Simultaneous confidence and prediction intervals for nonlinear regression models with application to a groundwater flow model

Abstract

Methods are presented for computing three types of simultaneous confidence and prediction intervals (exact, likelihood ratio, and linearized) on output from nonlinear regression models with normally distributed residuals. The confidence intervals can be placed on individual regression parameters or on the true regression function at any number of points in the domain of the independent variables, and the prediction intervals can be placed on any number of future observations. The confidence intervals are analogous to simultaneous Scheffè intervals for linear models and the prediction intervals are analogous to the prediction intervals of Hahn (1972). All three types of intervals can be computed efficiently by using the same straightforward Lagrangian optimization scheme. The prediction intervals can be treated in the same computational framework as the confidence intervals by including the random errors as pseudoparameters in the Lagrangian scheme. The methods are applied to a hypothetical groundwater model for flow to a well penetrating a leaky aquifer. Three different data sets are used to demonstrate the effect of sampling strategies on the intervals. For all three data sets, the linearized confidence intervals are inferior to the exact and likelihood ratio intervals, with the latter two being very similar; however, all three types of prediction intervals yielded similar results. The third data set (time drawdown data at only a single observation well) points out many of the problems that can arise from extreme nonlinear behavior of the regression model.