Geostatistical Analysis Through Spectral Techniques: Some Words of Caution

Tapering is a technique proposed in order to avoid the so-called edge-effects in spectral estimation. Recent literature has emphasised the importance of tapering for spectral methods applied to the analysis of spatial dependence. In this work we show, through applications and an extensive simulation study, that tapering can be very dangerous if it is not used with caution. An interesting aspect of spectral estimation arises in the presence of a nugget effect in the spatial structure.     

Geoadditive models

Summary. A study into geographical variability of reproductive health outcomes (e.g. birth weight) in Upper Cape Cod, Massachusetts, USA, benefits from geostatistical mapping or kriging . However, also observed are some continuous covariates (e.g. maternal age) that exhibit pronounced non-linear relationships with the response variable. To account for such effects properly we merge kriging with additive models to obtain what we call geoadditive models . The merging becomes effortless by expressing both as linear mixed models. The resulting mixed model representation for the geoadditive model allows for fitting and diagnosis using standard methodology and software.     

Residuals for the linear model with general covariance structure

A general theory is presented for residuals from the general linear model with correlated errors. It is demonstrated that there are two fundamental types of residual associated with this model, referred to here as the marginal and the conditional residual. These measure respectively the distance to the global aspects of the model as represented by the expected value and the local aspects as represented by the conditional expected value. These residuals may be multivariate. Some important dualities are developed which have simple implications for diagnostics. The results are illustrated by reference to model diagnostics in time series and in classical multivariate analysis with independent cases.     

Optimal predictive design augmentation for spatial generalised linear mixed models

A typical model for geostatistical data when the observations are counts is the spatial generalised linear mixed model. We present a criterion for optimal sampling design under this framework which aims to minimise the error in the prediction of the underlying spatial random effects. The proposed criterion is derived by performing an asymptotic expansion to the conditional prediction variance. We argue that the mean of the spatial process needs to be taken into account in the construction of the predictive design, which we demonstrate through a simulation study where we compare the proposed criterion against the widely used space-filling design. Furthermore, our results are applied to the Norway precipitation data and the rhizoctonia disease data.     

Modified linear projection for large spatial datasets

Recent developments in engineering techniques for spatial data collection such as geographic information systems have resulted in an increasing need for methods to analyze large spatial datasets. These sorts of datasets can be found in various fields of the natural and social sciences. However, model fitting and spatial prediction using these large spatial datasets are impractically time-consuming, because of the necessary matrix inversions. Various methods have been developed to deal with this problem, including a reduced rank approach and a sparse matrix approximation. In this article, we propose a modification to an existing reduced rank approach to capture both the large- and small-scale spatial variations effectively. We have used simulated examples and an empirical data analysis to demonstrate that our proposed approach consistently performs well when compared with other methods. In particular, the performance of our new method does not depend on the dependence properties of the spatial covariance functions.     

On spatiotemporal prediction for on-line monitoring data

Spatiotemporal prediction is of interest in many areas of applied statistics, especially in environmental monitoring with on-line data information. At first, this article reviews the approaches for spatiotemporal modeling in the context of stochastic processes and then introduces the new class of spatiotemporal dynamic linear models. Further, the methods for linear spatial data analysis, universal kriging and trend surface prediction, are related to the method of spatial linear Bayesian analysis. The Kalman filter is the preferred method for temporal linear Bayesian inferences. By combining the Kalman filter recursions with the trend surface predictor and universal kriging predictor, the prior and posterior spatiotemporal predictors for the observational process are derived, which form the main result of this article. The problem of spatiotemporal linear prediction in the case of unknown first and second order moments is treated as well.     

Spatial dependence estimation using FFT of biased covariances

One of the main problems in geostatistics is fitting a valid variogram or covariogram model in order to describe the underlying dependence structure in the data. The dependence between observations can be also modeled in the spectral domain, but the traditional methods based on the periodogram as an estimator of the spectral density may present some problems for the spatial case. In this work, we propose an estimation method for the covariogram parameters based on the fast Fourier transform (FFT) of biased covariances. The performance of this estimator for finite samples is compared through a simulation study with other classical methods stated in spatial domain, such as weighted least squares and maximum likelihood, as well as with other spectral estimators. Additionally, an example of application to real data is given.     

Strategies for Fitting Large,Geostatistical Data in MCMC Simulation

Models for geostatistical data introduce spatial dependence in the covariance matrix of location-specific random effects. This is usually defined to be a parametric function of the distances between locations. Bayesian formulations of such models overcome asymptotic inference and estimation problems involved in maximum likelihood-based approaches and can be fitted using Markov chain Monte Carlo (MCMC) simulation. The MCMC implementation, however, requires repeated inversions of the covariance matrix which makes the problem computationally intensive, especially for large number of locations. In the present work, we propose to convert the spatial covariance matrix to a sparse matrix and compare a number of numerical algorithms especially suited within the MCMC framework in order to accelerate large matrix inversion. The algorithms are assessed empirically on simulated datasets of different size and sparsity. We conclude that the band solver applied after ordering the distance matrix reduces the computational time in inverting covariance matrices substantially.