Stochastic market modeling with Gaussian Quadratures: Do rotations of Stroud's octahedron matter?

Abstract

Recently, stochastic applications of large-scale applied simulation models of agricultural markets have become more frequent. However, stochastic modeling with large market models comes with high computational and management costs for data storage, analysis and manipulation. Gaussian Quadratures (GQ), are efficient sampling methods requiring few points to approximate the central moments of the joint probability distribution of stochastic variables and therefore reduce computational costs. For symmetric regions of integration, the vertices of Stroud's n-octahedron (Stroud, 1957) are GQ formulas of degree 3 with a minimal number of points which can make the stochastic modeling with large economic models manageable. However, we have the conjecture that rotations of Stroud's n-octahedron have an effect on the accuracy of approximation of model results; thus, we test eight different rotations using the European Simulation Model (ESIM). It was found that the 45° rotation yields higher accuracy than the 0° rotation. With the 45° rotation and in models with large regions or variables which strongly determine the outcome of model results such as soft wheat in ESIM, the arrangement of the stochastic variables (A1 or A2) in the covariance matrix or the selected method to introduce correlation (via the Cholesky decomposition –C– or via the diagonalization method –D–) may have a significant effect on the accuracy of the quadratures. With the 45° rotation and with markets where the effect of the different regions or variables on model outcomes are more homogenous as in the case of rapeseed in ESIM, the selection of the arrangements A1 or A2 and the method of introducing correlation C or D may not have a significant effect on the accuracy of the quadratures.