2.4 Sensitivity and Sensitivity/Uncertainty Analysis

The aim of sensitivity analysis is to estimate the rate of change in the output of a model with respect to changes in model inputs. Such knowledge is important for (a) evaluating the applicability of the model, (b) determining parameters for which it is important to have more accurate values, and (c) understanding the behavior of the system being modeled. The choice of a sensitivity analysis method depends to a great extent on (a) the sensitivity measure employed, (b) the desired accuracy in the estimates of the sensitivity measure, and (c) the computational cost involved.

In general, the meaning of the term ``sensitivity analysis'' depends greatly on the sensitivity measure that is used. Table 2.3 presents some of the sensitivity measures that are often employed in the sensitivity analysis of a mathematical model of the form

$$\ensuremath{\mathcal{F}({\ensuremath{\textstyle\mbox{\boldmath${u}$ }}},{\ensuremath{\textstyle\mbox{\boldmath${k}$ }}}) = 0}$$ (2.5)

where ${\ensuremath{\textstyle\mbox{\boldmath${k}$ }} }$ is a set of $m$ parameters, and ${\ensuremath{\textstyle\mbox{\boldmath${u}$ }} }$ is a vector of $n$ output variables.

 

 

Table 2.3: A summary of sensitivity measures employed in sensitivity analysis, adapted from McRae et al., 1982

Sensitivity Measure	Definition
Response from arbitrary parameter variation	$${\ensuremath{\textstyle\mbox{\boldmath${u}$ }} }= {\ensuremath{\t... ...tyle\mbox{\boldmath${u}$ }} }({\ensuremath{\textstyle\mbox{\boldmath${k}$ }} })$$
Normalized Response	$$D_i = \frac{\delta u_i}{u_i(\overline{{\ensuremath{\textstyle\mbox{\boldmath${k}$ }} }})}$$
Average Response	$$\overline{u_{i}(\overline{{\ensuremath{\textstyle\mbox{\boldmath... ...${k}$ }} }}{\ldots}\!\!\int d{\ensuremath{\textstyle\mbox{\boldmath${k}$ }} }}$$
Expected Value	$\left< u_i({\ensuremath{\textstyle\mbox{\boldmath${k}$ }} }) \right> = \displa... ...tyle\mbox{\boldmath${k}$ }} })d{\ensuremath{\textstyle\mbox{\boldmath${k}$ }} }$
Variance	$\delta_i^2({\ensuremath{\textstyle\mbox{\boldmath${k}$ }} }) = \left< u_i({\en... ...\right> - \left< u_i({\ensuremath{\textstyle\mbox{\boldmath${k}$ }} })\right>^2$
Extrema	max [ $u_i({\ensuremath{\textstyle\mbox{\boldmath${k}$ }} })$], min [ $u_i({\ensuremath{\textstyle\mbox{\boldmath${k}$ }} })$]
Local Gradient Approximation	$$\delta{\ensuremath{\textstyle\mbox{\boldmath${u}$ }} }\approx [{... ...style\mbox{\boldmath${k}$ }} }\ ;\ \ S_{ij} = \frac{\partial u_i}{\partial k_j}$$
Normalized Gradient	$$S_{ij}^{n} = \frac{\overline{k}_j}{u_i(\overline{{\ensuremath{\textstyle\mbox{\boldmath${k}$ }} }})} \frac{\partial u_i}{\partial k_j}$$

Based on the choice of a sensitivity metric and the variation in the model parameters, sensitivity analysis methods can be broadly classified into the following categories:

• Variation of parameters or model formulation: In this approach, the model is run at a set of sample points (different combinations of parameters of concern) or with straightforward changes in model structure (e.g., in model resolution). Sensitivity measures that are appropriate for this type of analysis include the response from arbitrary parameter variation, normalized response and extrema. Of these measures, the extreme values are often of critical importance in environmental applications.
• Domain-wide sensitivity analysis: Here, the sensitivity involves the study of the system behavior over the entire range of parameter variation, often taking the uncertainty in the parameter estimates into account.
• Local sensitivity analysis: Here, the focus is on estimates of model sensitivity to input and parameter variation in the vicinity of a sample point. This sensitivity is often characterized through gradients or partial derivatives at the sample point.