objective analyses and mapping techniques for rainfall fields an objective comparison many methods have been proposed and applied individually to rainfall fields in order to estimate point or average values at ungauged sites this paper reviews some of the usual as well as some of the more recent techniques they are classified according to their fundamental principles optimality criteria and practical aspects some theoretical developments mostly for the statistically based techniques are offered and the practical aspects are mainly directed to the particular case of rainfall fields a case study is presented for a region with particularly high rainfall variability providing a good sample of events for testing the sensitivity of the estimation to the different methods used the identification of statistical structure functions required by certain methods using regionalized variable concepts is discussed in detail for the case considered method evaluation is based on a classical validation technique that divides the available stations into two separate networks i e a base network is used to perform estimations at the stations of a test network for a set of independent storm events careful attention is paid to result interpretation and visualisation finally conclusions are drawn as to which method should be implemented taking into account cost efficiency considerations

decomposition of besov and triebel lizorkin spaces on the sphere abstract a discrete system of almost exponentially localized elements needlets on the n dimensional unit sphere s n is constructed it shown that the needlet system can be used for decomposition of besov and triebel lizorkin spaces on the sphere as an application of besov spaces on s n a jackson estimate for nonlinear m term approximation from the needlet system is obtained

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

a new decoder for the optimum recovery of nonadditive watermarks watermark detection i e the detection of an invisible signal hidden within an image for copyright protection or data authentication has classically been tackled by means of correlation based techniques nevertheless when watermark embedding does not obey an additive rule or when the features the watermark is superimposed on do not follow a gaussian pdf correlation based decoding is not the optimum choice a new decoding algorithm is presented here which is optimum for nonadditive watermarks embedded in the magnitude of a set of full frame dft coefficients of the host image by relying on statistical decision theory the structure of the optimum is derived according to the neyman pearson criterion thus permitting to minimize the missed detection probability subject to a given false detection rate the validity of the optimum decoder has been tested thoroughly to assess the improvement it permits to achieve from a robustness perspective the results we obtained confirm the superiority of the novel algorithm with respect to classical correlation based decoding

analyzing nonstationary spatial data using piecewise gaussian processes in many problems in geostatistics the response variable of interest is strongly related to the underlying geology of the spatial location in these situations there is often little correlation in the responses found in different rock strata so the underlying covariance structure shows sharp changes at the boundaries of the rock types conventional stationary and nonstationary spatial methods are inappropriate because they typically assume that the covariance between points is a smooth function of distance in this article we propose a generic method for the analysis of spatial data with sharp changes in the underlying covariance structure our method works by automatically decomposing the spatial domain into disjoint regions within which the process is assumed to be stationary but the data are assumed independent across regions uncertainty in the number of disjoint regions their shapes and the model within regions is dealt with in a fully bayesian fashion we illustrate our approach on a previously

bayesian spatial prediction of random space time fields with application to mapping pm2 5 exposure this article presents a multivariate spatial prediction methodology in a bayesian framework the method is especially suited for use in environmetrics where vector valued responses are observed at a small set of ambient monitoring stations gauged sites at successive time points however the stations may have varying start up times so that the data have a staircase pattern monotone pattern in the terminology of rubin and shaffer the lowest step corresponds to the newest station in the monitoring network we base our approach on a hierarchical bayes prior involving a gaussian generalized inverted wishart model for given hyperparameters we derive the predictive distribution for currently gauged sites at times before their start up when no measurements were taken the resulting predictive distribution is a matric t distribution with appropriate covariance parameters and degrees of freedom we estimate the hyperparameters using the method of moments mom as an easy to implement alternative to t

nested sampling algorithm for subsurface flow model selection uncertainty quantification and nonlinear calibration 1 calibration of subsurface flow models is an essential step for managing ground water aquifers designing of contaminant remediation plans and maximizing recovery from hydrocarbon reservoirs we investigate an efficient sampling algorithm known as nested sampling ns which can simultaneously sample the posterior distribution for uncertainty quantification and estimate the bayesian evidence for model selection model selection statistics such as the bayesian evidence are needed to choose or assign different weights to different models of different levels of complexities in this work we report the first successful application of nested sampling for calibration of several nonlinear subsurface flow problems the estimated bayesian evidence by the ns algorithm is used to weight different parameterizations of the subsurface flow models prior model selection the results of the numerical evaluation implicitly enforced occam s razor where simpler models with fewer number of parameters are favored over complex models the proper level of model complexity was automatically determined based on the information content of the calibration data and the data mismatch of the calibrated model

bayesian estimation and experimental design in linear regression models estimation and design as a bayesian decision problem choice of a prior distribution conjugate prior distributions bayes estimation of the regression parameter optimality and robustness of the bayes estimator bayesian interpretation of estimators using non bayesian prior knowledge bayes estimation in case of prior ignorance further problems the design problem for the linear bayes estimator characterization of optimal designs construction of optimal continuous designs construction of exact optimal designs

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

bayesian hierarchical spatially correlated functional data analysis with application to colon carcinogenesis in this article we present new methods to analyze data from an experiment using rodent models to investigate the role of p27 an important cell cycle mediator in early colon carcinogenesis the responses modeled here are essentially functions nested within a two stage hierarchy standard functional data analysis literature focuses on a single stage of hierarchy and conditionally independent functions with near white noise however in our experiment there is substantial biological motivation for the existence of spatial correlation among the functions which arise from the locations of biological structures called colonic crypts this possible functional correlation is a phenomenon we term crypt signaling thus as a point of general methodology we require an analysis that allows for functions to be correlated at the deepest level of the hierarchy our approach is fully bayesian and uses markov chain monte carlo methods for inference and estimation analysis of this data set gives new insights into the structure of p27 expression in early colon carcinogenesis and suggests the existence of significant crypt signaling our methodology uses regression splines and because of the hierarchical nature of the data dimension reduction of the covariance matrix of the spline coefficients is important we suggest simple methods for overcoming this problem

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

bayesian geostatistical design task driven optimal site investigation when the geostatistical model is uncertain 1 geostatistical optimal design optimizes subsurface exploration for maximum information toward task specific prediction goals until recently most geostatistical design studies have assumed that the geostatistical description i e the mean trends covariance models and their parameters is given a priori this contradicts as emphasized by rubin and dagan 1987a the fact that only few or even no data at all offer support for such assumptions prior to the bulk of exploration effort we believe that geostatistical design should 1 avoid unjustified a priori assumptions on the geostatistical description 2 instead reduce geostatistical model uncertainty as secondary design objective 3 rate this secondary objective optimal for the overall prediction goal and 4 be robust even under inaccurate geostatistical assumptions bayesian geostatistical design follows these guidelines by considering uncertain covariance model parameters we transfer this concept from kriging like applications to geostatistical inverse problems we also deem it inappropriate to consider parametric uncertainty only within a single covariance model the matern family of covariance functions has an additional shape parameter controlling model shape by a parameter converts covariance model selection to parameter identification and resembles bayesian model averaging over a continuous spectrum of covariance models this is appealing since it generalizes bayesian model averaging from a finite number to an infinite number of models we illustrate how our approach fulfills the above four guidelines in a series of synthetic test cases the underlying scenarios are to minimize the prediction variance of 1 contaminant concentration or 2 arrival time at an ecologically sensitive location by optimal placement of hydraulic head and log conductivity measurements results highlight how both the impact of geostatistical model uncertainty and the sampling network design vary according to the choice of objective function

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

aberrant crypt foci and semiparametric modeling of correlated binary data motivated by the spatial modeling of aberrant crypt foci acf in colon carcinogenesis we consider binary data with probabilities modeled as the sum of a nonparametric mean plus a latent gaussian spatial process that accounts for short range dependencies the mean is modeled in a general way using regression splines the mean function can be viewed as a fixed effect and is estimated with a penalty for regularization with the latent process viewed as another random effect the model becomes a generalized linear mixed model in our motivating data set and other applications the sample size is too large to easily accommodate maximum likelihood or restricted maximum likelihood estimation reml so pairwise likelihood a special case of composite likelihood is used instead we develop an asymptotic theory for models that are sufficiently general to be used in a wide variety of applications including but not limited to the problem that motivated this work the splines have penalty parameters that must converge to zero asymptotically we derive theory for this along with a data driven method for selecting the penalty parameter a method that is shown in simulations to improve greatly upon standard devices such as likelihood crossvalidation finally we apply the methods to the data from our experiment acf we discover an unexpected location for peak formation of acf

modeling functional data with spatially heterogeneous shape characteristics summary we propose a novel class of models for functional data exhibiting skewness or other shape characteristics that vary with spatial or temporal location we use copulas so that the marginal distributions and the dependence structure can be modeled independently dependence is modeled with a gaussian or t copula so that there is an underlying latent gaussian process we model the marginal distributions using the skew t family the mean variance and shape parameters are modeled nonparametrically as functions of location a computationally tractable inferential framework for estimating heterogeneous asymmetric or heavy tailed marginal distributions is introduced this framework provides a new set of tools for increasingly complex data collected in medical and public health studies our methods were motivated by and are illustrated with a state of the art study of neuronal tracts in multiple sclerosis patients and healthy controls using the tools we have developed we were able to find those locations along the tract most affected by the disease however our methods are general and highly relevant to many functional data sets in addition to the application to one dimensional tract profiles illustrated here higher dimensional extensions of the methodology could have direct applications to other biological data including functional and structural magnetic resonance imaging mri

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

quantifying uncertainty on pareto fronts with gaussian process conditional simulations multi objective optimization algorithms aim at finding pareto optimal solutions recovering pareto fronts or pareto sets from a limited number of function evaluations are challenging problems a popular approach in the case of expensive to evaluate functions is to appeal to metamodels kriging has been shown efficient as a base for sequential multi objective optimization notably through infill sampling criteria balancing exploitation and exploration such as the expected hypervolume improvement here we consider kriging metamodels not only for selecting new points but as a tool for estimating the whole pareto front and quantifying how much uncertainty remains on it at any stage of kriging based multi objective optimization algorithms our approach relies on the gaussian process interpretation of kriging and bases upon conditional simulations using concepts from random set theory we propose to adapt the vorob ev expectation and deviation to capture the variability of the set of non dominated points numerical experiments illustrate the potential of the proposed workflow and it is shown on examples how gaussian process simulations and the estimated vorob ev deviation can be used to monitor the ability of kriging based multi objective optimization algorithms to accurately learn the pareto front

non gaussian bayesian geostatistical modeling sampling models for geostatistical data are usually based on gaussian processes however real data often display non gaussian features such as heavy tails in this article we propose a more flexible class of sampling models we start from the spatial linear model that has a spatial trend plus a stationary gaussian error process we extend the sampling model to non gaussianity by including a scale parameter at each location we make sure that we obtain a valid stochastic process the scale parameters are spatially correlated to ensure that the process is mean square continuous we derive expressions for the moments and the kurtosis of the process this more general stochastic process allows us to accommodate and identify observations that would be outliers under a gaussian sampling process for the spatial correlation structure we adopt the flexible matern class with unknown smoothness parameter furthermore a nugget effect is included in the model bayesian inference posterior and predictive is perf

analysis of the spatial structure of properties of selected aquifers data from 31 regional aquifers are analyzed to identify the horizontal spatial correlation structure of transmissivity hydraulic conductivity and storage coefficient three parameters are estimated the variance of small scale variability or nugget and the variance and integral scale range of an exponential covariance function the assumption of normality of the aforementioned geohydrologic parameters and their logarithmic transformations is also examined results are obtained which indicate that the logarithms of these properties generally pass normality tests

robust design optimization with quadratic loss derived from gaussian process models gaussian process models which include the class of linear models are widely employed for modeling responses as a function of control or noise factors using these models the average loss at control factor settings can be estimated and compared however robust design optimization is often performed based on the expected quadratic loss computed as if the posterior mean were the true response function this can give very misleading results we propose an expected quadratic loss criterion derived by taking expectation with respect to the noise factors and the posterior predictive process approximate but highly accurate credible intervals for the average quadratic loss are constructed via the numerical inversion of the lugannani rice saddlepoint approximation the accuracy of the lugannani rice intervals is compared with intervals constructed via moment matching techniques on real data this article has supplementary materials that are available online

random walk kernels and learning curves for gaussian process regression on random graphs we consider learning on graphs guided by kernels that encode similarity between vertices our focus is on random walk kernels the analogues of squared exponential kernels in euclidean spaces we show that on large locally treelike graphs these have some counter intuitive properties specifically in the limit of large kernel lengthscales we consider using these kernels as covariance functions of gaussian processes in this situation one typically scales the prior globally to normalise the average of the prior variance across vertices we demonstrate that in contrast to the euclidean case this generically leads to significant variation in the prior variance across vertices which is undesirable from a probabilistic modelling point of view we suggest the random walk kernel should be normalised locally so that each vertex has the same prior variance and analyse the consequences of this by studying learning curves for gaussian process regression numerical calculations as well as novel theoretical predictions for the learning curves using belief propagation show that one obtains distinctly different probabilistic models depending on the choice of normalisation our method for predicting the learning curves using belief propagation is significantly more accurate than previous approximations and should become exact in the limit of large random graphs

fused adaptive lasso for spatial and temporal quantile function estimation quantile functions are important in characterizing the entire probability distribution of a random variable especially when the tail of a skewed distribution is of interest this article introduces new quantile function estimators for spatial and temporal data with a fused adaptive lasso penalty to accommodate the dependence in space and time this method penalizes the difference among neighboring quantiles hence it is desirable for applications with features ordered in time or space without replicated observations the theoretical properties are investigated and the performances of the proposed methods are evaluated by simulations the proposed method is applied to particulate matter pm data from the community multiscale air quality cmaq model to characterize the upper quantiles which are crucial for studying spatial association between pm concentrations and adverse human health effects

nested polynomial trends for the improvement of gaussian process based predictors the role of simulation has kept increasing for the sensitivity analysis and the uncertainty quantification of complex systems such numerical procedures are generally based on the processing of a huge amount of code evaluations when the computational cost associated with one particular evaluation of the code is high such direct approaches based on the computer code only can be not affordable surrogate models have therefore to be introduced to interpolate the information given by a fixed set of code evaluations to the whole input space when confronted to deterministic mappings the gaussian process based regression gpr or kriging presents a good compromise between complexity efficiency and error control such a method considers the quantity of interest of the system as a particular realization of a gaussian stochastic process which mean and covariance functions have to be identified from the available code evaluations in this context this work proposes an innovative parameterization of this mean function which is based on the composition of two polynomials this approach is particularly relevant for the approximation of strongly non linear quantities of interest from very little information after presenting the theoretical basis of this method this work compares its efficiency to alternative approaches on a series of examples

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

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