imposing boundary and interface conditions in multi resolution wavelet galerkin method for numerical solution of helmholtz problems abstract in this paper we propose a new formulation for numerical solution of boundary value problems using mesh free multi resolution wavelet galerkin wg method a difficulty with the wg method is imposing boundary condition constraints single scale wavelet basis functions together with a jump function approach using cubic polynomial functions have been used in solving some boundary problems in the literature ramp and cubic polynomial functions have been also used in the jump function approach described in element free galerkin context and it is proven ramp functions exactly satisfy the desirable property for the derivative of jump but the cubic polynomial functions do not in this paper a multi resolution wg method is employed and both ramp and cubic polynomial jump functions are applied obtained results in one dimension are compared with analytical solutions results obtained with ramp jump functions in two dimensions are also reported the results obtained with the wg method manifest boundary and interface conditions are accurately imposed when the ramp jump functions are applied numerical results are in good agreement with accurate solutions

a wavelet based approach for the identification of linear time varying dynamical systems a novel procedure is developed for the identification of linear discrete models of dynamical systems from noisy data of particular interest is the application of the methodology to time varying systems the procedure is based on a representation of the governing differential equations with respect to a wavelet basis and the formulation of an inverse algebraic problem in the associated subspace the effect of noisy data is considered and numerical simulations demonstrating the applicability of the method to single and multi degree of freedom dynamical systems are presented

numerical solution of evolution equations by the haar wavelet method an efficient numerical method for solution of nonlinear evolution equations based on the haar wavelets approach is proposed the method is tested in the case of burgers and sine gordon equations numerical results obtained by computer simulation are compared with other available solutions these calculations demonstrate that the accuracy of the haar wavelet solutions is quite high even in the case of a small number of grid points

universal approximation using radial basis function networks there have been several recent studies concerning feedforward networks and the problem of approximating arbitrary functionals of a finite number of real variables some of these studies deal with cases in which the hidden layer nonlinearity is not a sigmoid this was motivated by successful applications of feedforward networks with nonsigmoidal hidden layer units this paper reports on a related study of radial basis function rbf networks and it is proved that rbf networks having one hidden layer are capable of universal approximation here the emphasis is on the case of typical rbf networks and the results show that a certain class of rbf networks with the same smoothing factor in each kernel node is broad enough for universal approximation

numerical solution of integro differential equations of high order by wavelet basis its algorithm and convergence analysis abstract this paper presents for the first time numerical solutions for this particular type of integro differential equations according to equations which will be introduced suitable wavelet galerkin method is provided using wavelet basis in the space c r l 2 r 0 that c r is the holder space of exponent this approach has two advantages first the wavelets basis are arbitrary it means that any differentiable wavelets basis can be used second the desired orders for this equation are the reasons for involving a wide variety of these types of equations the algorithm and convergence analysis of this scheme are described numerical examples plots and tablets of errors confirm the applicability and the validity of the proposed method

reduced order wavelet galerkin solution for the coupled nonlinear stochastic response of slender buildings in transient winds abstract a tall building is prone to wind induced stochastic vibration originating from complex fluid structure interaction dynamic coupling and nonlinear aerodynamic phenomena the loading induced by extreme wind events such as downburst storms hurricanes and tornadoes is naturally transient and nonstationary in comparison with the hypothesis of stationary wind loads used in both structural engineering research and practice time domain integration methods widely applied for solving nonlinear differential equations are hardly applicable to the analysis of coupled nonlinear and stochastic response of tall buildings under transient winds therefore the investigation of alternative and computationally efficient simulation methods is important this study employs the wavelet galerkin wg method to achieve this objective by examining the stochastic dynamic response of two tall building models subject to stationary and transient wind loads these are 1 a single degree of freedom equivalent model of a tall structure and 2 a multi degree of freedom reduced order full building model compactly supported daubechies wavelets are used as orthonormal basis functions in conjunction with the galerkin projection scheme to decompose and transform the coupled nonlinear differential equations of the two models into random algebraic equations in the wavelet domain methodology feasibility and applicability of the wg method are investigated in some special cases of stiffness nonlinearity duffing type and damping nonlinearity van der pol type for the single degree of freedom model for the reduced order tall building model the wg method is used to solve for dynamic coupling aerodynamics and transient wind load effects computation of connection coefficients effects of boundary conditions wavelet resolution and wavelet order are examined in order to adequately replicate the dynamic response realizations of multivariate stationary and transient wind loads for the building models are digitally simulated in the numerical computations

optimal combinations of diagnostic tests based on auc summary when several diagnostic tests are available one can combine them to achieve better diagnostic accuracy this article considers the optimal linear combination that maximizes the area under the receiver operating characteristic curve auc the estimates of the combination s coefficients can be obtained via a nonparametric procedure however for estimating the auc associated with the estimated coefficients the apparent estimation by re substitution is too optimistic to adjust for the upward bias several methods are proposed among them the cross validation approach is especially advocated and an approximated cross validation is developed to reduce the computational cost furthermore these proposed methods can be applied for variable selection to select important diagnostic tests the proposed methods are examined through simulation studies and applications to three real examples

observations on the behavior of radial basis function approximations near boundaries abstract rbf approximations would appear to be very attractive for approximating spatial derivatives in numerical simulations of pdes rbfs allow arbitrarily scattered data generalize easily to several space dimensions and can be spectrally accurate however accuracy degradations near boundaries in many cases severely limit the utility of this approach with that as motivation this study aims at gaining a better understanding of the properties of rbf approximations near the ends of an interval in 1 d and towards edges in 2 d

connection coefficients on an interval and wavelet solutions of burgers equation a definition of connection coefficients is introduced and techniques of computation are presented we use semi implicit time difference scheme to solve burgers equation by applying the evaluations of connection coefficients in calculating the integrals of the variational form comparisons of accuracy and robustness of numerical solutions are mentioned in the examples

large sample properties of simulations using latin hypercube sampling latin hypercube sampling mckay conover and beckman 1979 is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables the asymptotic variance of such an estimate is obtained the estimate is also shown to be asymptotically normal asymptotically the variance is less than that obtained using simple random sampling with the degree of variance reduction depending on the degree of additivity in the function being integrated a method for producing latin hypercube samples when the components of the input variables are statistically dependent is also described these techniques are applied to a simulation of the performance of a printer actuator

the weakly nonlocal limit of a one population wilson cowan model abstract we derive the weakly nonlocal limit of a one population neuronal field model of the wilson cowan type in one spatial dimension by transforming this equation to an equation in the firing rate variable it is shown that stationary periodic solutions exist by appealing to a pseudo potential analysis the solutions of the full nonlocal equation obey a uniform bound and the stationary periodic solutions in the weakly nonlocal limit satisfying the same uniform bound are characterized by finite ranges of pseudo energy constants the time dependent version of the model is reformulated as a ginzburg landau khalatnikov type of equation in the firing rate variable where the maximum minimum points correspond stable unstable homogeneous solutions of the weakly nonlocal limit based on this formulation it is also conjectured that the stationary periodic solutions are unstable we implement a numerical method for the weakly nonlocal limit of the wilson cowan type of model based on the wavelet galerkin approach we perform some numerical tests to illustrate the stability of homogeneous solutions and the evolution of the bumps

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

best combination of multiple diagnostic tests for screening purposes in a medical diagnostic testing problem multiple diagnostic tests are often available in distinguishing between diseased and nondiseased subjects different diagnostic tests are usually sensitive to different aspects of the disease a desirable approach is to combine multiple diagnostic tests so as to obtain an optimal composite diagnostic test with higher sensitivity and specificity that detects the presence of the disease more accurately to accomplish this it has been observed via signal detection theory developed in the 1950s and 1960s that the optimal combination of different diagnostic variables i e the diagnostic test results is determined by the likelihood ratio function for the diseased and nondiseased groups the conventional approach is to fit parametric models for the diseased and nondiseased groups separately and then to use the fitted likelihood ratio function for the best combination of test results however this approach is not so robust if the underlying distribution functions are misspecified since the optimal combination depends only on the likelihood ratio function it would be more appropriate to model this function directly a two sample semiparametric inference technique is applied to the model for the likelihood ratio function we consider the best combination of multiple diagnostic tests and study semiparametric likelihood estimation of the optimal receiver operating characteristic curve and the area under the curve we present a bootstrap procedure along with some results on simulation and on analysis of two real data sets

on the efficacy of the wavelet decomposition for high frequency vibration analyses abstract this paper reports the extraordinary ability of the wavelet decomposition for vibration analyses under the framework of rayleigh ritz method using a beam as an example daubechies wavelet scale functions are used as admissible functions for decomposing the flexural displacement of the structure along with the artificial springs at the boundary to predict vibration of an euler bernoulli beam in an extremely large frequency range it is shown that the use of wavelet basis allows reaching very high frequencies typically covering more than 1000 modes using conventional computational facility within the available numerical dynamics of the computers with no particular care needed for round off errors as a side benefit the use of spring boundary also allows handling any elastic boundary conditions through a dynamic contribution in the hamiltonian of the beam the wavelet decomposed approach combines the flexibility of the global methods and the accuracy of local methods by inheriting the versatility of the rayleigh ritz approach and the superior fitting ability of the wavelets numerical results on both free and forced vibrations are given in excellent agreement with predictions of classical methods

wavelet galerkin solutions for one dimensional partial differential equations in this paper we describe how wavelets may be used to solve partial differential equations these problems are currently solved by techniques such as finite differences finite elements and multi grid the wavelet method however offers several advantages over traditional methods wavelets have the ability to represent functions at different levels of resolution thereby providing a logical means of developing a hierarchy of solutions furthermore compactly supported wavelets such as those due to daubechies1 are localized in space which means that the solution can be refined in regions of high gradient e g stress concentrations without having to regenerate the mesh for the entire problem in order to demonstrate the wavelet technique we consider the one dimensional counterpart of helmholtz s equation by comparison with a simple finite difference solution to this problem with periodic boundary conditions we show how a wavelet technique may be efficiently developed dirichlet boundary conditions are then imposed using the capacitance matrix method described by proskurowski and widlund2 and others the convergence rates of the wavelet solutions are examined and they are found to compare extremely favourably to the finite difference solutions preliminary investigations also indicate that the wavelet technique is a strong contender to the finite element method at least for problems with simple geometries

evaluation of multifidelity surrogate modeling techniques to construct closure laws for drag in shock particle interactions abstract meta or surrogate models constructed from meso scale simulations can be used in place of empirical correlations to close macro scale equations in shocked particulate flows surrogate models for drag are constructed as functions of shock mach number ma particle volume fraction reynolds number re etc the computational cost of the high fidelity meso scale simulations is a challenge in construction of surrogates in such hierarchical multi scale frameworks here multifidelity surrogate modeling techniques are evaluated as inexpensive alternatives to high fidelity surrogate models for obtaining closure laws for drag in shock particle interactions preliminary surrogates for drag as a function of ma and are constructed from ensembles of low fidelity coarse grid mesoscale computations the low fidelity surrogates are subsequently corrected using only a few n h f high fidelity computations to obtain multifidelity surrogate models the paper evaluates three different methods for correcting an initial low fidelity surrogate space mapping sm radial basis functions rbf and modified bayesian kriging mbkg of these methods mbkg is found to provide the best multi fidelity surrogate model simultaneously minimizing the computational cost and error in the constructed surrogate

using radial basis functions to approximate a function and its error bounds a novel network called the validity index network vi net is presented the vi net derived from radial basis function networks fits functions and calculates confidence intervals for its predictions indicating local regions of poor fit and extrapolation

wavelets and the numerical solution of partial differential equations we present a new numerical method for the solution of partial differential equations in nonseparable domains the method uses a wavelet galerkin solver with a nontrivial adaptation of the standard capacitance matrix method the numerical solutions exhibit spectral convergence with regard to the order of the compactly supported daubechies wavelet basis furthermore the rate of convergence is found to be independent of the geometry we solve the helmholtz equation since for variations in the parameter the solutions have qualitative properties that well illustrate the applications of our method

an adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations in recent years there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary partial differential equations to this end the spectral stochastic finite element method ssfem is the most popular method due to its fast convergence rate recently the stochastic sparse grid collocation method has emerged as an attractive alternative to ssfem it approximates the solution in the stochastic space using lagrange polynomial interpolation the collocation method requires only repetitive calls to an existing deterministic solver similar to the monte carlo method however both the ssfem and current sparse grid collocation methods utilize global polynomials in the stochastic space thus when there are steep gradients or finite discontinuities in the stochastic space these methods converge very slowly or even fail to converge in this paper we develop an adaptive sparse grid collocation strategy using piecewise multi linear hierarchical basis functions hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region numerical examples especially for problems related to long term integration and stochastic discontinuity are presented comparisons with monte carlo and multi element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method

combining large number of weak biomarkers based on auc combining multiple biomarkers to improve diagnosis and or prognosis accuracy is a common practice in clinical medicine both parametric and non parametric methods have been developed for finding the optimal linear combination of biomarkers to maximize the area under the receiver operating characteristic curve auc primarily focusing on the setting with a small number of well defined biomarkers this problem becomes more challenging when the number of observations is not order of magnitude greater than the number of variables especially when the involved biomarkers are relatively weak such settings are not uncommon in certain applied fields the first aim of this paper is to empirically evaluate the performance of existing linear combination methods under such settings the second aim is to propose a new combination method namely the pairwise approach to maximize auc our simulation studies demonstrated that the performance of several existing methods can become unsatisfactory as the number of markers becomes large while the newly proposed pairwise method performs reasonably well furthermore we apply all the combination methods to real datasets used for the development and validation of mammaprint the implication of our study for the design of optimal linear combination methods is discussed copyright 2015 john wiley sons ltd

comments on the computation of wavelet galerkin approximation on a bounded interval in this letter we identify and correct the errors in int j numer meth eng 1996 39 2921 2944 and we also develop clearer procedures for the computation of the connection coefficients from the wavelet galerkin scheme

pitfalls of using a single criterion for selecting experimental designs for surrogate construction a good experimental design ed is essential to simultaneously reduce the effect of noise and bias errors however most eds cater to a single criterion and may lead to small gains in that criterion at the expense of large deteriorations in other criteria we use multiple criteria to assess the performance of different popular eds we demonstrate that these eds offer different trade offs and that use of a single criterion is indeed risky in addition we show that popular eds such as latin hypercube sampling lhs and d optimal designs often leave large regions of the design space unsampled even for moderate dimensions we discuss a few possible strategies to combine multiple criteria and illustrate them with examples we show that complementary criteria e g bias handling criterion for variance based designs and vice versa can be combined to improve the performance of eds we demonstrate improvements in the trade offs between noise and bias error by combining a model based criterion like the d optimality criterion and a geometry based criterion like lhs next we demonstrate that selecting an ed from three candidate eds using a suitable error based criterion helped eliminate potentially poor designs finally we show benefits from combining the multiple criteria based strategies that is generation of multiple eds using the d optimality and lhs criteria and selecting one design using a pointwise bias error criterion the encouraging results from the examples indicate that it may be worthwhile studying these strategies more rigorously and in more detail copyright 2007 john wiley sons ltd

a note on one point quadrature formula for daubechies scale function with partial support abstract this note is concerned with an efficient computation of integrals of products of a smooth function and daubechies scale function with partial support by using a one point quadrature rule the error estimate is obtained the rule is illustrated by considering an example from the literature

a stable algorithm for flat radial basis functions on a sphere when radial basis functions rbfs are made increasingly flat the interpolation error typically decreases steadily until some point when runge type oscillations either halt or reverse this trend because the most obvious method to calculate an rbf interpolant becomes a numerically unstable algorithm for a stable problem in the case of near flat basis functions there will typically also be a separate point at which disastrous ill conditioning enters we introduce here a new method rbf qr which entirely eliminates such ill conditioning and we apply it in the special case when the data points are distributed over the surface of a sphere this algorithm works even for thousands of node points and it allows the rbf shape parameter to be optimized without the limitations imposed by stability concerns since interpolation in the flat rbf limit on a sphere is found to coincide with spherical harmonics interpolation new insights are gained as to why the rbf approach with nonflat basis functions often is the more accurate of the two methods

solving volterra integral equations of the second kind by wavelet galerkin scheme in this paper we apply the wavelet galerkin method to obtain approximate solutions to linear volterra integral equations vies of the second kind daubechies wavelets are used to find such approximations in this approach we introduce some new connection coefficients and discuss their properties and propose algorithms to evaluate them these coefficients can be computed just once and applied for solving every linear vie of the second kind convergence and error analysis are discussed and numerical examples illustrate the efficiency of the method

multiscale approximation with hierarchical radial basis functions networks an approximating neural model called hierarchical radial basis function hrbf network is presented here this is a self organizing by growing multiscale version of a radial basis function rbf network it is constituted of hierarchical layers each containing a gaussian grid at a decreasing scale the grids are not completely filled but units are inserted only where the local error is over threshold this guarantees a uniform residual error and the allocation of more units with smaller scales where the data contain higher frequencies only local operations which do not require any iteration on the data are required this allows to construct the network in quasi real time through harmonic analysis it is demonstrated that although a hrbf cannot be reduced to a traditional wavelet based multiresolution analysis mra it does employ riesz bases and enjoys asymptotic approximation properties for a very large class of functions hrbf networks have been extensively applied to the reconstruction of three dimensional 3 13 models from noisy range data the results illustrate their power in denoising the original data obtaining an effective multiscale reconstruction of better quality than that obtained by mra

using the refinement equation for evaluating integrals of wavelets the wavelet galerkin method for solving partial differential equations leads to the problem of computing integrals of products of derivatives of wavelets this paper studies the problem from the point of view of stationary subdivision schemes one of the main results is to identify these integrals as components of the unique solution of a certain eigenvector moment problem associated with the coefficients of the refinement equation asymptotic expansions for the corresponding subdivision schemes form an important ingredient of our approach

selecting variables in non parametric regression models for binary response an application to the computerized detection of breast cancer in many biomedical applications interest lies in being able to distinguish between two possible states of a given response variable depending on the values of certain continuous predictors if the number of predictors p is high or if there is redundancy among them it then becomes important to decide on the selection of the best subset of predictors that will be able to obtain the models with greatest discrimination capacity with this aim in mind logistic generalized additive models were considered and receiver operating characteristic roc curves were applied in order to determine and compare the discriminatory capacity of such models this study sought to develop bootstrap based tests that allow for the following to be ascertained a the optimal number q or p of predictors and b the model or models including q predictors which display the largest auc area under the roc curve a simulation study was conducted to verify the behaviour of these tests finally the proposed method was applied to a computer aided diagnostic system dedicated to early detection of breast cancer

combining multiple biomarkers linearly to maximize the partial area under the roc curve it is now common in clinical practice to make clinical decisions based on combinations of multiple biomarkers in this paper we propose new approaches for combining multiple biomarkers linearly to maximize the partial area under the receiver operating characteristic curve pauc the parametric and nonparametric methods that have been developed for this purpose have limitations when the biomarker values for populations with and without a given disease follow a multivariate normal distribution it is easy to implement our proposed parametric approach which adopts an alternative analytic expression of the pauc when normality assumptions are violated a kernel based approach is presented which handles multiple biomarkers simultaneously we evaluated the proposed as well as existing methods through simulations and discovered that when the covariance matrices for the disease and nondisease samples are disproportional traditional methods such as the logistic regression are more likely to fail to maximize the pauc while the proposed methods are more robust the proposed approaches are illustrated through application to a prostate cancer data set and a rank based leave one out cross validation procedure is proposed to obtain a realistic estimate of the pauc when there is no independent validation set available

wavelet galerkin method for the free vibrations of an elastic cable carrying an attached mass a multilevel representation of daubechies compactly supported wavelet has been used to study the free vibrations of elastic catenary cables carrying an attached mass anti derivatives of wavelets are used to guarantee satisfaction of boundary conditions natural frequencies mode shapes and dynamic tensions are obtained and compared with the classical fourier series representation the localization feature of wavelets has been implemented to enter the singularity region that is produced by the attached mass more wavelets are used near the mass location and the spurious oscillations in the solution are minimized with few number of terms in the series however the fourier solution shows many oscillations along the cable length and gibbs phenomenon at the mass location in both methods reverting and swapping modes are discovered in which higher modes revert to lower modes and that the horizontal displacement components become greater than the vertical ones even for cables with small sag to span ratios

a dynamic multiscale lifting computation method using daubechies wavelet an important property of wavelet multiresolution analysis is the capability to represent functions in a dynamic multiscale manner so the solution in the wavelet domain enables a hierarchical approximation to the exact solution the typical problem that arises when using daubechies wavelets in numerical analysis especially in finite element analysis is how to calculate the connection coefficients an integral of products of wavelet scaling functions or derivative operators associated with these the method to calculate multiscale connection coefficients for stiffness matrices and load vectors is presented for the first time and the algorithm of multiscale lifting computation is developed the numerical examples are given to verify the effectiveness of such a method

evaluation of convergence behavior of metamodeling techniques for bridging scales in multi scale multimaterial simulation the effectiveness of several metamodeling techniques viz the polynomial stochastic collocation method adaptive stochastic collocation method a radial basis function neural network a kriging method and a dynamic kriging method is evaluated this is done with the express purpose of using metamodels to bridge scales between micro and macro scale models in a multi scale multimaterial simulation the rate of convergence of the error when used to reconstruct hypersurfaces of known functions is studied for sufficiently large number of training points stochastic collocation methods generally converge faster than the other metamodeling techniques while the dkg method converges faster when the number of input points is less than 100 in a two dimensional parameter space because the input points correspond to computationally expensive micro meso scale computations the dkg is favored for bridging scales in a multi scale solver

comparison of the solutions obtained by adomian decomposition and wavelet galerkin methods of boundary value problems abstract in this paper we will compare the performance of adomian decomposition method and the wavelet galerkin method applied to the solution of boundary value problems involving non homogeneous heat and wave equations it is shown that the adomian decomposition method in many instances gives better results in the wavelet galerkin solutions daubechies six wavelets are used because they give better results than those of lower degree wavelets the results are then compared with those obtained using the adomian decomposition method although the adomian decomposition solution required slightly more computational effort than the wavelet galerkin solution it resulted in more accurate results than the wavelet galerkin method

time dependent predictive accuracy in the presence of competing risks competing risks arise naturally in time to event studies in this article we propose time dependent accuracy measures for a marker when we have censored survival times and competing risks time dependent versions of sensitivity or true positive tp fraction naturally correspond to consideration of either cumulative or prevalent cases that accrue over a fixed time period or alternatively to incident cases that are observed among event free subjects at any select time time dependent dynamic specificity 1 false positive fp can be based on the marker distribution among event free subjects we extend these definitions to incorporate cause of failure for competing risks outcomes the proposed estimation for cause specific cumulative tp dynamic fp is based on the nearest neighbor estimation of bivariate distribution function of the marker and the event time on the other hand incident tp dynamic fp can be estimated using a possibly nonproportional hazards cox model for the cause specific hazards and riskset reweighting of the marker distribution the proposed methods extend the time dependent predictive accuracy measures of heagerty lumley and pepe 2000 biometrics 56 337 344 and heagerty and zheng 2005 biometrics 61 92 105

using a monotonic density ratio model to find the asymptotically optimal combination of multiple diagnostic tests abstractwith the advent of new technology new biomarker studies have become essential in cancer research to achieve optimal sensitivity and specificity one needs to combine different diagnostic tests the celebrated neyman pearson lemma enables us to use the density ratio to optimally combine different diagnostic tests in this article we propose a semiparametric model by directly modeling the density ratio between the diseased and nondiseased population as an unspecified monotonic nondecreasing function of a linear or nonlinear combination of multiple diagnostic tests this method is appealing in that it is not necessary to assume separate models for the diseased and nondiseased populations further the proposed method provides an asymptotically optimal way to combine multiple test results we use a pool adjacent violation algorithm to find the semiparametric maximum likelihood estimate of the receiver operating characteristic roc curve using modern empirical process theory we show cubic root n c

combining predictors for classification using the area under the receiver operating characteristic curve no single biomarker for cancer is considered adequately sensitive and specific for cancer screening it is expected that the results of multiple markers will need to be combined in order to yield adequately accurate classification typically the objective function that is optimized for combining markers is the likelihood function in this article we consider an alternative objective function the area under the empirical receiver operating characteristic curve auc we note that it yields consistent estimates of parameters in a generalized linear model for the risk score but does not require specifying the link function like logistic regression it yields consistent estimation with case control or cohort data simulation studies suggest that auc based classification scores have performance comparable with logistic likelihood based scores when the logistic regression model holds analysis of data from a proteomics biomarker study shows that performance can be far superior to logistic regression derived scores when the logistic regression model does not hold model fitting by maximizing the auc rather than the likelihood should be considered when the goal is to derive a marker combination score for classification or prediction

solution of second kind integral equation with cauchy type kernel using daubechies scale function a method based on the daubechies scale function has been devised to solve numerically a class of second kind integral equations with a cauchy type kernel the convergence associated to the method has been established the method is illustrated with integral equations in this class whose exact solutions are known the numerical results obtained by the present method are seen to differ negligibly from the exact results

evaluation of kriging based surrogate models constructed from mesoscale computations of shock interaction with particles abstract macro scale computations of shocked particulate flows require closure laws that model the exchange of momentum energy between the fluid and particle phases closure laws are constructed in this work in the form of surrogate models derived from highly resolved mesoscale computations of shock particle interactions the mesoscale computations are performed to calculate the drag force on a cluster of particles for different values of mach number and particle volume fraction two kriging based methods viz the dynamic kriging method dkg and the modified bayesian kriging method mbkg are evaluated for their ability to construct surrogate models with sparse data i e using the least number of mesoscale simulations it is shown that if the input data is noise free the dkg method converges monotonically convergence is less robust in the presence of noise the mbkg method converges monotonically even with noisy input data and is therefore more suitable for surrogate model construction from numerical experiments this work is the first step towards a full multiscale modeling of interaction of shocked particle laden flows

the computation of wavelet galerkin approximation on a bounded interval this paper describes exact evaluations of various finite integrals whose integrands involve products of daubechies compactly supported wavelets and their derivatives and or integrals these finite integrals play an essential role in the wavelet galerkin approximation of differential or integral equations on a bounded interval

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

overestimation of the receiver operating characteristic curve for logistic regression logistic regression is often used to find a linear combination of covariates which best discriminates between two groups or populations the roc receiver operating characteristic curve is a good way of assessing the performance of the resulting score but using the same data both to fit the score and to calculate its roc leads to an over optimistic estimate of the performance which the score would give if it were to be validated on a sample of future cases the paper studies the extent of this overestimation and suggests a shrinkage correction for the roc curve itself and for the area under the curve the correction is consistent with efron s formula for the bias in the error rate of a binary prediction rule two medical examples are discussed

wavelet calculus and finite difference operators this paper shows that the naturally induced discrete differentiation operators induced from a wavelet galerkin finite dimensional approximation to a standard function space approximates differentiation with an error of order 0 h2d 2 where d is the degree of the wavelet system the degree of a wavelet system is defined as one less than the degree of the lowest order nonvanishing moment of the fundamental wavelet we consider in this paper compactly supported wavelets of the type introduced by daubechies in 1988 the induced differentiation operators are described in terms of connection coefficients which are intrinsically defined functional invariants of the wavelet system defined as l2 inner products of derivatives of wavelet basis functions with the basis functions themselves these connection coefficients can be explicitly computed without quadrature and they themselves have key moment vanishing properties proved in this paper which are dependent upon the degree of the wavelet system this is the basis for the proof of the principal results concerning the degree of approximation of the differentiation operator by the wavelet galerkin discrete differentiation operator

orthogonal least squares learning algorithm for radial basis function networks the radial basis function network offers a viable alternative to the two layer neural network in many applications of signal processing a common learning algorithm for radial basis function networks is based on first choosing randomly some data points as radial basis function centers and then using singular value decomposition to solve for the weights of the network such a procedure has several drawbacks and in particular an arbitrary selection of centers is clearly unsatisfactory the authors propose an alternative learning procedure based on the orthogonal least squares method the procedure chooses radial basis function centers one by one in a rational way until an adequate network has been constructed in the algorithm each selected center maximizes the increment to the explained variance or energy of the desired output and does not suffer numerical ill conditioning problems the orthogonal least squares learning strategy provides a simple and efficient means for fitting radial basis function networks this is illustrated using examples taken from two different signal processing applications

forced vibrations of a turbine blade undergoing regularized unilateral contact conditions through the wavelet balance method summary the method of weighted residuals can efficiently enforce time periodic solutions of flexible structures experiencing unilateral contact the harmonic balance method hbm based on fourier expansion of the sought solution is a common formulation although wavelet bases that can sparsely define nonsmooth solutions may be superior this hypothesis is investigated using a full three dimensional blade with unilateral contact conditions on a set of nc discrete contact points located at its tip the unilateral contact conditions are first regularized and a distributional formulation in time is introduced allowing l2 s1 n trial functions to properly approximate in the time domain the solution to the governing equations the mixed wavelet petrov galerkin solutions are found to yield consistent or better results than hbm with higher convergence rates and seemingly more accurate contact force prediction copyright 2014 john wiley sons ltd

stable evaluation of gaussian radial basis function interpolants we provide a new way to compute and evaluate gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter i e for flat kernels this work is motivated by the fundamental ideas proposed earlier by bengt fornberg and his coworkers however following mercer s theorem an l 2 mathbb r d rho orthonormal expansion of the gaussian kernel allows us to come up with an algorithm that is simpler than the one proposed by fornberg larsson and flyer and that is applicable in arbitrary space dimensions d in addition to obtaining an accurate approximation of the radial basis function interpolant using many terms in the series expansion of the kernel we also propose and investigate a highly accurate least squares approximation based on early truncation of the kernel expansion

solution of cartesian and curvilinear quantum equations via multiwavelets on the interval abstract it is shown how orthogonal compact support multiwavelets may be used for the solution of quantum mechanical eigenvalue problems subject to specific boundary conditions special scaling functions and wavelets with convenient limiting behaviors at the edges of an interval are constructed in analogy to earlier work on single wavelet families all of the integrals required for hamiltonian matrix elements involving both regular and edge functions are calculated efficiently through use of recursion and quadrature methods it is demonstrated through accurate eigenvalue determination that both cartesian and curvilinear degrees of freedom are readily accommodated with such a basis using as examples the particle in a box and the hydrogen atom in spherical polar coordinates

a numerical algorithm for the solution of telegraph equations abstract in this paper we present a new competitive numerical scheme to solve nonlinear telegraph equations the method is based on rothe s approximation in time discretization and on the wavelet galerkin in the spatial discretization the approximate solutions converge in the space c 0 t l 2 l 2 0 t w 0 1 2 to the variational solution a full error analysis is performed and a numerical experiment is given to illustrate the good convergence behavior of the approximate solution

solving the 2 d heat equations using wavelet galerkin method with variable time step a wavelet galerkin method to solve nonhomogenous 2 d heat equations in finite rectangular domains is presented integrals involving nonhomogenity terms with scaling function are evaluated based on available connection coefficients variable time step technique is used to delay the error blow up hence obtaining better results for solutions that reach steady state

galerkin wavelet methods for two point boundary value problems anti derivatives of wavelets are used for the numerical solution of differential equations optimal error estimates are obtained in the applications to two point boundary value problems of second order the orthogonal property of the wavelets is used to construct efficient iterative methods for the solution of the resultant linear algebraic systems numerical examples are given