discriminant analysis by gaussian mixtures fisher rao linear discriminant analysis lda is a valuable tool for multigroup classification lda is equivalent to maximum likelihood classification assuming gaussian distributions for each class in this paper we fit gaussian mixtures to each class to facilitate effective classification in non normal settings especially when the classes are clustered low dimensional views are an important by product of lda our new techniques inherit this feature we can control the within class spread of the subclass centres relative to the between class spread our technique for fitting these models permits a natural blend with nonparametric versions of lda

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

shape functions and integration formulas for three dimensional finite element analysis shape functions and numerical integration formulas for three dimensional finite element analysis as found in most finite element reference books are incomplete for example shape functions and integration formulas for a pyramid with a quadrilateral base are missing it is also difficult to find symmetric higher order integration formulas for triangular and tetrahedral elements in general these shape functions and integration formulas cannot be satisfactorily derived as degenerate cases of shape functions and integration formulas for hexahedral brick elements in this paper we present c continuous quadrilateral pyramid elements and integration formulas for two and three dimensional elements

feature extraction by non parametric mutual information maximization we present a method for learning discriminative feature transforms using as criterion the mutual information between class labels and transformed features instead of a commonly used mutual information measure based on kullback leibler divergence we use a quadratic divergence measure which allows us to make an efficient non parametric implementation and requires no prior assumptions about class densities in addition to linear transforms we also discuss nonlinear transforms that are implemented as radial basis function networks extensions to reduce the computational complexity are also presented and a comparison to greedy feature selection is made

solving low density subset sum problems the subset sum problem is to decide whether or not the 0 l integer programming problem s n i l a i x i m i x i 0 or 1 has a solution where the a i and m are given positive integers this problem is np complete and the difficulty of solving it is the basis of public key cryptosystems of knapsack type an algorithm is proposed that searches for a solution when given an instance of the subset sum problem this algorithm always halts in polynomial time but does not always find a solution when one exists it converts the problem to one of finding a particular short vector v in a lattice and then uses a lattice basis reduction algorithm due to a k lenstra h w lenstra jr and l lovasz to attempt to find v the performance of the proposed algorithm is analyzed let the density d of a subset sum problem be defined by d n log 2 max i a i then for almost all problems of density d d n it is proved that the lattice basis reduction algorithm locates v extensive computational tests of the algorithm suggest that it works for densities d d c n where d c n is a cutoff value that is substantially larger than 1 n this method gives a polynomial time attack on knapsack public key cryptosystems that can be expected to break them if they transmit information at rates below d c n as n

a short note on a bernstein bezier basis for the pyramid we introduce a bernstein bezier basis for the pyramid whose restriction to the face reduces to the bernstein bezier basis on the triangle or quadrilateral the basis satisfies the standard positivity and partition of unity properties common to bernstein polynomials and spans the same space as nonpolynomial pyramid bases in the literature procedures for differentiation and integration of these basis functions are also discussed

code construction and decoding algorithms for semi quantitative group testing with nonuniform thresholds we analyze a new group testing scheme termed semi quantitative group testing which may be viewed as a concatenation of an adder channel and a discrete quantizer our focus is on non uniform quantizers with arbitrary thresholds for the most general semi quantitative group testing model we define three new families of sequences capturing the constraints on the code design imposed by the choice of the thresholds the sequences represent extensions and generalizations of b h and certain types of super increasing and lexicographically ordered sequences and they lead to code structures amenable for efficient recursive decoding we describe the decoding methods and provide an accompanying computational complexity and performance analysis

a comparison of high order interpolation nodes for the pyramid the use of pyramid elements is crucial to the construction of efficient hex dominant meshes m bergot g cohen and m durufle j sci comput 42 2010 pp 345 381 for conforming nodal finite element methods with mixed element types it is advantageous for nodal distributions on the faces of the pyramid to match those on the faces and edges of hexahedra and tetrahedra we adapt existing procedures for constructing optimized tetrahedral nodal sets for high order interpolation to the pyramid with constrained face nodes including two generalizations of the explicit warp and blend construction of nodes on the tetrahedron t warburton j engrg math 56 2006 pp 247 262 comparisons between nodal sets show that the lowest lebesgue constants are given by warp and blend nodes for order n leq 7 and fekete nodes for n 7 though numerical experiments show little variation in the conditioning and accuracy of all surveyed nonequidistant points

local and global mappings of topology representing networks as data analysis tasks often have to deal with complex data structures the nonlinear dimensionality reduction methods play an important role in exploratory data analysis in the literature a number of nonlinear dimensionality reduction techniques have been proposed e g sammon mapping locally linear embedding these techniques attempt to preserve either the local or the global geometry of the original data and they perform metric or non metric dimensionality reduction nevertheless it is difficult to apply most of them to large data sets there is a need for new algorithms that are able to combine vector quantisation and mapping methods in order to visualise the data structure in a low dimensional vector space in this paper we define a new class of algorithms to quantify and disclose the data structure that are based on the topology representing networks and apply different mapping methods to the low dimensional visualisation not only existing methods are combined for that purpose but also a novel group of mapping methods topology representing network map are introduced as a part of this class topology representing network maps utilise the main benefits of the topology representing networks and of the multidimensional scaling methods to disclose the real structure of the data set under study to determine the main properties of the topology representing network based mapping methods a detailed analysis of classical benchmark examples wine and optical recognition of handwritten digits data set is presented

newman polynomials with prescribed vanishing and integer sets with distinct subset sums we study the problem of determining the minimal degree d m of a polynomial that has all coefficients in 0 1 and a zero of multiplicity m at 1 we show that a greedy solution is optimal precisely when m 5 mirroring a result of boyd on polynomials with 1 coefficients we then examine polynomials of the form e e xe 1 where e is a set of m positive odd integers with distinct subset sums and we develop algorithms to determine the minimal degree of such a polynomial we determine that d m satisfies inequalities of the form 2m c1m d m 103 96 2m c2 last we consider the related problem of finding a set of m positive integers with distinct subset sums and minimal largest element and show that the conway guy sequence yields the optimal solution for m 9 extending some computations of lunnon

discriminative components of data a simple probabilistic model is introduced to generalize classical linear discriminant analysis lda in finding components that are informative of or relevant for data classes the components maximize the predictability of the class distribution which is asymptotically equivalent to 1 maximizing mutual information with the classes and 2 finding principal components in the so called learning or fisher metrics the fisher metric measures only distances that are relevant to the classes that is distances that cause changes in the class distribution the components have applications in data exploration visualization and dimensionality reduction in empirical experiments the method outperformed in addition to more classical methods a renyi entropy based alternative while having essentially equivalent computational cost

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

integer sets with distinct subset sums in section 1 we introduce the problem of finding minimal height sets of n natural numbers with distinct subset sums ssd and in section 2 review the well known conway guy sequence u conjectured to yield a minimal ssd set for every n we go on section 3 to prove that u certainly cannot be improved upon by any greedy sequence to verify numerically section 4 that it does yield ssd sets for n 80 and section 5 by direct search to show that these are minimal for n s 8 there is a brief interlude section 6 on the problem of decoding the subset from its sum in section 7 generalizations of u are constructed which are asymptotically smaller defining the limit ratio of a sequence w to be a 1imn wn 2l the atkinson negro santoro sequence v known to give ssd sets has a 0 6334 conwayguy conjectured to has a 0 4703 and our best generalization has a 0 4419 we also section 8 discuss when such sequences have the same a and section 9 how a may efficiently be computed to high accuracy 1 the distinct subset sum problem a well known problem in combinatorial number theory 2 pp 64 65 involves the construction for given n of a set p pi i 1 n of natural numbers possessing the property we shall call subset sum distinctness or ssd for short that is l l f pi p e pi s t c l n i s i t or distinct subsets of p have distinct sums evidently choosing pi 21 satisfies 1 1 the interest lies in how much the maximum element pn can be reduced below 2n 1 in order to explore the ssd property it is convenient to consider a more general concept we say that x has a representation by p with length k and signature 1 when for some s and t as before 1 2 x e pe po ies iet where s n t 0 si it k isi iti 1 that is x is the difference of two disjoint subset sums of which the first has 1 terms more than the second and the two together have k terms alternatively n 1 3 x eip i l received june 23 1986 revised october 20 1986 1980 mathematics subject classification 1985 revision primary 11b13 05 04 05a99 11j13 94a60 65b05

nonlinear embedding preserving multiple local linearities locally linear embedding lle is one of the effective and efficient algorithms for nonlinear dimensionality reduction this paper discusses the stability of lle focusing on the optimal weights for extracting local linearity behind the considered manifold it is proven that there are multiple sets of weights that are approximately optimal and can be used to improve the stability of lle a new algorithm using multiple weights is then proposed together with techniques for constructing multiple weights this algorithm is called as nonlinear embedding preserving multiple local linearities neml neml improves the preservation of local linearity and is more stable than lle a short analysis for neml is also given for isometric manifolds neml is compared with the local tangent space alignment ltsa in methodology since both of them adopt multiple local constraints numerical examples are given to show the improvement and efficiency of neml

a lowest order composite finite element exact sequence on pyramids abstract composite basis functions for pyramidal elements on the spaces h 1 h curl h div and l 2 are presented in particular we construct the lowest order composite pyramidal elements and show that they respect the de rham diagram i e we have an exact sequence and satisfy the commuting property moreover the finite elements are fully compatible with the standard finite elements for the lowest order raviart thomas nedelec sequence on tetrahedral and hexahedral elements that is to say the new elements have the same degrees of freedom on the shared interface with the neighbouring hexahedral or tetrahedra elements and the basis functions are conforming in the sense that they maintain the required level of continuity full tangential component normal component etc across the interface furthermore we study the approximation properties of the spaces as an initial partition consisting of tetrahedra hexahedra and pyramid elements are successively subdivided and show that the spaces result in the same optimal order of approximation in terms of the mesh size h as one would obtain using purely hexahedral or purely tetrahedral partitions

pyramid algorithms for bernstein bezier finite elements of high nonuniform order in any dimension the archetypal pyramid algorithm is the de casteljau algorithm which is a standard tool for the evaluation of bezier curves and surfaces pyramid algorithms replace an operation on a single high order polynomial by a recursive sequence of self similar affine combinations and are ubiquitous in computer aided geometric design for computations involving high order curves and surfaces pyramid algorithms have received no attention whatsoever from the high or low order finite element community we develop and analyze pyramid algorithms for the efficient handling of all of the basic finite element building blocks including the assembly of the element load vectors and element stiffness matrices the complexity of the algorithm for generating the element stiffness matrix is optimal a new nonuniform order variant of the de casteljau algorithm is developed that is applicable to the variable polynomial order case but incurs no additional complexity compared with the original algorithm the work provides the

rapid and brief communication nonlinear discriminant mapping using the laplacian of a graph in this paper an algorithm for nonlinear discriminant mapping ndm is presented which elegantly integrates the ideas of both linear discriminant analysis lda and isomap by using the laplacian of a graph the objective of ndm is to find a linear subspace project of nonlinear data set which preserves maximum difference between between class scatter and within class scatter

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

high order composite finite element exact sequences based on tetrahedral hexahedral prismatic pyramidal partitions abstract the combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and quadrilateral faces this paper presents high order exact sequences of finite element approximations in h 1 h curl h div and l 2 based on such kind of three dimensional mesh configurations the approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra the traces associated with triangular faces of these tetrahedral elements are constrained to match the quadrilateral shape functions on the quadrilateral face of the pyramid in order to maintain conformity with shared neighboring hexahedron or prism two classes of composite exact sequences are constructed one using classic nedelec spaces of first kind and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces projection based interpolants satisfying the commuting diagram property are presented in a general form for each type of element the interpolants are expressed as the sum of linearly independent contributions associated with vertices edges faces and volume according to the kind of traces appropriate to the space under consideration furthermore we study applications to the mixed formulation of darcy s problems based on compatible pairs of approximations in h div l 2 for such tetrahedral hexahedral prismatic pyramidal meshes an error analysis is outlined showing same optimal orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions enhanced accuracy for potential and flux divergence variables is obtained when enriched space configurations are applied the predicted convergence orders are verified for some test problems

feature extraction based on laplacian bidirectional maximum margin criterion maximum margin criterion mmc based feature extraction is more efficient than linear discriminant analysis lda for calculating the discriminant vectors since it does not need to calculate the inverse within class scatter matrix however mmc ignores the discriminative information within the local structures of samples and the structural information embedding in the images in this paper we develop a novel criterion namely laplacian bidirectional maximum margin criterion lbmmc to address the issue we formulate the image total laplacian matrix image within class laplacian matrix and image between class laplacian matrix using the sample similar weight that is widely used in machine learning the proposed lbmmc based feature extraction computes the discriminant vectors by maximizing the difference between image between class laplacian matrix and image within class laplacian matrix in both row and column directions experiments on the feret and yale face databases show the effectiveness of the proposed lbmmc based feature extraction method

rapid and brief communication laplacian linear discriminant analysis a novel linear discriminant criterion function is proved to be equal to fisher s criterion function the analysis of the function is linked to spectral decomposition of the laplacian of a graph moreover the function is maximized using two algorithms experimental results show the effectiveness and some specific characteristics of our algorithms

on the relevance of linear discriminative features linear discriminant analysis lda has been widely used to extract linear features for classification in real applications the usefulness of the extracted features usually needs to be confirmed using an error rate of classification embedded in a classifier little attention has been paid to whether and how discriminative features themselves can be interpreted as indicators of usefulness we refer to this as relevance i e the capability of discriminative features to characterize the contribution of the original variables to classification we approach the relevance by considering how it could be lost while extracting optimal discriminative features then the discrepancy between the relevance and optimality of discriminative features is shown to originate from the angle between the space spanned by eigenvectors of the within class scattering matrix and the primary space in which the original variables reside in particular for a given dataset the larger the angle the less evident is the relevance discovered from optimal discriminative features furthermore the relevance and optimality are regarded as two constraint conditions or a tradeoff in order to extract relevant discriminative features at last a simulated experiment is used to show how the relevance is lost when the angle is changed experimental results based on both usps handwritten digitals and pie face databases show that a maximum margin criterion is a reasonable compromise between the relevance and optimality since it approximates the averaged class margin using euclidean distance measured in the primary space

second order nedelec curl conforming prismatic element for computational electromagnetics a systematic approach to obtaining mixed order curl conforming basis functions for a triangular prism is presented focus is made on the second order case space of functions for the prism is given basis functions are obtained as the dual basis with respect to suitably discretized nedelec degrees of freedom functionals acting on elements of the space thus the linear independence of the basis functions is assured while the belonging of the basis to the a priori given space of functions is guaranteed different strategies for the finite element assembly of the basis are discussed numerical results showing the verification procedure of the correctness of the implemented basis functions are given numerical results about sensibility of the condition number of the basis obtained concerning the quality of the elements of the mesh are also shown comparison with other representative sets of basis functions for prisms is included

on the information and representation of non euclidean pairwise data two common data representations are mostly used in intelligent data analysis namely the vectorial and the pairwise representation pairwise data which satisfy the restrictive conditions of euclidean spaces can be faithfully translated into a euclidean vectorial representation by embedding non metric pairwise data with violations of symmetry reflexivity or triangle inequality pose a substantial conceptual problem for pattern recognition since the amount of predictive structural information beyond what can be measured by embeddings is unclear we show by systematic modeling of non euclidean pairwise data that there exists metric violations which can carry valuable problem specific information furthermore euclidean and non metric data can be unified on the level of structural information contained in the data stable component analysis selects linear subspaces which are particularly insensitive to data fluctuations experimental results from different domains support our pattern recognition strategy

sparsity optimized high order finite element functions for h div on simplices this paper deals with conforming high order finite element discretizations of the vector valued function space h div in 2 and 3 dimensions a new set of hierarchic basis functions on simplices with the following two main properties is introduced when working with an affine simplicial triangulation first the divergence of the basis functions is l 2 orthogonal and secondly the l 2 inner product of the interior basis functions is sparse with respect to the polynomial degree p the construction relies on a tensor product based construction with properly weighted jacobi polynomials as well as an explicit splitting of the higher order basis functions into solenoidal and non solenoidal ones the basis is suited for fast assembling strategies the general proof of the sparsity result is done by the assistance of computer algebra software tools several numerical experiments document the proved sparsity patterns and practically achieved condition numbers for the parameter dependent div div problem even on curved elements a tremendous improvement in condition numbers is observed the precomputed mass and stiffness matrix entries in general form are available online

bernstein bezier finite elements on tetrahedral hexahedral pyramidal partitions abstract a construction for high order continuous finite elements on partitions consisting of tetrahedra hexahedra and pyramids based on polynomial bernstein bezier shape functions is presented along with algorithms that allow the computation of the system matrices in optimal complexity o 1 per entry

hierarchic hp edge element families for maxwell s equations on hybrid quadrilateral triangular meshes we construct and study a set of hierarchic basis functions for the galerkin discretisation of the space h curl 169 suitable for hybrid meshes containing both quadrilateral and triangular elements with arbitrary non uniform order polynomial approximation we investigate the conditioning and dispersive behaviour of the elements in addition numerical examples are shown which demonstrate the accuracy of the space for computing solutions of the time harmonic maxwell s equations that have singularities

efficient and robust feature extraction by maximum margin criterion in pattern recognition feature extraction techniques are widely employed to reduce the dimensionality of data and to enhance the discriminatory information principal component analysis pca and linear discriminant analysis lda are the two most popular linear dimensionality reduction methods however pca is not very effective for the extraction of the most discriminant features and lda is not stable due to the small sample size problem in this paper we propose some new linear and nonlinear feature extractors based on maximum margin criterion mmc geometrically feature extractors based on mmc maximize the average margin between classes after dimensionality reduction it is shown that mmc can represent class separability better than pca as a connection to lda we may also derive lda from mmc by incorporating some constraints by using some other constraints we establish a new linear feature extractor that does not suffer from the small sample size problem which is known to cause serious stability problems for lda the kernelized nonlinear counterpart of this linear feature extractor is also established in the paper our extensive experiments demonstrate that the new feature extractors are effective stable and efficient

fast formation of isogeometric galerkin matrices by weighted quadrature abstract in this paper we propose an algorithm for the formation of matrices of isogeometric galerkin methods the algorithm is based on three ideas the first is that we perform the external loop over the rows of the matrix the second is that we calculate the row entries by weighted quadrature the third is that we exploit the local tensor product structure of the basis functions while all ingredients have a fundamental role for computational efficiency the major conceptual change of paradigm with respect to the standard implementation is the idea of using weighted quadrature the test function is incorporated in the integration weight while the trial function the geometry parametrization and the pdes coefficients form the integrand function this approach is very effective in reducing the computational cost while maintaining the optimal order of approximation of the method analysis of the cost is confirmed by numerical testing where we show that for p large enough the time required by the floating point operations is less than the time spent in unavoidable memory operations the sparse matrix allocation and memory write the proposed algorithm allows significant time saving when assembling isogeometric galerkin matrices for all the degrees of the test spline space and paves the way for a use of high degree k refinement in isogeometric analysis

ranking a random feature for variable and feature selection we describe a feature selection method that can be applied directly to models that are linear with respect to their parameters and indirectly to others it is independent of the target machine it is closely related to classical statistical hypothesis tests but it is more intuitive hence more suitable for use by engineers who are not statistics experts furthermore some assumptions of classical tests are relaxed the method has been used successfully in a number of applications that are briefly described

high order polygonal discontinuous petrov galerkin polydpg methods using ultraweak formulations abstract this work represents the first endeavor in using ultraweak formulations to implement high order polygonal finite element methods via the discontinuous petrov galerkin dpg methodology ultraweak variational formulations are nonstandard in that all the weight of the derivatives lies in the test space while most of the trial space can be chosen as copies of l 2 discretizations that have no need to be continuous across adjacent elements additionally the test spaces are broken along the mesh interfaces this allows one to construct conforming polygonal finite element methods termed here as polydpg methods by defining most spaces by restriction of a bounding triangle or box to the polygonal element the only variables that require nontrivial compatibility across elements are the so called interface or skeleton variables which can be defined directly on the element boundaries unlike other high order polygonal methods polydpg methods do not require ad hoc stabilization terms thanks to the crafted stability of the dpg methodology a proof of convergence of the form h p is provided and corroborated through several illustrative numerical examples these include polygonal meshes with n sided convex elements and with highly distorted concave elements as well as the modeling of discontinuous material properties along an arbitrary interface that cuts a uniform grid since polydpg methods have a natural a posteriori error estimator a polygonal adaptive strategy is developed and compared to standard adaptivity schemes based on constrained hanging nodes this work is also accompanied by an open source polydpg software supporting polygonal and conventional elements

new shape functions for triangular p fem using integrated jacobi polynomials in this paper the second order boundary value problem x y u f is discretized by the finite element method using piecewise polynomial functions of degree p on a triangular mesh on the reference element we define integrated jacobi polynomials as interior ansatz functions if is a constant function on each triangle and each triangle has straight edges we prove that the element stiffness matrix has not more than nonzero matrix entries an application for preconditioning is given numerical examples show the advantages of the proposed basis

penalized discriminant analysis fisher s linear discriminant analysis lda is a popular data analytic tool for studying the relationship between a set of predictors and a categorical response in this paper we describe a penalized version of lda it is designed for situations in which there are many highly correlated predictors such as those obtained by discretizing a function or the grey scale values of the pixels in a series of images in cases such as these it is natural efficient and sometimes essential to impose a spatial smoothness constraint on the coefficients both for improved prediction performance and interpretability we cast the classification problem into a regression framework via optimal scoring using this our proposal facilitates the use of any penalized regression technique in the classification setting the technique is illustrated with examples in speech recognition and handwritten character recognition

bernstein bezier finite elements of arbitrary order and optimal assembly procedures algorithms are presented that enable the element matrices for the standard finite element space consisting of continuous piecewise polynomials of degree n on simplicial elements in mathbb r d to be computed in optimal complexity mathcal o n 2d the algorithms i take into account numerical quadrature ii are applicable to nonlinear problems and iii do not rely on precomputed arrays containing values of one dimensional basis functions at quadrature points although these can be used if desired the elements are based on bernstein polynomials and are the first to achieve optimal complexity for the standard finite element spaces on simplicial elements

joint sampling distribution between actual and estimated classification errors for linear discriminant analysis error estimation must be used to find the accuracy of a designed classifier an issue that is critical in biomarker discovery for disease diagnosis and prognosis in genomics and proteomics this paper presents for what is believed to be the first time the analytical formulation for the joint sampling distribution of the actual and estimated errors of a classification rule the analysis presented here concerns the linear discriminant analysis lda classification rule and the resubstitution and leave one out error estimators under a general parametric gaussian assumption exact results are provided in the univariate case and a simple method is suggested to obtain an accurate approximation in the multivariate case it is also shown how these results can be applied in the computation of condition bounds and the regression of the actual error given the observed error estimate in contrast to asymptotic results the analysis presented here is applicable to finite training data in particular it applies in the small sample settings commonly found in genomics and proteomics applications numerical examples which include parameters estimated from actual microarray data illustrate the analysis throughout

adaptive nonlinear manifolds and their applications to pattern recognition dimensionality reduction has long been associated with retinotopic mapping for understanding cortical maps multisensory information is processed fused and mapped to an essentially 2 d cortex in an information preserving manner data processing and projection techniques inspired by this biological mechanism are playing an increasingly important role in pattern recognition computational intelligence data mining information retrieval and image recognition dimensionality reduction involves reduction of features or volume of data and has become an essential step of information processing in many fields the topic of manifold learning has recently attracted a great deal of attention and a number of advanced techniques for extracting nonlinear manifolds and reducing data dimensions have been proposed from statistics geometry theory and adaptive neural networks this paper provides an overview of this challenging and emerging topic and discusses various recent methods such as self organizing map som kernel pca principal manifold isomap local linear embedding and laplacian eigenmap many of them can be considered in a learning manifold framework the paper further elaborates on the biologically inspired som model and its metric preserving variant visom under the framework of adaptive manifold and their applications in dimensionality reduction with face recognition are investigated the experiments demonstrate that adaptive visom based methods produce markedly improved performance over the others due to their metric scaling and preserving properties along the nonlinear manifold

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements

the invertibility of the isoparametric mapping for pyramidal and prismatic finite elements we consider the isoparametric transformation which maps a given reference element onto a global element given by its vertices for multi linear finite elements on pyramids and prisms we present easily computable conditions on the position of the vertices which ensure that the isoparametric transformation is bijective

orientation embedded high order shape functions for the exact sequence elements of all shapes a unified construction of high order shape functions is given for all four classical energy spaces h1 h curl h div and l2 and for elements of all shapes segment quadrilateral triangle hexahedron tetrahedron triangular prism and pyramid the discrete spaces spanned by the shape functions satisfy the commuting exact sequence property for each element the shape functions are conforming hierarchical and compatible with other neighboring elements across shared boundaries so they may be used in hybrid meshes expressions for the shape functions are given in coordinate free format in terms of the relevant affine coordinates of each element shape the polynomial order is allowed to differ for each separate topological entity vertex edge face or interior in the mesh so the shape functions can be used to implement local p adaptive finite element methods each topological entity may have its own orientation and the shape functions can have that orientation embedded by a simple permutation of arguments

spline spaces on tr meshes with hanging vertices polynomial spline spaces defined on mixed meshes consisting of triangles and rectangles are studied for the c 0 case these include triangulations with hanging vertices as well as t meshes in addition to dimension formulae explicit basis functions are constructed and their supports and stability are discussed the approximation power of the spaces is also treated

formation of pyramid elements for hexahedra to tetrahedra transitions new algorithms are proposed for the modification of a mixed hexahedra tetrahedra element mesh to maintain compatibility by the insertion of pyramid elements several methods for generation of the pyramids are presented involving local tetrahedral transformations and or node insertion near the hex tet interface local smoothing and topological operations improve the quality of the transition region results show superior performance of the resulting elements in a commercial finite element code over non conforming interface conditions