uncorrelated discriminant vectors using a kernel method uncorrelated discriminant vectors using a kernel method are proposed in this paper in some sense kernel uncorrelated discriminant vectors extend jin s method and then several related theorems are stated most importantly the proposed method can deal with nonlinear problems finally experimental results on handwritten numeral characters show that the proposed method is effective and feasible

cohort based kernel visualisation with scatter matrices visualisation with good discrimination between data cohorts is important for exploratory data analysis and for decision support interfaces this paper proposes a kernel extension of the cluster based linear visualisation method described in lisboa et al 15 a representation of the data in dual form permits the application of the kernel trick so projecting the data onto the orthonormalised cohort means in the feature space the only parameters of the method are those for the kernel function the method is shown to obtain well discriminating visualisations of non linearly separable data with low computational cost the linearity of the visualisation was tested using nearest neighbour and linear discriminant classifiers achieving significant improvements in classification accuracy with respect to the original features especially for high dimensional data where 93 accuracy was obtained for the splice junction gene sequences data set from the uci repository

eigenfaces vs fisherfaces recognition using class specific linear projection we develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression taking a pattern classification approach we consider each pixel in an image as a coordinate in a high dimensional space we take advantage of the observation that the images of a particular face under varying illumination but fixed pose lie in a 3d linear subspace of the high dimensional image space if the face is a lambertian surface without shadowing however since faces are not truly lambertian surfaces and do indeed produce self shadowing images will deviate from this linear subspace rather than explicitly modeling this deviation we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation our projection method is based on fisher s linear discriminant and produces well separated classes in a low dimensional subspace even under severe variation in lighting and facial expressions the eigenface technique another method based on linearly projecting the image space to a low dimensional subspace has similar computational requirements yet extensive experimental results demonstrate that the proposed fisherface method has error rates that are lower than those of the eigenface technique for tests on the harvard and yale face databases

bayesian hybrid generative discriminative learning based on finite liouville mixture models recently hybrid generative discriminative approaches have emerged as an efficient knowledge representation and data classification engine however little attention has been devoted to the modeling and classification of non gaussian and especially proportional vectors our main goal in this paper is to discover the true structure of this kind of data by building probabilistic kernels from generative mixture models based on liouville family from which we develop the beta liouville distribution and which includes the well known dirichlet as a special case the beta liouville has a more general covariance structure than the dirichlet which makes it more practical and useful our learning technique is based on a principled purely bayesian approach which resulted models are used to generate support vector machine svm probabilistic kernels based on information divergence in particular we show the existence of closed form expressions of the kullback leibler and renyi divergences between two beta liouville distributions and then between two dirichlet distributions as a special case through extensive simulations and a number of experiments involving synthetic data visual scenes and texture images classification we demonstrate the effectiveness of the proposed approaches

perturbed collocation and runge kutta methods it is well known thatsome implicit runge kutta methods are equivalent to collocation methods this fact permits very short and natural proofs of order anda b an bn stability properties for this subclass of methods see 9 and 10 the present paper answers the natural question ifall rk methods can be considered as a somewhat perturbed collocation after having introduced this notion we give a proof on the order of the method and discuss their stability properties much of known theory becomes simple and beautiful

conservative numerical methods for x f x abstract two distinctly different numerical methods are developed for solving the conservative initial value problem x f x x o dot x o both methods conserve exactly the same total energy as does the given differential equation computer examples are described and

a modal based approach to the nonlinear vibration of strings against a unilateral obstacle simulations and experiments in the pointwise case this article is concerned with the vibration of a stiff linear string in the presence of a rigid obstacle a numerical method for unilateral and arbitrary shaped obstacles is developed based on a modal approach in order to take into account the frequency dependence of losses in strings the contact force of the barrier interaction is treated using a penalty approach while a conservative scheme is derived for time integration in order to ensure long term numerical stability in this way the linear behaviour of the string when not in contact with the barrier can be controlled via a mode by mode fitting so that the model is particularly well suited for comparisons with experiments an experimental configuration is used with a point obstacle either centered or near an extremity of the string in this latter case such a pointwise obstruction approximates the end condition found in the tanpura an indian stringed instrument the second polarisation of the string is also analysed and included in the model numerical results are compared against experiments showing good accuracy over a long time scale

on one leg multistep methods the properties and possibilities of one leg methods are presented in a manner that admits the generalization to smoothly varying step size a new definition is given of the local truncation error and of the order of consistency some data are given for the most accurate one leg methods i e for p k and p k 1 where p is the order of consistency and k is the step number of the method

one step collocation uniform superconvergence predictor corrector method local error estimate in this paper we consider the one step collocation method at gaussian points for the numerical solution of ivp s for ode s it is well known that this method has a superconvergence rate o h 2n at the nodes while it uniformly converges with the lower order o h n 1 we give a rule to reach the uniform superconvergence by some kind of idec method an interesting by product of this procedure is an accurate estimation of the local discretization error moreover we give a modified version of the method in a predictor corrector mode for which all the previous results keep on holding in particular we propose a new method of order four with step size control

median null sw based method for face feature recognition with the progress of science and technology artificial intelligence is being paid more and more attention people want to use computers to deal with complex practical problems so linear discriminant analysis lda is widely used as a dimensionality reduction technique in image and text recognition classification tasks however a weakness of lda model is that the class average vector in the formula completely depends on class sample average under special circumstances such as noise bright light some outliers will appear in the practical input databases therefore by employing several given practical samples the class sample average is not enough to estimate the class average accurately so the recognition performance of lda model will decline compared to human intelligence computers are far short of necessary fundamental knowledge of judgment which people normally acquire during the formative years of their lives in order to solve the problem and also to render lda model more robust we propose a within class scatter matrix null space median method m n s w which first transforms the original space by employing a basis of within class scatter matrix null space and then in the transformed space the maximum of between class scatter matrix is pursued in the second stage within class median vector is used in the traditional lda model experiments on orl feret and yale face data sets are performed to test and evaluate the effectiveness of the proposed method

breast cancer diagnosis using genetic programming generated feature this paper proposes a novel method for breast cancer diagnosis using the feature generated by genetic programming gp we developed a new feature extraction measure modified fisher linear discriminant analysis mflda to overcome the limitation of fisher criterion gp as an evolutionary mechanism provides a training structure to generate features a modified fisher criterion is developed to help gp optimize features that allow pattern vectors belonging to different categories to distribute compactly and disjoint regions first the mflda is experimentally compared with some classical feature extraction methods principal component analysis fisher linear discriminant analysis alternative fisher linear discriminant analysis second the feature generated by gp based on the modified fisher criterion is compared with the features generated by gp using fisher criterion and an alternative fisher criterion in terms of the classification performance the classification is carried out by a simple classifier minimum distance classifier finally the same feature generated by gp is compared with a original feature set as the inputs to multi layer perceptrons and support vector machine results demonstrate the capability of this method to transform information from high dimensional feature space into one dimensional space and automatically discover the relationship among data to improve classification accuracy

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

high order algebraically stable multistep runge kutta methods in this paper we examine some of the stability properties of a special class of multivalue methods called in k burrage order properties of multivalue methods ima j numer anal submitted multistep runge kutta methods and construct families of algebraically stable methods of arbitrarily high order

numerical simulations of piano strings i a physical model for a struck string using finite difference methods the first attempt to generate musical sounds by solving the equations of vibrating strings by means of finite difference methods fdm was made by hiller and ruiz j audio eng soc 19 462 472 1971 it is shown here how this numerical approach and the underlying physical model can be improved in order to simulate the motion of the piano string with a high degree of realism starting from the fundamental equations of a damped stiff string interacting with a nonlinear hammer a numerical finite difference scheme is derived from which the time histories of string displacement and velocity for each point of the string are computed in the time domain the interacting force between hammer and string as well as the force acting on the bridge are given by the same scheme the performance of the model is illustrated by a few examples of simulated string waveforms a brief discussion of the aspects of numerical stability and dispersion with reference to the proper choice of sampling parameters is also included

multistep collocation methods for volterra integro differential equations multistep collocation methods for volterra integro differential equations are derived and analyzed they increase the order of convergence of classical one step collocation methods at the same computational cost the numerical stability analysis is carried out and classes of a 0 stable methods are provided numerical experiments confirm theoretical expectations and make comparisons with the one step collocation methods

energy conserving schemes for the simulation of musical instrument contact dynamics abstract collisions are an innate part of the function of many musical instruments due to the nonlinear nature of contact forces special care has to be taken in the construction of numerical schemes for simulation and sound synthesis finite difference schemes and other time stepping algorithms used for musical instrument modelling purposes are normally arrived at by discretising a newtonian description of the system however because impact forces are non analytic functions of the phase space variables algorithm stability can rarely be established this way this paper presents a systematic approach to deriving energy conserving schemes for frictionless impact modelling the proposed numerical formulations follow from discretising hamilton s equations of motion generally leading to an implicit system of nonlinear equations that can be solved with newton s method the approach is first outlined for point mass collisions and then extended to distributed settings such as vibrating strings and beams colliding with rigid obstacles stability and other relevant properties of the proposed approach are discussed and further demonstrated with simulation examples the methodology is exemplified through a case study on tanpura string vibration with the results confirming the main findings of previous studies on the role of the bridge in sound generation with this type of string instrument

the stability function for multistep collocation methods c polynomials for rational approximation to the exponential function was introduced by norsett 7 to study stability properties of one step methods for one step collocation methods thec polynomial has a very simple form in this paper we studyc polynomials for multistep collocation methods and obtain results that generalize those in the one step case and provide a way to analyze linear stability of such methods

regularized generalized eigen decomposition with applications to sparse supervised feature extraction and sparse discriminant analysis we propose a general technique for obtaining sparse solutions to generalized eigenvalue problems and call it regularized generalized eigen decomposition rged for decades fisher s discriminant criterion has been applied in supervised feature extraction and discriminant analysis and it is formulated as a generalized eigenvalue problem thus rged can be applied to effectively extract sparse features and calculate sparse discriminant directions for all variants of fisher discriminant criterion based models particularly rged can be applied to matrix based and even tensor based discriminant techniques for instance 2d linear discriminant analysis 2d lda furthermore an iterative algorithm based on the alternating direction method of multipliers is developed the algorithm approximately solves rged with monotonically decreasing convergence and at an acceptable speed for results of modest accuracy numerical experiments based on four data sets of different types of images show that rged has competitive classification performance with existing multidimensional and sparse techniques of discriminant analysis highlightswe propose a new technique called regularized generalized eigen decomposition rged rged solves generalized eigenvalue problems and obtains sparse solutions it is easy and straightforward applying rged to sparse discriminant analysis and feature extraction an algorithm is developed to solve it with monotonically decreasing convergence rged has competitive classification performance comparing with other methods

novel fisher discriminant classifiers at the present several applications need to classify high dimensional points belonging to highly unbalanced classes unfortunately when the training set cardinality is small compared to the data dimensionality small sample size problem the classification performance of several well known classifiers strongly decreases similarly the classification accuracy of several discriminative methods decreases when non linearly separable and unbalanced classes are treated in this paper we firstly survey state of the art methods that employ improved versions of linear discriminant analysis lda to deal with the above mentioned problems secondly we propose a family of classifiers based on the fisher subspace estimation which efficiently deal with the small sample size problem non linearly separable classes and unbalanced classes the promising results obtained by the proposed techniques on benchmark datasets and the comparison with state of the art predictors show the efficacy of the proposed techniques

piano string excitation vi nonlinear modeling accurate modeling of the piano string hammer interaction requires that the nonlinearity of the force displacement relation for the hammer be recognized and included two models that accomplish this with numerical integration are described and sample results are presented the first model is valid for a perfectly flexible string where nondispersive traveling waves can be conveniently followed with a digital delay line the second model uses standing waves instead to describe the vibrations of a stiff string with the stiffness itself justifying truncation of what would otherwise be a poorly converging normal mode summation predictions with these models give significantly better agreement with data than did previous calculations with completely linear models

subspace learning for unsupervised feature selection via matrix factorization dimensionality reduction is an important and challenging task in machine learning and data mining feature selection and feature extraction are two commonly used techniques for decreasing dimensionality of the data and increasing efficiency of learning algorithms specifically feature selection realized in the absence of class labels namely unsupervised feature selection is challenging and interesting in this paper we propose a new unsupervised feature selection criterion developed from the viewpoint of subspace learning which is treated as a matrix factorization problem the advantages of this work are four fold first dwelling on the technique of matrix factorization a unified framework is established for feature selection feature extraction and clustering second an iterative update algorithm is provided via matrix factorization which is an efficient technique to deal with high dimensional data third an effective method for feature selection with numeric data is put forward instead of drawing support from the discretization process fourth this new criterion provides a sound foundation for embedding kernel tricks into feature selection with this regard an algorithm based on kernel methods is also proposed the algorithms are compared with four state of the art feature selection methods using six publicly available datasets experimental results demonstrate that in terms of clustering results the proposed two algorithms come with better performance than the others for almost all datasets we experimented with here highlightspropose a new feature selection based on matrix factorization present a fast convergent algorithm for matrix factorization on certain constraints incorporate kernel tricks into feature selection problems construct a unified framework for feature extraction feature selection and clustering

parameter fitting for piano sound synthesis by physical modeling a difficult issue in the synthesis of piano tones by physical models is to choose the values of the parameters governing the hammer string model in fact these parameters are hard to estimate from static measurements causing the synthesis sounds to be unrealistic an original approach that estimates the parameters of a piano model from the measurement of the string vibration by minimizing a perceptual criterion is proposed the minimization process that was used is a combination of a gradient method and a simulated annealing algorithm in order to avoid convergence problems in case of multiple local minima the criterion based on the tristimulus concept takes into account the spectral energy density in three bands each allowing particular parameters to be estimated the optimization process has been run on signals measured on an experimental setup the parameters thus estimated provided a better sound quality than the one obtained using a global energetic criterion both the sound s attack and its brightness were better preserved this quality gain was obtained for parameter values very close to the initial ones showing that only slight deviations are necessary to make synthetic sounds closer to the real ones

generalized discriminant analysis using a kernel approach we present a new method that we call generalized discriminant analysis gda to deal with nonlinear discriminant analysis using kernel function operator the underlying theory is close to the support vector machines svm insofar as the gda method provides a mapping of the input vectors into high dimensional feature space in the transformed space linear properties make it easy to extend and generalize the classical linear discriminant analysis lda to nonlinear discriminant analysis the formulation is expressed as an eigenvalue problem resolution using a different kernel one can cover a wide class of nonlinearities for both simulated data and alternate kernels we give classification results as well as the shape of the decision function the results are confirmed using real data to perform seed classification

essence of kernel fisher discriminant in this paper the method of kernel fisher discriminant kfd is analyzed and its nature is revealed i e kfd is equivalent to kernel principal component analysis kpca plus fisher linear discriminant analysis lda based on this result a more transparent kfd algorithm is proposed that is kpca is first performed and then lda is used for a second feature extraction in the kpca transformed space finally the effectiveness of the proposed algorithm is verified using the cenparmi handwritten numeral database

finite difference calculus invariant structure of a class of algorithms for the nonlinear klein gordon equation in a previous work the authors have presented a formalism for deriving systematically invariant symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities this formalism is herein cast in the context of exact finite difference calculus the algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus in the same sense as that of the corresponding differential equation being derivable from its associated energy function a conserved quantity a clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities which are the consistent counterpart of the invariant quantities in the continuous case results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism and that preserve energy or linear momentum are discussed in the context of finite difference calculus some

an energy preserving description of nonlinear beam vibrations in modal coordinates abstract conserved quantities are identified in the equations describing large amplitude free vibrations of beams projected onto their linear normal modes this is achieved by writing the geometrically exact equations of motion in their intrinsic or hamiltonian form before the modal transformation for nonlinear free vibrations about a zero force equilibrium it is shown that the finite dimensional equations of motion in modal coordinates are energy preserving even though they only approximate the total energy of the infinite dimensional system for beams with constant follower forces energy like conserved quantities are also obtained in the finite dimensional equations of motion via casimir functions the duality between space and time variables in the intrinsic description is finally carried over to the definition of a conserved quantity in space which is identified as the local cross sectional power numerical examples are used to illustrate the main results

modeling the dynamics of a vibrating string with a finite distributed unilateral constraint application to the sitar the free vibration response of an ideal string impacting a distributed parabolic obstacle located at its boundary has been analyzed the goal being to understand and simulate a sitar string the portion of the string in contact with the obstacle is governed by a different partial differential equation pde from the free portion represented by the classical string equation these two pdes and corresponding boundary conditions along with the transversality condition that governs the dynamics of the moving boundary are obtained using hamilton s principle a galerkin approximation is used to convert them into a system of nonlinear ordinary differential equations with lower mode shapes parametrized with respect to the location of the moving boundary as basis functions this system is solved numerically and the behavior of the string studied from simulations the advantages and disadvantages of the proposed method are discussed in comparison to the penalty approach for simulating wrapping contacts simulati

a family of methods for abel integral equations of the second kind abstract a class of methods depending on some parameters are introduced for the numerical solution of the abel integral equations of the second kind some bounds on the parameters are determined so that the corresponding methods have infinite stability intervals

class incremental generalized discriminant analysis generalized discriminant analysis gda is the nonlinear extension of the classical linear discriminant analysis lda via the kernel trick mathematically gda aims to solve a generalized eigenequation problem which is always implemented by the use of singular value decomposition svd in the previously proposed gda algorithms a major drawback of svd however is the difficulty of designing an incremental solution for the eigenvalue problem moreover there are still numerical problems of computing the eigenvalue problem of large matrices in this article we propose another algorithm for solving gda as for the case of small sample size problem which applies qr decomposition rather than svd a major contribution of the proposed algorithm is that it can incrementally update the discriminant vectors when new classes are inserted into the training set the other major contribution of this article is the presentation of the modified kernel gram schmidt mkgs orthogonalization algorithm for implementing the qr decomposition in the feature space which is more numerically stable than the kernel gram schmidt kgs algorithm we conduct experiments on both simulated and real data to demonstrate the better performance of the proposed methods

global discriminative based nonnegative spectral clustering based on spectral graph theory spectral clustering is an optimal graph partition problem it has been proven that the spectral clustering is equivalent to nonnegative matrix factorization nmf under certain conditions based on the equivalence some spectral clustering methods are proposed but the global discriminative information of the dataset is neglected in this paper based on the equivalence between spectral clustering and nmf we simultaneously maximize the between class scatter matrix and minimize the within class scatter matrix to enhance the discriminating power we integrate the geometrical structure and discriminative structure in a joint framework with a global discriminative regularization term added into the nonnegative matrix factorization framework we propose two novel spectral clustering methods named global discriminative based nonnegative and spectral clustering gdbnsc ncut and gdbnsc rcut these new spectral clustering algorithms can preserve both the global geometrical structure and global discriminative structure the intrinsic geometrical information of the dataset is detected and clustering quality is improved with enhanced discriminating power in addition the proposed algorithms also have very good abilities of handling out of sample data experimental results on real word data demonstrate that the proposed algorithms outperform some state of the art methods with good clustering qualities we maximize the between class scatter matrix meanwhile we minimize the within class scatter matrix we integrate the geometrical structure and discriminative structure in a framework a global discriminative regularization term added into the objective functions the new algorithms can preserve the global geometrical and discriminative structure

discriminative common vector method with kernels in some pattern recognition tasks the dimension of the sample space is larger than the number of samples in the training set this is known as the small sample size problem linear discriminant analysis lda techniques cannot be applied directly to the small sample size case the small sample size problem is also encountered when kernel approaches are used for recognition in this paper we attempt to answer the question of how should one choose the optimal projection vectors for feature extraction in the small sample size case based on our findings we propose a new method called the kernel discriminative common vector method in this method we first nonlinearly map the original input space to an implicit higher dimensional feature space in which the data are hoped to be linearly separable then the optimal projection vectors are computed in this transformed space the proposed method yields an optimal solution for maximizing a modified fisher s linear discriminant criterion discussed in the paper thus under certain conditions a 100 recognition rate is guaranteed for the training set samples experiments on test data also show that in many situations the generalization performance of the proposed method compares favorably with other kernel approaches

on the stability of symplectic and energy momentum algorithms for non linear hamiltonian systems with symmetry abstract this paper presents a detailed comparison of two implicit time integration schemes for a simple non linear hamiltonian system with symmetry the motion of a particle in a central force field the goal is to establish analytical and numerical results pertaining to the stability properties of the implicit mid point rule the proto typical implicit symplectic method and a particular energy momentum conserving scheme and to compare the two schemes with respect to accuracy while all results presented herein are within the context of a simple model problem the problem was constructed so as to exhibit key features typical of more complex systems with symmetry such as those arising in non linear solid mechanics namely the presence of large and relatively slow overall motions together with high frequency internal motions

a compact fourth order in space energy preserving method for riesz space fractional nonlinear wave equations in this work we investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the riesz type the model under consideration generalizes various nonlinear wave equations including the sine gordon and the nonlinear klein gordon models the system considered in this work is conservative when homogeneous dirichlet boundary conditions are imposed motivated by this fact we propose a finite difference method based on fractional centered differences that is capable of preserving the discrete energy of the system the method under consideration is a nonlinear implicit scheme which has various numerical properties among the most interesting numerical features we show that the methodology is consistent of second order in time and fourth order in space moreover we show that the technique is stable and convergent some numerical simulations show that the method is capable of preserving the energy of the discrete system this characteristic of the technique is in obvious agreement with the properties of its continuous counterpart