Multivariate Imputation of Unequally Sampled Geological Variables
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Multivariate Imputation of Unequally Sampled Geological Variables Ryan M. Barnett · Clayton V. Deutsch
Received: 24 July 2014 / Accepted: 26 December 2014 © International Association for Mathematical Geosciences 2015
Abstract Unequally sampled data pose a practical and significant problem for geostatistical modeling. Multivariate transformations are frequently applied in modeling workflows to reproduce the multivariate relationships of geological data. Unfortunately, these transformations may only be applied to data observations that sample all of the variables. In the case of unequal sampling, practitioners must decide between excluding incomplete observations and imputing (inferring) the missing values. While imputation is recommended by missing data theorists, the use of deterministic methods such as regression is generally discouraged. Instead, techniques such as multiple imputation (MI) are advocated to increase the accuracy, decrease the bias, and capture the uncertainty of imputed values. As missing data theory has received little attention within geostatistical literature and practice, MI has not been adapted from its conventional form to be suitable for geological data. To address this, geostatistical algorithms are integrated within an MI framework to produce parametric and non-parametric methods. Synthetic and geometallurgical case studies are used to demonstrate the feasibility of each method, where techniques that use both spatial and colocated information are shown to outperform the alternatives. Keywords
Missing data analysis · Statistics · Geostatistics · Modeling
1 Introduction Virtually every mining and petroleum project requires multiple regionalized variables to be characterized. The relationships between these variables may have a large impact
R. M. Barnett (B) · C. V. Deutsch Department of Civil and Environmental Engineering, Centre for Computational Geostatistics, University of Alberta, 3-133 NREF Building, Edmonton, AL T6G 2W2, Canada e-mail: [email protected]
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Math Geosci
on technical tasks such as project design and forecasting. For example, the joint distribution of a mining resource and contaminant variables will often dictate blending and plant design. As geostatistical models are a critical input to such tasks, multivariate relationships should be carefully reproduced in simulated realizations. Multivariate transformations are commonly used for modeling multiple variables, where the objective is to align the data with assumptions of the simulation algorithm. This may involve decorrelating the data or removing multivariate complexity such as heteroscedasticity, non-linearity, and constraints. Back-transformations will then return the original correlation or complexity to simulated realizations. Some examples include decorrelation transformations such as principal component analysis (Davis and Greenes 1983; Hotelling 1933) and minimum/maximum autocorrelation factors (Desbarats and Dimitrakopoulos 2000; Switzer and Green 1984), as well as multivariate Gaussian (multiGaussia
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