extended nearest neighbor chain induced instance weights for svms prior knowledge plays an important role in increasing the performance of support vector machine svm one way to utilize prior knowledge is to assign different weight to each sample which is called weighted support vector machine wsvm in this paper a novel instance weighted method is proposed for support vector machines the weight is determined by the distances between a sequence of the nearest heterogeneous samples named the extended nearest neighbor chain the samples near the decision plane also locate near the heterogeneous samples thus the samples near the decision plane can be found by the extended nearest neighbor chain before svms training and then assigned higher weights meanwhile the samples far away from the decision plane are assigned lower weights this is consistent with the fact that the samples near the decision plane are more important for svms moreover this strategy can also assign lower weights to the samples which locate in the wrong classes the results of the experiments performed on both artificial synthetic datasets and real world benchmark datasets demonstrate that the proposed instance weighted method can further improve the performance of svms a novel instance weighted strategy is proposed for svms the samples near decision plane are assigned higher weights via the proposed method the sample near decision plane can be found by a chain named extended nearest neighbor chain the proposed method makes svm become more robust for noises

aspects of an adaptive finite element method for the fractional laplacian a priori and a posteriori error estimates efficient implementation and multigrid solver abstract we develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations on non trivial domains in d 1 dimensions our main approach consists of taking tools that have been shown to be effective for adaptive boundary element methods and where necessary modifying them so that they can be applied to the fractional pde case improved a priori error estimates are derived for the case of quasi uniform meshes which are seen to deliver sub optimal rates of convergence owing to the presence of singularities attention is then turned to the development of an a posteriori error estimate and error indicators which are suitable for driving an adaptive refinement procedure we assume that the resulting refined meshes are locally quasi uniform and develop efficient methods for the assembly of the resulting linear algebraic systems and their solution using iterative methods including the multigrid method the storage of the dense matrices along with efficient techniques for computing the dense matrix vector products needed for the iterative solution is also considered the performance and efficiency of the resulting algorithm is illustrated for a variety of examples

the sturm liouville equation with time dependent boundary conditions where a b denotes some x interval a p are real valued continuously differentiable functions on 0 t and t denotes some fixed positive number and q x t e c a b x 0 t in section 2 we show that the eigenvalues a are continuously differentiable functions of t if we drop the dependence of q x t on x then da dt can be given explicitly moreover a t d l dt is uniformly bounded in n e n and t e 0 t this is the content of section 3 in the section 4 the previous results are used to show that p t a l t is an operator in l2 a b holdercontinuous in t where a t is the operator defined by i 2 3 and p t represents a specific closed linear perturbation operator this type of assumption is made in connection with evolution equations see e g 6 p 1221 this result can be utilized to establish the existence of a solution of a certain boundary value problem connected with a non linear partial differential equation see 21 22 it is well known of course how the eigenvalues a change as the coefficient function q x varies and the boundary conditions are kept constant cf titchmarsh io for the singular case one can conclude with courant and hilbert 3 p 4 191 that the 12 th eigenvalue l depends continuously on the functions a p in 2 3 and q in l but the change depends on n 215 0022 247x 84 3 00

posterior probability support vector machines for unbalanced data this paper proposes a complete framework of posterior probability support vector machines ppsvms for weighted training samples using modified concepts of risks linear separability margin and optimal hyperplane within this framework a new optimization problem for unbalanced classification problems is formulated and a new concept of support vectors established furthermore a soft ppsvm with an interpretable parameter spl nu is obtained which is similar to the spl nu svm developed by scho spl uml lkopf et al and an empirical method for determining the posterior probability is proposed as a new approach to determine spl nu the main advantage of an ppsvm classifier lies in that fact that it is closer to the bayes optimal without knowing the distributions to validate the proposed method two synthetic classification examples are used to illustrate the logical correctness of ppsvms and their relationship to regular svms and bayesian methods several other classification experiments are conducted to demonstrate that the performance of ppsvms is better than regular svms in some cases compared with fuzzy support vector machines fsvms the proposed ppsvm is a natural and an analytical extension of regular svms based on the statistical learning theory

fuzzy regularized generalized eigenvalue classifier with a novel membership function supervised classification of data affected by noise or error with unknown probability distribution is a challenging task to this extend we propose the fuzzy regularized eigenvalue classifier based on a recent technique to classify data in two or more classes we compare the execution time and accuracy of the classifier with other de facto standard methods with the adoption of a novel membership function the classifier is capable to produce more accurate models that well compare with results obtained by other methods and fuzzy weighting functions on benchmark datasets

rapid and brief communication fuzzy discriminant analysis with kernel methods a novel fuzzy nonlinear classifier called kernel fuzzy discriminant analysis kfda is proposed to deal with linear non separable problem with kernel methods kfda can perform efficient classification in kernel feature space through some nonlinear mapping the input data can be mapped implicitly into a high dimensional kernel feature space where nonlinear pattern now appears linear different from fuzzy discriminant analysis fda which is based on euclidean distance kfda uses kernel induced distance theoretical analysis and experimental results show that the proposed classifier compares favorably with fda

the theoretical foundations of statistical learning theory based on fuzzy number samples statistical learning theory based on real valued random samples has been regarded as a better theory on statistical learning with small sample the key theorem of learning theory and bounds on the rate of convergence of learning processes are important theoretical foundations of statistical learning theory in this paper the theoretical foundations of the statistical learning theory based on fuzzy number samples are discussed the concepts of fuzzy expected risk functional fuzzy empirical risk functional and fuzzy empirical risk minimization principle are redefined the key theorem of learning theory based on fuzzy number samples is proved furthermore the bounds on the rate of convergence of learning processes based on fuzzy number samples are discussed

color image watermark extraction based on support vector machines abstract this work proposes a novel watermarking technique called svm based color image watermarking sciw based on support vector machines svms for the authentication of color images to protect the copyright of a color image a signature a watermark which is represented by a sequence of binary data is embedded in the color image the watermark extraction issue can be treated as a classification problem involving binary classes the sciw method constructs a set of training patterns with the use of binary labels by employing three image features which are the differences between a local image statistic and the luminance value of the center pixel in a sliding window with three distinct shapes this set of training patterns is gathered from a pair of images an original image and its corresponding watermarked image in the spatial domain a quasi optimal hyperplane a binary classifier can be realized by an svm the sciw method utilizes this set of training patterns to train the svm and then applies the trained svm to classify a set of testing patterns following the results produced by the classifier the trained svm the sciw method retrieves the hidden signature without the original image during watermark extraction experimental results have demonstrated that the sciw method is sufficiently robust against several color image manipulations and that it outperforms other proposed methods considered in this work

a numerical scheme for impact problems i the one dimensional case we consider a mechanical system with impact and one degree of freedom the system is not necessarily lagrangian the representative point is subject to the constraint u t in er for all t we assume that at impact the velocity is reversed and multiplied by a given coefficient of restitution e in 0 1 we define a numerical scheme which enables us to approximate the solutions of the cauchy problem this is an ad hoc scheme which does not require a systematic search for the times of impact we prove the convergence of this numerical scheme to a solution many of the features of this proof will be reused in the nonconvex multidimensional case written in generalized coordinates given in the companion paper l paoli and m schatzman siam j numer anal 40 2002 pp 734 768 we present some numerical results obtained with the scheme for a spring dashpot system and we compare them to the results obtained by impact detection and penalization

periodic solutions of semilinear equations of evolution of compact type abstract the existence of solutions in a weak sense of x a b t x x f t x x 0 x t is established under the conditions that a generates a semigroup of compact type on a hilbert space h b t x is a bounded linear operator and f t x a function with values in h for each square integrable t the problem with b t t and f t t in place of b t x and f t x has a unique solution and b and f satisfy certain boundedness and continuity conditions

a regularized monotonic fuzzy support vector machine model for data mining with prior knowledge incorporating prior knowledge into data mining is an interesting but challenging problem and this study proposes a novel fuzzy support vector machine svm model to explore this issue it considers the fact that in many applications each input point may not be exactly labeled as one particular class and thus it applies a fuzzy membership to each input point it also utilizes expert knowledge concerning the monotonic relations between the response and predictor variables which is represented in the form of monotonicity constraints we formulate the classification problem of a monotonically constrained fuzzy svm called a monotonic fsvm derive its dual optimization problem and theoretically analyze its monotonic property the tikhonov regularization method is further applied to ensure that the solution is unique and bounded a new measure i e the frequency monotonicity rate is proposed to evaluate the ability of the model to retain the monotonicity the results of the experiments on real world and synthetic datasets show that this method which considers different contributions of each data and the prior knowledge of the monotonicity has a number of advantages with regard to predictive ability and retaining monotonicity over the original fsvm and svm models when applied to classification problems

time like minimal submanifolds as singular limits of nonlinear wave equations abstract we consider the sharp interface limit 0 of the semilinear wave equation u w u 2 0 in r 1 n where u takes values in r k k 1 2 and w is a double well potential if k 1 and vanishes on the unit circle and is positive elsewhere if k 2 for fixed 0 we find some special solutions constructed around minimal surfaces in r n in the general case under some additional assumptions we show that the solutions converge to a radon measure supported on a time like k codimensional minimal submanifold of the minkowski space time this result holds also after the appearance of singularities and enforces the observation made by j neu that this semilinear equation can be regarded as an approximation of the born infeld equation

nonlinear wave equations as limits of convex minimization problems proof of a conjecture by de giorgi we prove a conjecture by de giorgi which states that global weak solutions of nonlinear wave equations such as w jwj p 2 w 0 can be obtained as limits of functions that minimize suitable functionals of the calculus of variations these functionals which are integrals in space time of a convex lagrangian contain an exponential weight with a parameter and the initial data of the wave equation serve as boundary conditions as tends to zero the minimizers v converge up to subsequences to a solution of the nonlinear wave equation there is no restriction on the nonlinearity exponent and the method is easily extended to more general equations

a study on relationship between generalization abilities and fuzziness of base classifiers in ensemble learning we investigate essential relationships between generalization capabilities and fuzziness of fuzzy classifiers viz the classifiers whose outputs are vectors of membership grades of a pattern to the individual classes the study makes a claim and offers sound evidence behind the observation that higher fuzziness of a fuzzy classifier may imply better generalization aspects of the classifier especially for classification data exhibiting complex boundaries this observation is not intuitive with a commonly accepted position in traditional pattern recognition the relationship that obeys the conditional maximum entropy principle is experimentally confirmed furthermore the relationship can be explained by the fact that samples located close to classification boundaries are more difficult to be correctly classified than the samples positioned far from the boundaries this relationship is expected to provide some guidelines as to the improvement of generalization aspects of fuzzy classifiers

the probabilistic constraints in the support vector machine in this paper a new support vector machine classifier with probabilistic constrains is proposed which presence probability of samples in each class is determined based on a distribution function noise is caused incorrect calculation of support vectors thereupon margin can not be maximized in the proposed method constraints boundaries and constraints occurrence have probability density functions which it help for achieving maximum margin experimental results show superiority of the probabilistic constraints support vector machine pc svm relative to standard svm

the theoretical fundamentals of learning theory based on fuzzy complex random samples statistical learning theory based on real valued random samples has been regarded as one of the influential developments for small samples statistical estimation and learning the key theorem of learning theory and the bounds on the rate of convergence of learning process are the most important theoretical fundamentals of the statistical learning theory in this paper we discuss a statistical learning theory based on fuzzy complex random samples firstly the definition of fuzzy complex numbers is introduced and the fuzzy complex random variables along with their numeric characteristic are investigated secondly we carry out further research focused on a special type of fuzzy complex number namely rectangular fuzzy complex number and establish some properties and develop important theorems we also prove the strong law of large numbers based on fuzzy complex random variables thirdly the definitions of the fuzzy complex expected risk functional the fuzzy complex empirical risk functional the fuzzy complex empirical risk minimization principle and the consistency are provided and discussed finally the key theorem of learning theory and the bounds on the rate of convergence of learning process based on fuzzy complex random samples are discussed

centered kernel alignment inspired fuzzy support vector machine abstract support vector machine svm is a theoretically well motivated algorithm developed from statistical learning theory which has shown impressive performance in many fields in spite of its success it still suffers from the noise sensitivity problem originating from the assumption that each training point has equal importance or weight in the training process to relax this problem the svm was extended to the fuzzy svm fsvm by applying a fuzzy membership to each training point such that different training points can make different contributions to the learning of the decision surface although well determined fuzzy memberships can improve classification performance there are no general guidelines for their construction in this paper inspired by the centered kernel alignment cka which measures the degree of similarity between two kernels or kernel matrices we propose a new fuzzy membership function calculation method in which a heuristic function derived from the cka is used to calculate the dependence between a data point and its associated label although the cka induced fsvm is similar to the kernel target alignment kta induced fsvm there is actually a critical difference without that centering the definition of alignment does not correlate well with the performance of learning machines extensive experiments are performed on real world data sets from the uci benchmark repository and the application domain of computational biology which validate the superiority of the proposed fsvm model in terms of several classification performance measures

single object tracking with fuzzy least squares support vector machine single object tracking in which a target is often initialized manually in the first frame and then is tracked and located automatically in the subsequent frames is a hot topic in computer vision the traditional tracking by detection framework which often formulates tracking as a binary classification problem has been widely applied and achieved great success in single object tracking however there are some potential issues in this formulation for instance the boundary between the positive and negative training samples is fuzzy and the objectives of tracking and classification are inconsistent in this paper we attempt to address the above issues from the fuzzy system perspective and propose a novel tracking method by formulating tracking as a fuzzy classification problem first we introduce the fuzzy strategy into tracking and propose a novel fuzzy tracking framework which can measure the importance of the training samples by assigning different memberships to them and offer more strict spatial constraints second we develop a fuzzy least squares support vector machine fls svm approach and employ it to implement a concrete tracker in particular the primal form dual form and kernel form of fls svm are analyzed and the corresponding closed form solutions are derived for efficient realizations besides a least squares regression model is built to control the update adaptively retaining the robustness of the appearance model the experimental results demonstrate that our method can achieve comparable or superior performance to many state of the art methods

mathematical modelling of surface vibration with volume constraint and its analysis abstract a hyperbolic problem with volume constraint is investigated by means of a minimizing method called the discrete morse flow a weak solution is introduced in a constructive way and results of numerical computations are shown

fuzzy support vector machines a support vector machine svm learns the decision surface from two distinct classes of the input points in many applications each input point may not be fully assigned to one of these two classes in this paper we apply a fuzzy membership to each input point and reformulate the svms such that different input points can make different contributions to the learning of decision surface we call the proposed method fuzzy svms fsvms

constructing l2 svm based fuzzy classifiers in high dimensional space with automatic model selection and fuzzy rule ranking in this paper a new scheme for constructing parsimonious fuzzy classifiers is proposed based on the l2 support vector machine l2 svm technique with model selection and feature ranking performed simultaneously in an integrated manner in which fuzzy rules are optimally generated from data by l2 svm learning in order to identify the most influential fuzzy rules induced from the svm learning two novel indexes for fuzzy rule ranking are proposed and named as alpha values and omega values of fuzzy rules in this paper the alpha values are defined as the lagrangian multipliers of the l2 svm and adopted to evaluate the output contribution of fuzzy rules while the omega values are developed by considering both the rule base structure and the output contribution of fuzzy rules as a prototype based classifier the l2 svm based fuzzy classifier evades the curse of dimensionality in high dimensional space in the sense that the number of support vectors which equals the number of induced fuzzy rules is not related to the dimensionality experimental results on high dimensional benchmark problems have shown that by using the proposed scheme the most influential fuzzy rules can be effectively induced and selected and at the same time feature ranking results can also be obtained to construct parsimonious fuzzy classifiers with better generalization performance than the well known algorithms in literature

fuzzy passive aggressive classification a robust and efficient algorithm for online classification problems fuzzy weighting which is designed to reduce the effects of outliers for batch classification problems might generate unreasonable membership grades especially for the samples following an input outlier when incorporated into online classification algorithms directly in this paper a generalized framework for online fuzzy weighting is presented which incrementally calculates the membership of each incoming sample by taking into account the membership grades of previous samples in a pairwise manner the advocated pairwise distance based scheme can not only identify possible outliers but also show good adaptation to the sequentially received samples in the online setting we apply it to online passive aggressive pa algorithm in a direct way the resulting fuzzy passive aggressive fpa algorithm achieves comparable classification accuracy with benchmark incremental svm while still enjoying the time efficiency of simple pa which is a perceptron like algorithm besides fpa exhibits the best performance among pa family which makes it a robust and efficient alternative to pa in order to deal with unavoidable outliers in large scale or high dimensional real datasets the study is supported by a series of experiments with ida benchmark repository as well as two real world problems namely place recognition and radar emitter recognition

a vector valued support vector machine model for multiclass problem in this paper a new model named multiclass support vector machines with vector valued decision m svms vvd or vvd is proposed the basic idea is to separate 2 a classes by a svm hyperplanes in the feature space induced by certain kernels where a is a finite positive integer we start from a 2 a class problem and extend it to any class problem by applying a hierarchical decomposition procedure compared with the existing svm based multiclass methods the vvd model has two advantages first it reduces the computational complexity by using a small number of classifiers second the feature space partition induced by the hyperplanes effectively eliminates the unclassifiable regions urs that may affect the classification performance of the algorithm experimental comparisons with several state of the art multiclass methods demonstrate that vvd maintains a comparable testing accuracy while it improves the classification efficiency with less classifiers a smaller number of support vectors svs and shorter testing time

fuzzy support vector machine for imbalanced data with borderline noise abstract this work is an extension of the fuzzy support vector machines for class imbalance learning fsvm cil method proposed by rukshan batuwita and vasile palade for fsvms a very important part is the fuzzy function transforming different distance measures to membership values between 0 and 1 the larger the membership value the more important the corresponding training data point although various variants have been proposed recently few have discussed proper fuzzy functions this work first shows the limitations of fuzzy functions in original fsvm cil for imbalanced data with noise around the between class borderline noted as borderline noise in this paper and then a new fuzzy function named the gaussian fuzzy function is proposed and explained in detail modifications are also made to the current distance measures experiments on several public imbalanced datasets show the effectiveness of the proposed methods through the comparison with fsvm cil and several other popular approaches for imbalanced data

support vector machines with genetic fuzzy feature transformation for biomedical data classification abstract in this paper we present a genetic fuzzy feature transformation method for support vector machines svms to do more accurate data classification given data are first transformed into a high feature space by a fuzzy system and then svms are used to map data into a higher feature space and then construct the hyperplane to make a final decision genetic algorithms are used to optimize the fuzzy feature transformation so as to use the newly generated features to help svms do more accurate biomedical data classification under uncertainty the experimental results show that the new genetic fuzzy svms have better generalization abilities than the traditional svms in terms of prediction accuracy

a parsimony fuzzy rule based classifier using axiomatic fuzzy set theory and support vector machines this paper proposes a classification method that is based on easily interpretable fuzzy rules and fully capitalizes on the two key technologies namely pruning the outliers in the training data by svms support vector machines i e eliminating the influence of outliers on the learning process finding a fuzzy set with sound linguistic interpretation to describe each class based on afs axiomatic fuzzy set theory compared with other fuzzy rule based methods the proposed models are usually more compact and easily understandable for the users since each class is described by much fewer rules the proposed method also comes with two other advantages namely each rule obtained from the proposed algorithm is simply a conjunction of some linguistic terms there are no parameters that are required to be tuned the proposed classification method is compared with the previously published fuzzy rule based classifiers by testing them on 16 uci data sets the results show that the fuzzy rule based classifier presented in this paper offers a compact understandable and accurate classification scheme a balance is achieved between the interpretability and the accuracy

a kernel fuzzy c means clustering based fuzzy support vector machine algorithm for classification problems with outliers or noises the support vector machine svm has provided higher performance than traditional learning machines and has been widely applied in real world classification problems and nonlinear function estimation problems unfortunately the training process of the svm is sensitive to the outliers or noises in the training set in this paper a common misunderstanding of gaussian function based kernel fuzzy clustering is corrected and a kernel fuzzy c means clustering based fuzzy svm algorithm kfcm fsvm is developed to deal with the classification problems with outliers or noises in the kfcm fsvm algorithm we first use the fcm clustering to cluster each of two classes from the training set in the high dimensional feature space the farthest pair of clusters where one cluster comes from the positive class and the other from the negative class is then searched and forms one new training set with membership degrees finally we adopt fsvm to induce the final classification results on this new training set the computational complexity of the kfcm fsvm algorithm is analyzed a set of experiments is conducted on six benchmarking datasets and four artificial datasets for testing the generalization performance of the kfcm fsvm algorithm the results indicate that the kfcm fsvm algorithm is robust for classification problems with outliers or noises

a variational scheme for hyperbolic obstacle problems abstract we consider an obstacle problem for possibly non local wave equations and we prove existence of weak solutions through a convex minimization approach based on a time discrete approximation scheme we provide the corresponding numerical implementation and raise some open questions

a variational approach to a constrained hyperbolic free boundary problem abstract this paper concerns the construction of a weak solution to a constrained hyperbolic equation in one space dimension the problem includes a nonlocal term corresponding to a volume constraint and the presence of a free boundary we approach our problem by means of a variational method known as the discrete morse flow

regularity of solutions to the parabolic fractional obstacle problem in recent years there has been an increasing interest in studying constrained variational problems with a fractional diffusion one of the motivations comes from mathematical finance jumpdiffusion processes where incorporated by merton 14 into the theory of option evaluation to introduce discontinuous paths in the dynamics of the stock s prices in contrast with the classical lognormal diffusion model of black and scholes 2 these models allow to take into account large price changes and they have become increasingly popular for modeling market fluctuations both for risk management and option pricing purposes let us recall that an american option gives its holder the right to buy a stock at a given price prior but not later than a given time t 0 if v x represents the rational price of an american option with a payoff at time t 0 then v will solve in the viscosity sense the following obstacle problem min lv v 0 v t

a new fuzzy support vector machine to evaluate credit risk due to recent financial crises and regulatory concerns financial intermediaries credit risk assessment is an area of renewed interest in both the academic world and the business community in this paper we propose a new fuzzy support vector machine to discriminate good creditors from bad ones because in credit scoring areas we usually cannot label one customer as absolutely good who is sure to repay in time or absolutely bad who will default certainly our new fuzzy support vector machine treats every sample as both positive and negative classes but with different memberships by this way we expect the new fuzzy support vector machine to have more generalization ability while preserving the merit of insensitive to outliers as the fuzzy support vector machine svm proposed in previous papers we reformulate this kind of two group classification problem into a quadratic programming problem empirical tests on three public datasets show that it can have better discriminatory power than the standard support vector machine and the fuzzy support vector machine if appropriate kernel and membership generation method are chosen

regularity of the obstacle problem for the parabolic biharmonic equation we consider the obstacle problem for the parabolic biharmonic equation we study the problem via an implicit time discretization we prove the existence of a unique solution and discuss its regularity property

fuzzy linear discriminant analysis guided maximum entropy fuzzy clustering algorithm linear discriminant analysis lda is a classical statistical approach for supervised feature extraction and dimensionality reduction hard c means hcm is a classical unsupervised learning algorithm for clustering based on the analysis of the relationship between lda and hcm linear discriminant analysis guided adaptive subspace hard c means clustering algorithm lda hcm had been proposed lda hcm combines lda and hcm into a coherent framework and can adaptively reduce the dimension of data while performing data clustering simultaneously seeing that lda hcm is still a hard clustering algorithm we consider the fuzzy extension version of lda hcm in this paper to this end firstly we propose a new optimization criterion of fuzzy linear discriminant analysis flda by extending the value of membership function in classical lda from binary 0 or 1 into closed interval 0 1 in the meantime we present an efficient algorithm for the proposed flda secondly we show the close relationship between flda and maximum entropy fuzzy clustering algorithm mefca they both are maximizing fuzzy between class scatter and minimizing within class scatter simultaneously finally based on the above analysis combining flda and mefca into a joint framework we propose fuzzy linear discriminant analysis guided maximum entropy fuzzy clustering algorithm flda mefca lda mefca is a natural and effective fuzzy extension of lda hcm due to the introduction of soft decision strategy flda mefca can yield fuzzy partition of data set and is more flexible than lda hcm we also give the convergence proof of flda mefca extensive experiments on a collection of benchmark data sets are presented to show the effectiveness of the proposed algorithm

ssc eke semi supervised classification with extensive knowledge exploitation we introduce a new semi supervised classification method that extensively exploits knowledge the method has three steps first the manifold regularization mechanism adapted from the laplacian support vector machine lapsvm is adopted to mine the manifold structure embedded in all training data especially in numerous label unknown data meanwhile by converting the labels into pairwise constraints the pairwise constraint regularization formula pcrf is designed to compensate for the few but valuable labelled data second by further combining the pcrf with the manifold regularization the precise manifold and pairwise constraint jointly regularized formula mpcjrf is achieved third by incorporating the mpcjrf into the framework of the conventional svm our approach referred to as semi supervised classification with extensive knowledge exploitation ssc eke is developed the significance of our research is fourfold 1 the mpcjrf is an underlying adjustment with respect to the pairwise constraints to the graph laplacian enlisted for approximating the potential data manifold this type of adjustment plays the correction role as an unbiased estimation of the data manifold is difficult to obtain whereas the pairwise constraints converted from the given labels have an overall high confidence level 2 by transforming the values of the two terms in the mpcjrf such that they have the same range with a trade off factor varying within the invariant interval 0 1 the appropriate impact of the pairwise constraints to the graph laplacian can be self adaptively determined 3 the implication regarding extensive knowledge exploitation is embodied in ssc eke that is the labelled examples are used not only to control the empirical risk but also to constitute the mpcjrf moreover all data both labelled and unlabelled are recruited for the model smoothness and manifold regularization 4 the complete framework of ssc eke organically incorporates multiple theories such as joint manifold and pairwise constraint based regularization smoothness in the reproducing kernel hilbert space empirical risk minimization and spectral methods which facilitates the preferable classification accuracy as well as the generalizability of ssc eke

application of multi class fuzzy support vector machine classifier for fault diagnosis of wind turbine this paper presents an approach for fault diagnosis of wind turbine wt based on multi class fuzzy support vector machine fsvm classifier in this method empirical mode decomposition is adopted to extract fault feature vectors from vibration signals fsvm is used for solving classification problem with outliers or noises where kernel fuzzy c means clustering algorithm and particle swarm optimization algorithm are applied to calculate fuzzy membership and optimize the parameters of kernel function of fsvm respectively in addition to study the performance of the proposed approach another two widely used methods named back propagation neural network and standard support vector machine are studied and compared discrete wavelet transform is also used to extract fault feature vectors to validate the proposed approach a direct drive wt test rig is constructed and the experiments are carried out the experimental results show that the proposed approach is an effective fault diagnosis method for wt which has a better performance and can achieve higher diagnostic accuracy a multi class fsvm classifier constructed by one against other method is used for fault diagnosis of wind turbine kfcm algorithm is extended to calculate fuzzy membership values of training samples for a multi class fsvm classifier pso algorithm is applied to optimize the parameters of the kernel function of fsvm fault diagnosis steps are presented and the effectiveness of the proposed method is validated

a rough margin based support vector machine by introducing the rough set theory into the support vector machine svm a rough margin based svm rmsvm is proposed to deal with the overfitting problem due to outliers similar to the classical svm the rmsvm searches for the separating hyper plane that maximizes the rough margin defined by the lower and upper margin in this way more data points are adaptively considered rather than the few extreme value points used in the classical svm in addition different support vectors may have different effects on the learning of the separating hyper plane depending on their positions in the rough margin points in the lower margin have more effects than those in the boundary of the rough margin from experimental results on six benchmark datasets the classification accuracy of this algorithm is improved without additional computational expense compared with the classical n svm

variational approach to evolutionary free boundary problems abstract the analysis of evolutionary nonlocal problems represented by volume constrained motion is discussed in the physical sense these problems can describe the motion of droplets on a solid plane the variational approach called the discrete morse flow combined with a fluid dynamics model appears to be a suitable tool

a construction of a lipschitz continuous minimizer of a free boundary problem abstract the free boundary problem of the variational type studied by alt and caffarelli is investigated here especially treated is the quadratic functional having the coefficient of diagonal form and the characteristic function term the main result is to construct a minimizer with the lipschitz continuity which will play an essential role for proving the regularity of the free boundary this is established by treating the regularized functional which approximates the original functional in the sense of convergence the most important part of this paper is to show the uniform lipschitz continuity of the minimizers of the regularized functionals taking advantage of the minimality not resorting to the euler lagrange equation the minimizers turn out to belong to degiorgi class uniformly the uniform holder estimate is obtained and as a result the lipschitz estimate is attained with the aid of the uniform harnack inequality

on support vector regression machines with linguistic interpretation of the kernel matrix initially the idea of approximate reasoning using generalized modus ponens and a fuzzy implication is recalled next a fuzzy system based on logical interpretation of if then rules and with parametric conclusions is presented then it is shown that global and local e insensitive learning of the above fuzzy system may be presented as the learning of a support vector regression machine with a special type of a kernel matrix obtained from clustering the kernel matrix may be interpreted in terms of linguistic values based on the premises of if then rules a new method of obtaining a fuzzy system by means of a support vector machine svm with a data dependent kernel matrix is introduced this paper contains examples of a svm used to design fuzzy models of real life data simulation results show an improvement in the generalization ability of a fuzzy system learned by the new method compared with traditional learning methods

robust kernel discriminant analysis using fuzzy memberships linear discriminant analysis lda is a simple but widely used algorithm in the area of pattern recognition however it has some shortcomings in that it is sensitive to outliers and limited to linearly separable cases to solve these problems in this paper a non linear robust variant of lda called robust kernel fuzzy discriminant analysis rkfda is proposed rkfda uses fuzzy memberships to reduce the effect of outliers and adopts kernel methods to accommodate non linearly separable cases there have been other attempts to solve the problems of lda including attempts using kernels however rkfda encompassing previous methods is the most general one furthermore theoretical analysis and experimental results show that rkfda is superior to other existing methods in solving the problems

support vector based fuzzy neural network for pattern classification fuzzy neural networks fnns for pattern classification usually use the backpropagation or c cluster type learning algorithms to learn the parameters of the fuzzy rules and membership functions from the training data however such kinds of learning algorithms usually cannot minimize the empirical risk training error and expected risk testing error simultaneously and thus cannot reach a good classification performance in the testing phase to tackle this drawback a support vector based fuzzy neural network svfnn is proposed for pattern classification in this paper the svfnn combines the superior classification power of support vector machine svm in high dimensional data spaces and the efficient human like reasoning of fnn in handling uncertainty information a learning algorithm consisting of three learning phases is developed to construct the svfnn and train its parameters in the first phase the fuzzy rules and membership functions are automatically determined by the clustering principle in the second phase the parameters of fnn are calculated by the svm with the proposed adaptive fuzzy kernel function in the third phase the relevant fuzzy rules are selected by the proposed reducing fuzzy rule method to investigate the effectiveness of the proposed svfnn classification it is applied to the iris vehicle dna satimage ijcnn1 datasets from the uci repository statlog collection and ijcnn challenge 2001 respectively experimental results show that the proposed svfnn for pattern classification can achieve good classification performance with drastically reduced number of fuzzy kernel functions

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

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

robust support vector machines based on the rescaled hinge loss function abstract the support vector machine svm is a popular classifier in machine learning but it is not robust to outliers in this paper based on the correntropy induced loss function we propose the rescaled hinge loss function which is a monotonic bounded and nonconvex loss that is robust to outliers we further show that the hinge loss is a special case of the proposed rescaled hinge loss then we develop a new robust svm based on the rescaled hinge loss after using the half quadratic optimization method we find that the new robust svm is equivalent to an iterative weighted svm which can help explain the robustness of iterative weighted svm from a loss function perspective experimental results confirm that the new robust svm not only performs better than svm and the existing robust svms on the datasets that have outliers but also presents better sparseness than svm

regularity of a free boundary in parabolic potential theory we study the regularity of the free boundary in a stefan type problem 1u t u in d rn r u u 0 on d with no sign assumptions on u and the time derivative t u

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

credit risk evaluation using multi criteria optimization classifier with kernel fuzzification and penalty factors abstract with the fast development of financial products and services bank s credit departments collected large amounts of data which risk analysts use to build appropriate credit scoring models to evaluate an applicant s credit risk accurately one of these models is the multi criteria optimization classifier mcoc by finding a trade off between overlapping of different classes and total distance from input points to the decision boundary mcoc can derive a decision function from distinct classes of training data and subsequently use this function to predict the class label of an unseen sample in many real world applications however owing to noise outliers class imbalance nonlinearly separable problems and other uncertainties in data classification quality degenerates rapidly when using mcoc in this paper we propose a novel multi criteria optimization classifier based on kernel fuzzification and penalty factors kfp mcoc firstly a kernel function is used to map input points into a high dimensional feature space then an appropriate fuzzy membership function is introduced to mcoc and associated with each data point in the feature space and the unequal penalty factors are added to the input points of imbalanced classes thus the effects of the aforementioned problems are reduced our experimental results of credit risk evaluation and their comparison with mcoc support vector machines svm and fuzzy svm show that kfp mcoc can enhance the separation of different applicants the efficiency of credit risk scoring and the generalization of predicting the credit rank of a new credit applicant

constructing a solution in time semidiscretization method to an equation of vibrating string with an obstacle this article treats a problem which describs movement of a string that hits to an obstacle in 10 m schatzman solves this problem in a slightly classical way in 5 6 k maruo constructs a solution to this problem by the use of yosida approximation the purpose of this article is to construct a solution to this problem in time semidiscretization method in general approximation by time semidiscretization method is dierent from yosida approximation a simple example is presented in appendix

fuzzy classification boundaries against adversarial network attacks abstract adversarial machine learning copes with the development of methods to prevent machine learning algorithms from being misled by malicious users this field is especially relevant for applications where machine learning lies in the core of security systems in the field of network security adversarial samples are actually novel network attacks or old attacks with tuned properties this paper proposes to blur classification boundaries in order to enhance machine learning robustness and improve the detection of adversarial samples that exploit learning weaknesses we test this concept by an experimental setup with network traffic in which linear decision trees are wrapped by a one class membership scoring algorithm we benchmark our proposal with plain linear decision trees and fuzzy decision trees results show that evasive attacks i e false negatives tend to be ranked with low class membership levels meaning that they are located in zones close to classification thresholds in addition classification performances improve when membership scores are added as new features using fuzzy class boundaries is highly consistent with the interpretation of many network traffic features used for malware detection moreover it prevents network attackers from exploiting classification boundaries as attack objectives

complex system fault diagnosis based on a fuzzy robust wavelet support vector classifier and an adaptive gaussian particle swarm optimization this paper proposes a robust loss function that penalizes hybrid noise i e gaussian noise singularity points and larger magnitude noise in a complex fuzzy fault diagnosis system a mapping relationship between fuzzy numbers and crisp real numbers that allows a fuzzy sample set to be transformed into a crisp real sample set is also presented furthermore the paper proposes a novel fuzzy robust wavelet support vector classifier frwsvc based on a wavelet base function and develops an adaptive gaussian particle swarm optimization agpso algorithm to seek the optimal unknown parameter of the frwsvc the results of experiments that apply the hybrid diagnosis model based on the frwsvc and the agpso algorithm to fault diagnosis demonstrate that it is both feasible and effective tests comparing the method proposed in this paper against other fuzzy support vector classifier fsvc machines show that it outperforms them

fuzzy one class support vector machines in one class classification the problem is to distinguish one class of data from the rest of the feature space it is important in many applications where one of the classes is characterized well while no measurements are available for the other class scholkopf et al first introduced a method of adapting the support vector machine svm methodology to the one class classification problem called one class svm in this paper we incorporate the concept of fuzzy set theory into the one class svm we apply a fuzzy membership to each input point and reformulate the one class svm such that different input points can make different contributions to the learning of decision surface besides the parameters to be identified in one class svm such as the components within the weight vector and the bias term are fuzzy numbers this integration preserves the benefits of svm learning theory and fuzzy set theory where the svm learning theory characterizes the properties of learning machines which enable them to effectively generalize the unseen data and the fuzzy set theory might be very useful for finding a fuzzy structure in an evaluation system

face recognition using total margin based adaptive fuzzy support vector machines this paper presents a new classifier called total margin based adaptive fuzzy support vector machines taf svm that deals with several problems that may occur in support vector machines svms when applied to the face recognition the proposed taf svm not only solves the overfitting problem resulted from the outlier with the approach of fuzzification of the penalty but also corrects the skew of the optimal separating hyperplane due to the very imbalanced data sets by using different cost algorithm in addition by introducing the total margin algorithm to replace the conventional soft margin algorithm a lower generalization error bound can be obtained those three functions are embodied into the traditional svm so that the taf svm is proposed and reformulated in both linear and nonlinear cases by using two databases the chung yuan christian university cycu multiview and the facial recognition technology feret face databases and using the kernel fisher s discriminant analysis kfda algorithm to extract discriminating face features experimental results show that the proposed taf svm is superior to svm in terms of the face recognition accuracy the results also indicate that the proposed taf svm can achieve smaller error variances than svm over a number of tests such that better recognition stability can be obtained

tsk fuzzy modeling based on spl epsiv insensitive learning in this paper new learning methods tolerant to imprecision are introduced and applied to fuzzy modeling based on the takagi sugeno kang fuzzy system the fuzzy modeling has an intrinsic inconsistency it may perform thinking tolerant to imprecision but learning methods are zero tolerant to imprecision the proposed methods make it possible to exclude this intrinsic inconsistency of a fuzzy modeling where zero tolerance learning is used to obtain fuzzy model tolerant to imprecision these new methods can be called spl epsiv insensitive learning or spl epsiv learning where in order to fit the fuzzy model to real data the spl epsiv insensitive loss function is used this leads to a weighted or fuzzified version of vapnik s support vector regression machine this paper introduces two approaches to solving the spl epsiv insensitive learning problem the first approach leads to the quadratic programming problem with bound constraints and one linear equality constraint the second approach leads to a problem of solving a system of linear inequalities two computationally efficient numerical methods for the spl epsiv insensitive learning are proposed the spl epsiv insensitive learning leads to a model with the minimal vapnik chervonenkis dimension which results in an improved generalization ability of this model and its outliers robustness finally numerical examples are given to demonstrate the validity of the introduced methods

a novel technique for subpixel image classification based on support vector machine this paper presents a novel support vector machine classifier designed for subpixel image classification pixel spectral unmixing the proposed classifier generalizes the properties of svms to the identification and modeling of the abundances of classes in mixed pixels by using fuzzy logic this results in the definition of a fuzzy input fuzzy output support vector machine f2svm classifier that can 1 process fuzzy information given as input to the classification algorithm for modeling the subpixel information in the learning phase of the classifier and 2 provide a fuzzy modeling of the classification results allowing a relation many to one between classes and pixels the presented binary f2svm can address multicategory problems according to two strategies the fuzzy one against all foaa and the fuzzy one against one foao strategies these strategies generalize to the fuzzy case techniques based upon ensembles of binary classifiers used for addressing multicategory problems in crisp classification problems the effectiveness of the proposed f2svm classifier is tested on three problems related to image classification in presence of mixed pixels having different characteristics experimental results confirm the validity of the proposed subpixel classification method