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

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

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

existence and nonexistence results of an optimal control problem by using relaxed control relaxed controls have proved to be very useful in studying the existence of optimal controls in optimal control theory many positive results have been obtained in the literature however negative results have also made their rare appearances the optimal control problem considered in this paper looks quite simple yet by treating such a problem we can get interesting results substantiating our idea as to whether an optimal control exists or not in our opinion the method used in the paper can be applied to more generalized cases

adaptive control variates for finite horizon simulation adaptive monte carlo methods are simulation efficiency improvement techniques designed to adaptively tune simulation estimators most of the work on adaptive monte carlo methods has been devoted to adaptively tuning importance sampling schemes we instead focus on adaptive methods based on control variate schemes we introduce two adaptive control variate methods and develop their asymptotic properties the first method uses stochastic approximation to adaptively tune control variate estimators it is easy to implement but it requires some nontrivial tuning of parameters the second method is based on sample average approximation tuning is no longer required but it can be computationally expensive numerical results for the pricing of barrier options are presented to demonstrate the methods

adaptive importance sampling monte carlo simulation for general multivariate probability laws we establish a parametric adaptive importance sampling variance reduction method for general multivariate probability laws employing the principle of bypass distributions makes it possible to develop adaptive algorithms without relying on particular properties of the target and proposal laws both of which in the proposed framework are as general as the uniform law on the unit hypercube without changing the sampling distribution at each iteration we establish the asymptotic normality of the estimator of the desired mean and of the importance sampling parameter as the number of observations tends to infinity although implementation of the proposed methodology requires a small amount of initial work it has the potential to yield substantial improvements in estimator efficiency in various general problem settings to illustrate the applicability and effectiveness we provide numerical results throughout in which we apply exponential and normal bypass distributions as well as demonstrate that well known adaptive importance sampling formulations in the literature can be easily rewritten in the proposed framework

bayesian inference for the spatial random effects model spatial statistical analysis of massive amounts of spatial data can be challenging because computation of optimal procedures can break down the spatial random effects sre model uses a fixed number of known but not necessarily orthogonal multiresolutional spatial basis functions which gives a flexible family of nonstationary covariance functions results in dimension reduction and yields optimal spatial predictors whose computations are scalable by modeling spatial data in a hierarchical manner with a process model that includes the sre model the choice is whether to estimate the sre model s parameters or to take a bayesian approach and put a prior distribution on them in this article we develop bayesian inference for the sre model when the spatial basis functions are multiresolutional then the covariance matrix of the random effects decomposes naturally in terms of givens angles and eigenvalues for which a new class of prior distributions is developed this approach to prior specification of

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

compressing an ensemble with statistical models an algorithm for global 3d spatio temporal temperature one of the main challenges when working with modern climate model ensembles is the increasingly larger size of the data produced and the consequent difficulty in storing large amounts of spatio temporally resolved information many compression algorithms can be used to mitigate this problem but since they are designed to compress generic scientific datasets they do not account for the nature of climate model output and they compress only individual simulations in this work we propose a different statistics based approach that explicitly accounts for the space time dependence of the data for annual global three dimensional temperature fields in an initial condition ensemble the set of estimated parameters is small compared to the data size and can be regarded as a summary of the essential structure of the ensemble output therefore it can be used to instantaneously reproduce the temperature fields in an ensemble with a substantial saving in storage and time the statistical model exploits the gri

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

adaptive importance sampling and control variates abstract we construct and investigate an adaptive variance reduction framework in which both importance sampling and control variates are employed the three lines monte carlo averaging and two variance reduction parameter search lines run in parallel on a common sequence of uniform random vectors on the unit hypercube given that these two variance reduction techniques are effective often in a complementary way their combined application is well expected to widen the applicability of adaptive variance reduction we derive convergence rates of the theoretical estimator variance towards its minimum as a fixed computing budget increases when stochastic approximation runs with optimal constant learning rates we derive sufficient conditions for the proposed algorithm to attain the minimal estimator variance in the limit by stochastic approximation with decreasing learning rates or by sample average approximation when computing budget is unlimitedly available numerical results support our theoretical findings and illustrate the effectiveness of the proposed framework

fitting continuous arma models to unequally spaced spatial data abstract methods for fitting continuous spatial autoregressive moving average arma models to unequally spaced observations in two dimensions are reviewed and extended these are models with rational two dimensional spectra assuming gaussian input noise and observational errors maximum likelihood methods are used to estimate the arma parameters and the regression coefficients of the deterministic trend when the number of observations is too large for exact maximum likelihood estimation approximate maximum likelihood estimation is used based on nearest neighbors comparisons of nearest neighbor methods with exact likelihood methods are presented predictions of the height of the field at unobserved points can be calculated with confidence intervals

on a class of random field models which allows long range dependence summary in this note it is shown how a convenient class of random field models with rational spectral densities can be augmented to incorporate the possibility of long range dependence

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

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

nonparametric importance sampling abstract importance sampling is a widely used variance reduction simulation technique for the evaluation of high dimensional integrals a key step in the implementation of importance sampling is to choose a proper distribution function from which pseudorandom numbers are generated parametric sampling distributions if available at all are often inadequate for high dimensional integrals over irregular regions one possible remedy is to use a nonparametric method to estimate the unknown optimal sampling function we show that the nonparametric approach yields integral estimates that converge faster than estimates obtained from parametric approaches we also demonstrate that an adaptive method which has been used successfully in parametric settings does not yield better results than simple one step methods in the nonparametric setting

regionalization of multiscale spatial processes by using a criterion for spatial aggregation error summary the modifiable areal unit problem and the ecological fallacy are known problems that occur when modelling multiscale spatial processes we investigate how these forms of spatial aggregation error can guide a regionalization over a spatial domain of interest by regionalization we mean a specification of geographies that define the spatial support for areal data this topic has been studied vigorously by geographers but has been given less attention by spatial statisticians thus we propose a criterion for spatial aggregation error which we minimize to obtain an optimal regionalization to define the criterion we draw a connection between spatial aggregation error and a new multiscale representation of the karhunen loeve expansion this relationship between the criterion for spatial aggregation error and the multiscale karhunen loeve expansion leads to illuminating theoretical developments including connections between spatial aggregation error squared prediction error spatial variance and a novel extension of obled creutin eigenfunctions the effectiveness of our approach is demonstrated through an analysis of two data sets one using the american community survey and one related to environmental ocean winds

more efficient approximation of smoothing splines via space filling basis selection we consider the problem of approximating smoothing spline estimators in a nonparametric regression model when applied to a sample of size formula see text the smoothing spline estimator can be expressed as a linear combination of formula see text basis functions requiring formula see text computational time when the number formula see text of predictors is two or more such a sizeable computational cost hinders the broad applicability of smoothing splines in practice the full sample smoothing spline estimator can be approximated by an estimator based on formula see text randomly selected basis functions resulting in a computational cost of formula see text it is known that these two estimators converge at the same rate when formula see text is of order formula see text where formula see text depends on the true function and formula see text depends on the type of spline such a formula see text is called the essential number of basis functions in this article we develop a more efficient basis selection method by selecting basis functions corresponding to approximately equally spaced observations the proposed method chooses a set of basis functions with great diversity the asymptotic analysis shows that the proposed smoothing spline estimator can decrease formula see text to around formula see text when formula see text applications to synthetic and real world datasets show that the proposed method leads to a smaller prediction error than other basis selection methods

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

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

kriging nonstationary data abstract spatial data modeled to have come from a random function with a nonstationary mean are considered the spatial prediction method known as kriging exploits second order spatial correlation structure to obtain minimum variance predictions of certain average values of the random function but to do so it must be assumed that either the mean function the drift is known up to a constant or the second order structure the variogram is known exactly knowledge of the drift allows the stationary variogram to be estimated and leads to ordinary kriging knowledge of the variogram allows the drift to be estimated and leads to universal kriging more usually neither is known this article shows how median polish of gridded spatial data provides a resistant and relatively bias free way of kriging in the presence of drift yet yields results as good as the mathematically optimal but operationally difficult universal kriging comparisons are performed on two data sets

a fast optimal spatial prediction method for massive datasets this article considers a class of multiresolution tree structured models that are spatially shifted versions of each other and proposes a new spatial prediction method that averages over the optimal spatial predictors produced from members of this class of models as a consequence the resulting predicted surface is smooth even when the predictors generated separately from individual multiresolution tree structured models are not we call the new predictor the multiresolution spatial murs predictor and develop a computationally efficient algorithm for it the algorithm can handle massive datasets even when some observations are missing moreover the murs predictor can be shown to be the minimum mean squared error predictor for a large class of covariance functions a simulation example for massive datasets shows that the murs method consistently outperforms two commonly used filtering methods total column ozone data remotely sensed from a satellite are analyzed using the new methodology

smooth fitting of geophysical data using continuous global surfaces continuous global surfaces cgs are a general framework for interpolation and smoothing of geophysical data the first of two smoothing techniques we consider in this paper is generalized cross validation gcv which is a bootstrap measure of the predictive error of a surface that requires no prior knowledge of noise levels the second smoothing technique is to define the cgs surface with fewer centers than data points and compute the fit by least squares lsqr the noise levels are implicitly estimated by the number and placement of the centers relative to the data points we show that both smoothing methods can be implemented using extensions to the existing fast framework for interpolation so that it is now possible to construct realistic smooth fits to the very large data sets typically collected in geophysics thin plate spline and kriging surfaces with gcv smoothing appear to produce realistic fits to noisy radiometric data the resulting surfaces are similar yet the thin plate spline required

minimum norm quadratic estimation of spatial variograms abstract the estimation of spatial variograms a measure of spatial correlation is a critical problem in the implementation of kriging a method for interpolating random fields we consider the use of minimum norm quadratic estimators of the variogram when it is specified up to a finite number of linear parameters we investigate the asymptotic behavior of such estimators for gaussian processes as the number of observations within some bounded region increases the basic conclusion is that we can estimate consistently those functions of the parameters and only those functions that have a nonnegligible impact asymptotically on the kriging procedure in general the behavior of the variogram over relatively short distances is the only aspect of the variogram that is asymptotically important as an example consider a gaussian process z on the real line with unknown constant mean and where is known as the semivariogram of the process and is a finite vector of unknown parameters suppose for 0

simulation of moisture deficits and areal interpolation by universal cokriging simulation models for calculating moisture deficits for areas of land require interpolation procedures to arrive from point observations to area covering statements we use both 1 calculate first interpolate later ci procedures which interpolate calculated model results for test locations and 2 procedures which interpolate basic soil data toward test locations followed by model calculations interpolate first calculate later ic procedures in this study several ci and ic procedures which simulate moisture deficits are compared by means of a test set of 100 observations yielding the mean standard error mse ci procedures consistently produced lower mse values than ic procedures parameters of the pseudocovariance function pcf which models the spatial structure of bivariate increments in universal cokriging were estimated by means of the restricted maximum likelihood procedure compared to universal kriging universal cokriging yielded comparable mse values but a lower mean variance of the prediction error best results in this study were obtained by pointwise simulation model calculations followed by statistical interpolations

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

modeling nonstationarity in space and time we propose to model a spatio temporal random field that has nonstationary covariance structure in both space and time domains by applying the concept of the dimension expansion method in bornn et al 2012 simulations are conducted for both separable and nonseparable space time covariance models and the model is also illustrated with a streamflow dataset both simulation and data analyses show that modeling nonstationarity in both space and time can improve the predictive performance over stationary covariance models or models that are nonstationary in space but stationary in time

statistical methods for regular monitoring data summary meteorological and environmental data that are collected at regular time intervals on a fixed monitoring network can be usefully studied combining ideas from multiple time series and spatial statistics particularly when there are little or no missing data this work investigates methods for modelling such data and ways of approximating the associated likelihood functions models for processes on the sphere crossed with time are emphasized especially models that are not fully symmetric in space time two approaches to obtaining such models are described the first is to consider a rotated version of fully symmetric models for which we have explicit expressions for the covariance function the second is based on a representation of space time covariance functions that is spectral in just the time domain and is shown to lead to natural partially nonparametric asymmetric models on the sphere crossed with time various models are applied to a data set of daily winds at 11 sites in ireland over 18 years spectral and space time domain diagnostic procedures are used to assess the quality of the fits the spectral in time modelling approach is shown to yield a good fit to many properties of the data and can be applied in a routine fashion relative to finding elaborate parametric models that describe the space time dependences of the data about as well

a note on composite likelihood inference and model selection a composite likelihood consists of a combination of valid likelihood objects usually related to small subsets of data the merit of composite likelihood is to reduce the computational complexity so that it is possible to deal with large datasets and very complex models even when the use of standard likelihood or bayesian methods is not feasible in this paper we aim to suggest an integrated general approach to inference and model selection using composite likelihood methods in particular we introduce an information criterion for model selection based on composite likelihood we also describe applications to the modelling of time series of counts through dynamic generalised linear models and to the analysis of the well known old faithful geyser dataset

use of kriging models to approximate deterministic computer models the use of kriging models for approximation and metamodel based design and optimization has been steadily on the rise in the past decade the widespread usage of kriging models appears to be hampered by 1 the lack of guidance in selecting the appropriate form of the kriging model 2 computationally efficient algorithms for estimating the model s parameters and 3 an effective method to assess the resulting model s quality in this paper we compare 1 maximum likelihood estimation mle and cross validation cv parameter estimation methods for selecting a kriging model s parameters given its form and 2 and an r 2 of prediction and the corrected akaike information criterion for assessing the quality of the created kriging model permitting the comparison of different forms of a kriging model these methods are demonstrated with six test problems finally different forms of kriging models are examined to determine if more complex forms are more accurate and easier to fit than simple forms of kriging models for approximating computer models

fourier analysis of irregularly spaced data on rd summary the purpose of the paper is to propose a frequency domain approach for irregularly spaced data on rd we extend the original definition of a periodogram for time series to that for irregularly spaced data and define non parametric and parametric spectral density estimators in a way that is similar to the classical approach introduction of the mixed asymptotics which are one of the asymptotics for irregularly spaced data makes it possible to provide asymptotic theories to the spectral estimators the asymptotic result for the parametric estimator is regarded as a natural extension of the classical result for regularly spaced data to that for irregularly spaced data empirical studies are also included to illustrate the frequency domain approach in comparisons with the existing spatial and frequency domain approaches

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

the method of normalized correlations a fast parameter estimation method for random processes and isotropic random fields that focuses on short range dependence we focus on the inference of covariance parameters from spatial or temporal data using the method of normalized correlations monc we compare the monc with maximum likelihood estimation and weighted least squares using simulated data as well as gridded and scattered real data for the synthetic experiments none of the methods considered performed uniformly better than the others in terms of estimation accuracy and precision however for regularly spaced data the monc was significantly faster and practically insensitive to sample size for scattered data the monc s speed was reduced but still competitive potential applications of the monc include those involving large data sets and fast preconditioning

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

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 deriving the inverse of a sum of matrices available expressions are reviewed and new ones derived for the inverse of the sum of two matrices one of them being nonsingular particular attention is given to text bf a text bf ubv 1 where textbf a is nonsingular and textbf u b and textbf v may be rectangular generalized inverses of textbf a textbf ubv are also considered several statistical applications are discussed

acceleration on adaptive importance sampling with sample average approximation we construct and analyze acceleration techniques for adaptive monte carlo simulations for general multivariate probability laws when the sample average approximation is employed for optimal parameter search our goal is to accelerate the adaptive monte carlo estimation by leading the parameter search line based on the sample average approximation to reach a nearly optimal realm at its small sample size stage first we introduce an auxiliary parameter into the parameter search line and update it wisely to aim for small sample convergence all three lines of the algorithm the monte carlo averaging the importance sampling parameter search and the auxiliary parameter updating run concurrently in a fully automated manner with a common set of random vectors without ad hoc tuning of the simulation system we next propose and examine various criteria for the auxiliary parameter updating all under which the asymptotic normality of the estimator of the desired mean and of the importance sampling parameter hol

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

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

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

local adaptive importance sampling for multivariate densities with strong nonlinear relationships abstract we consider adaptive importance sampling techniques that use kernel density estimates at each iteration as importance sampling functions these can provide more nearly constant importance weights and more precise estimates of quantities of interest than the sampling importance resampling algorithm when the initial importance sampling function is diffuse relative to the target we propose a new method that adapts to the varying local structure of the target when the target has unusual structure such as strong nonlinear relationships between variables this method provides estimates with smaller mean squared error than alternative methods

the reconstruction approach from interpolation to regression this article introduces an interpolation based method called the reconstruction approach for nonparametric regression based on the fact that interpolation usually has negligible errors compared

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

preaveraged localized orthogonal polynomial estimators for surface smoothing and partial differentiation abstract we propose a multivariate smoothing method based on products of localized orthogonal polynomial series estimators for a smooth regression surface in the fixed design regression model the estimation of partial derivatives is included the proposed method provides for automatic and efficient boundary modifications near the edges of the surface assuming that the boundary of the support of the regression function satisfies some regularity conditions by allowing for a preaveraging step the corresponding algorithms are speeded up considerably and are easy to implement computation of special boundary kernels as required by the kernel method to avoid edge effects is not necessary it is shown that under sufficient smoothness assumptions the global average mean squared error has the same optimal rate of convergence as the mean squared error at an interior point that is the boundary correction is asymptotically effective the method depends on two smoothing parameters one determining the amount

efficient gaussian process regression for large datasets gaussian processes are widely used in nonparametric regression classification and spatiotemporal modelling facilitated in part by a rich literature on their theoretical properties however one of their practical limitations is expensive computation typically on the order of n3 where n is the number of data points in performing the necessary matrix inversions for large datasets storage and processing also lead to computational bottlenecks and numerical stability of the estimates and predicted values degrades with increasing n various methods have been proposed to address these problems including predictive processes in spatial data analysis and the subset of regressors technique in machine learning the idea underlying these approaches is to use a subset of the data but this raises questions concerning sensitivity to the choice of subset and limitations in estimating fine scale structure in regions that are not well covered by the subset motivated by the literature on compressive sensing we propose an alternative approach that involves linear projection of all the data points onto a lower dimensional subspace we demonstrate the superiority of this approach from a theoretical perspective and through simulated and real data examples

speeding up neighborhood search in local gaussian process prediction recent implementations of local approximate gaussian process models have pushed computational boundaries for nonlinear nonparametric prediction problems particularly when deployed as emulators for computer experiments their flavor of spatially independent computation accommodates massive parallelization meaning that they can handle designs two or more orders of magnitude larger than previously however accomplishing that feat can still require massive computational horsepower here we aim to ease that burden we study how predictive variance is reduced as local designs are built up for prediction we then observe how the exhaustive and discrete nature of an important search subroutine involved in building such local designs may be overly conservative rather we suggest that searching the space radially that is continuously along rays emanating from the predictive location of interest is a far thriftier alternative our empirical work demonstrates that ray based search yields predictors with accur

non nested testing of spatial correlation we develop non nested tests in a general spatial spatio temporal or panel data context the spatial aspect can be interpreted quite generally in either a geographical sense or employing notions of economic distance or when parametric modelling arises in part from a common factor or other structure in the former case observations may be regularly spaced across one or more dimensions as is typical with much spatio temporal data or irregularly spaced across all dimensions both isotropic models and non isotropic models can be considered and a wide variety of correlation structures in the second case models involving spatial weight matrices are covered such as spatial autoregressive models the setting is sufficiently general to potentially cover other parametric structures such as certain factor models and vector valued observations and here our preliminary asymptotic theory for parameter estimates is of some independent value the test statistic is based on a gaussian pseudo likelihood ratio and is shown to have an asymptotic standard normal distribution under the null hypothesis that one of the two models is correct this limit theory rests strongly on a central limit theorem for the gaussian pseudo maximum likelihood parameter estimates a small monte carlo study of finite sample performance is included

bayesian hierarchical models with conjugate full conditional distributions for dependent data from the natural exponential family we introduce a bayesian approach for analyzing possibly high dimensional dependent data that are distributed according to a member from the natural exponential family of distributions this probl

a new method of prediction for spatial regression models with correlated errors summary this paper deals with minimum mean squared error unbiased linear interpolation of a continuous domain spatial process based on a sparse set of irregularly spaced observations the process is assumed to be governed by a linear regression model with errors that follow a second order stationary gaussian random field a new method of prediction is developed that is compatible with the parameter estimation procedures of vecchia the result is a new likelihood based method for joint parameter estimation and prediction that can be applied to large or small data sets with irregularly spaced data simulated and observed data sets are analysed to illustrate the methods

distributed control of systems governed by a general class of quasilinear elliptic equations abstract in this paper we are concerned with optimal control problems governed by a general class of quasilinear elliptic equations our aim is to prove existence of solution and derive optimality conditions in order to deal with this last question we begin investigating differentiability of state solution of equation with respect to the control second term of equation obtaining it only in certain cases a perturbation method is used to treat other cases as a consequence of the optimality system regularity of optimal control and state is establised

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

dynamic models for spatiotemporal data we propose a model for non stationary spatiotemporal data to account for spatial variability we model the mean function at each time period as a locally weighted mixture of linear regressions to incorporate temporal variation we allow the regression coefficients to change through time the model is cast in a gaussian state space framework which allows us to include temporal components such as trends seasonal effects and autoregressions and permits a fast implementation and full probabilistic inference for the parameters interpolations and forecasts to illustrate the model we apply it to two large environmental data sets tropical rainfall levels and atlantic ocean temperatures

resolution adaptive fixed rank kriging abstractthe spatial random effects model is flexible in modeling spatial covariance functions and is computationally efficient for spatial prediction via fixed rank kriging frk however the model depends on a class of basis functions which if not selected properly may result in unstable or undesirable results additionally the maximum likelihood ml estimates of the model parameters are commonly computed using an expectation maximization em algorithm which further limits its applicability when a large number of basis functions are required in this research we propose a class of basis functions extracted from thin plate splines the functions are ordered in terms of their degrees of smoothness with higher order functions corresponding to larger scale features and lower order ones corresponding to smaller scale details leading to a parsimonious representation of a nonstationary spatial covariance function with the number of basis functions playing the role of spatial resolution the proposed

accelerating asymptotically exact mcmc for computationally intensive models via local approximations abstractwe construct a new framework for accelerating markov chain monte carlo in posterior sampling problems where standard methods are limited by the computational cost of the likelihood or of numerical models embedded therein our approach introduces local approximations of these models into the metropolis hastings kernel borrowing ideas from deterministic approximation theory optimization and experimental design previous efforts at integrating approximate models into inference typically sacrifice either the sampler s exactness or efficiency our work seeks to address these limitations by exploiting useful convergence characteristics of local approximations we prove the ergodicity of our approximate markov chain showing that it samples asymptotically from the exact posterior distribution of interest we describe variations of the algorithm that employ either local polynomial approximations or local gaussian process regressors our theoretical results reinforce the key observation underlying this

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

stochastic partial differential equation based modelling of large space time data sets summary increasingly larger data sets of processes in space and time ask for statistical models and methods that can cope with such data we show that the solution of a stochastic advection diffusion partial differential equation provides a flexible model class for spatiotemporal processes which is computationally feasible also for large data sets the gaussian process defined through the stochastic partial differential equation has in general a non separable covariance structure its parameters can be physically interpreted as explicitly modelling phenomena such as transport and diffusion that occur in many natural processes in diverse fields ranging from environmental sciences to ecology to obtain computationally efficient statistical algorithms we use spectral methods to solve the stochastic partial differential equation this has the advantage that approximation errors do not accumulate over time and that in the spectral space the computational cost grows linearly with the dimension the total computational cost of bayesian or frequentist inference being dominated by the fast fourier transform the model proposed is applied to post processing of precipitation forecasts from a numerical weather prediction model for northern switzerland in contrast with the raw forecasts from the numerical model the post processed forecasts are calibrated and quantify prediction uncertainty moreover they outperform the raw forecasts in the sense that they have a lower mean absolute error

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

dimension reduction and alleviation of confounding for spatial generalized linear mixed models summary non gaussian spatial data are very common in many disciplines for instance count data are common in disease mapping and binary data are common in ecology when fitting spatial regressions for such data one needs to account for dependence to ensure reliable inference for the regression coefficients the spatial generalized linear mixed model offers a very popular and flexible approach to modelling such data but this model suffers from two major shortcomings variance inflation due to spatial confounding and high dimensional spatial random effects that make fully bayesian inference for such models computationally challenging we propose a new parameterization of the spatial generalized linear mixed model that alleviates spatial confounding and speeds computation by greatly reducing the dimension of the spatial random effects we illustrate the application of our approach to simulated binary count and gaussian spatial data sets and to a large infant mortality data set

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

aggregation cokriging for highly multivariate spatial data best linear unbiased prediction of spatially correlated multivariate random processes often called cokriging in geostatistics requires the solution of a large linear system based on the covariance and cross covariance matrix of the observations for many problems of practical interest it is impossible to solve the linear system with direct methods we propose an efficient linear unbiased predictor based on a linear aggregation of the covariables the primary variable together with this single meta covariable is used to perform cokriging we discuss the optimality of the approach under different covariance structures and use it to create reanalysis type high resolution historical temperature fields

maximum principle of optimal control for degenerate quasi linear elliptic equations optimal control problems governed by degenerate quasi linear partial differential equations of elliptic type are considered the optimal control systems considered may lack cesari type conditions and therefore the corresponding approximate optimal control problem may have no solution to yield the maximum principle of optimal pairs relaxed controls are used to overcome the difficulties occurring when considering approximate problems the relaxed controls used are defined by finite additive measures so that the case of the control set being noncompact can be treated

tukey g and h random fields abstractwe propose a new class of transgaussian random fields named tukey g and h tgh random fields to model non gaussian spatial data the proposed tgh random fields have extremely flexible marginal distributions possibly skewed and or heavy tailed and therefore have a wide range of applications the special formulation of the tgh random field enables an automatic search for the most suitable transformation for the dataset of interest while estimating model parameters asymptotic properties of the maximum likelihood estimator and the probabilistic properties of the tgh random fields are investigated an efficient estimation procedure based on maximum approximated likelihood is proposed and an extreme spatial outlier detection algorithm is formulated kriging and probabilistic prediction with tgh random fields are developed along with prediction confidence intervals the predictive performance of tgh random fields is demonstrated through extensive simulation studies and an application to a dataset

a unifying view of sparse approximate gaussian process regression we provide a new unifying view including all existing proper probabilistic sparse approximations for gaussian process regression our approach relies on expressing the effective prior which the methods are using this allows new insights to be gained and highlights the relationship between existing methods it also allows for a clear theoretically justified ranking of the closeness of the known approximations to the corresponding full gps finally we point directly to designs of new better sparse approximations combining the best of the existing strategies within attractive computational constraints

on semiparametric inference of geostatistical models via local karhunen loeve expansion summary we develop a semiparametric approach to geostatistical modelling and inference in particular we consider a geostatistical model with additive components where the form of the covariance function of the spatial random error is not prespecified and thus is flexible a novel local karhunen loeve expansion is developed and a likelihood based method is devised for estimating the model parameters and statistical inference a simulation study demonstrates sound finite sample properties and a real data example is given for illustration finally the theoretical properties of the estimates are explored and in particular consistency results are established

an optimal control problem governed by quasi linear variational inequalities an optimal control problem governed by quasi linear variational inequality is considered the cost functional to be minimized contains the solution of a quasi linear variational inequality to get the existence regularity and necessary condition for the optimal pair a new related control problem is introduced by proving the existence of an optimal pair to such a new problem the existence and regularity of the optimal pair to the original problem are obtained it turns out that the regularity obtained is sharp in general some necessary conditions of the optimal pair are also obtained