Abstract
Abstract Background Nonlinear relationships are common in the environmental discipline. Spreadsheet packages such as Microsoft Excel come with an add-on for nonlinear regression, but parameter uncertainty estimates are not yet available. The purpose of this paper is to use Monte Carlo and bootstrap methods to estimate nonlinear parameter uncertainties with a Microsoft Excel spreadsheet. As an example, uncertainties of two parameters (α and n) for a soil water retention curve are estimated. Results The fitted parameters generally do not follow a normal distribution. Except for the upper limit of α using the bootstrap method, the lower and upper limits of α and n obtained by these two methods are slightly greater than those obtained using the SigmaPlot software which linearlizes the nonlinear model. Conclusions Since the linearization method is based on the assumption of normal distribution of parameter values, the Monte Carlo and bootstrap methods may be preferred to the linearization method.
Highlights
Nonlinear relationships are common in the environmental discipline
The predicted θ value can be calculated with the van Genuchten soil water retention curve model (Eq 5) using suction pressure and all parameter values
Because the linearization method is based on the assumption of normal distribution of parameters and linearity at the vicinity of the estimated parameter value, and it is more complicated in terms of calculation, the Monte Carlo and bootstrap methods may be preferred to the linearization method to calculate the parameter uncertainties in spreadsheets
Summary
Nonlinear relationships are common in the environmental discipline. Spreadsheet packages such as Microsoft Excel come with an add-on for nonlinear regression, but parameter uncertainty estimates are not yet available. Nonlinear regression programs usually give the parameter uncertainty by calculating the standard error of the mean, and assuming linear relationship between variables in the vicinity of the estimated parameter values and normal distribution of parameter values This method usually involves evaluating a Hessian matrix (a square matrix of second-order partial derivatives of a scalar-valued function to describe the local curvature of a function of many variables) or an inequality, which makes it more complicated and time demanding (Brown 2001). More general methods such as Monte Carlo and bootstrap simulation can be used to estimate the parameter uncertainties.
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