Sensitivity Analysis—Stability and Conditionality
Sensitivity analysis is an important tool for systematic evaluation of mathematical models (Lehr et al., 1994). Sensitivity analysis can be use for various purposes, including model validation, evaluating model behavior, estimating model uncertainties, decision making using uncertain models, and determining potential areas of research (Lehr et al., 1994). Sensitivity analysis provides information about the effect of the errors and/or variations in the variables and/or parameters and models on the predicted behavior. Sensitivity of a model to changes in its input data determines the condition of the model (Chapra and Canale, 1998). The sensitivity of a system's outcome or response to changes in a variable is defined by the partial derivative (Lehr et al., 1994):
Relative sensitivity (Lehr et al., 1994) or the condition number (Chapra and Canale, 1998) is defined as the ratio of the relative change or error in the function to the relative change or error in the variable or parameter value. Thus, for a single parameter function, the relative sensitivity can be expressed by means of Lehr et al., 1994; (Chapra and Canale, 1998):
Thus, the condition number or relative sensitivity can be used as a criteria to evaluate the effect of an uncertainty in the x variable on the condition of a system as (Chapra and Canale, 1998):
Given the differential equations of a model, the sensitivity equations can be formulated for determining the sensitivity trajectory. The following
example by Lehr et al. (1994) illustrates the process. Consider a mathematical model given by an ordinary differential equation as:
A differentiation of the g(f,x,t) function with respect to the variable (or parameters) x leads to:
Substituting Eqs. 17-34 and 37 into Eq. 17-38 and rearranging yield the following sensitivity trajectory equation:
One of the practical applications of the sensitivity analysis is to determine the critical parameters, which strongly effect the predictions of models (Lehr et al., 1994). Lehr et al. (1994) studied the sensitivity of an oil spill evaporation model.
Lehr et al. (1994) depict the sensitivity of the fractional oil evaporation,f, from an oil spill with respect to the initial bubble point, TB, and the rate of bubble point variation by the fraction of oil evaporated (the slope of the evaporation curve), TG =d2TB/dtdf, respectively.
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