Estimation of parameters in monte carlo modelling

Abstract

This paper presents a general method for estimating model parameters from experimental data when the model relating the parameters and input variables to the output responses is a Monte Carlo simulation. From a statistical point of view a Bayesian approach is used in which the distribution of the parameters is handled in discretized form as elements of an array in computer storage. The stochastic nature of the Monte Carlo model allows only an estimate of the distribution to be calculated from which the true distribution must then be estimated. For this purpose an exponentiated polynomial function has been found to be useful. The method provides point estimates as well as joint probability regions. Marginal distributions and distributions of functions of the parameters can also be handled. The motivation for exploring this alternative parameter estimation technique comes from the recognition that for some systems, particularly when the underlying process is stochastic in nature, Monte Carlo simulation often is the most suitable way of modelling. As such, the Monte Carlo approach increases the range of problems which can be handled by mathematical modelling.The technique is applied to the modelling of binary copolymerization. Two models, the Mayo-Lewis and the Penultimate Group Effects models, are considered and a method for discriminating between these models in the light of sequence distribution data is proposed. ~ ~~~~~On prksente dans cet article une mkthode gentrale pour I'estimation de paramktres de modele B partir de donnees experimentales lorsque le modkle reliant les parametres et les variables d'entree aux reactions de sortie est une simulation de Monte Carlo. D'un point de vue statistique, on utilise une methode bayesienne, dans laquelle la distribution des parametres est traitee sous forme discrkte comme elements d'un tableau informatique. La nature stochastique du modele Monte Carlo permet uniquement l'estirnation de la distribution B calculer a partir de laquelle la veritable distribution peut Ctre estimbe. A cette fin, on a trouvk utile une fonction polynomiale exponentialisee. La methode donne des estimations de points ainsi que des zones de probabilitk communes. On peut egalement trait6 les distributions marginales et les distributions des fonctions de ces paramktres. L'exploration de cette autre technique d'estimation des paramktres est encouragee par le fait que pour certains systkmes, particulitrement lorsque le procedt sous-jacent est de nature stochastique, la simulation de Monte Carlo est souvent la meilleure facon de modeliser. Ainsi, la methode de Monte Carlo augmente I'kentail des problemes qui peuvent Ctre traitbs par la modelisation mathematique.La technique est appliquee a la modelisation de la copolymerisation binaire. Deux modeles, celui de Mayo-Lewis et celui des effets du groupe pCnulti&me, sont pris en consideration, et on propose une mtthode pour la discrimination entre ces modkles a la lurnikre des donnees de distribution des sequences.

Monte Carlo Simulation of Peroxide Initiated Degradation of Polypropylene

“…The solution methods of the stochastic formulation of chemical kinetics range from analytical to numerical ones. Because of the complicated nature of chemical systems, the analytical methods are not tractable for most practical cases and computer-oriented Monte Carlo techniques are often more preferable in practical applications (Kotliar, 1963;McDermott and Klein, 1986;McDermott eta!., 1990;Train and Klein, 1988;Duever eta!., 1988;Duever and Reilly, 1990).…”

Kinetic Parameter Estimation in Peroxide Initiated Degradation of Polypropylene

“…This error has been termed "shimmer" (Duever and Reilly, 1990). Incorporating shimmer leads to a model of the form (12) This model is not easy to apply here since the joint distribution of y and g_ is difficult to assess.…”

“…Duever and Reilly (1990) show that under the assumption that the error vector S.u is distributed as the v-variate normal NvCQ., ~). where ~ is the vxv covariance matrix, the posterior distribution under the assumption that~ is known is given by (14) Unlike the deterministic approach, the parameters lt can not be estimated by an Box and Cox (1964).…”

“…Here, the statistical inference of the parameters will be based on the appropriate posterior probability density function and corresponding joint confidence regions developed using the procedure described by Duever and Reilly (1990). In what follows the approach of Duever and Reilly (1990) will first be briefly outlined, followed by a description of the application of this method to the particular estimation problem solved here. The Monte Carlo simulation algorithm used is described in a previous paper (Huang et al, 1995).…”

“…Each of these methods will require different approaches to parameter estimation. Examples of kinetic parameter estimation using stochastic models have been given by Schaad (1963), Plate et al (1981) and Duever et al (1988) and Duever and Reilly (1990).…”