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

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

linear preservers between matrix modules over connected commutative rings abstract let r be a connected commutative ring with identity 1 r contains no idempotents except 0 and 1 and let m n r be the r module of all n n matrices over r r is said to be idempotence diagonalizable if every idempotent matrix over r is similar to a diagonal matrix for two arbitrary positive integers n and m we characterize a linear maps from m n r to m m r preserving tripotence when r is any idempotence diagonalizable ring with the units 2 and 3 and b linear maps from m n r to m m r preserving inverses respectively drazin inverses group inverses 1 inverses 2 inverses and 1 2 inverses when r is either any idempotence diagonalizable ring with the units 2 and 3 or any commutative principal ideal domain with at least one unit except for 1 and 2 these characterizations are completed by using an idempotence preserving result obtained by cao linear maps preserving idempotence on matrix modules over some rings j natur sci heilongjiang univ 16 1 1999 1 4 moreover we also give a simple proof of cao s result

hierarchical spatial process models for multiple traits in large genetic trials this article expands upon recent interest in bayesian hierarchical models in quantitative genetics by developing spatial process models for inference on additive and dominance genetic variance within the context of large spatially referenced trial datasets of multiple traits of interest direct application of such multivariate models to large spatial datasets is often computationally infeasible because of cubic order matrix algorithms involved in estimation the situation is even worse in markov chain monte carlo mcmc contexts where such computations are performed for several thousand iterations here we discuss approaches that help obviate these hurdles without sacrificing the richness in modeling for genetic effects we demonstrate how an initial spectral decomposition of the relationship matrices negates the expensive matrix inversions required in previously proposed mcmc methods for spatial effects we discuss a multivariate predictive process that reduces the computational burden by projecting th

a unified framework for fitting bayesian semiparametric models to arbitrarily censored survival data including spatially referenced data a comprehensive unified approach to modeling arbitrarily censored spatial survival data is presented for the three most commonly used semiparametric models proportional hazards proportional odds

a likelihood approximation for locally stationary processes a new approximation to the gaussian likelihood of a multivariate locally stationary process is introduced it is based on an approximation of the inverse of the covariance matrix of such processes the new quasi likelihood is a generalisation of the classical whittle likelihood for stationary processes for parametric models asymptotic normality and efficiency of the resulting estimator are proved since the likelihood has a special local structure it can be used for nonparametric inference as well this is briefly sketched for different estimates

modeling tangential vector fields on a sphere abstractphysical processes that manifest as tangential vector fields on a sphere are common in geophysical and environmental sciences these naturally occurring vector fields are often subject to physical constraints such as being curl free or divergence free we start with constructing parametric models for curl free and divergence free vector fields that are tangential to the unit sphere through applying the surface gradient or the surface curl operator to a scalar random potential field on the unit sphere using the helmholtz hodge decomposition we then construct a class of simple but flexible parametric models for general tangential vector fields which are represented as a sum of a curl free and a divergence free components we propose a likelihood based parameter estimation procedure and show that fast computation is possible even for large data sets when the observations are on a regular latitude longitude grid characteristics and practical utility of the proposed methodology are illustrated th

joint composite estimating functions in spatiotemporal models summary modelling of spatiotemporal processes has received considerable attention in recent statistical research however owing to the high dimensionality of the data the joint modelling of spatial and temporal processes presents a great computational challenge in both likelihood based and bayesian approaches we propose a joint composite estimating function approach to estimating spatiotemporal covariance structures this substantially reduces the computational complexity and is more efficient than existing composite likelihood methods the novelty of the proposed joint composite estimating function is rooted in the construction of three sets of estimating functions from spatial temporal and spatiotemporal cross pairs which results in overidentified estimating functions thus we form a joint inference function in a spirit that is similar to hansen s generalized method of moments we show that under practical scenarios the estimator proposed is consistent and asymptotically normal simulation studies prove that our method performs well in finite samples finally we illustrate the joint composite estimating function method by estimating the spatiotemporal dependence structure of airborne particulates pm10 in the north eastern usa over a 32 month period

a bayesian analysis of kriging this article is concerned with predicting for gaussian random fields in a way that appropriately deals with uncertainty in the covariance function to this end we analyze the best linear unbiased prediction procedure within a bayesian framework particular attention is paid to the treatment of parameters in the covariance structure and their effect on the quality both real and perceived of the prediction these ideas are implemented using topographical data from davis

modelling the distribution of white matter hyperintensities due to ageing on mri images using bayesian inference white matter hyperintensities wmh also known as white matter lesions are localised white matter areas that appear hyperintense on mri scans wmh commonly occur in the ageing population and are often associated with several factors such as cognitive disorders cardiovascular risk factors cerebrovascular and neurodegenerative diseases despite the fact that some links between lesion location and parametric factors such as age have already been established the relationship between voxel wise spatial distribution of lesions and these factors is not yet well understood hence it would be of clinical importance to model the distribution of lesions at the population level and quantitatively analyse the effect of various factors on the lesion distribution model in this work we compare various methods including our proposed method to generate voxel wise distributions of wmh within a population with respect to various factors our proposed bayesian spline method models the spatio temporal distribution of wmh with respect to a parametric factor of interest in this case age within a population our probabilistic model takes as input the lesion segmentation binary maps of subjects belonging to various age groups and provides a population level parametric lesion probability map as output we used a spline representation to ensure a degree of smoothness in space and the dimension associated with the parameter and formulated our model using a bayesian framework we tested our algorithm output on simulated data and compared our results with those obtained using various existing methods with different levels of algorithmic and computational complexity we then compared the better performing methods on a real dataset consisting of 1000 subjects of the uk biobank divided in two groups based on hypertension diagnosis finally we applied our method on a clinical dataset of patients with vascular disease on simulated dataset the results from our algorithm showed a mean square error mse value of 7 27 10 5 which was lower than the mse value reported in the literature with the advantage of being robust and computationally efficient in the uk biobank data we found that the lesion probabilities are higher for the hypertension group compared to the non hypertension group and further verified this finding using a statistical t test finally when applying our method on patients with vascular disease we observed that the overall probability of lesions is significantly higher in later age groups which is in line with the current literature

spline based emulators for radiative shock experiments with measurement error radiation hydrodynamics and radiative shocks are of fundamental interest in the high energy density physics research due to their importance in understanding astrophysical phenomena such as supernovae in the laboratory experiments can produce shocks with fundamentally similar physics on reduced scales however the cost and time constraints of the experiment necessitate use of a computer algorithm to generate a reasonable number of outputs for making valid inference we focus on modeling emulators that can efficiently assimilate these two sources of information accounting for their intrinsic differences the goal is to learn how to predict the breakout time of the shock given the information on associated parameters such as pressure and energy under the framework of the kennedy o hagan model we introduce an emulator based on adaptive splines depending on the preference of having an interpolator for the computer code output or a computationally fast model a couple of different variants are proposed

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

covariance tapering for likelihood based estimation in large spatial data sets maximum likelihood is an attractive method of estimating covariance parameters in spatial models based on gaussian processes but calculating the likelihood can be computationally infeasible for large data sets requiring o n3 calculations for a data set with n observations this article proposes the method of covariance tapering to approximate the likelihood in this setting in this approach covariance matrixes are tapered or multiplied element wise by a sparse correlation matrix the resulting matrixes can then be manipulated using efficient sparse matrix algorithms we propose two approximations to the gaussian likelihood using tapering one of these approximations simply replaces the model covariance with a tapered version whereas the other is motivated by the theory of unbiased estimating equations focusing on the particular case of the matern class of covariance functions we give conditions under which estimators maximizing the tapering approximations are like the maximum likelihood estimat

on multimodality of the likelihood in the spatial linear model summary a popular covariance scheme used for the spatial linear model in geostatistics has spherical form however the likelihood is not twice differentiable with respect to the range parameter and this raises some questions regarding the unimodality of the likelihood we compare the likelihoods of the spatial linear model for small samples under this scheme and the doubly geometric scheme also a power covariance with range parameter is proposed in view of potential multimodal likelihoods for small samples for this model a convenient profile likelihood is introduced and studied

inference for deterministic simulation models the bayesian melding approach abstract deterministic simulation models are used in many areas of science engineering and policy making typically these are complex models that attempt to capture underlying mechanisms in considerable detail and they have many user specified inputs the inputs are often specified by some form of trial and error approach in which plausible values are postulated the corresponding outputs inspected and the inputs modified until plausible outputs are obtained here we address the issue of more formal inference for such models a probabilistic approach called bayesian synthesis was shown to suffer from the borel paradox according to which the results can depend on the parameterization of the model we propose a modified approach called bayesian melding which takes into full account information and uncertainty about both inputs and outputs to the model while avoiding the borel paradox this is done by recognizing the existence of two priors one implicit and one explicit on each input and output

permutation and grouping methods for sharpening gaussian process approximations abstractvecchia s approximate likelihood for gaussian process parameters depends on how the observations are ordered which has been cited as a deficiency this article takes the alternative standpoint that the ordering can be tuned to sharpen the approximations indeed the first part of the article includes a systematic study of how ordering affects the accuracy of vecchia s approximation we demonstrate the surprising result that random orderings can give dramatically sharper approximations than default coordinate based orderings additional ordering schemes are described and analyzed numerically including orderings capable of improving on random orderings the second contribution of this article is a new automatic method for grouping calculations of components of the approximation the grouping methods simultaneously improve approximation accuracy and reduce computational burden in common settings reordering combined with grouping reduces kullback leibler divergence from the target model by more th

on polynomial expansions for likelihoods of spatial data with finite range correlation function summary we seek to establish a framework for providing computationally attractive forms for quantities in the likelihood of the spatial linear model this framework permits the study of correlation schemes with finite range parameter and can be extended to include a nugget effect parameter we study and obtain results for one simple example and indicate how generalizations to this work will operate

statistical estimation of polynomial generalized covariance functions and hydrologic applications minimum variance unbiased linear estimation theory has found many applications in the analysis of spatial rainfall and hydrogeologic data estimation of point values or areal averages calculation of variances of estimation error and network design however the important problem of inferring the spatial structure or interdependence of the function of interest has received relatively less attention three methods for the estimation of the parameters of matheron s polynomial generalized covariance function a useful model for nonstationary functions are presented and evaluated maximum likelihood ml minimum variance unbiased quadratic mvuq and minimum norm least squares the estimation error covariance matrix of the parameters is calculated with each of these methods ml and mvuq estimation have very good properties and for reasonably large samples give very similar results minimum norm estimation is appropriate for preliminary data analysis the application of these methodologies is illustrated with a set of rainfall data and problems of implementation are discussed it is shown that the estimation of the parameters of the covariance function involves sizeable errors it is suggested that in practice it may be rather difficult to identify the parameters of all but simple models it is important that future research evaluate the effects of parameter uncertainty in the circulation of estimation error variances of point or areal averages and in the design of hydrologic data collection networks

quasi likelihood and its application a general approach to optimal parameter estimation the general framework an alternative approach e sufficiency asymptotic confidence zones of minimum size asymptotic quasi likelihood combining estimating functions projected quasi likelihood bypassing the likelihood hypothesis testing infinite dimensional problems miscellaneous applications consistency and asymptotic normality for estimating functions complements and strategies for application

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

a multi resolution approximation for massive spatial datasets abstractautomated sensing instruments on satellites and aircraft have enabled the collection of massive amounts of high resolution observations of spatial fields over large spatial regions if these datasets can be efficiently exploited they can provide new insights on a wide variety of issues however traditional spatial statistical techniques such as kriging are not computationally feasible for big datasets we propose a multi resolution approximation m ra of gaussian processes observed at irregular locations in space the m ra process is specified as a linear combination of basis functions at multiple levels of spatial resolution which can capture spatial structure from very fine to very large scales the basis functions are automatically chosen to approximate a given covariance function which can be nonstationary all computations involving the m ra including parameter inference and prediction are highly scalable for massive datasets crucially the inference algorithms can also be parallelize

bayesian nonparametric covariance regression capturing predictor dependent correlations amongst the elements of a multivariate response vector is fundamental to numerous applied domains including neuroscience epidemiology and finance although there is a rich literature on methods for allowing the variance in a univariate regression model to vary with predictors relatively little has been done in the multivariate case as a motivating example we consider the google flu trends data set which provides indirect measurements of influenza incidence at a large set of locations over time our predictor to accurately characterize temporally evolving influenza incidence across regions it is important to develop statistical methods for a time varying covariance matrix importantly the locations provide a redundant set of measurements and do not yield a sparse nor static spatial dependence structure we propose to reduce dimensionality and induce a flexible bayesian nonparametric covariance regression model by relating these location specific trajectories to a lower dimensional subspace through a latent factor model with predictor dependent factor loadings these loadings are in terms of a collection of basis functions that vary nonparametrically over the predictor space such low rank approximations are in contrast to sparse precision assumptions and are appropriate in a wide range of applications our formulation aims to address three challenges scaling to large p domains coping with missing values and allowing an irregular grid of observations the model is shown to be highly exible while leading to a computationally feasible implementation via gibbs sampling the ability to scale to large p domains and cope with missing values is fundamental in analyzing the google flu trends data

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

fixed domain asymptotic properties of tapered maximum likelihood estimators when the spatial sample size is extremely large which occurs in many environmental and ecological studies operations on the large covariance matrix are a numerical challenge covariance tapering is a technique to alleviate the numerical challenges under the assumption that data are collected along a line in a bounded region we investigate how the tapering affects the asymptotic efficiency of the maximum likelihood estimator mle for the microergodic parameter in the matern covariance function by establishing the fixed domain asymptotic distribution of the exact mle and that of the tapered mle our results imply that under some conditions on the taper the tapered mle is asymptotically as efficient as the true mle for the microergodic parameter in the matern model

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

prediction of reservoir variables based on seismic data and well observations offshore petroleum reservoirs are usually explored by acquisition of vast amounts of seismic reflection data and observations along a small number of wells drilled through the reservoirs the seismic data are indirect measurements of reservoir characteristics and have good spatial coverage but low precision the well observations are sparse but precise information from these sources of data are integrated to characterize the reservoir variables of interest reservoir porosity in the current study critical parameters in the acquisition procedure are unknown and are simultaneously estimated this process termed seismic inversion constitutes a spatial multivariate ill posed inverse problem the problem is cast in a bayesian framework with a focus on sampling from the posterior model high dimensionality nonlinear components and unknown parameters in the likelihood model along with a complex design of the conditioning data complicate the problem a fast sequential sampling algorithm containing severa

problems with likelihood estimation of covariance functions of spatial gaussian processes summary maximum likelihood has frequently been suggested as a way to estimate covariance parameters in spatial gaussian processes the results of some simple examples based on simulated data are discussed it is shown by an example with real data that the profile likelihood can be multimodal and that the global maximum may not correspond to a sensible value of the parameters mardia marshall 1984 worked in the following framework we have a gaussian process z x i x e x with mean given by a regression equation e z x f x t for a p dimensional vector the covariance function cov z x z y c x y 0 is also specified by a q dimensional vector 0 this process is observed at n points xi xn and from these observations we wish to estimate 3 and 0 let zn z xi z xn t be the vector of observations and k c xi xj 0 be its covariance matrix finally let f be the matrix with rows f xi t then zn has a multivariate normal distribution with mean f18 dispersion matrix k and the log likelihood is up to an additive constant

the role of the range parameter for estimation and prediction in geostatistics summary two canonical problems in geostatistics are estimating the parameters in a specified family of stochastic process models and predicting the process at new locations we show that asymptotic results for a gaussian process over a fixed domain with mat ern covariance function previously proven only in the case of a fixed range parameter can be extended to the case of jointly estimating the range and the variance of the process moreover we show that intuition and approximations derived from asymptotics using a fixed range parameter can be problematic when applied to finite samples even for large sample sizes in contrast we show via simulation that performance is improved and asymptotic approximations are applicable for smaller sample sizes when the parameters are jointly estimated these effects are particularly apparent when the process is mean square differentiable or the effective range of spatial correlation is small

a full scale approximation of covariance functions for large spatial data sets summary gaussian process models have been widely used in spatial statistics but face tremendous computational challenges for very large data sets the model fitting and spatial prediction of such models typically require o n3 operations for a data set of size n various approximations of the covariance functions have been introduced to reduce the computational cost however most existing approximations cannot simultaneously capture both the large and the small scale spatial dependence a new approximation scheme is developed to provide a high quality approximation to the covariance function at both the large and the small spatial scales the new approximation is the summation of two parts a reduced rank covariance and a compactly supported covariance obtained by tapering the covariance of the residual of the reduced rank approximation whereas the former part mainly captures the large scale spatial variation the latter part captures the small scale local variation that is unexplained by the former part by combining the reduced rank representation and sparse matrix techniques our approach allows for efficient computation for maximum likelihood estimation spatial prediction and bayesian inference we illustrate the new approach with simulated and real data sets

convolution roots of radial positive definite functions with compact support a classical theorem of boas kac and krein states that a characteristic function with x 0 for x t admits a representation of the form x u y u y x dy x r where the convolution root u l 2 r is complex valued with u x 0 for x 2 the result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type this paper examines the boas kac representation under additional constraints if is real valued and even can the convolution root u be chosen as a real valued and or even function a complete answer in terms of the zeros of the fourier transform of is obtained furthermore the analogous problem for radially symmetric functions defined on r d is solved perhaps surprisingly there are compactly supported radial positive definite functions that do not admit a convolution root with half support however under the additional assumption of nonnegativity radially symmetric convolution roots with half support exist further results in this paper include a characterization of extreme points pointwise and integral bounds turan s problem and a unified solution to a minimization problem for compactly supported positive definite functions specifically if f is a probability density on r d whose characteristic function vanishes outside the unit ball then x 2 f x dx 0 4j 2 d 2 2 where j v denotes the first positive zero of the bessel function j v and the estimate is sharp applications to spatial moving average processes geostatistical simulation crystallography optics and phase retrieval are noted in particular a real valued half support convolution root of the spherical correlation function in r 2 does not exist

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

on abstract data types presented by multiequations equational presentation of abstract data types is generalized to presentation by multiequations i e exclusive or s of equations in order to capture parametric data types such as array or set multiinitial algebra semantics for such data types is introduced classes of algebras described by multiequations are characterized

estimating space and space time covariance functions for large data sets a weighted composite likelihood approach in this article we propose two methods for estimating space and space time covariance functions from a gaussian random field based on the composite likelihood idea the first method relies on the maximization of a weighted version of the composite likelihood function while the second one is based on the solution of a weighted composite score equation this last scheme is quite general and could be applied to any kind of composite likelihood an information criterion for model selection based on the first estimation method is also introduced the methods are useful for practitioners looking for a good balance between computational complexity and statistical efficiency the effectiveness of the methods is illustrated through examples simulation experiments and by analyzing a dataset on ozone measurements

scaling multidimensional inference for structured gaussian processes exact gaussian process gp regression has cal o n 3 runtime for data size n making it intractable for large n many algorithms for improving gp scaling approximate the covariance with lower rank matrices other work has exploited structure inherent in particular covariance functions including gps with implied markov structure and inputs on a lattice both enable cal o n or cal o n log n runtime however these gp advances have not been well extended to the multidimensional input setting despite the preponderance of multidimensional applications this paper introduces and tests three novel extensions of structured gps to multidimensional inputs for models with additive and multiplicative kernels first we present a new method for inference in additive gps showing a novel connection between the classic backfitting method and the bayesian framework we extend this model using two advances a variant of projection pursuit regression and a laplace approximation for non gaussian observations lastly for multiplicative kernel structure we present a novel method for gps with inputs on a multidimensional grid we illustrate the power of these three advances on several data sets achieving performance equal to or very close to the naive gp at orders of magnitude less cost

computing f a b via least squares polynomial approximations given a certain function f various methods have been proposed in the past for addressing the important problem of computing the matrix vector product f a b without explicitly computing the matrix f a such methods were typically developed for a specific function f a common case being that of the exponential this paper discusses a procedure based on least squares polynomials that can in principle be applied to any continuous function f the idea is to start by approximating the function by a spline of a desired accuracy then a particular definition of the function inner product is invoked that facilitates the computation of the least squares polynomial to this spline function since the function is approximated by a polynomial the matrix a is referenced only through a matrix vector multiplication in addition the choice of the inner product makes it possible to avoid numerical integration as an important application we consider the case when f t sqrt t and a is a sparse symmetric positive definite matrix which arises in sampling from a gaussian process distribution the covariance matrix of the distribution is defined by using a covariance function that has a compact support at a very large number of sites that are on a regular or irregular grid we derive error bounds and show extensive numerical results to illustrate the effectiveness of the proposed technique

a resampling based stochastic approximation method for analysis of large geostatistical data the gaussian geostatistical model has been widely used in modeling of spatial data however it is challenging to computationally implement this method because it requires the inversion of a large covariance matrix particularly when there is a large number of observations this article proposes a resampling based stochastic approximation method to address this challenge at each iteration of the proposed method a small subsample is drawn from the full dataset and then the current estimate of the parameters is updated accordingly under the framework of stochastic approximation since the proposed method makes use of only a small proportion of the data at each iteration it avoids inverting large covariance matrices and thus is scalable to large datasets the proposed method also leads to a general parameter estimation approach maximum mean log likelihood estimation which includes the popular maximum log likelihood estimation mle approach as a special case and is expected to play an important role

maximum likelihood estimation of models for residual covariance in spatial regression we describe the maximum likelihood method for fitting the linear model when residuals are correlated and when the covariance among the residuals is determined by a parametric model containing unknown parameters observations are assumed to be gaussian we give conditions which ensure consistency and asymptotic normality of the estimators our main concern is with the analysis of spatial data and in this context we describe some simulation experiments to assess the small sample behaviour of estimators we also discuss an application of the spectral approximation to the likelihood for processes on a lattice

fixed domain asymptotics for spatial periodograms abstract the periodogram for a spatial process observed on a lattice is often used to estimate the spectral density the bases for such estimators are two asymptotic properties that periodograms commonly possess 1 the periodogram at a particular frequency is approximately unbiased for the spectral density and 2 the correlation of the periodogram at distinct frequencies is approximately zero for spatial data it is often appropriate to use fixed domain asymptotics in which the observations get increasingly dense in some fixed region as their number increases using fixed domain asymptotics this article shows that standard asymptotic results for periodograms do not apply and that using the periodogram of the raw data can yield highly misleading results but by appropriately filtering the data before computing the periodogram it is possible to obtain results similar to the standard asymptotic results for spatial periodograms

compressed gaussian process for manifold regression nonparametric regression for large numbers of features p is an increasingly important problem if the sample size n is massive a common strategy is to partition the feature space and then separately apply simple models to each partition set this is not ideal when n is modest relative to p and we propose an alternative approach relying on random compression of the feature vector combined with gaussian process regression the proposed approach is particularly motivated by the setting in which the response is conditionally independent of the features given the projection to a low dimensional manifold conditionally on the random compression matrix and a smoothness parameter the posterior distribution for the regression surface and posterior predictive distributions are available analytically running the analysis in parallel for many random compression matrices and smoothness parameters model averaging is used to combine the results the algorithm can be implemented rapidly even in very large p and moderately large n nonparametric regression has strong theoretical justification and is found to yield state of the art predictive performance