Efficient approximation for building error budgets for process models

Error budgets of process models allow us to partition the uncertainty (estimation error) in model projections caused by propagation of uncertainty in model inputs. Orthogonal polynomials, which are often employed to empirically represent unknown and possibly very complex relationships, have been applied to the development of error budgets by fitting the variance of the projection as a function of the standard errors of the parameter estimates of the process model inputs. Data generation approaches to fit those polynomials have involved some type of factorial design. However, that strategy may become unworkable for process models with many model inputs. In this paper, we propose an efficient method for building error budgets to not only overcome the limitations of factorial arrangements, but also to approximate the results obtainable through those designs. The proposed data generation scheme consists of: (a) repeatedly sampling random, equally-spaced, and equally-probable levels of standard error of the model input parameter estimates to characterize the level of error of their probability distributions; and (b) for each set of standard errors, conducting a number of Monte Carlo runs by sampling random values of input parameter estimates from each of those distributions to obtain projections of the process model and thus estimate the variance of the projection. The variance of the projection is then regressed on the orthogonally-transformed values of the input standard errors. The terms of the orthogonal polynomial are then ranked in decreasing order with respect to their explanatory importance in the function. The characteristics of this sampling scheme are analyzed through intensive supercomputer simulations under different combinations of number of model inputs and sample sizes. An example is conducted for a process forest growth model based on the pipe model theory and the self-thinning rule applied to a stand of red pine (Pinus resinosa Ait.) growing in the Great Lakes region of North America. The error budget resulting from this example closely resembles the error budget obtained by another study conducted for the same process model and forest stand using a fractional factorial design. It is concluded that the method proposed here provides an efficient strategy for building error budgets for process models with many model inputs.