# Kriging Software VERIFIED

The Kriging function implemented in XLSTAT-R allows you to create gstat objects, generate a variogram model and fit a Variogram model to a sample Variogram. Three types of kriging can be applied: the ordinary, the simple and the universal kriging. This XLSTAT-R function calls the gstat, vgm and fit.variogram functions from the gstat package in R (Edzer Pebesma, Benedikt Graeler).

## Kriging Software

I have a point dataset which I'd like to Krige, ideally using an open-source software package. If possible, I'd also like to choose the semi-variogram model during the process to improve the estimation.

GSLIB (Geostatistical Software Library) is top-notch file/command-driven software developed from Stanford University and released in the 1990s, with some maintenance last decade. The source code can be freely downloaded and compiled on Linux/Windows using a Fortran compiler. There are online resources and a book available.

Kriging uses variograms to compute optimal interpolations with known variance. Cokriging uses variograms for both a primary attribute and a more heavily sampled covariate attribute to interpolate the primary data. GS+ provides Kriging methods that fit various needs:

Kriging will in general not be more effective than simpler methods of interpolation if there is little spatial autocorrelation among the sampled data points (that is, if the values do not co-vary in space). If there is at least moderate spatial autocorrelation, however, kriging can be a helpful method to preserve spatial variability that would be lost using a simpler method (for an example, see Auchincloss 2007, below).

The two main assumptions for kriging to provide best linear unbiased prediction are those of stationarity and isotropy, though there are various forms and methods of kriging that allow the strictest form of each of these assumptions to be relaxed.

Ordinary kriging, for which the assumption of stationarity (that the mean and variance of the values is constant across the spatial field) must be assumed. This is one of the simplest forms of kriging, but the stationarity assumption is not often met in applications relevant to environmental health, such as air pollution distributions.

Cokriging, in which additional observed variables (which are often correlated with each other and the variable of interest) are used to enhance the precision of the interpolation of the variable of interest at each location.

Similarly, the assumptions of the kriging model (e.g. that of second-order stationarity) may be difficult to meet in the context of many environmental exposures. Some newer methods (e.g. Bayesian approaches) have thus been developed to try and surmount these obstacles.

In general, the accuracy of interpolation by kriging will be limited if the number of sampled observations is small, the data is limited in spatial scope, or the data are in fact not amply spatially correlated. In these cases, a sample variogram is hard to generate, and methods such as land-use regression may prove preferable to kriging for spatial prediction.

Waller, Lance A ., and Carol A. Gotway. Applied spatial statistics for public health data. Vol. 368. John Wiley & Sons, 2004. Available here. (This textbook is an excellent resource that describes the application of spatial statistical methods to public health data. It serves as an introduction to epidemiologic research and GIS. All chapters are very useful and several topics were covered. For example, fundamental principals of epidemiologic research are described as well as management, mapping and reporting of spatial data. However, Chapter 8 discusses spatial interpolation in general and section 8.3, focuses on kriging and describes a few of the kriging methods.)

Zimmerman, D., Pavlik, C., Ruggles, A., & Armstrong, M. P. (1999). An experimental comparison of ordinary and universal kriging and inverse distance weighting.Mathematical Geology, 31(4), 375-390. (This article compares different interpolation methods (ordinary kriging, universal kriging, and inverse squared-distance weighting) using simulated data. The effects of the interpolation methods were tested for statistical significance).

Ali, M., Goovaerts, P., Nazia, N., Haq, M.Z., Yunus, M., Emch, M. Application of Poisson kriging to the mapping of cholera and dysentery incidence in an endemic area of Bangladesh. Int J Health Geogr 5:45 (2006). (Aside from exposure estimation, another common application of kriging in the health sciences is in modeling disease incidence. Here Poisson kriging is used to map cholera and dysentery in Bangladesh).

Goovaerts,P., & Gebreab, S. (2008). How does Poisson kriging compare to the popular BYM model for mapping disease risks?. International journal of health geographics, 7(1), 6. (The article compares Bayesian spatial models with Poisson kriging first using lung and cervix cancer mortality rates from 118 counties and then using simulated data.)

Spatial Analysis Techniques in R taught by Professor David Unwin. The course is offered by Statictics.com and its main objective is to introduce R for geographic information analysis using R. it covers point pattern analysis, lattice objects and geostatistical data (including but not limited to kriging).

Applied Spatial Statistics at the Yale School of Forestry and Environmental Studies (Topics include spatial sampling, visualizing spatial data, quantifying spatial association and autocorrelation, interpolation methods, fittingvariograms, kriging, and related modeling techniques for spatially correlated data.)

Applications of the program include generating yield maps (Whelan et al., 1996), interpolation of digital elevation models, and assessment of topsoil salinity problems (Walter et al., 1999). The program also allows conventional kriging with a whole area variogram, with options for manual adjustment and fitting of the whole-area variogram. The user-friendly interface permits the creation of a field boundary and generation of an interpolation grid. Future development will incorporate local regression kriging into the program with more advanced guidance for users incorporated with the interface.

VESPER is a PC-Windows program developed by the Australian Centre for Precision Agriculture (ACPA) for spatial prediction that is capable of performing kriging with local variograms (Haas, 1990). Kriging with local variograms involves searching for the closest neighbourhood for each prediction site, estimating the variogram from the neighbourhood, fitting a variogram model to the data and predicting the value and its uncertainty. The local variogram is modelled in the program by fitting a variogram model automatically through the nonlinear least-squares method. Several variogram models are available, namely spherical, exponential, Gaussian and linear with sill. Punctual and block kriging is available as interpolation options. This program adapts itself spatially in the presence of distinct differences in local structure over the whole field.

This article briefly discusses statistical interpolation features and then provides some details about the empirical Bayesian kriging (EBK) model implemented in ArcGIS 10.1 Geostatistical Analyst. Extensive testing using a large variety of data showed that EBK is a reliable automatic interpolator. This kriging model is also available as a geoprocessing tool that can be used in ModelBuilder and Python scripts.

When kriging predictors are applied to the analysis of radioactive contamination, they can answer questions such as, What is the probability that food contamination exceeds the radioecological standard at the specified location? and provide estimates of average and total contamination in specified areas.

There are some statistical assumptions behind kriging. The main assumption is stationarity (spatial homogeneity). If data is stationary, the data mean and the semivariogram are the same at all locations in the data extent. If this assumption is held, just a few kriging model parameters have to be estimated from the data to make optimal predictions and valid statistical inferences.

If the data distribution is Gaussian, the best predictor is one that uses a linear combination of the nearby data values. For other distributions, however, the best predictor is often nonlinear and, therefore, more complex. The data can be transformed to follow a Gaussian distribution. Then it is possible to accurately back transform kriging predictions to the original data scale, which can be done in ArcGIS Geostatistical Analyst.

Classical kriging also assumes that the estimated semivariogram is the true semivariogram of the observed data. This means the data was generated from Gaussian distribution with the correlation structure defined by the estimated semivariogram. This is a very strong assumption, and it rarely holds true in practice. Hence, action should be taken to make the statistical model more realistic.

EBK differs from classical kriging methods by accounting for the error introduced by estimating the semivariogram model. This is done by estimating, and then using, many semivariogram models rather than a single semivariogram. This process entails the following steps:

The default kriging model in EBK is called the intrinsic random function of order 0, and the spatial correlation model is the power model where b, c, and α (the allowed value of the power value α is between 0 and 2) are the model parameters. This correlation model corresponds to fractional Brownian motion, also known as the random walk process. It consists of steps in a random direction and filters out a moderate trend in the data.

Empirical Bayesian kriging as implemented in the ArcGIS 10.1 Geostatistical Analyst extension provides both a straightforward and robust method of data interpolation. For more information on using EBK, see the online help for the ArcGIS Geostatistical Analyst extension. To learn more about spatial statistics, read Spatial Statistical Data Analysis for GIS Users published by Esri Press. 041b061a72