Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30127
Variogram Fitting Based on the Wilcoxon Norm

Authors: Hazem Al-Mofleh, John Daniels, Joseph McKean

Abstract:

Within geostatistics research, effective estimation of the variogram points has been examined, particularly in developing robust alternatives. The parametric fit of these variogram points which eventually defines the kriging weights, however, has not received the same attention from a robust perspective. This paper proposes the use of the non-linear Wilcoxon norm over weighted non-linear least squares as a robust variogram fitting alternative. First, we introduce the concept of variogram estimation and fitting. Then, as an alternative to non-linear weighted least squares, we discuss the non-linear Wilcoxon estimator. Next, the robustness properties of the non-linear Wilcoxon are demonstrated using a contaminated spatial data set. Finally, under simulated conditions, increasing levels of contaminated spatial processes have their variograms points estimated and fit. In the fitting of these variogram points, both non-linear Weighted Least Squares and non-linear Wilcoxon fits are examined for efficiency. At all levels of contamination (including 0%), using a robust estimation and robust fitting procedure, the non-weighted Wilcoxon outperforms weighted Least Squares.

Keywords: Non-Linear Wilcoxon, robust estimation, Variogram estimation.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1127080

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 629

References:


[1] F. R. Hampel, “Robust estimation: A condensed partial survey,” Zeitschrift fr Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 27, no. 2, pp. 87–104, 1998.
[2] P. J. Huber, Robust Statistical Procedures, 2nd ed. Philadelphia: Society for Industrial and Applied Mathematics, 1996.
[3] G. Matheron, Trait de gostatistique applique, ser. M´emoires du Bureau de Recherches G´eologiques et Mini`eres. ´ Editions Technip, 1963, no. v. 2.
[4] N. Cressie and D. M. Hawkins, “Robust estimation of the variogram: I,” Mathematical Geology, vol. 12, no. 2, pp. 115–125, 1980.
[5] N. Cressie, Statistics for Spatial Data. New York: John Wiley and Sons, Ltd, 1991.
[6] M. G. Genton, “Highly robust variogram estimation,” Mathematical Geology, vol. 30, no. 2, pp. 213–221, 1998.
[7] K. V. Mardia and R. J. Marshall, “Maximum likelihood estimation of models for residual covariance in spatial regression,” Biometrika, vol. 71, no. 1, pp. 135–146, 1984.
[8] H. Patterson and R. Thompson, “Recovery of inter-block information when block sizes are unequal,” Biometrika, vol. 58, no. 3, pp. 545–554, 1971.
[9] “Maximum likelihood estimation of components of variance.” Constanta, Romania: 8th International Biometrics Conference, 1974, pp. 197–207.
[10] C. R. Rao, “Minqe theory and its relation to ml and mml estimation of variance components,” Sankhy: The Indian Journal of Statistics, Series B (1960-2002), vol. 41, no. 3/4, pp. 138–153, 1979.
[11] N. Cressie, “Fitting variogram models by weighted least squares,” Journal of the International Association for Mathematical Geology, vol. 17, no. 5, pp. 563–586, 1985.
[12] T. P. Hettmansperger and J. W. McKean, Robust Nonparametric Statistical Methods, 2nd ed. New York: Chapman-Hall, 2011.
[13] J. W. McKean and J. D. Kloke, “Efficient and adaptive rank-based fits for linear models with skew-normal errors,” Journal of Statistical Distributions and Applications, vol. 1, no. 1, pp. 1–18, 2014.
[14] A. Abebe and J. W. McKean, “Highly efficient nonlinear regression based on the wilcoxon norm,” in D. Umbach (ed.), Festschrift in Honor of Mir Masoom Ali on the Occasion of his Retirement, Ball State University, Muncie, IN, 2007, pp. 340–357.
[15] H. L. Koul, G. L. Sievers, and J. W. McKean, “An estimator of the scale parameter for the rank analysis of linear models under general score functions,” Scandinavian Journal of Statistics, vol. 14, no. 2, pp. 131–141, 1987.
[16] J. D. Kloke and J. W. McKean, “Rfit: Rank-based estimation for linear models,” The R Journal, vol. 4, no. 2, pp. 57–64, 2012.
[17] R. G. Clark and S. Allingham, “Robust resampling confidence intervals for empirical variograms,” Mathematical Geosciences, vol. 43, no. 2, pp. 243–259, 2011.
[18] R Core Team, R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria, 2015, r package version 3.2.2. (Online). Available: https://www.R-project.org/
[19] L. Tanga, W. R. Schucany, W. A. Woodwardb, and R. F. Gunstb, “A parametric spatial bootstrap,” Southern Methodist University, Dallas, Texas, Tech. Rep. SMU-TR-337, 2006.
[20] H. M. Al-Mofleh, J. E. Daniels, and J. W. McKean, “Robust variogram fitting using non-linear rank-based estimators,” International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, vol. 10, no. 2, pp. 68 – 77, 2016. (Online). Available: http://waset.org/Publications?p=110
[21] J.-P. Chiles and P. Delfiner, Geostatistics: Modelling Spatial Uncertainty. New York: John Wiley and Sons, Ltd, 1999.
[22] S. Garrigues, D. Allard, F. Baret, and M. Weiss, “Quantifying spatial heterogeneity at the landscape scale using variogram models,” Remote Sensing of Environment, vol. 103, no. 1, pp. 81–96, 2006.
[23] O. Atteia, J.-P. Dubois, and R. Webster, “Geostatistical analysis of soil contamination in the swiss jura,” Environmental Pollution, vol. 86, no. 3, pp. 315–327, 1994.
[24] P. Goovaerts, Geostatistics for Natural Resources Evaluation. New York: Oxford University Press, 1997.
[25] M. Schlather, A. Malinowski, P. J. Menck, M. Oesting, and K. Strokorb, “Analysis, simulation and prediction of multivariate random fields with package RandomFields,” Journal of Statistical Software, vol. 63, no. 8, pp. 1–25, 2015. (Online). Available: http://www.jstatsoft.org/v63/i08/
[26] A. G. Journel and C. J. Huijbregts, Mining Geostatistics, reprint ed. New York: The Blackburn Press, 2003.