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

nonparametric bayes modelling of count processes data on count processes arise in a variety of applications including longitudinal spatial and imaging studies measuring count responses the literature on statistical models for dependent count data is dominated by models built from hierarchical poisson components the poisson assumption is not warranted in many applied contexts and hierarchical poisson models make restrictive assumptions about overdispersion in marginal distributions in this article we propose a class of nonparametric bayes count process models constructed through rounding real valued underlying processes the proposed class of models accommodates situations in which separate count valued functional data are observed for each subject under study theoretical results on large support and posterior consistency are established and computational algorithms are developed based on markov chain monte carlo simulation the methods are evaluated via simulation and illustrated by application to longitudinal tumour counts and to asthma inhaler usage copyright 2013 oxford university press

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

adaptive nonparametric bayesian inference using location scale mixture priors we study location scale mixture priors for nonparametric statistical problems including multivariate regression density estimation and classification we show that a rate adaptive procedure can be obtained if the prior is properly constructed in particular we show that adaptation is achieved if a kernel mixture prior on a regression function is constructed using a gaussian kernel an inverse gamma bandwidth and gaussian mixing weights

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

spatially balanced sampling of natural resources the spatial distribution of a natural resource is an important consideration in designing an efficient survey or monitoring program for the resource generally sample sites that are spatially balanced that is more or less evenly dispersed over the extent of the resource are more efficient than simple random sampling we review a unified strategy for selecting spatially balanced probability samples of natural resources the technique is based on creating a function that maps two dimensional space into one dimensional space thereby defining an ordered spatial address we use a restricted randomization to randomly order the addresses so that systematic sampling along the randomly ordered linear structure results in a spatially well balanced random sample variable inclusion probability proportional to an arbitrary positive ancillary variable is easily accommodated the basic technique selects points in a two dimensional continuum but is also applicable to sampling finite populations or one dimension

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

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

butler groups of infinite rank ii a torsion free abelian group g is called a butler group if bext g t 0 for any torsion group t we show that every butler group g of cardinality ltl is a b2 group i e g is a union of a smooth ascending chain of pure subgroups g0 where g ga b0 b 0 a butler group of finite rank assuming the validity of the continuum hypothesis ch we show that every butler group of cardinality not exceeding n is a b2 group moreover we are able to prove that the derived functor bext2 a t 0 for any torsion group t and any torsion free a with jai this implies that under ch all balanced subgroups of completely decomposable groups of cardinality nt are b2 groups

nonconjugate bayesian estimation of covariance matrices and its use in hierarchical models abstract the problem of estimating a covariance matrix in small samples has been considered by several authors following early work by stein this problem can be especially important in hierarchical models where the standard errors of fixed and random effects depend on estimation of the covariance matrix of the distribution of the random effects we propose a set of hierarchical priors hps for the covariance matrix that produce posterior shrinkage toward a specified structure here we examine shrinkage toward diagonality we then address the computational difficulties raised by incorporating these priors and nonconjugate priors in general into hierarchical models we apply a combination of approximation gibbs sampling possibly with a metropolis step and importance reweighting to fit the models and compare this hybrid approach to alternative markov chain monte carlo methods our investigation involves three alternative hps the first works with the spectral decomposition of the covariance matrix an

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

non gaussian spatiotemporal modelling through scale mixing we construct non gaussian processes that vary continuously in space and time with nonseparable covariance functions starting from a general and flexible way of constructing valid nonseparable covariance functions through mixing over separable covariance functions the resulting models are generalized by allowing for outliers as well as regions with larger variances we induce this through scale mixing with separate positive valued processes smooth mixing processes are applied to the underlying correlated processes in space and in time thus leading to regions in space and time of increased spread an uncorrelated mixing process on the nugget effect accommodates outliers posterior and predictive bayesian inference with these models is implemented through a markov chain monte carlo sampler an application to temperature data in the basque country illustrates the potential of this model in the identification of outliers and regions with inflated variance and shows that this improves the predictive performance

variational fourier features for gaussian processes this work brings together two powerful concepts in gaussian processes the variational approach to sparse approximation and the spectral representation of gaussian processes this gives rise to an approximation that inherits the benefits of the variational approach but with the representational power and computational scalability of spectral representations the work hinges on a key result that there exist spectral features related to a finite domain of the gaussian process which exhibit almost independent covariances we derive these expressions for matern kernels in one dimension and generalize to more dimensions using kernels with specific structures under the assumption of additive gaussian noise our method requires only a single pass through the data set making for very fast and accurate computation we fit a model to 4 million training points in just a few minutes on a standard laptop with non conjugate likelihoods our mcmc scheme reduces the cost of computation from o nm2 for a sparse gaussian process to o nm per iteration where n is the number of data and m is the number of features

multivariate convex regression with adaptive partitioning we propose a new nonparametric method for multivariate regression subject to convexity or concavity constraints on the response function convexity constraints are common in economics statistics operations research financial engineering and optimization but there is currently no multivariate method that is stable and computationally feasible for more than a few thousand observations we introduce convex adaptive partitioning cap which creates a globally convex regression model from locally linear estimates fit on adaptively selected covariate partitions cap is a computationally efficient consistent method for convex regression we demonstrate empirical performance by comparing the performance of cap to other shape constrained and unconstrained regression methods for predicting weekly wages and value function approximation for pricing american basket options

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

approximate inference for disease mapping with sparse gaussian processes gaussian process gp models are widely used in disease mapping as they provide a natural framework for modeling spatial correlations their challenges however lie in computational burden and memory requirements in disease mapping models the other difficulty is inference which is analytically intractable due to the non gaussian observation model in this paper we address both these challenges we show how to efficiently build fully and partially independent conditional fic pic sparse approximations for the gp in two dimensional surface and how to conduct approximate inference using expectation propagation ep algorithm and laplace approximation la we also propose to combine fic with a compactly supported covariance function to construct a computationally efficient additive model that can model long and short length scale spatial correlations simultaneously the benefit of these approximations is computational the sparse gps speed up the computations and reduce the memory requirements the posterior inference via ep and laplace approximation is much faster and is practically as accurate as via markov chain monte carlo

on moore penrose inverses of quasi kronecker structured matrices abstract the moore penrose inverse and generalized inverse of a x 1 x 2 where a x 1 x 2 are complex matrices are given under various assumptions we use the result to derive the moore penrose inverse and inverse for bdiag a k uv e with p complex matrices a k two complex p vectors u and v and a complex matrix e such block structured matrices occur in hierarchical modeling of multivariate spatial or space time gaussian processes for the latter we also give expressions of the determinant and of conditional variances

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

cellular interference alignment interference alignment promises that in gaussian interference channels each link can support half of a degree of freedom dof per pair of transmit receive antennas however in general this result requires to precode the data bearing signals over a signal space of asymptotically large diversity e g over an infinite number of dimensions for time frequency varying fading channels or over an infinite number of rationally independent signal levels in the case of time frequency invariant channels in this paper we consider a wireless cellular system scenario where the promised optimal dofs are achieved with linear precoding in one shot i e over a single time frequency slot we focus on the uplink of a symmetric cellular system where each cell is split into three sectors with orthogonal intrasector multiple access in our model interference is local i e it is due to transmitters in neighboring cells only we consider a noniterative local cooperation scheme where base stations pass to their neighbors their decoded messages such that interference from already decoded messages can be canceled therefore for a given decoding order the interference between sectors is described by a directed locally connected graph the problem consists of maximizing the per sector dofs over all possible decoding orders and precoding schemes in particular we provide a decoding order and a one shot interference alignment scheme able to achieve optimal per sector dofs up to an additive gap due to boundary effects that vanishes as the size of the network becomes large then we extend our treatment by considering the case of intersector interference with joint processing of the three sector at each cell site in order to avoid signaling schemes relying on the strength of interference we further introduce the notion of topologically robust schemes which are able to guarantee a minimum rate or dofs irrespectively of the strength of the interfering links toward this end we design a different decoding order and alignment scheme which is topologically robust and still achieves the same optimum dofs finally we provide a new scheme for the downlink based on local base station cooperation where base stations pass to their neighbors a quantized version of their dirty paper coded signals for the proposed downlink scheme we can prove a dofs duality result showing that for an appropriate choice of the precoding order and of the alignment beamforming vectors it can achieve the same per sector dofs of the corresponding uplink schemes

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

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

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

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

degrees of freedom of interference channels with comp transmission and reception we study the degrees of freedom dof of the k user interference channel with coordinated multipoint comp transmission and reception each message is jointly transmitted by m t successive transmitters and is jointly received by m r successive receivers we refer to this channel as the comp channel with a transmit cooperation order of m t and receive cooperation order of m r since the channel has a total of k transmit antennas and k receive antennas the maximum possible dof is equal to k we show that the comp channel has k dof if and only if m t m r geq k 1 the key idea is that the zero forcing of the interference corresponding to the i rm th message at the decoder of the j rm th message where j ne i can be viewed as a shared responsibility between the m t transmitters carrying the i rm th message and the m r receivers decoding the j rm th message for the general case we derive an outer bound that states that the dof is bounded above by left lceil k m t m r 2 2 right rceil for the special case with only comp transmission i e m r 1 we propose a scheme that can achieve k m t 1 2 dof for all k and conjecture that the result holds true for all k in the proposed coding scheme the m t transmitters carrying each message are used to cancel the interference introduced by this message at the first m t 1 receivers thereby allowing each of these receivers to enjoy 1 dof and asymptotic interference alignment is used to align the interfering signals at each other receiver to occupy half the signal space the achievability proofs are based on the notion of algebraic independence from algebraic geometry

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

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

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

analysis of computer experiments with functional response this article is motivated by a computer experiment conducted for optimizing residual stresses in the machining of metals although kriging is widely used in the analysis of computer experiments it cannot be easily applied to model the residual stresses because they are obtained as a profile the high dimensionality caused by this functional response introduces severe computational challenges in kriging it is well known that if the functional data are observed on a regular grid the computations can be simplified using an application of kronecker products however the case of irregular grid is quite complex in this article we develop a gibbs sampling based expectation maximization algorithm which converts the irregularly spaced data into a regular grid so that the kronecker product based approach can be employed for efficiently fitting a kriging model to the functional data supplementary materials are available online

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

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

joint uplink downlink cooperative interference management with flexible cell associations we study information theoretic models of interference networks that consist of k base station bs mobile terminal mt pairs bs i is connected to the mts with indices in the set i i 1 ldots i l we fix the value of l and study the per user degrees of freedom pudof in large networks we assume that each mt can be associated with n c bss and these associations are determined by a cloud based controller that has a global view of the network an mt has to be associated with a bs for the bs to transmit its message in the downlink or have its decoded message in the uplink we propose pudof inner bounds for arbitrary values of l for the uplink and prove their optimality when only zero forcing schemes are allowed we then introduce new achievable average uplink downlink pudof values and show their optimality for the range when n c leq frac l 2 and when we restrict our attention to zero forcing schemes additionally for the remaining range we characterize the optimal downlink scheme when the uplink optimal associations are used finally we show that the proposed scheme is information theoretically optimal for wyner s linear interference network

shannon theoretic approach to a gaussian cellular multiple access channel we obtain shannon theoretic limits for a very simple cellular multiple access system in our model the received signal at a given cell site is the sum of the signals transmitted from within that cell plus a factor spl alpha 0 spl les spl alpha spl les 1 times the sum of the signals transmitted from the adjacent cells plus ambient gaussian noise although this simple model is scarcely realistic it nevertheless has enough meat so that the results yield considerable insight into the workings of real systems we consider both a one dimensional linear cellular array and the familiar two dimensional hexagonal cellular pattern the discrete time channel is memoryless we assume that n contiguous cells have active transmitters in the one dimensional case and that n sup 2 contiguous cells have active transmitters in the two dimensional case there are k transmitters per cell most of our results are obtained for the limiting case as n spl rarr spl infin the results include the following 1 we define c sub n c spl circ sub n as the largest achievable rate per transmitter in the usual shannon theoretic sense in the one and two dimensional cases respectively assuming that all signals are jointly decoded we find expressions for limn spl rarr spl infin c sub n and limn spl rarr spl infin c spl circ sub n 2 as the interference parameter spl alpha increases from 0 c sub n and c spl circ sub n increase or decrease according to whether the signal to noise ratio is less than or greater than unity 3 optimal performance is attainable using tdma within the cell but using tdma for adjacent cells is distinctly suboptimal 4 we suggest a scheme which does not require joint decoding of all the users and is in many cases close to optimal

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

spatial modeling with spatially varying coefficient processes in many applications the objective is to build regression models to explain a response variable over a region of interest under the assumption that the responses are spatially correlated in nearly all of this work the regression coefficients are assumed to be constant over the region however in some applications coefficients are expected to vary at the local or subregional level here we focus on the local case although parametric modeling of the spatial surface for the coefficient is possible here we argue that it is more natural and flexible to view the surface as a realization from a spatial process we show how such modeling can be formalized in the context of gaussian responses providing attractive interpretation in terms of both random effects and explaining residuals we also offer extensions to generalized linear models and to spatio temporal setting we illustrate both static and dynamic modeling with a dataset that attempts to explain log selling price of single family houses

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

spatial homogeneity pursuit of regression coefficients for large datasets spatial regression models have been widely used to describe the relationship between a response variable and some explanatory variables over a region of interest taking into account the spatial de

fast prediction of deterministic functions using sparse grid experimental designs random field models have been widely employed to develop a predictor of an expensive function based on observations from an experiment the traditional framework for developing a predictor with random field models can fail due to the computational burden it requires this problem is often seen in cases where the input of the expensive function is high dimensional while many previous works have focused on developing an approximative predictor to resolve these issues this article investigates a different solution mechanism we demonstrate that when a general set of designs is employed the resulting predictor is quick to compute and has reasonable accuracy the fast computation of the predictor is made possible through an algorithm proposed by this work this article also demonstrates methods to quickly evaluate the likelihood of the observations and describes some fast maximum likelihood estimates for unknown parameters of the random field the computational savings can be several orders of magnitude whe

smoothing spline anova models for large data sets with bernoulli observations and the randomized gacv we propose the randomized generalized approximate cross validation rangacv method for choosing multiple smoothing parameters in penalized likelihood estimates for bernoulli data the method is intended for application with penalized likelihood smoothing spline anova models in addition we propose a class of approximate numerical methods for solving the penalized likelihood variational problem which in conjunction with the rangacv method allows the application of smoothing spline anova models with bernoulli data to much larger data sets than previously possible these methods are based on choosing an approximating subset of the natural representer basis functions for the variational problem simulation studies with synthetic data including synthetic data mimicking demographic risk factor data sets is used to examine the properties of the method and to compare the approach with the grkpack code of wang 1997c bayesian confidence intervals are obtained for the fits and are shown in the simulation studies to have the across the function property usually claimed for these confidence intervals finally the method is applied to an observational data set from the beaver dam eye study with scientifically interesting results

efficient computation of gaussian process regression for large spatial data sets by patching local gaussian processes this paper develops an efficient computational method for solving a gaussian process gp regression for large spatial data sets using a collection of suitably defined local gp regressions the conventional local gp approach first partitions a domain into multiple non overlapping local regions and then fits an independent gp regression for each local region using the training data belonging to the region two key issues with the local gp are 1 the prediction around the boundary of a local region is not as accurate as the prediction at interior of the local region and 2 two local gp regressions for two neighboring local regions produce different predictions at the boundary of the two regions creating undesirable discontinuity in the prediction we address these issues by constraining the predictions of local gp regressions sharing a common boundary to satisfy the same boundary constraints which in turn are estimated by the data the boundary constrained local gp regressions are solved by a finite element method our approach shows competitive performance when compared with several state of the art methods using two synthetic data sets and three real data sets

generalizations of the stacked bases theorem let h be a subgroup of the free abelian group g in order for there to exist a basis xi iei of g for which h e3iei nixi for suitable nonnegative integers ni it is obviously necessary for g h to be a direct sum of cyclic groups in the 1950 s kaplansky raised the question of whether this condition on g h is sufficient for the existence of such a basis j cohen and h gluck demonstrated in 1970 that the answer is yes their result is known as the stacked bases theorem and it extends the classical and well known invariant factor theorem for finitely generated abelian groups in this paper we develop a theory that contains and in fact generalizes in several directions the stacked bases theorem our work includes a complete classification using numerical invariants of the various free resolutions of any abelian group

bayesian uncertainty quantification for subsurface inversion using a multiscale hierarchical model we consider a bayesian approach to nonlinear inverse problems in which the unknown quantity is a random field spatial or temporal the bayesian approach contains a natural mechanism for regularization in the form of prior information can incorporate information from heterogeneous sources and provide a quantitative assessment of uncertainty in the inverse solution the bayesian setting casts the inverse solution as a posterior probability distribution over the model parameters the karhunen loeve expansion is used for dimension reduction of the random field furthermore we use a hierarchical bayes model to inject multiscale data in the modeling framework in this bayesian framework we show that this inverse problem is well posed by proving that the posterior measure is lipschitz continuous with respect to the data in total variation norm computational challenges in this construction arise from the need for repeated evaluations of the forward model e g in the context of mcmc and are compounded by

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

dynamic multiscale spatiotemporal models for gaussian areal data summary we introduce a new class of dynamic multiscale models for spatiotemporal processes arising from gaussian areal data specifically we use nested geographical structures to decompose the original process into multiscale coefficients which evolve through time following state space equations our approach naturally accommodates data that are observed on irregular grids as well as heteroscedasticity moreover we propose a multiscale spatiotemporal clustering algorithm that facilitates estimation of the nested geographical multiscale structure in addition we present a singular forward filter backward sampler for efficient bayesian estimation our multiscale spatiotemporal methodology decomposes large data analysis problems into many smaller components and thus leads to scalable and highly efficient computational procedures finally we illustrate the utility and flexibility of our dynamic multiscale framework through two spatiotemporal applications the first example considers mortality ratios in the state of missouri whereas the second example examines agricultural production in espirito santo state brazil

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

full scale multi output gaussian process emulator with nonseparable auto covariance functions gaussian process emulator with separable covariance function has been utilized extensively in modeling large computer model outputs the assumption of separability imposes constraints on the emulator and may negatively affect its performance in some applications where separability may not hold we propose a multi output gaussian process emulator with a nonseparable auto covariance function to avoid limitations of using separable emulators in addition to facilitate the computation of nonseparable emulator we introduce a new computational method referred to as the full scale approximation method with block modulating function fsa block approach the fsa block is an effective and accurate covariance approximation method to reduce computations for gaussian process models which applies to both nonseparable and partially separable covariance models we illustrate the effectiveness of our method through simulation studies and compare it with emulators with separable covariances we also apply our method to a real computer code of the carbon capture system

model calibration with censored data abstractthe purpose of model calibration is to make the model predictions closer to reality the classical kennedy o hagan approach is widely used for model calibration which can account for the inadequacy of the computer model while simultaneously estimating the unknown calibration parameters in many applications the phenomenon of censoring occurs when the exact outcome of the physical experiment is not observed but is only known to fall within a certain region in such cases the kennedy o hagan approach cannot be used directly and we propose a method to incorporate the censoring information when performing model calibration the method is applied to study the compression phenomenon of liquid inside a bottle the results show significant improvement over the traditional calibration methods especially when the number of censored observations is large

writing on dirty paper corresp a channel with output y x s z is examined the state s sim n 0 qi and the noise z sim n 0 ni are multivariate gaussian random variables i is the identity matrix the input x in r n satisfies the power constraint l n sum i 1 n x i 2 leq p if s is unknown to both transmitter and receiver then the capacity is frac 1 2 ln 1 p n q nats per channel use however if the state s is known to the encoder the capacity is shown to be c ast frac 1 2 ln 1 p n independent of q this is also the capacity of a standard gaussian channel with signal to noise power ratio p n therefore the state s does not affect the capacity of the channel even though s is unknown to the receiver it is shown that the optimal transmitter adapts its signal to the state s rather than attempting to cancel it

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

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

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

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

tensor product space anova models to deal with the curse of dimensionality in high dimensional nonparametric problems we consider using tensor product space anova models which extend the popular additive models and are able to capture interactions of any order the multivariate function is given an anova decomposition that is it is expressed as a constant plus the sum of functions of one variable main effects plus the sum of functions of two variables two factor interactions and so on we assume the interactions to be in tensor product spaces we show in both regression and white noise settings the optimal rate of convergence for the tps anova model is within a log factor of the one dimensional optimal rate and that the penalized likelihood estimator in tps anova achieves this rate of convergence the quick optimal rate of the tps anova model makes it very preferable in high dimensional function estimation many properties of the tensor product space of sobolev hilbert spaces are also given

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

restricted spatial regression methods implications for inference the issue of spatial confounding between the spatial random effect and the fixed effects in regression analyses has been identified as a concern in the statistical literature multiple authors have

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

joint estimation of quantile planes over arbitrary predictor spaces abstractin spite of the recent surge of interest in quantile regression joint estimation of linear quantile planes remains a great challenge in statistics and econometrics we propose a novel parameterization that characterizes any collection of noncrossing quantile planes over arbitrarily shaped convex predictor domains in any dimension by means of unconstrained scalar vector and function valued parameters statistical models based on this parameterization inherit a fast computation of the likelihood function enabling penalized likelihood or bayesian approaches to model fitting we introduce a complete bayesian methodology by using gaussian process prior distributions on the function valued parameters and develop a robust and efficient markov chain monte carlo parameter estimation the resulting method is shown to offer posterior consistency under mild tail and regularity conditions we present several illustrative examples where the new method is compared against existing approaches and is found to o

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

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

a dimension reduced approach to space time kalman filtering summary many physical biological processes involve variability over both space and time as a result of difficulties caused by large datasets and the modelling of space time and spatiotemporal interactions traditional space time methods are limited in this paper we present an approach to space time prediction that achieves dimension reduction and uses a statistical model that is temporally dynamic and spatially descriptive that is it exploits the unidirectional flow of time in an autoregressive framework and is spatially descriptive in that the autoregressive process is spatially coloured with the inclusion of a measurement equation this formulation naturally leads to the development of a spatio temporal kalman filter that achieves dimension reduction in the analysis of large spatio temporal datasets unlike other recent space time kalman filters our model also allows a nondynamic spatial component the method is applied to a dataset of near surface winds over the topical pacific ocean spatial predictions with this dataset are improved by considering the additional non dynamic spatial process the improvement becomes more pronounced as the signal to noise ratio decreases