random attractors for stochastic porous media equations perturbed by space time linear multiplicative noise unique existence of solutions to porous media equations driven by continuous linear multiplicative space time rough signals is proven for initial data in l 1 o on bounded domains o the generation of a continuous order preserving random dynamical system rds on l 1 o and the existence of a random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained the random attractor is shown to be compact and attracting in l o norm uniform l bounds and uniform space time continuity of the solutions is shown general noise including fractional brownian motion for all hurst parameters is treated a pathwise wong zakai result for driving noise given by a continuous semimartingale is obtained for fast diffusion equations driven by continuous linear multiplicative space time rough signals existence of solutions is proven for initial data in l m 1 o

inconsistent estimation and asymptotically equal interpolations in model based geostatistics it is shown that in model based geostatistics not all parameters in the matern class can be estimated consistently if data are observed in an increasing density in a fixed domain regardless of the estimation methods used nevertheless one quantity can be estimated consistently by the maximum likelihood method and this quantity is more important to spatial interpolation the results are established by using the properties of equivalence and orthogonality of probability measures some sufficient conditions are provided for both gaussian and non gaussian equivalent measures and necessary conditions are provided for gaussian equivalent measures two simulation studies are presented that show that the fixed domain asymptotic properties can explain some finite sample behavior of both interpolation and estimation when the sample size is moderately large

first order intrinsic autoregressions and the de wijs process summary we discuss intrinsic autoregressions for a first order neighbourhood on a two dimensional rectangular lattice and give an exact formula for the variogram that extends known results to the asymmetric case we obtain a corresponding asymptotic expansion that is more accurate and more general than previous ones and use this to derive the de wijs variogram under appropriate averaging a result that can be interpreted as a two dimensional spatial analogue of brownian motion obtained as the limit of a random walk in one dimension this provides a bridge between geostatistics where the de wijs process was once the most popular formulation and markov random fields and also explains why statistical analysis using intrinsic autoregressions is usually robust to changes of scale we briefly describe corresponding calculations in the frequency domain including limiting results for higher order autoregressions the paper closes with some practical considerations including applications to irregularly spaced data

approximate likelihood for large irregularly spaced spatial data likelihood approaches for large irregularly spaced spatial datasets are often very difficult if not infeasible to implement due to computational limitations even when we can assume normality exact calculations of the likelihood for a gaussian spatial process observed at n locations requires o n3 operations we present a version of whittle s approximation to the gaussian log likelihood for spatial regular lattices with missing values and for irregularly spaced datasets this method requires o nlog2n operations and does not involve calculating determinants we present simulations and theoretical results to show the benefits and the performance of the spatial likelihood approximation method presented here for spatial irregularly spaced datasets and lattices with missing values we apply these methods to estimate the spatial structure of sea surface temperatures using satellite data with missing values

edge effects and efficient parameter estimation for stationary random fields summary we consider the estimation of the parameters of a stationary random field on d dimensional lattice by minimizing the classical whittle approximation to the gaussian log likelihood if the usual biased sample covariances are used the estimate is efficient only in one dimension to remove this edge effect we introduce data tapers and show that the resulting modified estimate is efficient also in two and three dimensions this avoids the use of the unbiased sample covariances which are in general not positive definite

levy ornstein uhlenbeck and subordination spectral vs jump description unlike brownian motion which propagates diffusively and whose sample path trajectories are continuous non brownian levy motions propagate via jumps flights and their sample path trajectories are purely discontinuous when analyzing systems involving non brownian levy motions the common practice is to use either spectral or fractional calculus methods in this manuscript we suggest an alternative analytical approach using the poisson superposition jump structure of non brownian levy motions we demonstrate this approach in two exemplary topics i systems governed by l evy driven ornstein uhlenbeck dynamics and ii systems subject to temporal levy subordination we show that this approach yields answers and insights that are not attainable using spectral methods alone

spline smoothing over difficult regions summary it is occasionally necessary to smooth data over domains in ir2 with complex irregular boundaries or interior holes traditional methods of smoothing which rely on the euclidean metric or which measure smoothness over the entire real plane may then be inappropriate this paper introduces a bivariate spline smoothing function defined as the minimizer of a penalized sum of squares functional the roughness penalty is based on a partial differential operator and is integrated only over the problem domain by using finite element analysis the method is motivated by and applied to two sample smoothing problems and is compared with the thin plate spline

on conditional and intrinsic autoregressions summary gaussian conditional autoregressions have been widely used in spatial statistics and bayesian image analysis where they are intended to describe interactions between random variables at fixed sites in euclidean space the main appeal of these distributions is in the markovian interpretation of their full conditionals intrinsic autoregressions are limiting forms that retain the markov property despite being improper they can have advantages over the standard autoregressions both conceptually and in practice for example they often avoid difficulties in parameter estimation without apparent loss or exhibit appealing invariances as in texture analysis however on small arrays and in nonlattice applications both forms of autoregression can lead to undesirable second order characteristics either in the variables themselves or in contrasts among them this paper discusses standard and intrinsic autoregressions and describes how the problems that arise can be alleviated using dempster s 1972 algorithm or an appropriate modification the approach represents a partial synthesis of standard geostatistical and gaussian markov random field formulations some nonspatial applications are also mentioned

splines as local smoothers a smoothing spline is a nonparametric curve estimate that is defined as the solution to a minimization problem one problem with this representation is that it obscures the fact that a spline like most other nonparametric estimates is a local weighted average of the observed data this property has been used extensively to study the limiting properties of kernel estimates and it is advantageous to apply similar techniques to spline estimates although equivalent kernels have been identified for a smoothing spline these functions are either not accurate enough for asymptotic approximations or are restricted to equally spaced points this paper extends this previous work to understand a spline estimate s local properties it is shown that the absolute value of the spline weight function decreases exponentially away from its center this result is not asymptotic the only requirement is that the empirical distribution of the observation points be sufficiently close to a continuous distribution with a strictly positive density function these bounds are used to derive the asymptotic form for the bias and variance of a first order smoothing spline estimate the arguments leading to this result can be easily extended to higher order splines

an approach to producing space time covariance functions on spheres for space time processes on global or large scales it is critical to use models that respect the earth s spherical shape the covariance functions of such processes should be not only positive definite on sphere time but also capable of capturing the dynamics of the processes well we develop space time covariance functions on sphere time that are flexible in producing space time interactions especially space time asymmetries our idea is to consider a sum of independent processes in which each process is obtained by applying a first order differential operator to a fully symmetric process on sphere time the resulting covariance functions can produce various types of space time interactions and give different covariance structures along different latitudes our approach yields explicit expressions for the covariance functions which has great advantages in computation moreover it applies equally well to generating asymmetric space time covariance functions on flat or other spatial domains we

statistical analyses of brain surfaces using gaussian random fields on 2 d manifolds interest in the morphometric analysis of the brain and its subregions has recently intensified because growth or degeneration of the brain in health or illness affects not only the volume but also the shape of cortical and subcortical brain regions and new image processing techniques permit detection of small and highly localized perturbations in shape or localized volume with remarkable precision an appropriate statistical representation of the shape of a brain region is essential however for detecting localizing and interpreting variability in its surface contour and for identifying differences in volume of the underlying tissue that produce that variability across individuals and groups of individuals our statistical representation of the shape of a brain region is defined by a reference region for that region and by a gaussian random field grf that is defined across the entire surface of the region we first select a reference region from a set of segmented brain images of healthy individuals the grf is then estimated as the signed euclidean distances between points on the surface of the reference region and the corresponding points on the corresponding region in images of brains that have been coregistered to the reference correspondences between points on these surfaces are defined through deformations of each region of a brain into the coordinate space of the reference region using the principles of fluid dynamics the warped coregistered region of each subject is then unwarped into its native space simultaneously bringing into that space the map of corresponding points that was established when the surfaces of the subject and reference regions were tightly coregistered the proposed statistical description of the shape of surface contours makes no assumptions other than smoothness about the shape of the region or its grf the description also allows for the detection and localization of statistically significant differences in the shapes of the surfaces across groups of subjects at both a fine and coarse scale we demonstrate the effectiveness of these statistical methods by applying them to study differences in shape of the amygdala and hippocampus in a large sample of normal subjects and in subjects with attention deficit hyperactivity disorder adhd

principal warps thin plate splines and the decomposition of deformations the decomposition of deformations by principal warps is demonstrated the method is extended to deal with curving edges between landmarks this formulation is related to other applications of splines current in computer vision how they might aid in the extraction of features for analysis comparison and diagnosis of biological and medical images in indicated

maximum likelihood estimation using composite likelihoods for closed exponential families in certain multivariate problems the full probability density has an awkward normalizing constant but the conditional and or marginal distributions may be much more tractable in this paper we investigate the use of composite likelihoods instead of the full likelihood for closed exponential families both are shown to be maximized by the same parameter values for any number of observations examples include log linear models and multivariate normal models in other cases the parameter estimate obtained by maximizing a composite likelihood can be viewed as an approximation to the full maximum likelihood estimate an application is given to an example in directional data based on a bivariate von mises distribution

random attractors for singular stochastic evolution equations abstract the existence of random attractors for singular stochastic evolution equations see perturbed by general additive noise is proven the drift is only assumed to satisfy the standard assumptions of the variational approach to see with compact embeddings in the gelfand triple and singular coercivity for ergodic monotone contractive random dynamical systems it is proven that the random attractor consists of a single random point in case of real linear multiplicative noise finite time extinction is obtained applications include stochastic generalized fast diffusion equations and stochastic generalized singular p laplace equations perturbed by levy noise with jump measure having finite first and second moments

going off grid computationally efficient inference for log gaussian cox processes this paper introduces a new method for performing computational inference on log gaussian cox processes the likelihood is approximated directly by making novel use of a continuously specified gaussian random field we show that for sufficiently smooth gaussian random field prior distributions the approximation can converge with arbitrarily high order while an approximation based on a counting process on a partition of the domain only achieves first order convergence the given results improve on the general theory of convergence of the stochastic partial differential equation models introduced by lindgren et al 2011 the new method is demonstrated on a standard point pattern data set and two interesting extensions to the classical log gaussian cox process framework are discussed the first extension considers variable sampling effort throughout the observation window and implements the method of chakraborty et al 2011 the second extension constructs a log gaussian cox process on the world s oceans the analysis is performed using integrated nested laplace approximation for fast approximate inference

classes of nonseparable spatio temporal stationary covariance functions abstract suppose that a random process z s t indexed in space and time has spatio temporal stationary covariance c h u where h d d 1 is a spatial lag and u is a temporal lag separable spatio temporal covariances have the property that they can be written as a product of a purely spatial covariance and a purely temporal covariance their ease of definition is counterbalanced by the rather limited class of random processes to which they correspond in this article we derive a new approach that allows one to obtain many classes of nonseparable spatio temporal stationary covariance functions and fit several such classes to spatio temporal data on wind speed over a region in the tropical western pacific ocean

parameter estimation for a stationary process on a d dimensional lattice summary we study asymptotic properties of various estimation procedures for a general stationary regular process on a d dimensional lattice differences between d 1 time series and d 2 spatial processes are pointed out we suppose that the process is observed on a set pn with cn points which tends to infinity with the same speed in all directions the relative edge effect is of order cn lid increasing with d this effect is without statistical importance if d 1 but is important if d 2 we give a lcn consistent asymptotically normal estimator of the underlying parameter the procedure being constructed by a modification of whittle s approximation to the log likelihood in the gaussian case this procedure is asymptotically efficient

multivariate type g matern stochastic partial differential equation random fields for many applications with multivariate data random field models capturing departures from gaussianity within realizations are appropriate for this reason we formulate a new class of multivariate non gaussian models based on systems of stochastic partial differential equations with additive type g noise whose marginal covariance functions are of matern type we consider four increasingly flexible constructions of the noise where the first two are similar to existing copula based models in contrast with these the last two constructions can model non gaussian spatial data without replicates computationally efficient methods for likelihood based parameter estimation and probabilistic prediction are proposed and the flexibility of the models suggested is illustrated by numerical examples and two statistical applications less

weak solutions to the stochastic porous media equation via kolmogorov equations the degenerate case a stochastic version of the porous medium equation with coloured noise is studied the corresponding kolmogorov equation is solved in the space l 2 h where is an infinitesimally excessive measure then a weak solution is constructed

logarithmic diffusion and porous media equations a unified description in this work we present the logarithmic diffusion equation as a limit case when the index that characterizes a nonlinear fokker planck equation in its diffusive term goes to zero a linear drift and a source term are considered in this equation its solution has a lorentzian form consequently this equation characterizes a superdiffusion like a levy kind in addition an equation that unifies the porous media and the logarithmic diffusion equations including a generalized diffusion equation in fractal dimension is obtained this unification is performed in the nonextensive thermostatistics context and increases the possibilities about the description of anomalous diffusive processes

nonseparable stationary covariance functions for space time data geostatistical approaches to spatiotemporal prediction in environmental science climatology meteorology and related fields rely on appropriate covariance models this article proposes general classes of nonseparable stationary covariance functions for spatiotemporal random processes the constructions are directly in the space time domain and do not depend on closed form fourier inversions the model parameters can be associated with the data s spatial and temporal structures respectively and a covariance model with a readily interpretable space time interaction parameter is fitted to wind data from ireland

matern cross covariance functions for multivariate random fields we introduce a flexible parametric family of matrix valued covariance functions for multivariate spatial random fields where each constituent component is a matern process the model parameters are interpretable in terms of process variance smoothness correlation length and colocated correlation coefficients which can be positive or negative both the marginal and the cross covariance functions are of the matern type in a data example on error fields for numerical predictions of surface pressure and temperature over the north american pacific northwest we compare the bivariate matern model to the traditional linear model of coregionalization

on a random scaled porous media equation abstract it is shown that a random scaled porous media equation arising from a stochastic porous media equation with linear multiplicative noise through a random transformation is well posed in l in the fast diffusion case we show existence in l p

stochastic porous media equations and self organized criticality the existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and wiener forcing term is proven the finite time extinction of solutions with high probability is also proven in 1 d the results are relevant for self organized criticality behavior of stochastic nonlinear diffusion equations with critical states

spde in hilbert space with locally monotone coefficients the aim of this paper is to extend the usual framework of spde with monotone coecients to include a large class of cases with merely locally monotone coecients this new framework is conceptually not more involved than the classical one but includes many more fundamental examples not included previously thus our main result can be applied to various types of spdes such as stochastic reaction diusion equations stochastic burgers type equation stochastic 2 d navier stokes equation stochastic p laplace equation and stochastic porous media equation with non monotone perturbations

comment getting into space with a weight problem this article will focus its discussion on accounting for the complex design when analyzing survey data and on spatial modeling with respect to the former this article considers two approaches direct use of the weights and a model based approach using a spatial model to acknowledge clustering with respect to spatial modeling this article describes the stochastic partial differential equations spdes lindgren et al 2011 approach to modeling

an explicit link between gaussian fields and gaussian markov random fields the stochastic partial differential equation approach continuously indexed gaussian fields gfs are the most important ingredient in spatial statistical modelling and geostatistics the specification through the covariance function gives an intuitive interpretation of the field properties on the computational side gfs are hampered with the big n problem since the cost of factorizing dense matrices is cubic in the dimension although computational power today is at an all time high this fact seems still to be a computational bottleneck in many applications along with gfs there is the class of gaussian markov random fields gmrfs which are discretely indexed the markov property makes the precision matrix involved sparse which enables the use of numerical algorithms for sparse matrices that for fields in r 2 only use the square root of the time required by general algorithms the specification of a gmrf is through its full conditional distributions but its marginal properties are not transparent in such a parameterization we show that using an approximate stochastic weak solution to linear stochastic partial differential equations we can for some gfs in the matern class provide an explicit link for any triangulation of r d between gfs and gmrfs formulated as a basis function representation the consequence is that we can take the best from the two worlds and do the modelling by using gfs but do the computations by using gmrfs perhaps more importantly our approach generalizes to other covariance functions generated by spdes including oscillating and non stationary gfs as well as gfs on manifolds we illustrate our approach by analysing global temperature data with a non stationary model defined on a sphere less

specification and testing of some modified count data models abstract this paper explores the specification and testing of some modified count data models these alternatives permit more flexible specification of the data generating process dgp than do familiar count data models e g the poisson and provide a natural means for modeling data that are over or underdispersed by the standards of the basic models in the cases considered the familiar forms of the distributions result as parameter restricted versions of the proposed modified distributions accordingly score tests of the restrictions that use only the easily computed ml estimates of the standard models are proposed the tests proposed by hausman 1978 and white 1982 are also considered the tests are then applied to count data models estimated using survey microdata on beverage consumption

random attractors for a class of stochastic partial differential equations driven by general additive noise abstract the existence of random attractors for a large class of stochastic partial differential equations spde driven by general additive noise is established the main results are applied to various types of spde as e g stochastic reaction diffusion equations the stochastic p laplace equation and stochastic porous media equations besides classical brownian motion we also include space time fractional brownian motion and space time levy noise as admissible random perturbations moreover cases where the attractor consists of a single point are also investigated and bounds for the speed of attraction are obtained

a generalization of levy s concentration variance inequality sharp lower bounds are found for the concentration of a probability distribution as a function of the expectation of any given convex symmetric function in the case x x c 2 wherec is the expected value of the distribution these bounds yield the classical concentration variance inequality of levy an analogous sharp inequality is obtained in a similar linear search setting where a sharp lower bound for the concentration is found as a function of the maximum probability swept out from a fixed starting point by a path of given length

a variational approach to stochastic nonlinear parabolic problems abstract this work is concerned with an optimal control approach to stochastic nonlinear parabolic diffusion equations with monotonically increasing nonlinearity this approach leads to sharper existence and uniqueness results under minimal growth conditions on nonlinear diffusion coefficients

a generalization of gruss s inequality in inner product spaces and applications a generalization of the classical gruss s integral inequality in inner product spaces is given some applications for positive linear functionals and integrals are also pointed out

a regularization approach for surface reconstruction from point clouds abstract ill posed inverse problems arise in many branches of science and engineering in the typical situation one is interested in recovering a whole function given a finite number of noisy measurements on functionals regularization approach is one of the solutions for this kind of problems among which we find surface reconstruction from point clouds in this article we present a classification of methods for surface reconstruction and its solution in the framework of classical regularization in terms of linear combinations of the reproducing kernel in certain proper space for reconstruction we evaluate the quality of this solution to approximate 3d data and some examples for computational implementation are also shown in order to complete a user s guide for an efficient employment of kernel methods in the reconstruction of multivariate functions

bayesian inference of survival probabilities under stochastic ordering constraints abstract in the statistical analysis of survival data arising from two populations it often happens that the analyst knows a priori that the life lengths in one population are stochastically shorter than those in the other nevertheless survival probability estimates if determined separately from the corresponding samples may not be consistent with this prior assumption because of inherent statistical variability in the observations this problem has been considered in a number of papers during the past decade by adopting a generalized maximum likelihood approach our approach is bayesian and in essence nonparametric the a priori assumption regarding stochastic ordering is formulated naturally in terms of a joint prior distribution defined for pairs of survival functions nonparametric specification of the model based on hazard rates and using a few hyperparameters allows for sufficient flexibility in practical applications the numerical computations are based on a coupled version of the m

nonparametric estimation of nonstationary spatial covariance structure abstract estimation of the covariance structure of spatial processes is a fundamental prerequisite for problems of spatial interpolation and the design of monitoring networks we introduce a nonparametric approach to global estimation of the spatial covariance structure of a random function z x t observed repeatedly at times ti i 1 t at a finite number of sampling stations xi i 1 2 n in the plane our analyses assume temporal stationarity but do not assume spatial stationarity or isotropy we analyze the spatial dispersions var z xi t z xj t as a natural metric for the spatial covariance structure and model these as a general smooth function of the geographic coordinates of station pairs xi xj the model is constructed in two steps first using nonmetric multidimensional scaling mds we compute a two dimensional representation of the sampling stations for which a monotone function of interpoint distances ij approximates the spatial dispersions mds transforms the problem

objective bayesian analysis of spatially correlated data spatially varying phenomena are often modeled using gaussian random fields specified by their mean function and covariance function the spatial correlation structure of these models is commonly specified to be of a certain form e g spherical power exponential rational quadratic or matern with a small number of unknown parameters we consider objective bayesian analysis of such spatial models when the mean function of the gaussian random field is specified as in a linear model it is thus necessary to determine an objective or default prior distribution for the unknown mean and covariance parameters of the random field we first show that common choices of default prior distributions such as the constant prior and the independent jeffreys prior typically result in improper posterior distributions for this model next the reference prior for the model is developed and is shown to yield a proper posterior distribution a further attractive property of the reference prior is that it can be used

generalized hermite interpolation and positive definite kernels on a riemannian manifold abstract in this paper we introduce a class of positive definite kernels defined on a closed compact riemannian manifold these kernels provide a grid free method for solving uniquely a generalized version of the hermite interpolation problem a version in which one fits a smooth surface to multi dimensional scattered data including data generated by derivatives fluxes or any other quantity one can obtain by integrating a function against a compactly supported distribution the positive definite kernels introduced here include the c members of the class of spherical positive definite functions introduced by schoenberg duke math j 9 1942 96 108 and shown to be strictly positive definite by xu and cheney proc amer math sec 116 1992 977 981 thus as a consequence of our results the generalized hermite interpolation problem is well poised i e a solution exists and is unique for the class of interpolants constructed out of c spherical positive definite functions we also provide similar results when the underlying manifold is the m dimensional torus

stochastic generalized porous media and fast diffusion equations abstract we present a generalization of krylov rozovskii s result on the existence and uniqueness of solutions to monotone stochastic differential equations as an application the stochastic generalized porous media and fast diffusion equations are studied for finite reference measures where the drift term is given by a negative definite operator acting on a time dependent function which belongs to a large class of functions comparable with the so called n functions in the theory of orlicz spaces

approximating likelihoods for large spatial data sets summary likelihood methods are often difficult to use with large irregularly sited spatial data sets owing to the computational burden even for gaussian models exact calculations of the likelihood for n observations require o n3 operations since any joint density can be written as a product of conditional densities based on some ordering of the observations one way to lessen the computations is to condition on only some of the past observations when computing the conditional densities we show how this approach can be adapted to approximate the restricted likelihood and we demonstrate how an estimating equations approach allows us to judge the efficacy of the resulting approximation previous work has suggested conditioning on those past observations that are closest to the observation whose conditional density we are approximating through theoretical numerical and practical examples we show that there can often be considerable benefit in conditioning on some distant observations as well

fixed rank kriging for very large spatial data sets summary spatial statistics for very large spatial data sets is challenging the size of the data set n causes problems in computing optimal spatial predictors such as kriging since its computa tional cost is of order a73 in addition a large data set is often defined on a large spatial domain so the spatial process of interest typically exhibits non stationary behaviour over that domain a flexible family of non stationary covariance functions is defined by using a set of basis functions that is fixed in number which leads to a spatial prediction method that we call fixed rank kriging specifically fixed rank kriging is kriging within this class of non stationary covariance functions it relies on computational simplifications when n is very large for obtaining the spatial best linear unbiased predictor and its mean squared prediction error for a hidden spatial process a method based on minimizing a weighted frobenius norm yields best estimators of the covari ance function parameters which are then substituted into the fixed rank kriging equations the new methodology is applied to a very large data set of total column ozone data observed over the entire globe where n is of the order of hundreds of thousands

a trotter type result for the stochastic porous media equations abstract this paper proves the continuous dependence with respect to diffusivity of the solutions to stochastic porous media equations with noncoercive monotone diffusivity functions and multiplicative noise

gaussian predictive process models for large spatial data sets with scientific data available at geocoded locations investigators are increasingly turning to spatial process models for carrying out statistical inference over the last decade hierarchical models implemented through markov chain monte carlo methods have become especially popular for spatial modelling given their flexibility and power to fit models that would be infeasible with classical methods as well as their avoidance of possibly inappropriate asymptotics however fitting hierarchical spatial models often involves expensive matrix decompositions whose computational complexity increases in cubic order with the number of spatial locations rendering such models infeasible for large spatial data sets this computational burden is exacerbated in multivariate settings with several spatially dependent response variables it is also aggravated when data are collected at frequent time points and spatiotemporal process models are used with regard to this challenge our contribution is to work with what we call predictive process models for spatial and spatiotemporal data every spatial or spatiotemporal process induces a predictive process model in fact arbitrarily many of them the latter models project process realizations of the former to a lower dimensional subspace thereby reducing the computational burden hence we achieve the flexibility to accommodate non stationary non gaussian possibly multivariate possibly spatiotemporal processes in the context of large data sets we discuss attractive theoretical properties of these predictive processes we also provide a computational template encompassing these diverse settings finally we illustrate the approach with simulated and real data sets

on the stochastic cahn hilliard equation with a singular double well potential abstract we prove well posedness and regularity for the stochastic pure cahn hilliard equation under homogeneous neumann boundary conditions with both additive and multiplicative wiener noise in contrast with great part of the literature the double well potential is treated as generally as possible its convex part being associated to a multivalued maximal monotone graph everywhere defined on the real line on which no growth nor smoothness assumptions are assumed the regularity result allows to give appropriate sense to the chemical potential and to write a natural variational formulation of the problem the proofs are based on suitable monotonicity and compactness arguments in a generalized variational framework

pathwise mild solutions for quasilinear stochastic partial differential equations abstract stochastic partial differential equations spdes have become a key modeling tool in applications yet there are many classes of spdes where the existence and regularity theory for solutions is not completely developed here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear cauchy problems including among others cross diffusion systems as a key application our solutions are local in time and are derived via a fixed point argument in suitable function spaces the key idea is to combine in a suitable way the classical theory of deterministic quasilinear parabolic partial differential equations pdes with recent theory of evolution semigroups we also show how to apply our theory to the shigesada kawasaki teramoto skt model furthermore we provide examples of blow up and ill posed operators which can occur after finite time showing that solutions can only be local in time for general quasilinear spdes while they might be global in time for special subclasses of problems

well posedness for a class of doubly nonlinear stochastic pdes of divergence type abstract we prove well posedness for doubly nonlinear parabolic stochastic partial differential equations of the form d x t div x t d t x t d t b t x t d w t where and are the two nonlinearities assumed to be multivalued maximal monotone operators everywhere defined on r d and r respectively and w is a cylindrical wiener process using variational techniques suitable uniform estimates both pathwise and in expectation and some compactness results well posedness is proved under the classical leray lions conditions on and with no restrictive smoothness or growth assumptions on the operator b is assumed to be hilbert schmidt and to satisfy some classical lipschitz conditions in the second variable

a survey on some inequalities for expectation and variance inequalities for continuous random variables having the probability density function defined on a finite interval obtained recently by the authors are surveyed

bayesian inference for non stationary spatial covariance structure via spatial deformations summary in geostatistics it is common practice to assume that the underlying spatial process is stationary and isotropic i e the spatial distribution is unchanged when the origin of the index set is translated and under rotation about the origin however in environmental problems such assumptions are not realistic since local influences in the correlation structure of the spatial process may be found in the data the paper proposes a bayesian model to address the anisot ropy problem following sampson and guttorp we define the correlation function of the spatial process by reference to a latent space denoted by d where stationarity and isotropy hold the space where the gauged monitoring sites lie is denoted by g we adopt a bayesian approach in which the mapping between g and d is represented by an unknown function d a gaussian process prior distribution is defined for d unlike the sampson guttorp approach the mapping of both gauged and ungauged sites is handled in a single framework and predictive inferences take explicit account of uncertainty in the mapping markov chain monte carlo methods are used to obtain samples from the posterior distributions two examples are discussed a simulated data set and the solar radiation data set that also was analysed by sampson and guttorp

deformable templates using large deformation kinematics a general automatic approach is presented for accommodating local shape variation when mapping a two dimensional 2 d or three dimensional 3 d template image into alignment with a topologically similar target image local shape variability is accommodated by applying a vector field transformation to the underlying material coordinate system of the template while constraining the transformation to be smooth globally positive definite jacobian smoothness is guaranteed without specifically penalizing large magnitude deformations of small subvolumes by constraining the transformation on the basis of a stokesian limit of the fluid dynamical navier stokes equations this differs fundamentally from quadratic penalty methods such as those based on linearized elasticity or thin plate splines in that stress restraining the motion relaxes over time allowing large magnitude deformations kinematic nonlinearities are inherently necessary to maintain continuity of structures during large magnitude deformations and are included in all results after initial global registration final mappings are obtained by numerically solving a set of nonlinear partial differential equations associated with the constrained optimization problem automatic regridding is performed by propagating templates as the nonlinear transformations evaluated on a finite lattice become singular application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability

spatial gliding temporal trapping and anomalous transport abstract we introduce and study a model of stochastic dynamics governed by markovian laws of motion in a phase space filled with random traps the resulting non markovian stochastic trapped motion alternates intermittently between periods of spatial gliding where the motion glides the underlying markovian dynamics and periods of temporal trapping where the motion is halted in random traps we investigate the asymptotics and scaling limits of this model we prove that when the random trappings are heavy tailed then mittag leffler functions and probability laws emerge and govern the functional structure and statistics of the system and that the time flow has a random fractal structure whose fractal exponent dimension is determined by the heaviness of the trappings we study the effect of random trapping on general levy dynamics we prove that subjecting levy dynamics to heavy tailed trapping will always result in i sub diffusive behavior when the underlying levy dynamics are of finite variance and ii space time fractal behavior when the underlying levy dynamics are scale invariant furthermore we explore the issue of first exit times to that end a general feynman kac framework for trapped processes is developed and a method of transforming trapped feynman kac equations to standard ones is established the study of first exit times enables us to quantitatively connect macroscopic observations to microscopic behavior in general markovian dynamics subjected to random trapping in the case of levy dynamics first exit times from balls are computed and the relationships between their statistics and the statistics of the trapped motion are derived

nonlinear fokker planck equations driven by gaussian linear multiplicative noise abstract existence of a strong solution in h 1 r d is proved for the stochastic nonlinear fokker planck equation d x div d x d t x d t x d w in 0 t r d x 0 x respectively for a corresponding random differential equation here d 1 w is a wiener process in h 1 r d d c 1 r d r d and is a continuous monotonically increasing function satisfying some appropriate sublinear growth conditions which are compatible with the physical models arising in statistical mechanics the solution exists for x l 1 l and preserves positivity if is locally lipschitz the solution is unique pathwise lipschitz continuous with respect to initial data in h 1 r d stochastic fokker planck equations with nonlinear drift of the form d x div a x d t x d t x d w are also considered for lipschitzian continuous functions a r r d

estimation and model identification for continuous spatial processes summary formal parameter estimation and model identification procedures for continuous domain spatial processes are introduced the processes are assumed to be adequately described by a linear model with residuals that follow a second order stationary gaussian random field and data are assumed to consist of noisy observations of the process at arbitrary sampling locations a general class of two dimensional rational spectral density functions with elliptic contours is used to model the spatial covariance function an iterative estimation procedure alleviates many of the computational difficulties of conventional maximum likelihood estimation for non lattice data the procedure is applied to several generated data sets and to an actual ground water data set

bayesian inference of random fields represented with the karhunen loeve expansion abstract the integration of data into engineering models involving uncertain and spatially varying parameters is oftentimes key to obtaining accurate predictions bayesian inference is effective in achieving such an integration uncertainties related to spatially varying parameters are typically represented through random fields discretized into a finite number of random variables the prior correlation length and variance of the field as well as the number of terms in the random field discretization have a considerable impact on the outcome of the bayesian inference which has received little attention in the literature here we investigate the implications of different choices in the prior random field model on the solution of bayesian inverse problems we employ the karhunen loeve expansion for the representation of random fields we show that a higher order karhunen loeve discretization is required in bayesian inverse problems as compared to standard prior random field representations since the updated fields are non homogeneous furthermore the smoothing effect of the forward operator has a large influence on the posterior solution particularly if we are analyzing quantities of interest that are sensitive to local random fluctuations of the inverse quantity this is also reflected in posterior predictions such as the estimation of rare event probabilities we illustrate these effects analytically through a 1d cantilever beam with spatially varying flexibility and numerically using a 2d linear elasticity example where the young s modulus is spatially variable

finite time extinction of solutions to fast diffusion equations driven by linear multiplicative noise abstract in this paper one surveys and sharpens some recent results obtained by the authors on the finite time extinction property of solutions to stochastic diffusion equations of the form d x x m d t x d w and d x sgn x d t x d w where 0 m 1 0 these equations arise as models for nonlinear diffusion processes in porous media plasma and self organized criticality under stochastic gaussian perturbation these equations can be also viewed as control systems governed by fast diffusion porous media equations with a stochastic feedback controller x d w

studies in the history of probability and statistics xlix on the matern correlation family summary handcock stein 1993 introduced the matern family of spatial correlations into statistics as a flexible parametric class with one parameter determining the smoothness of the paths of the underlying spatial field we document the varied history of this family which includes contributions by eminent physical scientists and statisticians

well posedness of nonlinear diffusion equations with nonlinear conservative noise we prove the pathwise well posedness of stochastic porous media and fast diffusion equations driven by nonlinear conservative noise as a consequence the generation of a random dynamical system is obtained this extends results of the second author and souganidis who considered analogous spatially homogeneous and first order equations and earlier works of lions perthame and souganidis

a finite element method for density estimation with gaussian process priors a variational problem characterizing the density estimator defined by the maximum a posteriori method with gaussian process priors is derived it is shown that this problem is well posed and can be solved with newton s method numerically the solution is approximated by a galerkin finite element method with piecewise multilinear functions on uniform grids error bounds for this method are given and numerical experiments are performed for one two and three dimensional examples

stochastic porous media equations and self organized criticality convergence to the critical state in all dimensions if x x t is the solution to the stochastic porous media equation in mathcal o subset mathbf r d 1 le d le 3 modelling the self organized criticality barbu et al in commun math phys 285 901 923 2009 and x c is the critical state then it is proved that int infty 0m mathcal o setminus mathcal o t 0 dt infty mathbb p hbox a s and lim t to infty int mathcal o x t x c d xi ell infty mathbb p hbox a s here m is the lebesgue measure and mathcal o t c is the critical region xi in mathcal o x t xi x c xi and x c x 0 a e xi in mathcal o if the stochastic gaussian perturbation has only finitely many modes but is still function valued lim t to infty int k x t x c d xi 0 exponentially fast for all compact k subset mathcal o with probability one if the noise is sufficiently strong we also recover that in the deterministic case l 0

on a system of two dimensional recurrence equations summary the solution of a system of two dimensional recurrence equations is discussed particularly in the context of the auto correlation structure of a stationary first order autonormal scheme on the doubly infinite rectangular lattice a simple approximation in the region of primary interest is described

existence of strong solutions for stochastic porous media equation under general monotonicity conditions this paper addresses existence and uniqueness of strong solutions to stochastic porous media equations dx x dt b x dw t in bounded domains of r d with dirichlet boundary conditions here is a maximal monotone graph in r r possibly multivalued with

fast sampling of gaussian markov random fields this paper demonstrates how gaussian markov random fields conditional autoregressions can be sampled quickly by using numerical techniques for sparse matrices the algorithm is general and efficient and expands easily to various forms for conditional simulation and evaluation of normalization constants we demonstrate its use by constructing efficient block updates in markov chain monte carlo algorithms for disease mapping

bivariate binomial spatial modeling of loa loa prevalence in tropical africa we present a state of the art application of smoothing for dependent bivariate binomial spatial data to loa loa prevalence mapping in west africa this application starts with the nonspatial calibration of survey instruments continues with the spatial model building and assessment and ends with robust tested software intended for use by field workers for online prevalence map updating from a statistical perspective we address several important methodological issues building spatial models that are sufficiently complex to capture the structure of the data but remain computationally usable reducing the computational burden in the handling of very large covariate data sets and devising methods for comparing spatial prediction methods for a given exceedance policy threshold

space time covariance functions this work considers a number of properties of space time covariance functions and how these relate to the spatial temporal interactions of the process first it examines how the smoothness away from the origin of a space time covariance function affects for example temporal correlations of spatial differences models that are not smoother away from the origin than they are at the origin such as separable models have a kind of discontinuity to certain correlations that one might wish to avoid in some circumstances smoothness away from the origin of a covariance function is shown to follow from the corresponding spectral density having derivatives with finite moments these results are used to obtain a parametric class of spectral densities whose corresponding space time covariance functions are infinitely differentiable away from the origin and that allows for essentially arbitrary and possibly different degrees of smoothness for the process in space and time second this work considers models that

stochastic models that separate fractal dimension and the hurst effect fractal behavior and long range dependence have been observed in an astonishing number of physical biological geological and socioeconomic systems time series profiles and surfaces have been characterized by their fractal dimension a measure of roughness and by the hurst coefficient a measure of long memory dependence both phenomena have been modeled and explained by self affine random functions such as fractional gaussian noise and fractional brownian motion the assumption of statistical self affinity implies a linear relationship between fractal dimension and hurst coefficient and thereby links the two phenomena this article introduces stochastic models that allow for any combination of fractal dimension and hurst coefficient associated software for the synthesis of images with arbitrary prespecified fractal properties and power law correlations is available the new models suggest a test for self affinity that assesses coupling and decoupling of local and global behavior

approximate bayesian inference for latent gaussian models by using integrated nested laplace approximations summary structured additive regression models are perhaps the most commonly used class of models in statistical applications it includes among others generalized linear models generalized additive models smoothing spline models state space models semiparametric regression spatial and spatiotemporal models log gaussian cox processes and geostatistical and geoadditive models we consider approximate bayesian inference in a popular subset of structured additive regression models latent gaussian models where the latent field is gaussian controlled by a few hyperparameters and with non gaussian response variables the posterior marginals are not available in closed form owing to the non gaussian response variables for such models markov chain monte carlo methods can be implemented but they are not without problems in terms of both convergence and computational time in some practical applications the extent of these problems is such that markov chain monte carlo sampling is simply not an appropriate tool for routine analysis we show that by using an integrated nested laplace approximation and its simplified version we can directly compute very accurate approximations to the posterior marginals the main benefit of these approximations is computational where markov chain monte carlo algorithms need hours or days to run our approximations provide more precise estimates in seconds or minutes another advantage with our approach is its generality which makes it possible to perform bayesian analysis in an automatic streamlined way and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged

singular stochastic allen cahn equations with dynamic boundary conditions abstract we prove a well posedness result for stochastic allen cahn type equations in a bounded domain coupled with generic boundary conditions the nonlinear flux at the boundary aims at describing the interactions with the hard walls and is motivated by some recent literature in physics the singular character of the drift part allows for a large class of maximal monotone operators generalizing the usual double well potentials one of the main novelties of the paper is the absence of any growth condition on the drift term of the evolution neither on the domain nor on the boundary a well posedness result for variational solutions of the system is presented using a priori estimates as well as monotonicity and compactness techniques a vanishing viscosity argument for the dynamic on the boundary is also presented

model based geostatistics for prevalence mapping in low resource settings abstractin low resource settings prevalence mapping relies on empirical prevalence data from a finite often spatially sparse set of surveys of communities within the region of interest possibly supplemented by remotely sensed images that can act as proxies for environmental risk factors a standard geostatistical model for data of this kind is a generalized linear mixed model with binomial error distribution logistic link and a combination of explanatory variables and a gaussian spatial stochastic process in the linear predictor in this article we first review statistical methods and software associated with this standard model then consider several methodological extensions whose development has been motivated by the requirements of specific applications these include methods for combining randomized survey data with data from nonrandomized and therefore potentially biased surveys spatio temporal extensions and spatially structured zero inflation throughout we illustrate the methods with

an inequality of ostrowski gruss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules abstract in this paper we derive a new inequality of ostrowski gruss type and apply it to the estimation of error bounds for some special means and for some numerical quadrature rules

choosing the sample size of a computer experiment a practical guide we provide reasons and evidence supporting the informal rule that the number of runs for an effective initial computer experiment should be about 10 times the input dimension our arguments quantify two key characteristics of computer codes that affect the sample size required for a desired level of accuracy when approximating the code via a gaussian process gp the first characteristic is the total sensitivity of a code output variable to all input variables the second corresponds to the way this total sensitivity is distributed across the input variables specifically the possible presence of a few prominent input factors and many impotent ones i e effect sparsity both measures relate directly to the correlation structure in the gp approximation of the code in this way the article moves toward a more formal treatment of sample size for a computer experiment the evidence supporting these arguments stems primarily from a simulation study and via specific codes modeling climate and ligand activa

nonparametric estimation of diffusions a differential equations approach we consider estimation of scalar functions that determine the dynamics of diffusion processes it has been recently shown that nonparametric maximum likelihood estimation is ill posed in this context we adopt a probabilistic approach to regularize the problem by the adoption of a prior distribution for the unknown functional a gaussian prior measure is chosen in the function space by specifying its precision operator as an appropriate differential operator we establish that a bayesian gaussian conjugate analysis for the drift of one dimensional nonlinear diffusions is feasible using high frequency data by expressing the loglikelihood as a quadratic function of the drift with sufficient statistics given by the local time process and the end points of the observed path computationally efficient posterior inference is carried out using a finite element method we embed this technology in partially observed situations and adopt a data augmentation approach whereby we iteratively generate missing data paths and draws from the unknown functional our methodology is applied to estimate the drift of models used in molecular dynamics and financial econometrics using high and low frequency observations we discuss extensions to other partially observed schemes and connections to other types of nonparametric inference

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

soap film smoothing summary conventional smoothing methods sometimes perform badly when used to smooth data over complex domains by smoothing inappropriately across boundary features such as peninsulas solutions to this smoothing problem tend to be computationally complex and not to provide model smooth functions which are appropriate for incorporating as components of other models such as generalized additive models or mixed additive models we propose a class of smoothers that are appropriate for smoothing over difficult regions of 2 which can be represented in terms of a low rank basis and one or two quadratic penalties the key features of these smoothers are that they do not smooth across boundary features that their representation in terms of a basis and penalties allows straightforward incorporation as components of generalized additive models mixed models and other non standard models that smoothness selection for these model components is straightforward to accomplish in a computationally efficient manner via generalized cross validation akaike s information criterion or restricted maximum likelihood for example and that their low rank means that their use is computationally efficient

existence and uniqueness of the solution for stochastic super fast diffusion equations with multiplicative noise abstract in this paper we prove an existence and uniqueness result for the stochastic porous media equation with very singular diffusion and multiplicative noise by using monotonicity techniques the multiplicative gaussian noise is essential in the proof of existence

spectral density estimation for random fields via periodic embeddings we introduce methods for estimating the spectral density of a random field on a formula see text dimensional lattice from incomplete gridded data data are iteratively imputed onto an expanded lattice according to a model with a periodic covariance function the imputations are convenient computationally in that circulant embedding and preconditioned conjugate gradient methods can produce imputations in formula see text time and formula see text memory however these so called periodic imputations are motivated mainly by their ability to produce accurate spectral density estimates in addition we introduce a parametric filtering method that is designed to reduce periodogram smoothing bias the paper contains theoretical results on properties of the imputed data periodogram and numerical and simulation studies comparing the performance of the proposed methods to existing approaches in a number of scenarios we present an application to a gridded satellite surface temperature dataset with missing values

bayesian calibration of computer models we consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models such models implemented as computer codes are often generic in the sense that by a suitable choice of some of the model s input parameters the code can be used to predict the behaviour of the system in a variety of specific applications however in any specific application the values of necessary parameters may be unknown in this case physical observations of the system in the specific context are used to learn about the unknown parameters the process of fitting the model to the observed data by adjusting the parameters is known as calibration calibration is typically effected by ad hoc fitting and after calibration the model is used with the fitted input values to predict the future behaviour of the system we present a bayesian calibration technique which improves on this traditional approach in two respects first the predictions allow for all sources of uncertainty including the remaining uncertainty over the fitted parameters second they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best fitting parameter values the method is illustrated by using data from a nuclear radiation release at tomsk and from a more complex simulated nuclear accident exercise

levy flights from a continuous time process l evy flight dynamics can stem from simple random walks in a system whose operational time number of steps n typically grows superlinearly with physical time t thus this process is a kind of continuous time random walk ctrw dual to the typical scher montroll model in which n grows sublinearly with t models in which l evy flights emerge due to a temporal subordination allow one easily to discuss the response of a random walker to a weak outer force which is shown to be nonlinear on the other hand the relaxation of an ensemble of such walkers in a harmonic potential follows a simple exponential pattern and leads to a normal boltzmann distribution mixed models describing normal ctrw s in superlinear operational time and l evy flights under the operational time of subdiffusive ctrw s lead to a paradoxical diffusive behavior similar to the one found in transport on polymer chains the relaxation to the boltzmann distribution in such models is slow and asymptotically follows a power law

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures