Computational Statistics

, Volume 20, Issue 4, pp 623–642

# A comparison of approaches for valid variogram achievement

• Raquel Menezes
• Pilar Garcia-Soidán
• Manuel Febrero-Bande

## Summary

Variogram estimation is a major issue for statistical inference of spatially correlated random variables. Most natural empirical estimators of the variogram cannot be used for this purpose, as they do not achieve the conditional negative-definite property. Typically, this problem’s resolution is split into three stages:empirical variogram estimation;valid model selection; andmodel fitting. To accomplish these tasks, there are several different approaches strongly defended by their authors. Our work’s main purpose was to identify these approaches and compare them based on a numerical study, covering different kind of spatial dependence situations. The comparisons are based on the integrated squared errors of the resulting valid estimators. Additionally, we propose an easily implementable empirical method to compare the main features of the estimated variogram function.

### Keywords

Spatial dependence Empirical variogram Valid model Fitting criteria Non-parametric estimation

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### References

1. Cherry, S., Banfield, J. and Quimby, W. (1996), ‘An evaluation of a nonparametric method of estimating semivariograms of isotropic spatial processes’,Journal of Applied Statistics v.17, 563–586.
2. Christakos, G. (1984), ‘On the problem of permissible covariance and variogram models’,Water Resources Res. 20, 251–265.
3. Cressie, N. (1985), ‘Fitting variogram models by weighted least squares’,Journal of Int. Association for Mathematical Geology 17, n.5, 563–586.
4. Cressie, N. (1993),Statistics for Spatial Data, John Wiley and Sons Inc., New York.Google Scholar
5. Cressie, N. and Hawkins, D. (1980), ‘Robust estimation of the variogram’,Journal of Int. Association for Mathematical Geology 12, n.2, 115–125.
6. Garcia-Soidán, P., Febrero-Bande, M. and Gonzalez-Manteiga, W. (2004), ‘Nonparametric kernel estimation of an isotropic variogram’,J. Statist. Plann. Inference 121, 65–92.
7. Genton, M. (1998a), ‘Highly robust variogram estimation’,Journal of Int. Association for Mathematical Geology 30, n.2, 213–221.
8. Genton, M. (1998), ‘Variogram fitting by generalized least squares using an explicit formula for the covariance structure’,Journal of Int. Association for Mathematical Geology 30, n.4, 323–345.
9. Gorsich, D. and Genton, M. (2000), ‘Variogram model selection via nonparametric derivate estimation’,Journal of Int. Association for Mathematical Geology 32, n.3, 249–270.
10. Gribov, A., Krivoruchko, K. and Ver Hoef, J. (2000), ‘Modified weighted least squares semivariogram and covariance model fitting algorithm’,Stochastic Modeling and Geostatistics. AAPG Computer Applications in Geology 2.Google Scholar
11. Journel, A. and Huijbregts, C. (1978),Mining Geostatistics, Academic Press, London.Google Scholar
12. Kyung-Joon, C. and Shucany, W. (1998), ‘Nonparametric kernel regression estimation near endpoints’,J. Statist. Plann. Inference 66, 289–304.
13. Maglione, D. and Diblasi, A. (2001), ‘Choosing a valid model for the Variogram of an isotropic spatial process’,2001 Annual Conference of Int. Association for Mathematical Geology.Google Scholar
14. Matheron, G. (1963), ‘Principles of geostatistics’,Economic Geology 58, 1246–1266.Google Scholar
15. Ribeiro Jr, P. and Diggle, P. (2001), ‘geoR: A package for geostatistical analysis’,R-NEWS vol1, n.2, ISSN 1609–3631.Google Scholar
16. Shapiro, A. and Botha, J. (1991), ‘Variogram fitting with a general class of conditionally nonnegative definite functions’,Computational Statistics and Data Analysis 11, 87–96.
17. Stein, M. (1998),Interpolation of Spatial Data-Some Theory for Kriging, Springer.Google Scholar
18. Zimmerman, D. and Zimmerman, M. (1991), ‘A comparison of spatial semivariogram estimators and corresponding ordinary kriging predictors’,Technometrics 33, n.l, 77–91.

## Authors and Affiliations

• Raquel Menezes
• 1
• Pilar Garcia-Soidán
• 2
• Manuel Febrero-Bande
• 3
1. 1.Department of Mathematics for Science and TechnologyUniversity of MinhoGuimarãesPortugal
2. 2.Department of Statistics and O.R.University of VigoPontevedraSpain
3. 3.Department of Statistics and O.R.University of Santiago de CompostelaSantiago de CompostelaSpain