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Spatial Models
Published in Virgilio Gómez-Rubio, Bayesian Inference with INLA, 2020
The sill and range of the variogram are very similar to the estimates obtained for the Matérn covariance computed using SPDE. The nugget effect in the variogram can be regarded as a measurement error or the variance in the Gaussian likelihood. A rough comparison can be done between the sill of the nugget effect in the variogram and the inverse of the mode of the precision of the Gaussian likelihood: ## [1] 13.08
Functional regression concurrent model with spatially correlated errors: application to rainfall ground validation
Published in Journal of Applied Statistics, 2019
Johann Ospína-Galindez, Ramón Giraldo, Mercedes Andrade-Bejarano
The modeling of spatial functional data (SFD) has had an increasing development in the last years [7,18,27]. In this context, the estimation of the spatial dependence structure is crucial. Similarly as occuring in univariate and multivariate spatial data analyses, the nature of the spatial domain allows to classify SFD as geostatistical functional data (GFD), functional point patterns or functional areal data [7]. Specifically GFD appear when this domain is a fixed subset of n sites (7]. In a geostatistical analysis of functional data [12], the cornerstone is the estimation, under some conditions of stationarity and isotropy, of the trace-variogram function [12]. This one can be seen as the extension to a functional scenario of the classical variogram function [31], widely used in univariate geostatistics to estimate the spatial correlation of the random field under study. In Equations (8) and (9), we define this function and its estimation method for the particular case of the problem treated. A more general review of its properties can be done in [14].
Linking biofilm spatial structure to real-time microscopic oxygen decay imaging
Published in Biofouling, 2018
S. Rubol, A. Freixa, X. Sanchez-Vila, A. M. Romaní
The CLSM images were interpreted by variography of the binary images to determine selected spatial characteristic sizes and statistical parameters of the biofilm compartments analyzed. The variogram associated to a stochastic process is a powerful tool to describe how the variable of interest correlates as a function of distance. The sample variogram is then fitted by one of the standard stationary models. In the analyses, the number of structures required to model the experimental variograms of the 10× images for algae, cyanobacteria, and EPS was studied. For each of them, a fitted range was obtained (ie the distance beyond which data are not spatially correlated), equivalent to the average size of the microbial aggregates for each component; also, each structure displays a sill, that represents the fraction of the total variance explained.
Efficient design of geographically-defined clusters with spatial autocorrelation
Published in Journal of Applied Statistics, 2022
We can consider the effective sample size when the covariance between observations is instead determined by the spatial autocorrelation. We adopt the standard geostatistical assumptions that each observation with the cluster is a point sample measured with error from a latent continuous process across the area of interest [10]. Under these assumptions, the variogram is a widely used tool for estimating the relationship between distance between sampled locations and the degree of spatial autocorrelation [7]. In particular, it is defined as Y at location vector x and d is a spatial ‘lag’ or distance. The Euclidean distance is frequently used for these applications Chilès and Delfiner [7], which we assume throughout, although other measures could be used such as travel time, Manhattan distance, or kernal distance (e.g. [34]). The empirical variogram can be defined by three parameters: the nugget, range, and sill. The sill (Y, the range r is the smallest value of d at which 17] describes, the variogram form of Equation (1) is given by: i and j, and