Variance modeling for nonstationary spatial processes with temporal replications (original) (raw)

Spatial models for spatial statistics: some unification

Journal of Vegetation Science, 1993

AbstracL A general statistical framework is proposed for comparing linear models of spatial process and pattern. A spatial linear model for nested analysis of variance can be based on either fixed effects or random effects. originally used a fixed effects model, but there are also examples of random effects models in Lhe soil science literature. Assuming imrinsic stationarity for a linear model, the expectations of a spatial nested ANOVA lllld (wo teon local variance (lTLV, Hill 1973) are funclions of the variogram, and several examples are given. Paired quadrat variance (PQV. Ludwig & Goodall 1978) is a variogram estimator which can be used 10 approximate TIl..V. and we provide an example from ecological data. BOIh nested ANOVA and TILV can be seen as weighted lag-I variogram estimators that are functions of support, rather than distance. We show that there are two unbiased estimators for the variogram under aggregation, and computer simulation shows that the estimator with smaller variance depends on Ihe process autocorrelation.

Dynamical non-Gaussian modelling of spatial processes

2021

Spatio-temporal processes in environmental applications are often assumed to follow a Gaussian model, possibly after some transformation. However, heterogeneity in space and time might have a pattern that will not be accommodated by transforming the data. In this scenario, modelling the variance laws is an appealing alternative. This work adds flexibility to the usual Multivariate Dynamic Gaussian model by defining the process as a scale mixture between a Gaussian and log-Gaussian processes. The scale is represented by a process varying smoothly over space and time which is allowed to depend on covariates. State-space equations define the dynamics over time for both mean and variance processes resulting in feasible inference and prediction. Analysis of artificial datasets show that the parameters are identifiable and simpler models are well recovered by the general proposed model. The analyses of two important environmental processes, maximum temperature and maximum ozone, illustrat...

Developments in the Modelling of Nonstationary Spatial Covariance Structure from Space-Time Monitoring Data

1997

This paper presents the most recent methodological developments for an approachto modelling nonstationary spatial covariance structure through deformations ofthe geographic coordinate system that was first introduced in a technical report 10years ago (Sampson, 1986).We address primarily the problem of estimating the spatial covariance structurein levels of an environmental process at arbitrary locations (both monitored andunmonitored), based on records from N point monitoring sites x 1 ;...

Spatial process modelling for univariate and multivariate dynamic spatial data

Environmetrics, 2005

There is a considerable literature in spatiotemporal modelling. The approach adopted here applies to the setting where space is viewed as continuous but time is taken to be discrete. We view the data as a time series of spatial processes and work in the setting of dynamic models, achieving a class of dynamic models for such data. We seek rich, flexible, easy-to-specify, easy-to-interpret, computationally tractable specifications which allow very general mean structures and also non-stationary association structures. Our modelling contributions are as follows. In the case where univariate data are collected at the spatial locations, we propose the use of a spatiotemporally varying coefficient form. In the case where multivariate data are collected at the locations, we need to capture associations among measurements at a given location and time as well as dependence across space and time. We propose the use of suitable multivariate spatial process models developed through coregionalization. We adopt a Bayesian inference framework. The resulting posterior and predictive inference enables summaries in the form of tables and maps, which help to reveal the nature of the spatiotemporal behaviour as well as the associated uncertainty. We illuminate various computational issues and then apply our models to the analysis of climate data obtained from the National Center for Atmospheric Research to analyze precipitation and temperature measurements obtained in Colorado in 1997.

A Generalized Convolution Model for Multivariate Nonstationary Spatial Processes

Statistica Sinica

We propose a flexible class of nonstationary stochastic models for mul-tivariate spatial data. The method is based on convolutions of spatially varying covariance kernels and produces mathematically valid covariance structures. This method generalizes the convolution approach suggested by Majumdar and Gelfand (2007) to extend multivariate spatial covariance functions to the nonstationary case. A Bayesian method for estimation of the parameters in the covariance model based on a Gibbs sampler is proposed, then applied to simulated data. Model comparison is performed with the coregionalization model of Wackernagel (2003) that uses a stationary bivariate model. Based on posterior prediction results, the performance of our model appears to be considerably better.

GENERATING INTER-CORRELATED OBSERVATIONS UNDER A SPECIFIED SPATIAL MODEL

he requirement to generate this random process needs only to define the variance-covariance matrix of the random process. Since the random process is defined in three dimensional space, then we can use a spatial model. One of the spatial model to define the random process is in the form of variogram, which is a function of distance between pairs of observations. The variance-covariance matrix may be determined in relation with two other properties, those are correlogram and covariogram. The simulation process was started by generating a random points within a particular shape of region. The locations are uniformly distributed within the region. Lets V is a variance-covariance matrix of the random process Y[L]. The random process Y[L] may be defined by the semivariogram model γ(d ij ). The d ij is a Cartesian distance between two different individual within domain D of boundary B. The distribution-based approaches can be applied to generate random observations using Choleski decomposition.

“The Problem of Spatial Autocorrelation” and Local Spatial Statistics

Geographical Analysis, 2009

This article examines the relationship between spatial dependency and spatial heterogeneity, two properties unique to spatial data. The property of spatial dependence has led to a large body of research into spatial autocorrelation and also, largely independently, into geostatistics. The property of spatial heterogeneity has led to a growing awareness of the limitation of global statistics and the value of local statistics and local statistical models. The article concludes with a discussion of how the two properties can be accommodated within the same modelling framework.

Computational Issues in Fitting Spatial Deformation Models for Heterogeneous Spatial Correlation

Environmental monitoring networks are recording pollutant levels, weather, and a myriad of other factors. It is often of interest to estimate these values at locations where records are not available. Many spatial estimation procedures rely on spatial covariance models. Assumptions of spatial isotropy or stationarity may be violated due to factors such as topography and local emissions structures. In this paper we discuss computational issues for a heterogeneous (spatially non-stationary) model for spatial correlations between point monitoring sites. The modeling procedure involves deforming the geographic space into a new space (D-space) where inter-site correlations depend only on distances. Correlations between unmonitored sites are then estimated as a function of distance in the D-space. The estimation of the D-space locations of the monitoring sites, and of the parameters of the isotropic D-space variogram model is a di cult multidimensional problem. The dimensionality increases with the number of monitoring sites. We use examples to review the modeling approach and illustrate computational complexities. We indicate directions for future work necessary for modeling massive data sets.

On the Validity and Identifiability of Spatial Deformation Models for Heterogeneous Spatial Correlation Structure

Many environmental processes are heterogeneous in space (spatially non-stationary), due to factors such as topography, local pollutant emissions, and meteorology. Much of the commonly used spatial statistical methodology depends on simplifying assumptions such as spatial isotropy. Violations of these assumptions can cause problems, including incorrect error assessment of spatial estimates. This paper demonstrates important properties of the spatial deformation model of and for heterogeneous anisotropic spatial correlation structure.