On Stationary Processes in the Plane

Whittle, Peter

doi:10.1093/biomet/41.3-4.434

Cited by 1,289 publications

(559 citation statements)

References 6 publications

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“…Whittle [38,39] demonstrated how spatial covariance models could be computed for variables driven by diffusion from random sources in one, two or three dimensions. In this context we can infer the form of the covariance model for a process from understanding of the process's physical basis.…”

Section: Soil Knowledge In the Random Effects?mentioning

confidence: 99%

“…In some respects this is scientifically weaker than, for example, the selection of a physically-based model for gravimetric data. However, it is reasonable for the soil scientist to argue that the variables we study are often complex, and cannot be accounted for by the relatively simple models that lie behind the gravimetric form of the Cauchy model, or the models proposed by Whittle [38,39] for diffusion processes. In fact, reflection on the comment by Whittle [39] that the good fit of the diffusion model in three dimensions to the yield data of Fairfield Smith [42] could be explained by the dependence of yield on the concentrations of nutrients that diffuse in the 3-D volume of soil illustrates this.…”

Section: Soil Knowledge In the Random Effects?mentioning

confidence: 99%

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Towards soil geostatistics

Lark

2012

Spatial Statistics

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In a brief survey of some issues in the application of geostatistics in soil science it is shown how the recasting of classical geostatistical methods in the linear mixed model (LMM) framework has allowed the more effective integration of soil knowledge (classifications, covariates) with statistical spatial prediction of soil properties. The LMM framework has also allowed the development of models in which the spatial covariance need not be assumed to be stationary. Such models are generally more plausible than stationary ones from a pedological perspective, and when applied to soil data they have been found to give prediction error variances that better describe the uncertainty of predictions at validation sites. Finally consideration is given to how scientific understanding of variable processes in the soil might be used to infer the likely statistical form of the observed soil variation.

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Section: Soil Knowledge In the Random Effects?mentioning

confidence: 99%

Section: Soil Knowledge In the Random Effects?mentioning

confidence: 99%

Towards soil geostatistics

Lark

2012

Spatial Statistics

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“…It was introduced by Lindgren et al in 2011 and has since been extended and applied in various contexts Lindgren, 2011, 2013;Simpson et al, 2012a,b;Cameletti et al, 2013). However, the original idea dates back to the work of Whittle (1954;1963), where it is shown that the solution to the following SPDE (1) is a Gaussian random field with Matérn covariance function. The innovation process W on the right hand side of (1) is spatial Gaussian white noise, and ∆ is the Laplacian.…”

Section: Introductionmentioning

confidence: 99%

Spatial models with explanatory variables in the dependence structure

2014

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Please cite this article as: Ingebrigtsen, R., Lindgren, F., Steinsland, I., Spatial models with explanatory variables in the dependence structure. Spatial Statistics (2013), http://dx.doi.org/10. 1016/j.spasta.2013.06.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractGeostatistical models have traditionally been stationary. However, physical knowledge about underlying spatial processes often require models with nonstationary dependence structures. Thus, there has been an interest in the literature to provide flexible models and computationally efficient methods for non-stationary phenomena. In this work, we demonstrate that the stochastic partial differential equation (SPDE) approach to spatial modelling provides a flexible class of non-stationary models where explanatory variables easily can be included in the dependence structure. In addition, the SPDE approach enables computationally efficient Bayesian inference with integrated nested Laplace approximations (INLA) available through the R-package r-inla. We illustrate the suggested modelling framework with a case study of annual precipitation in southern Norway, and compare a non-stationary model with dependence structure governed by elevation to a stationary model. Further, we use a simulation study to explore the annual precipitation models. We investigate identifiability of model parameters and whether the deviance information criterion (DIC) is able to distinguish datasets from the non-stationary and stationary models.

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“…In 1973, Cliff and Ord (1973) give an general presentation on spatial econometrics models and introduce the STAR (SpaceTime AutoRegressive) and the Generalized Space-Time AuRegressive (GSTAR) models. The literature on spatial models is relatively abundant, we can also cite the Simultaneous AutoRegression model, SAR (Whittle, 1954), the Conditional AutoRegression model, CAR (Bartlett, 1971;Besag, 1974), the moving average model (Haining, 1978) or the unilateral models (Basu and Reinsel, 1993) among others. Spatial models are currently investigated in many research fields like meteorology (Lim et al, 2002), oceanography (Illig, 2006), agronomy (Whittle, 1954;Lambert et al, 2003), geology (Cressie, 1973), epidemiology (Marshall, 1991), image processing (Jain, 1981), econometrics (Anselin, 1988) and many others in which the data of interest are collected across space.…”

Section: Introductionmentioning

confidence: 99%

Statistical properties of the Seasonal Fractionally Integrated Separable Spatial Autoregressive Model

Cissé¹,

Diongue²,

Guégan³

2016

jas

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Abstract. In this paper we introduce a new model called Fractionally Integrated Separable Spatial Autoregressive processes with Seasonality and denoted Seasonal FISSAR. We focus on the class of separable spatial models whose correlation structure can be expressed as a product of correlations. This new modelling allows taking into account the seasonality patterns observed in spatial data. We investigate the properties of this new model providing stationary conditions, some explicit form of the autocovariance function and the spectral density. We also establish the asymptotic behaviour of the spectral density function near the seasonal frequencies. Résumé. On introduit une nouvelle classe de processus appelée Processus autoregressif spatiaux, fractionnaires, intégrés et séparables avec saisonnalité. On considère la classe des modè les spatiaux dont la structure de corrélation peutêtre exprimée comme produit de fonctions de corrélations. Cette nouvelle modé lisation permet de prendre en compte le phénomène de saisonnalité observé dans des données spatiales, bi-dimensionnelles. Nousétudions les propriétés statistiques du modèle proposé telles que les conditions de stationnarité, la fonction d'autocovariance (deux formes) et de la fonction de densité spectrale. Nousétablissons aussi l'approximation asymptotique de la fonction de densité spectrale au niveau des fréquences saisonnières. 902

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On Stationary Processes in the Plane

Cited by 1,289 publications

References 6 publications

Towards soil geostatistics

Towards soil geostatistics

Spatial models with explanatory variables in the dependence structure

Statistical properties of the Seasonal Fractionally Integrated Separable Spatial Autoregressive Model

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