Abstract
In many geostatistical applications, a transformation to standard normality is a first step in order to apply standard algorithms in two-point geostatistics. However, in the case of a set of collocated variables, marginal normality of each variable does not imply multivariate normality of the set, and a joint transformation is required. In addition, current methods are not affine equivariant, as should be required for multivariate regionalized data sets without a unique, canonical representation (e.g., vector-valued random fields, compositional random fields, layer cake models). This contribution presents an affine equivariant method of Gaussian anamorphosis based on a flow deformation of the joint sample space of the variables. The method numerically solves the differential equation of a continuous flow deformation that would transform a kernel density estimate of the actual multivariate density of the data into a standard multivariate normal distribution. Properties of the flow anamorphosis are discussed for a synthetic application, and the implementation is illustrated via two data sets derived from Western Australian mining contexts.
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Aitchison J (1986) The statistical analysis of compositional data. Chapman & Hall, London
Bandarian EM, Mueller U (2008) Reformulation of MAF as a generalised eigenvalue problem. In Ortiz JM, Emery X, Geostats2008, pp 1173–1178. Santiago.
Barnett RM, Manchuk JG, Deutsch CV (2014) Projection pursuit multivariate transform. Math Geosci 46(2):337–360
Filzmoser P, Hron K (2008) Outlier detection for compositional data using robust methods. Math Geosci 40:233–248
Korkmaz S, Goksuluk D, Zararsiz G (2014) MVN: an R package for assessing multivariate normality. R J 151–162. Retrieved from http://journal.r-project.org/archive/2014-2/korkmaz-goksuluk-zararsiz.pdf
Leuangthong O, Deutsch CV (2003) Stepwise conditional transformation for simulation of multiple variables. Math Geol 35(2):155–173
Szekely GJ, Rizzo ML (2013) Energy statistics: a class of statistics based on distances. J Stat Plan Infer 143(8):1249–1272
Tercan A (1999) The importance of orthogonalization algorithm in modeling conditional distributions by orthogonal transformed indicator methods. Math Geol 31(2):155–173
van den Boogaart KG, Mueller U, Tolosana Delgado R (2015) An affine equivariant anamorphosis for compositional data. In: Schaeben H, Tolosana Delgado R, van den Boogaart KG, van den Boogaart R (eds) Proceedings of IAMG 2015, Freiberg (Saxony) Germany, September 5–13, 2015. IAMG, Freiberg, pp 1302–1311
van den Boogaart KG, Mueller U, Tolosana-Delgado R (2016) An affine equivariant multivariate normal score transform for compositional data. Math Geosci. doi:10.1007/s11004-016-9645-y
Acknowledgments
The authors acknowledge financial support through the ECU CES Travel Scheme 2014, the DAAD-UA Grant CodaBlockKriging, and the Perth Convention Bureau Aspire Professional Development Grant 2016. Clint Ward, Cliffs Resources, and Matt Cobb, Consolidated Minerals, are thanked for provision of the data on which the example studies are based.
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Mueller, U., van den Boogaart, K.G., Tolosana-Delgado, R. (2017). A Truly Multivariate Normal Score Transform Based on Lagrangian Flow. In: Gómez-Hernández, J., Rodrigo-Ilarri, J., Rodrigo-Clavero, M., Cassiraga, E., Vargas-Guzmán, J. (eds) Geostatistics Valencia 2016. Quantitative Geology and Geostatistics, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-319-46819-8_7
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DOI: https://doi.org/10.1007/978-3-319-46819-8_7
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