boundary reconstruction in binary images using splines in image analysis it is often required to reconstruct the boundary of an object in a noisy image this paper presents a new method which relies on flexibility and computational simplicity of b spline curves to reconstruct a smooth connected boundary in a noisy binary image boundary inference is based on oriented distance functions yielding the estimator which is interpreted as a posterior expected boundary of the underlying random set the performance of the method and its dependence on the image quality and model specification are studied on simulated data the method is applied to reconstruct the skin air boundary in digitised analogue mammogram images highlightswe present a new method for boundary reconstruction in noisy images using b spline curves the method is bayesian and uses oriented distance functions for inference the resulting estimator can be interpreted as a posterior expected boundary we study the performance of the method on simulated data we apply the method to reconstruct the skin air boundary in mammogram images

adaptive radial basis function and hermite function pseudospectral methods for computing eigenvalues of the prolate spheroidal wave equation for very large bandwidth parameter asymptotic approximations show that the lowest modes of the prolate spheroidal wave equation are concentrated with an o 1 c length scale where c is the bandwidth parameter of the prolate differential equation accurate computation of the ground state eigenvalue by the long known legendre galerkin method requires roughly 3 8 c legendre polynomials some studies have therefore applied a grid with 20 000 points in conjunction with high order finite differences to achieve c 10 7 here we show that by adaptively applying either hermite functions or gaussian radial basis functions rbfs it is never necessary to use more than eighty degrees of freedom to calculate the lowest dozen eigenvalues of each symmetry class for small c the eigenmodes are not confined to a small portion of the domain 2 2 in latitude but are global we show that by periodizing the basis functions via imbricate series it is possible to apply hermite and rbfs even in the limit c 0 the legendre method is probably a little more efficient in this limit since the prolate functions tend to legendre polynomials in this limit a sideband truncation restricts the discretization to a small block taken from the large hermite galerkin matrix we show that sideband truncation with blocks as small as 5 5 is a very efficient way to compute high order modes in an appendix we prove a rigorous convergence theorem for the periodized hermite expansion

nonparametric estimation of the variogram and its spectrum in the study of intrinsically stationary spatial processes a new nonparametric variogram estimator is proposed through its spectral representation the methodology is based on estimation of the variogram s spectrum by solving a regularized inverse problem through quadratic programming the estimated variogram is guaranteed to be conditionally negative definite simulation shows that our estimator is flexible and generally has smaller mean integrated squared error than the parametric estimator under model misspecification our methodology is applied to a spatial dataset of decadal temperature changes copyright 2011 oxford university press

the calculation of singular coefficients abstract we consider in this paper the calculation of the singular coefficients associated with the solution of an elliptic partial differential equation near a singular point a re entrant corner or a crack tip etc these are the coefficients in the relevant singular expansion of the solution near the point of singularity they often have physical relevance and it is of interest to be able to calculate them accurately we consider the problem in the context of the global element method this is a variable order finite element method designed to be capable of producing highly accurate solutions for singular problems even in the neighbourhood of the singularity if the values of the singular coefficients are needed these must be extracted from the computed solution we show in this paper that a suitably defined least squares fitting procedure allows the calculation of values for the leading singular coefficients which are as accurate as the underlying solution

smoothly mixing regressions this paper extends the conventional bayesian mixture of normals model by permitting state probabilities to depend on observed covariates the dependence is captured by a simple multinomial probit model a conventional and rapidly mixing mcmc algorithm provides access to the posterior distribution at modest computational cost this model is competitive with existing econometric models as documented in the paper s illustrations the first illustration studies quantiles of the distribution of earnings of men conditional on age and education and shows that smoothly mixing regressions are an attractive alternative to nonbayesian quantile regression the second illustration models serial dependence in the s p 500 return and shows that the model compares favorably with arch models using out of sample likelihood criteria

mixture of regression models with varying mixing proportions a semiparametric approach in this article we study a class of semiparametric mixtures of regression models in which the regression functions are linear functions of the predictors but the mixing proportions are smoothing functions of a covariate we propose a one step backfitting estimation procedure to achieve the optimal convergence rates for both regression parameters and the nonparametric functions of mixing proportions we derive the asymptotic bias and variance of the one step estimate and further establish its asymptotic normality a modified expectation maximization type em type estimation procedure is investigated we show that the modified em algorithms preserve the asymptotic ascent property numerical simulations are conducted to examine the finite sample performance of the estimation procedures the proposed methodology is further illustrated via an analysis of a real dataset

the complex wavenumber eigenvalues of laplace s tidal equations for oceans bounded by meridians the complex wavenumber eigenvalues of laplace s tidal equations are determined for an ocean of constant depth bounded by meridians a galerkin method is used to expand the tide height and velocities in series of associated legendre functions a homogeneous system of equations results from the continuity and momentum equations the frequency and depth are fixed so that the meridional wavenumbers are the eigenvalues this gives rise to a generalized eigenvalue problem that must be solved numerically by iteration the eigenvalues are not integers and represent inertia gravity wave solutions at the specified tidal forcing frequency that can be excited by the presence of meridional boundaries those complex eigenvalues represent solutions that decay away from meridional boundaries the eigenvalue spectrum is investigated for the semi diurnal fortnightly and monthly tides one complex wavenumber for the semi diurnal tide explains the amphidromic systems within 20 of a north south coastline the fortnightly and monthly tides have only real wavenumber eigenvalues the basin scale deviation of these tides from equilibrium is attributed to low wavenumber divergent inertia gravity waves

a chebyshev rational chebyshev spectral method for the helmholtz equation in a sector on the surface of a sphere defeating corner singularities when the boundaries of a domain meet at an angle the solutions to an elliptic partial differential equation will usually be singular at the corner using the example of the helmholtz equation on the surface of a sphere in a domain bounded by meridians we show how corner singularities can be defeated by mapping the corner to infinity by applying a chebyshev series in longitude and a rational chebyshev series in the mercator coordinate y arctanh cos colatitude we obtain an exponential rate of convergence despite the corner singularities

a bayesian discovery procedure we discuss a bayesian discovery procedure for multiple comparison problems we show that under a coherent decision theoretic framework a loss function combining true positive and false positive counts leads to a decision rule based on a threshold of the posterior probability of the alternative under a semi parametric model for the data we show that the bayes rule can be approximated by the optimal discovery procedure odp recently introduced by storey 2007a improving the approximation leads us to a bayesian discovery procedure bdp which exploits the multiple shrinkage in clusters implied by the assumed nonparametric model we compare the bdp and the odp estimates in a simple simulation study and in an assessment of differential gene expression based on microarray data from tumor samples we extend the setting of the odp by discussing modifications of the loss function that lead to different single thresholding statistics finally we provide an application of the previous arguments to dependent spatial data

a fast 3d poisson solver of arbitrary order accuracy we present a direct solver for the poisson and laplace equations in a 3d rectangular box the method is based on the application of the discrete fourier transform accompanied by a subtraction technique which allows reducing the errors associated with the gibbs phenomenon and achieving any prescribed rate of convergence the algorithm requireso n3logn operations wherenis the number of grid points in each direction we show that our approach allows accurate treatment of singular cases which arise when the boundary function is discontinuous or incompatible with the differential equation

traps and snares in eigenvalue calculations with application to pseudospectral computations of ocean tides in a basin bounded by meridians we make several observations about eigenvalue problems using as examples laplace s tidal equations and the differential equation satisfied by the associated legendre functions whatever the discretization only some of the eigenvalues of then dimensional matrix eigenvalue problem will be good approximations to those of the differential equation usuallythen 2 eigenvalues of smallest magnitude for the tidal problem however the good eigenvalues are scattered so our first point is it is important to plot the drift of eigenvalues with changes in resolution we suggest plotting the difference between a low resolution eigenvalue and the nearest high resolution eigenvalue divided by the magnitude of the eigenvalue or the intermodal separation whichever is smaller second as a final safeguard it is important to look at the chebyshev coefficients of the mode we show a numerically computed anti kelvin wave which has little eigenvalue drift but is completely spurious as is obvious from its spectral series third inverting the roles of parameters can drastically modify the spectrum legendre s equation may have either an infinite number of discrete modes or only a handful depending on which parameter is the eigenvalue fourth when the modes are singular but decay to zero at the endpoints as is true of tides a tanh mapping can retrieve the usual exponential accuracy of spectral methods fifth the pseudospectral method is more reliable than deriving a banded galerkin matrix by means of recurrence relations the pseudospectral code is simple to check whereas it is easy to make an untestable mistake with the intricate algebra required for the galerkin method sixth we offer a brief cautionary tale about overlooked modes all these cautions are applicable toallforms of spatial discretization including finite difference and finite element methods however we limit our illustrations to spectral schemes where these difficulties are most easily resolved with a bit of care the pseudospectral method is a very robust and efficient method for solving differential eigenproblems even with singular eigenmodes

finite mixture varying coefficient models for analyzing longitudinal heterogenous data this paper aims to develop a mixture model to study heterogeneous longitudinal data on the treatment effect of heroin use from a california civil addict program each component of the mixture is characterized by a varying coefficient mixed effect model we use the bayesian p splines approach to approximate the varying coefficient functions we develop markov chain monte carlo algorithms to estimate the smooth functions unknown parameters and latent variables in the model we use modified deviance information criterion to determine the number of components in the mixture a simulation study demonstrates that the modified deviance information criterion selects the correct number of components and the estimation of unknown quantities is accurate we apply the proposed model to the heroin treatment study furthermore we identify heterogeneous longitudinal patterns copyright 2011 john wiley sons ltd

a bayesian framework to identify principal intravoxel diffusion profiles based on diffusion weighted mr imaging in this paper we introduce a new method to characterize the intravoxel anisotropy based on diffusion weighted imaging dwi the proposed solution under a fully bayesian formalism deals with the problem of joint bayesian model selection and parameter estimation to reconstruct the principal diffusion profiles or primary fiber orientations in a voxel we develop an efficient stochastic algorithm based on the reversible jump markov chain monte carlo rjmcmc method in order to perform the bayesian computation rjmcmc is a good choice for this problem because of its ability to jump between models of different dimensionality this methodology provides posterior estimates of the parameters of interest fiber orientation diffusivities etc unconditional of the model assumed it also gives an empirical posterior distribution of the number of primary nerve fiber orientations given the dwi data different probability maps can be assessed using this methodology 1 the intravoxel fiber orientation map or orientational distribution function that gives the probability of finding a fiber in a particular spatial orientation 2 a three dimensional map of the probability of finding a particular number of fibers in each voxel 3 a three dimensional maxpro maximum probability map that provides the most probable number of fibers for each voxel in order to study the performance and reliability of the presented approach we tested it on synthetic data an ex vivo phantom of intersecting capillaries and dwi data from a human subject

a spectral solution of the sturm liouville equation comparison of classical and nonclassical basis sets abstract this paper considers a spectral method of solution of the sturm liouville equation and the associated schroedinger equation the main objective is to develop a collocation method based on quadrature collocation points generated from nonclassical polynomials the polynomials and associated quadrature points are calculated with gautschi s stieltjes procedure from some specified weight function the particular spectral method used here with nonclassical basis sets is referred to as the quadrature discretization method qdm the qdm and a related weighted qdm are applied to several sturm liouville and schroedinger equations and the results are compared with the traditional spectral methods based on chebyshev and legendre quadrature points the results are also compared with the results of other workers wherever available the qdm was found to give the most rapid convergence relative to other methods for the problems studied

bayesian object identification this paper addresses the task of locating and identifying an unknown number of objects of different types in an image baddeley van lieshout 1993 advocate marked point processes as object priors whereas grenander miller 1994 use deformable template models in this paper elements of both approaches are combined to handle scenes containing variable numbers of objects of different types using reversible jump markov chain monte carlo methods for inference green 1995 the naive application of these methods here leads to slow mixing and we adapt the model and algorithm in tandem in proposing three strategies to deal with this the first two expand the model space by introducing an additional unknown object type and the idea of a variable resolution template the third strategy utilising the first two augments the algorithm with classes of updates which provide intuitive transitions between realisations containing different numbers of cells by splitting or merging nearby objects

the asymptotic chebyshev coefficients for functions with logarithmic endpoint singularities mappings and singular basis functions abstract when a function is singular at the ends of its expansion interval its chebyshev coefficients a n converge very poorly we analyze three numerical strategies for coping with such singularities of the form 1 x k log 1 x and in the process make some modest additions to the theory of chebyshev expansions the first two numerical methods are the convergence improving changes of coordinate x sin 2 y and x tanh ly 1 y 2 1 2 we derive the asymptotic chebyshev coefficients in the limit n for both mappings and for the original untransformed chebyshev series for the original function the asymptotic approximation for general k is augmented by the exact chebyshev coefficients for integer k numerical tests show that the sine mapping is excellent for k 1 increasing the rate of convergence to b n o 1 n 4 k 1 although the tanh transformation is guaranteed to be better for sufficiently large n we offer both theoretical and numerical evidence to explain why the sine mapping is usually better in practice sufficiently large n is usually huge instead of mapping one may use a third strategy supplementing the chebyshev polynomials with singular basis functions simple experiments show that this approach is also successful

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

diagnostic measures for model criticism abstract we discuss the problem of model criticism with emphasis on developing summary diagnostic measures we approach model criticism by identifying possible troublesome features of the currently entertained model embedding the model in an elaborated model and measuring the value of elaborating this requires three elements a model elaboration a prior distribution and a utility function each triplet generates a different diagnostic measure we focus primarily on the measure given by a kullback leibler divergence between the marginal prior and posterior distributions on the elaboration parameter we also develop a linearized version of this diagnostic and use it to show that our procedure is related to other tools commonly used for model diagnostics such as bayes factors and the score function one attraction of this approach is that it allows model criticism to be performed jointly with parameter inference and prediction also this diagnostic approach aims at maintaining an exploratory nature t

optimal scaling of discrete approximations to langevin diffusions we consider the optimal scaling problem for proposal distributions in hastings metropolis algorithms derived from langevin diffusions we prove an asymptotic diffusion limit theorem and show that the relative efficiency of the algorithm can be characterized by its overall acceptance rate independently of the target distribution the asymptotically optimal acceptance rate is 0 574 we show that as a function of dimension n the complexity of the algorithm is o n1 3 which compares favourably with the o n complexity of random walk metropolis algorithms we illustrate this comparison with some example simulations

boundary detection through dynamic polygons a method for the bayesian restoration of noisy binary images portraying an object with constant grey level on a background is presented the restoration performed by fitting a polygon with any number of sides to the object s outline is driven by a new probabilistic model for the generation of polygons in a compact subset of r 2 which is used as a prior distribution for the polygon some measurability issues raised by the correct specification of the model are addressed the simulation from the prior and the calculation of the a posteriori mean of grey levels are carried out through reversible jump markov chain monte carlo computation whose implementation and convergence properties are also discussed one example of restoration of a synthetic image is presented and compared with existing pixel based methods

on the stability of linear systems with uncertain delay this paper focuses on the stability of some class of delay systems including uncertainty in the delays more precisely we are interested in guaranteeing the stability of perturbed delay systems by assuming the stability of the nominal system if the delay perturbation is constant necessary and sufficient conditions are derived in terms of generalized eigenvalue distribution of some finite dimensional constant matrix pencil if the delay perturbation is time varying some sufficient stability conditions are derived using exact lyapunov krasovskii functionals illustrative examples are also included

the complex wavenumber eigenvalues of laplace s tidal equations for oceans bounded by meridians corrigendum an error in the article o connor 1995 was brought to my attention by charles sozou this changes the values of the eigenvalues calculated in table 1 and the geophysical conclusions are modified the introduction and 2 on laplace s tidal equations are correct the mistake occured in 3 when the legendre function expansion was substituted into the set of laplace s tidal equations the term a in the expansion for the tide height in equation 12 must be zero otherwise equation 7 cannot be satisfied i did not see this at the time because i neglected to include the a term when substituting the expansions 12 into equation 7 this is why no a term appears in equation 18 the corrections are to omit the a terms in equations 12 and 17

a rational spectral method for the kdv equation on the half line a petrov galerkin method using orthogonal rational functions is proposed for the korteweg de vries kdv equation on the half line with initial boundary values the nonlinear term and the right hand side term are treated by chebyshev rational interpolation explicitly and the linear terms are computed with the galerkin method implicitly such an approach is applicable using fast algorithms numerical results are presented for problems with both exponentially and algebraically decaying solutions respectively highlighting the performance of the proposed method

an image analysis problem in electron microscopy abstract this article considers the problem of identifying the irregular boundary of a magnetic domain in a thin multilayer film using data in the form of an electron micrograph two approaches are illustrated both of which are based on the concept of two smooth intensity surfaces one corresponding to the domain and one corresponding to the background so that the objective is to determine the boundary between the parts of the image where the two surfaces are present in the first a hierarchical bayesian approach priors are assumed for the set of domain background states and for the two intensity surfaces and restoration is carried out using a version of besag s icm procedure it is shown that it is important to initialize the method efficiently the second approach uses a template like model for the boundary and also relies on a bayesian approach the results from the second approach can be used as an end in themselves or as a way of initializing the more flexible first approach several illustrati

localization of shapes using statistical models and stochastic optimization in this paper we present a new model for deformations of shapes a pseudolikelihood is based on the statistical distribution of the gradient vector field of the gray level the prior distribution is based on the probabilistic principal component analysis ppca we also propose a new model based on mixtures of ppca that is useful in the case of greater variability in the shape a criterion of global or local object specificity based on a preliminary color segmentation of the image is included into the model the localization of a shape in an image is then viewed as minimizing the corresponding gibbs field we use the exploration selection e s stochastic algorithm in order to find the optimal deformation this yields a new unsupervised statistical method for localization of shapes in order to estimate the statistical parameters for the gradient vector field of the gray level we use an iterative conditional estimation ice procedure the color segmentation of the image can be computed with an exploration selection estimation ese procedure

a fast poisson solver of arbitrary order accuracy in rectangular regions in this paper we propose a direct method for the solution of the poisson equation in rectangular regions it has an arbitrary order accuracy and low cpu requirements which makes it practical for treating large scale problems the method is based on a pseudospectral fourier approximation and a polynomial subtraction technique fast convergence of the fourier series is achieved by removing the discontinuities at the corner points using polynomial subtraction functions these functions have the same discontinuities at the corner points as the sought solution in addition to this they satisfy the laplace equation so that the subtraction procedure does not generate nonperiodic nonhomogeneous terms the solution of a boundary value problem is obtained in a series form in o n log n floating point operations where n2 is the number of grid nodes evaluating the solution at all n2 interior points requires o n2 log n operations

numerical analysis of singularities in two dimensions part 1 computation of eigenpairs a numerical method is described for the computation of eigenpairs which characterize the exact solution of linear second order elliptic partial differential equations in two dimensions in the vicinity of singular points the singularities may be caused by re entrant corners and abrupt changes in boundary conditions or material properties such singularities are of great interest from the point of view of failure initiation the eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes for this reason failure theories directly or indirectly involve the eigenpairs and their amplitudes this paper addresses the problem of determining the eigenpairs numerically on the basis of the steklov formulation numerical results are presented for several cases importantly the method is applicable to three dimensional cases

pseudospectral methods on a semi infinite interval with application to the hydrogen atom a comparison of the mapped fourier sine method with laguerre series and rational chebyshev expansions the fourier sine with mapping pseudospectral algorithm of fattal et al phys rev e 53 1996 1217 has been applied in several quantum physics problems here we compare it with pseudospectral methods using laguerre functions and rational chebyshev functions we show that laguerre and chebyshev expansions are better suited for solving problems in the interval r 0 for example the coulomb schrodinger equation than the fourier sinemapping scheme all three methods give similar accuracy for the hydrogen atom when the scaling parameter l is optimum but the laguerre and chebyshev methods are less sensitive to variations in l we introduce a new variant of rational chebyshev functions which has a more uniform spacing of grid points for large r and gives somewhat better results than the rational chebyshev functions of boyd j comp phys 70 1987 63

prediction of spatial cumulative distribution functions using subsampling abstract the spatial cumulative distribution function scdf is a random function that provides a statistical summary of a random field over a spatial domain of interest in this article we develop a spatial subsampling method for predicting an scdf based on observations made on a hexagonal grid similar to the one used in the environmental monitoring and assessment program of the u s environmental protection agency we show that under quite general conditions the proposed subsampling method provides accurate data based approximations to the sampling distributions of various functionals of the scdf predictor in particular it produces estimators of different population characteristics such as the quantiles and weighted mean integrated squared errors of the empirical predictor as an illustration we apply the subsampling method to construct large sample prediction bands for the scdf of an ecological index for foliage condition of red maple trees in the state of maine

summary of sinc numerical methods sinc approximation methods excel for problems whose solutions may have singularities or infinite domains or boundary layers this article summarizes results obtained to date on sinc numerical methods of computation sinc methods provide procedures for function approximation over bounded or unbounded regions encompassing interpolation approximation of derivatives approximate definite and indefinite integration solving initial value ordinary differential equation problems approximation and inversion of fourier and laplace transforms approximation of hilbert transforms and approximation of indefinite convolutions the approximate solution of partial differential equations and the approximate solution of integral equations methods for constructing conformal maps and methods for analytic continuation indeed sinc are ubiquitous for approximating every operation of calculus

orthogonal rational functions on a semi infinite interval abstract by applying a mapping to the chebyshev polynomials we define a new spectral basis the rational chebyshev functions on the semi infinite interval denoted by tl n y continuing earlier work by the author and by grosch and orszag we show that these rational functions inherit most of the good numerical characteristics of the chebyshev polynomials orthogonality completeness exponential or infinite order convergence matrix sparsity for equations with polynomial coefficients and simplicity seven numerical examples illustrate their versatility the charney stability problem of meteorology for example is solved to show the feasibility of applying spectral methods to a semi infinite atmosphere for functions that are singular at both endpoints such as k 1 y one may combine rational chebyshev functions with a preliminary mapping to obtain a single exponentially convergent expansion on y 0 finally we successfully generalize the wkb method to obtain for the j 0 bessel function an amplitude phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y even the origin

numerical solution of singular poisson problems via the sinc galerkin method the galerkin method is used with sinc functions for the basis elements to approximate the solution of the poisson problem particular attention is paid to singular problems a first or higher order partial derivative of the solution is undefined on the boundary the linear system is derived and a spectral decomposition technique of solution is used explicit parameter selections that enhance the performance of the method are given the method is thoroughly tested with the numerical results presented

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

loss function approaches to predict a spatial quantile and its exceedance region in the analysis of spatial data it is common to predict a spatial exceedance and its associated exceedance region this is scientifically important because unusual events tend to strongly affect the environment we use classes of loss functions based on image metrics e g baddeley s loss function to predict the spatial exceedance region we then propose a joint loss to predict a spatial quantile and its exceedance region the optimal predictor is obtained by minimizing the posterior expected loss given the process parameters which we achieve by simulated annealing various predictors are compared through simulation this methodology is applied to a spatial data set of temperature change over the americas

on the semi uniform global attractors of locally asymptotically equivalent non autonomous dynamical systems abstract in this paper we prove that locally asymptotically equivalent non autonomous dynamical systems share the same semi uniform global attractor then we apply this abstract result to nonlinear differential equations in particular we show that the semi uniform global behavior of differential equations with time varying delays is robust with respect to certain types of perturbations in time delays

maximum likelihood estimation for stochastic differential equations using sequential gaussian process based optimization abstractstochastic differential equations sdes are used as statistical models in many disciplines however intractable likelihood functions for sdes make inference challenging and we need to resort to simulation based techniques to estimate and maximize the likelihood function while importance sampling methods have allowed for the accurate evaluation of likelihoods at fixed parameter values there is still a question of how to find the maximum likelihood estimate in this article we propose an efficient gaussian process based method for exploring the parameter space using estimates of the likelihood from an importance sampler our technique accounts for the inherent monte carlo variability of the estimated likelihood and does not require knowledge of gradients the procedure adds potential parameter values by maximizing the so called expected improvement leveraging the fact that the likelihood function is assumed to be smooth our simulations demonstrate that our method has significant computation

computational and inferential difficulties with mixture posterior distributions abstract this article deals with both exploration and interpretation problems related to posterior distributions for mixture models the specification of mixture posterior distributions means that the presence of k modes is known immediately standard markov chain monte carlo mcmc techniques usually have difficulties with well separated modes such as occur here the mcmc sampler stays within a neighborhood of a local mode and fails to visit other equally important modes we show that exploration of these modes can be imposed using tempered transitions however if the prior distribution does not distinguish between the different components then the posterior mixture distribution is symmetric and standard estimators such as posterior means cannot be used we propose alternatives for bayesian inference for permutation invariant posteriors including a clustering device and alternative appropriate loss functions

marginal likelihood from the gibbs output abstract in the context of bayes estimation via gibbs sampling with or without data augmentation a simple approach is developed for computing the marginal density of the sample data marginal likelihood given parameter draws from the posterior distribution consequently bayes factors for model comparisons can be routinely computed as a by product of the simulation hitherto this calculation has proved extremely challenging our approach exploits the fact that the marginal density can be expressed as the prior times the likelihood function over the posterior density this simple identity holds for any parameter value an estimate of the posterior density is shown to be available if all complete conditional densities used in the gibbs sampler have closed form expressions to improve accuracy the posterior density is estimated at a high density point and the numerical standard error of resulting estimate is derived the ideas are applied to probit regression and finite mixture models

practical bayesian density estimation using mixtures of normals abstract mixtures of normals provide a flexible model for estimating densities in a bayesian framework there are some difficulties with this model however first standard reference priors yield improper posteriors second the posterior for the number of components in the mixture is not well defined if the reference prior is used third posterior simulation does not provide a direct estimate of the posterior for the number of components we present some practical methods for coping with these problems finally we give some results on the consistency of the method when the maximum number of components is allowed to grow with the sample size

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

attractors for differential equations with variable delays using the relatively new concept of a pullback attractor we present some results on the existence of attractors for differential equations with variable delay we give a variety of examples to which our result applies

bayesian inference based on stationary fokker planck sampling a novel formalism for bayesian learning in the context of complex inference models is proposed the method is based on the use of the stationary fokker planck sfp approach to sample from the posterior density stationary fokker planck sampling generalizes the gibbs sampler algorithm for arbitrary and unknown conditional densities by the sfp procedure approximate analytical expressions for the conditionals and marginals of the posterior can be constructed at each stage of sfp the approximate conditionals are used to define a gibbs sampling process which is convergent to the full joint posterior by the analytical marginals efficient learning methods in the context of artificial neural networks are outlined offline and incremental bayesian inference and maximum likelihood estimation from the posterior are performed in classification and regression examples a comparison of sfp with other monte carlo strategies in the general problem of sampling from arbitrary densities is also presented it is shown that sfp is able to jump large low probability regions without the need of a careful tuning of any step size parameter in fact the sfp method requires only a small set of meaningful parameters that can be selected following clear problem independent guidelines the computation cost of sfp measured in terms of loss function evaluations grows linearly with the given model s dimension

prolate spheroidal wavefunctions as an alternative to chebyshev and legendre polynomials for spectral element and pseudospectral algorithms prolate spheroidal functions of order zero are generalizations of legendre polynomials which when the bandwidth parameter c 0 oscillate more uniformly on x 1 1 than either chebyshev or legendre polynomials this suggests that compared to these polynomials prolate functions give more uniform spatial resolution further when used as the spatial discretization for time dependent partial differential equations in combination with explicit time marching prolate functions allow a longer stable timestep than legendre polynomials we show that these advantages are real and further that it is almost trivial to modify existing pseudospectral and spectral element codes to use the prolate basis however improvements in spatial resolution are at most a factor of 2 approached slowly as n the timestep can be lengthened by several times but not by a factor that grows rapidly with n the prolate basis is not likely to radically expand the range of problems that can be done on a workstation however for production runs on the bleeding edge edge of arithmurgy such as numerical weather prediction the rewards for switching to a prolate basis may be considerable

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

on bayesian analysis of mixtures with an unknown number of components with discussion new methodology for fully bayesian mixture analysis is developed making use of reversible jump markov chain monte carlo methods that are capable of jumping between the parameter subspaces corresponding to different numbers of components in the mixture a sample from the full joint distribution of all unknown variables is thereby generated and this can be used as a basis for a thorough presentation of many aspects of the posterior distribution the methodology is applied here to the analysis of univariate normal mixtures using a hierarchical prior model that offers an approach to dealing with weak prior information while avoiding the mathematical pitfalls of using improper priors in the mixture context

confidence regions for spatial excursion sets from repeated random field observations with an application to climate abstractthe goal of this article is to give confidence regions for the excursion set of a spatial function above a given threshold from repeated noisy observations on a fine grid of fixed locations given an asymptotically gaussian estimator of the target function a pair of data dependent nested excursion sets are constructed that are sub and super sets of the true excursion set respectively with a desired confidence asymptotic coverage probabilities are determined via a multiplier bootstrap method not requiring gaussianity of the original data nor stationarity or smoothness of the limiting gaussian field the method is used to determine regions in north america where the mean summer and winter temperatures are expected to increase by mid 21st century by more than 2 degrees celsius

relabeling and summarizing posterior distributions in signal decomposition problems when the number of components is unknown this paper addresses the problems of relabeling and summarizing posterior distributions that typically arise in a bayesian framework when dealing with signal decomposition problems with an unknown number of components such posterior distributions are defined over union of subspaces of differing dimensionality and can be sampled from using modern monte carlo techniques for instance the increasingly popular rj mcmc method no generic approach is available however to summarize the resulting variable dimensional samples and extract from them component specific parameters we propose a novel approach named variable dimensional approximate posterior for relabeling and summarizing vapors to this problem which consists in approximating the posterior distribution of interest by a simple but still variable dimensional parametric distribution the distance between the two distributions is measured using the kullback leibler divergence and a stochastic em type algorithm driven by the rj mcmc sampler is proposed to estimate the parameters two signal decomposition problems are considered to show the capability of vapors both for relabeling and for summarizing variable dimensional posterior distributions the classical problem of detecting and estimating sinusoids in white gaussian noise on the one hand and a particle counting problem motivated by the pierre auger project in astrophysics on the other hand

new loss functions in bayesian imaging abstract unlike the development of more accurate prior distributions for use in bayesian imaging the design of more sensible estimators through loss functions has been neglected in the literature we discuss the design of loss functions with a local structure that depend only on a binary misclassification vector the proposed approach is similar to modeling with a markov random field the bayes estimate is calculated in a two step algorithm using markov chain monte carlo and simulated annealing algorithms we present simulation experiments with the ising model where the observations are corrupted with gaussian and flip noise