Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 31442
Robust Variogram Fitting Using Non-Linear Rank-Based Estimators

Authors: Hazem M. Al-Mofleh, John E. Daniels, Joseph W. McKean

Abstract:

In this paper numerous robust fitting procedures are considered in estimating spatial variograms. In spatial statistics, the conventional variogram fitting procedure (non-linear weighted least squares) suffers from the same outlier problem that has plagued this method from its inception. Even a 3-parameter model, like the variogram, can be adversely affected by a single outlier. This paper uses the Hogg-Type adaptive procedures to select an optimal score function for a rank-based estimator for these non-linear models. Numeric examples and simulation studies will demonstrate the robustness, utility, efficiency, and validity of these estimates.

Keywords: Asymptotic relative efficiency, non-linear rank-based, robust, rank estimates, variogram.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1112318

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1301

References:


[1] Abebe A., and McKean J., (2013), Weighted Wilcoxon Estimators in Nonlinear Regression, Australian & New Zealand Journal of Statistics, v. 55, no. 4, pp. 401–420.
[2] Chiles, J. P., and Delfiner, P., (1999). Geostatistics: Modelling Spatial Uncertainty, New York: John Wiley and Sons Ltd.
[3] Clark, R.G., and Allingham, S., (2011). Robust Resampling Confidence Intervals for Empirical Variograms, Mathematical Geosciences, v. 43, no. 2, pp. 243-259.
[4] Cressie N., (1985), Fitting variogram models by weighted least squares. Journal of the International Association for Mathematical Geology, v. 17, no. 5, pp. 563-586.
[5] Cressie N., (1991). Statistics for spatial data, New York: John Wiley and Sons Ltd.
[6] Cressie N., and Hawkins D. (1980), Robust estimation of the variogram, I: Math. Geology, v. 12, no. 2, pp. 115-125.
[7] Garrigues, S., and Allard, D., Baret, F., Weiss, M., (2006), Quantifying spatial heterogeneity at the landscape scale using variogram models. Remote Sensing of Environment, vol. 103, no. 1, pp.81-96.
[8] Genton M., (1998), Highly robust variogram estimation, Mathematical Geology, v. 30, no. 2, pp. 213-221.
[9] Hampel F., (1973), Robust estimation, a condensed partial survey: Zeitschrift fur Wahrschein-lichkeitstheorie und verwandte Gebiete, vol. 27, no. 2, pp. 87-104.
[10] Hettmansperger, T., and McKean J. W., (2011). Robust Nonparametric Statistical Methods, 2nd Ed., New York: Chapman-Hall.
[11] Hogg, R. V., Fisher, D. M., and Randles, R. H. (1975), A two-sample adaptive distribution-free test, Journal of the American Statistical Association, v. 70, pp. 656-661.
[12] Hogg, R. V., McKean, J. W., and Craig, A. T. (2013). Introduction to Mathematical Statistics, 7th Ed., Boston: Pearson.
[13] Huber, P. J., (1996). Robust statistical procedures, 2nd Ed., Philadelphia: Society for Industrial and Applied Mathematics, 56 p.
[14] Journal, A., and Huijbregts, C., (2003). Mining Geostatistics, reprint Ed., New York: Academic Press.
[15] Kloke, J., and McKean, J. W., (2012), Rfit: Rank-based estimation for linear models, R Journal, v. 4, no. 2, pp. 57-64.
[16] Kloke, J., McKean, J., (2015, September 1), Rfit: Rank Estimation for Linear Models. R package version 0.22.0. Retrieved from http://CRAN.R-project.org/package=Rfit.
[17] Kloke, J., McKean, J. W., (2014). Nonparametric statistical methods using R, FL: Chapman-Hall.
[18] Koul, H. L., Sievers, G., and McKean, J. W., (1987), An Estimator of the Scale Parameter forthe Rank Analysis of Linear Models under General Score Functions. Scandinavian Journal of Statistics, v. 14, no. 2, pp. 131-141.
[19] Mardia, K., and Marshall, R., (1984), Maximum likelihood estimation of models for residual covariance in spatial regression, Biometrika. v. 71, no. 1, pp. 135-146.
[20] Matheron, G., (1962), Traite de geostatistique appliquee, Tome I: Memories du Bureau de Recherches Geologiques et Minieres, no. 14, Editions Technip, Paris, 333 p.
[21] McBratney, A. B., Webster, R., and Burgess, T. M., (1981), The design of optimal sampling schemes for local estimation and mapping of regional variables-I: theory and method, Computers and Geosciences, v. 7, pp. 331-334.
[22] McKean, J. W., and Kloke, J., (2014), Efficient and Adaptive Rank-Based Fits for Linear Models with Skewed-Normal Errors. Journal of Statistical Distributions and Applications, v. 1, no. 1, pp. 1-18.
[23] Mercer, W. B. and Hall, A. D. (1911), The experimental error of field trials. Journal of Agricultural. Science 4, pp. 107-132.
[24] Patterson H., and Thompson, R., (1971), Recovery of inter-block information when block sizes are unequal. Biometrika, v. 58, no. 3, pp. 545-54.
[25] Patterson, H., and Thompson, R., (1974), Paper presented at: 8th International Biometrics Conference, Maximum likelihood estimation of components of variance. Constanta, Romania, pp. 197-207.
[26] R Core Team, (2015, September 1), R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Retrieved from http://www.R-project.org/.
[27] Rao, C., (1979), MINQE Theory and its Relation to ML and MML Estimation of Variance Components, The Indian Journal of Statistics, v. 41, Series B, Pts. 3 and 4, pp. 138-153.
[28] Schlather, M., Malinowski, A., Oesting, M., Boecker, D., Strokorb, K., Engelke, S., Martini, J., Ballani, F., Moreva, O., Menck, PJ., Gross, S., Ober, U., Berreth, C., Burmeister, K., Manitz, J., Morena, O., Ribeiro, P., Singleton, R., Pfaff, B., and R Core Team, (2015, August 1). RandomFields: Simulation and Analysis of Random Fields. R package version 3.1.1. Retrieved from http://CRAN.R-project.org/package=RandomFields (accessed September 1, 2015).
[29] Shomrani, A., (2003), A Comparison of Different Schemes for Selecting and Estimating Score Functions Based on Residuals, unpublished Ph.D. dissertation, Western Michigan University, Kalamazoo, Michigan.
[30] Tang, L., Schucany, W., Woodward W, and Gunst R., (2006), A Parametric Spatial Bootstrap, Technical Report SMU-TR-337, Southern Methodist University, Dallas, Texas.