Therefore, we copy the simulated θ data to a new sheet by right-clicking “Paste Special” and selecting “Values” in the dialogue of “Paste Special”. Note that the new data set of θ will change during optimization, which will result in errors in fitting.
Use the same procedure as introduced before to conduct nonlinear regression for each new data set of θ. For this, we use as an example, a common soil physical property - soil water retention curve, which has been widely used in soil, hydrological, and environmental communities.Ģ.
In addition, the influences of number of simulation on uncertainty estimates are also discussed. The objective of this paper is to apply the Monte Carlo and bootstrap simulations to obtain parameter uncertainties with a Microsoft Excel spreadsheet. These advantages are likely to make the use of spreadsheets to quantify parameter uncertainties more desirable. Microsoft Excel spreadsheets have other advantages including their general facility for data input and management, ease in implementing calculations, and often advanced graphics and reporting capabilities (Wraith and Or 1998). However, parameter uncertainties estimation in spreadsheets using the Monte Carlo and bootstrap methods has been rarely discussed.īoth nonlinear parameter values and their associated uncertainties are important for decision making and thus should be implemented in spreadsheet program like Excel. 2011) and estimating uncertainty of greenhouse gas emissions using the bootstrap simulation (Tong et al. Among numerous related applications are testing fire ignition selectivity of different landscape characteristics using the Monte Carlo simulation (Conedera et al. Both methods have their own advantages: while the Monte Carlo method is based on a theoretical probability distribution of a variable, the bootstrap method has no assumption on the probability distribution of a variable and thus has no limits on sampling size. More general methods such as Monte Carlo and bootstrap simulation can be used to estimate the parameter uncertainties. Furthermore, 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). 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. Parameter uncertainty can be obtained exactly by assuming normal distribution of a parameter in linear regression, but not in nonlinear regression. However, relatively less work has focused on the nonlinear parameter uncertainty estimates using spreadsheet packages. For these reasons, estimation of parameter uncertainties is significant for nonlinear parameter estimates. Parameter uncertainty can be used to judge the degree of reliability of the parameter estimates, which is important to making decisions for environmental management. It is also expressed as the standard error of the mean by assuming normal distribution of parameter values. It is usually expressed as an interval of parameter values at a certain confidence level, say, 95%. Different observations are usually obtained when experiments are repeated, resulting in different values of parameters. Parameter uncertainty refers to lack of knowledge regarding the exact true value of a quantity (Tong et al.
For these reasons, spreadsheets such as Microsoft Excel are widely suggested to make nonlinear parameter estimation (Harris 1998 Smith et al. In addition, spreadsheets have the merits of wide accessibility and powerful computation in terms of fitting nonlinear models. However, spreadsheet techniques are easier to learn than other specialized mathematical programs for nonlinear parameter estimation, because no programming skills are needed in spreadsheets to develop their own parameter estimation routines (Wraith and Or 1998).
Excel linear regression read output confidence interval software#
As a result, there are many software packages (such as SAS and MathCAD) that implement nonlinear parameter estimation. Nonlinear relationships are common in natural and environmental sciences (Wraith and Or 1998 Luo et al.