High-dimensional simulation-based estimation

Abstract

This paper describes a simulation-based technique for estimating the parameters of a high-dimensional stochastic model. The central idea is to find parameters which make the distribution of simulated multidimensional points Y identical to the distribution of the multidimensional points X observed in experiments. To do this, we minimize a criterion based on the heuristic that the univariate distribution of distances between the Ys and the Xs should be the same as the univariate distribution of distances among the replicated Xs themselves. The direction of random local searches in the parameter space for the minimizing value are guided by 1. (i) the degree of success of recent searches, and 2. (ii) a multiple regression fit of the recently investigated portion of the criterion's response surface to a deterministic approximation of the stochastic model which can be rapidly investigated. This approximation is most likely to be used when it is most valid, that is when R 2 is close to one. To guard against entrapment at a local minimum, the algorithm at random times will search the global parameter space to look for promising other portions to investigate. Unlike simulated annealing, where the criterion function can be evaluated exactly, our algorithm must deal with the fact that the observed value of the criterion for a given set of parameters is itself based on simulation and thus subject to variability. This difficulty is handled through a cross-validation procedure which examines the distribution of the criterion at the last successful point. The methodology is applied to a detailed stochastic predator-prey model originally described in [1].