Model Selection for Random Functions with Bounded Range: Applications in Science and Engineering

Abstract

Differential, integral, algebraic, and other equations can be used to describe the many types of systems encountered in applied science and engineering. Because of uncertainty, the specification of these equations often requires probabilistic models to describe the uncertainty in input and/or system properties. Since the available information on input and system properties is typically limited, there may be more than one model that is consistent with the available information. The collection of these models is referred to as the collection of candidate models $\mathcal{C}$. A main objective in model selection is the identification of the member of $\mathcal{C}$ which is optimal in some sense. Methods are developed for finding optimal models for random functions under limited information. The available information consists of: (a) one or more samples of the function and (b) knowledge that the function takes values in a bounded set, but whose actual boundary may or may not be known. In the latter case, the boundary of the set must be estimated from the available samples. The methods are developed and applied to the special case of non-Gaussian random functions referred to as translation random functions. Numerical examples are presented to illustrate the utility of the proposed approach for model selection, including optimal continuous time stochastic processes for structural reliability, and optimal random fields for representing material properties for applications in mechanical engineering.