Multivariate Stochastic Simulation with Subjective Multivariate Normal Distributions1

-In many applications of Monte Carlo simulation in forestry or forest products, it may be known that some variables are correlated. However, for simplicity, in most simulations it has been assumed that random variables are independently distributed. This report describes an alternative Monte Carlo simulation technique for subjectively assessed multivariate normal distributions. The method requires subjective estimates of the 99-percent confidence interval for the expected value of each random variable and of the partial correlations among the variables. The technique can be used to generate pseudorandom data corresponding to the specified distribution. If the subjective parameters do not yield a positive definite covariance matrix, the technique determines minimal adjustments in variance assumptions needed to restore positive definiteness. The method is validated and then applied to a capital investment simulation for a new papermaking technology. In that example, with ten correlated random variables, no significant difference was detected between multivariate stochastic simulation results and results that ignored the correlation. In general, however, data correlation could affect results of stochastic simulation, as shown by the validation results. INTRODUCTION MONTE CARLO TECHNIQUE Stochastic simulation is a practical approach to prediction because estimating a likelihood distribution for many variables in a model is often easier than estimating their precise values. In this paper we shall first review classical stochastic (Monte Carlo) simulation techniques, then suggest a method to take into account the subjective correlations among variables, validate the method, and apply it to a specific case study. has been largely ignored (e.g., Engelhard and Anderson, 1983). Generally, a mathematical model is used in stochastic simulation studies. In addition to randomness, correlation may exist among the variables or parameters of such models. In the case of forest ecosystems, for example, growth can be influenced by correlated variables, such as temperatures and precipitations. Similarly, in complex forest product technologies, predicted production, or returns, may depend on several engineering and economic variables, some of which may be correlated. However, in most applications of Monte Carlo simulation in forestry or forest products, data correlation The Monte Carlo simulation technique utilizes three essential elements: (1) a mathematical model to calculate a discrete numerical result or outcome as a function of one or more discrete variables, (2) a sequence of random (or pseudorandom) numbers to represent random probabilities, and (3) probability density functions or cumulative distribution functions for the random variables of the model. The mathematical model is used repetitively to calculate a large sample of outcomes from different values assigned to the random variables. The sequence of random numbers is generally drawn independently from the uniform distribution on the unit interval (0,1) and thus represents a sequence of socalled uniform deviates. The “distribution” of a continuous random variable refers to the probability of its occurrence over its domain or “distribution space.” The distribution of a random variable is represented mathematically by its probability density function, which gives the probability that the random variable will occur within any subspace of the distribution space. Examples of probability 1 For presentation at Symposium on Systems Analysis in Forest Resources, Charleston, South Carolina, March 3-7, 1991. Research supported by the USDA Forest Service, Forest Products Laboratory, and by the School of Natural Resources, University of Wisconsin, Madison The Forest Products Laboratory is maintained in cooperation with the University of Wisconsin. This article was written and prepared by U.S. Government 2 employees on official time, and it is therefore in the public domain and not subject to copyright. 3 USDA Forest Service, Forest Products Laboratory, Madison, WI 53705. Department of Forestry, University of Wisconsin, Madison, WI 53706.