Addressing Order Relation Issues with Constrained Radial Basis Functions and Consistent Indicator Variograms

• Author / Creator

  Sanchez Villar, Sebastian

• Quantifying uncertainty is a critical task of resource delineation in the mining industry. Uncertainty is used to assess risk in economic evaluation and for classification in resource reporting. The inference of local distributions from conditioning data is key to quantifying uncertainty. Multiple indicator Kriging (MIK) is a well-established non-parametric local distribution inference technique that does not assume a prior distribution. The local conditional cumulative distribution functions (CCDF) are estimated directly from indicators defined from thresholds. MIK is flexible since allows the addition of soft data and accounts for different spatial correlations related to different thresholds. These advantages can be eclipsed by the fact that MIK CCDFs almost always present order relation issues, that is, probabilities below zero, above 1 and decreasing for increasing thresholds. The aim of this work is to understand the origin of order relations issues and to find ways to reduce or eliminate them. It explores their relation with the negative weights of Kriging. It also focuses on analytical and practical techniques that help to avoid order relations issues. One technique explores the internal consistency of the indicator variograms used in MIK and its connection to order relation issues. The other proposes an interpolation methodology using Radial Basis Functions (RBF) with inequality constraints to produce CCDFs without order relation issues.
  There are four main contributions of this research. First, it presents different tests and examples that show how negative weights influence order relation issues and their distributions. It also shows the relation between negative weights and indicator variograms. Second, it explains the internal consistency of indicator variograms related to a bivariate distribution and shows novel equations for the calculation of probabilities of the internal bivariate distribution. Additionally, it proposes a workflow to use the equations as a tool to aid the indicator variogram modeling process. Third, it proposes a new methodology of MIK that uses the RBF framework to add inequality constraints to the estimator. The constraints force the MIK estimates to comply with a licit CCDF. Finally, the equations to calculate bivariate probabilities and the RBF methodology are combined into a single workflow for MIK. The RBF framework is proven to work with a similar performance to classic MIK.
  These contributions aid to understand the different modeling processes involved in MIK, from the intrinsic characteristics of Kriging and RBF to the internal consistency of indicator variogram models. The understanding of this helps to build more consistent models in the future.

• Subjects / Keywords

  • Geostatistics
  • Multiple Indicator Kriging
  • Radial Basis Functions
  • Bivariate Distribution Probabilities
  • Indicator Variograms

• Graduation date

  Fall 2023

• Type of Item

  Thesis

• Degree

  Master of Science

• DOI

  https://doi.org/10.7939/r3-gyzk-8v20

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.