular norm t min it is proved that the measure of fuzziness of the generalized fuzzy rough set is equal to zero if and only if the set is crisp and definable

rule learning for classification based on neighborhood covering reduction rough set theory has been extensively discussed in the domain of machine learning and data mining pawlak s rough set theory offers a formal theoretical framework for attribute reduction and rule learning from nominal data however this model is not applicable to numerical data which widely exist in real world applications in this work we extend this framework to numerical feature spaces by replacing partition of universe with neighborhood covering and derive a neighborhood covering reduction based approach to extracting rules from numerical data we first analyze the definition of covering reduction and point out its advantages and disadvantages then we introduce the definition of relative covering reduction and develop an algorithm to compute it given a feature space we compute the neighborhood of each sample and form a neighborhood covering of the universe and then employ the algorithm of relative covering reduction to the neighborhood covering thus derive a minimal covering rule set some numerical experiments are presented to show the effectiveness of the proposed technique

twofold fuzzy sets and rough sets some issues in knowledge representation abstract this paper deals with the representation of sets where the membership of some elements may be ill known rather than just a matter of degree as in a fuzzy set the notion of a twofold fuzzy set is introduced when the relevant information for determining the membership status is incomplete a twofold fuzzy set is made of a nested pair of fuzzy sets the one which gathers the elements which more or less necessarily belong and the one which gathers the elements which more or less possibly belong twofold fuzzy sets are compared from a frontal and from a semantical point of view with other proposals and particularly with the notion of a rough set recently introduced by pawlak set operations of twofold fuzzy sets are discussed and the cardinality of a twofold fuzzy set is defined twofold fuzzy relations are also introduced finally various applications of twofold fuzzy sets in knowledge representation are briefly discussed

differential equations in the spectral parameter darboux transformations and a hierarchy of master symmetries for kdv we study a certain family of schrodinger operators whose eigenfunctions satisfy a differential equation in the spectral parameter of the formb x we show that the flows of a hierarchy of master symmetries for kdv are tangent to the manifolds that compose the strata of this class ofbispectral potentials this extends and complements a result of duistermaat and grunbaum concerning a similar property for the adler and moser potentials and the flows of the kdv hierarchy

fuzzy rough regression with application to wind speed prediction accurate wind speed prediction is a prerequisite of large scale wind power generation there are several uncertain factors which degrade the performance of the current wind speed prediction systems fuzzy rough sets are considered as a powerful tool to deal with uncertainty and have been widely discussed and applied in classification learning in this work we describe a regression algorithm based on fuzzy rough sets consisting of fuzzy partition fuzzy approximation and estimation of regression values in this algorithm the training set is divided into k fuzzy classes with fuzzy partition and then the predicted values of test samples are determined in the finite intervals with fuzzy rough approximation finally they are estimated with lower and upper limits of the intervals numerical experiments on uci data sets and wind speed prediction show the effectiveness of the proposed algorithm

stationary problems for equation of the kdv type and dynamical r matrices we study a new quite general family of dynamical r matrices for an auxiliary loop algebra l su 2 related to restricted flows for equations of the kdv type this underlying r matrix structure allows us to reconstruct lax representations and to find variables of separation for a wide set of the integrable natural hamiltonian systems

relationships between generalized rough sets based on covering and reflexive neighborhood system rough set theory is an important tool to deal with uncertainty granularity and incompleteness of knowledge the covering based rough sets is one of the most important extensions of the classical pawlak rough sets in covering based rough set theory many types of rough sets were already defined in the literature to find out the relationships among different types of covering based rough sets and give an axiomatic definition is a very important issue the generalized rough sets in neighborhood system is another important extension of the classical pawlak rough sets in this paper we investigate relationships between covering based rough sets and generalized rough sets in neighborhood system then twenty three types of the element based the granule based and the subsystem based definitions of covering approximation operators are unified under the framework of generalized approximation operators in neighborhood systems moreover properties of the generalized approximation operators in neighborhood systems are presented

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