On the construction and analysis of stochastic models: Characterization and propagation of the errors associated with limited data

This paper investigates the predictive accuracy of stochastic models. In particular, a formulation is presented for the impact of data limitations associated with the calibration of parameters for these models, on their overall predictive accuracy. In the course of this development, a new method for the characterization of stochastic processes from corresponding experimental observations is obtained. Specifically, polynomial chaos representations of these processes are estimated that are consistent, in some useful sense, with the data. The estimated polynomial chaos coefficients are themselves characterized as random variables with known probability density function, thus permitting the analysis of the dependence of their values on further experimental evidence. Moreover, the error in these coefficients, associated with limited data, is propagated through a physical system characterized by a stochastic partial differential equation (SPDE). This formalism permits the rational allocation of resources in view of studying the possibility of validating a particular predictive model. A Bayesian inference scheme is relied upon as the logic for parameter estimation, with its computational engine provided by a Metropolis-Hastings Markov chain Monte Carlo procedure.