High-Order Sequential Simulation via Statistical Learning in Reproducing Kernel Hilbert Space

Abstract

The present work proposes a new high-order simulation framework based on statistical learning. The training data consist of the sample data together with a training image, and the learning target is the underlying random field model of spatial attributes of interest. The learning process attempts to find a model with expected high-order spatial statistics that coincide with those observed in the available data, while the learning problem is approached within the statistical learning framework in a reproducing kernel Hilbert space (RKHS). More specifically, the required RKHS is constructed via a spatial Legendre moment (SLM) reproducing kernel that systematically incorporates the high-order spatial statistics. The target distributions of the random field are mapped into the SLM-RKHS to start the learning process, where solutions of the random field model amount to solving a quadratic programming problem. Case studies with a known data set in different initial settings show that sequential simulation under the new framework reproduces the high-order spatial statistics of the available data and resolves the potential conflicts between the training image and the sample data. This is due to the characteristics of the spatial Legendre moment kernel and the generalization capability of the proposed statistical learning framework. A three-dimensional case study at a gold deposit shows practical aspects of the proposed method in real-life applications.

Training Image Free High-Order Stochastic Simulation Based on Aggregated Kernel Statistics

“…The Legendre polynomial expansion series can approximate arbitrary piecewise continuous function and are used for approximation of probability density function in high-order simulations (Mustapha and Dimitrakopoulos 2010a). The spatial Legendre moment reproducing kernel (SLM-kernel) (Yao et al 2020) is derived from a new computational model for high-order simulation (Yao et al 2018) based on the Legendre polynomial series. The SLM-kernel carries the information of high-order spatial statistics so that the density estimation in the high-order sequential simulation could be achieved by a statistical learning process in kernel space.…”

“…( 21) can be expanded to a quadratic programming problem by noticing that the inner products can be expressed as kernel functions. The details to solve the problem given p as a convex combination of certain prototype distributions is established in Yao et al (2020) and thus will not be repeated here. It should be noted that although Eq.…”

“…The high-order simulation methods provide a new framework to simulate complex spatial patterns, addressing the drawbacks in MPS methods as discussed in the related publications (Dimitrakopoulos et al 2010;Dimitrakopoulos 2010a, 2011;Dimitrakopoulos 2016, 2017;Minniakhmetov et al 2018;Yao et al 2018Yao et al , 2020de Carvalho et al 2019). The high-order simulation methods equip the multiple-point spatial structures with well-defined mathematical entities, such as spatial cumulants or high-order spatial moments (Dimitrakopoulos et al 2010;Mustapha and Dimitrakopoulos 2010b;Minniakhmetov et al 2018).…”

“…Yao et al (2020) propose a statistical learning framework of high-order simulation in kernel space by constructing a so-called spatial Legendre moment kernel from a new computational model of high-order simulation based on spatial Legendre moments (Yao et al 2018). The proposed statistical learning framework in Yao et al (2020) demonstrates the advantage of its generalization capacity with regards to improving of the numerical stability, as compared with the previous highorder simulation methods. This generalization capacity also mitigates the statistical conflicts between the samples and the TI.…”

“…This is due to the fact that the high-order spatial statistics are adjusted to the target probability distribution through the learning process, as opposed to directly being incorporated into the coefficients of polynomial expansion series as with the other methods. The simulation under a statistical learning framework (Yao et al 2020) proceeds sequentially according to a random path based on the sequential decomposition of the multivariate distribution of the random field model (Rosenblatt 1952;Journel 1994). Specifically, the replicates are mapped onto the spatial Legendre moment space and the empirical kernel statistics are computed thereafter.…”