Simulating Large Gaussian Random Vectors Subject to Inequality Constraints by Gibbs Sampling
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Simulating Large Gaussian Random Vectors Subject to Inequality Constraints by Gibbs Sampling Xavier Emery · Daisy Arroyo · María Peláez
Received: 28 January 2013 / Accepted: 5 October 2013 © International Association for Mathematical Geosciences 2013
Abstract The Gibbs sampler is an iterative algorithm used to simulate Gaussian random vectors subject to inequality constraints. This algorithm relies on the fact that the distribution of a vector component conditioned by the other components is Gaussian, the mean and variance of which are obtained by solving a kriging system. If the number of components is large, kriging is usually applied with a moving search neighborhood, but this practice can make the simulated vector not reproduce the target correlation matrix. To avoid these problems, variations of the Gibbs sampler are presented. The conditioning to inequality constraints on the vector components can be achieved by simulated annealing or by restricting the transition matrix of the iterative algorithm. Numerical experiments indicate that both approaches provide realizations that reproduce the correlation matrix of the Gaussian random vector, but some conditioning constraints may not be satisfied when using simulated annealing. On the contrary, the restriction of the transition matrix manages to satisfy all the constraints, although at the cost of a large number of iterations. Keywords Kriging neighborhood · Gaussian random field · Gibbs sampler · Markov chain · Simulated annealing · Restriction of transition matrix
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X. Emery ( ) Department of Mining Engineering, University of Chile, Santiago, Chile e-mail: [email protected] X. Emery Advanced Mining Technology Center, University of Chile, Santiago, Chile D. Arroyo · M. Peláez Department of Mathematics, Northern Catholic University, Antofagasta, Chile D. Arroyo e-mail: [email protected] M. Peláez e-mail: [email protected]
Math Geosci
1 Introduction The simulation of Gaussian random vectors subject to inequality constraints arises in the analysis of spatial data, when all or part of the data values are interval constraints rather than single numbers. In a mineral resources evaluation, this situation occurs when the depth of a geological horizon is greater than the depth at which drilling has been stopped, when a measured grade is smaller than a detection limit, or when working with soft data defined by lower and upper bounds (Journel 1986). Interval constraints are also encountered when simulating continuous variables represented by chi-square random fields, as well as indicators and categorical variables represented by truncated Gaussian or plurigaussian random fields (Matheron et al. 1987; Le Loc’h et al. 1994; Bárdossy 2006; Emery 2005, 2007a, 2007b; Armstrong et al. 2011). For instance, let d be a positive integer and consider an indicator variable obtained by truncating a stationary Gaussian random field Y = {Y (x) : x ∈ Rd } at a given threshold y ∈ R. The following procedure can be used to simulate the indicator (Lantuéjoul 2002): (1) Simulate Y a
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