A comparison of algorithms for global characterization of confidence regions for nonlinear models

Abstract

Environmental models are often highly nonlinear, and parameters have to be estimated from noisy data. The standard approach of locally linearizing the model, which leads to ellipsoid confidence regions, is inappropriate in this situation. A straightforward technique to characterize arbitrary-shaped confidence regions is to calculate model output on a grid of parameter values. Each parameter value P results in a goodness of fit G(P), which allows delineation of the set of parameters corresponding to G(P) < Gc, with Gc some threshold level (e.g., 5% probability). This approach is impractical and time-consuming for complex models, however. This article aims at finding an efficient alternative. It is first shown that the most general approach is to generate parameter values uniformly covering the set G(P) < Gc rather than finding the boundary G(P) = Gc. It is argued that the most efficient method of generating a uniform cover is by a (theoretical) algorithm known as pure adaptive search (PAS); the presently proposed method (uniform covering by probabilistic rejection; UCPR) is shown to be a good approximation to PAS. The UCPR is compared with alternative methods for a number of test problems. It is illustrated that for complex models (where model run time dominates total computer time) UCPR is considerably faster and its cover of Gc more uniform than existing alternatives. An intrinsic problem common to all methods is that the amount of work increases at least quadratically with the number of parameters considered, making them of limited use for high-dimensional problems.