Goals and use

The goal of sensitivity analysis is to understand the quantitative sources of uncertainty in model calculations and to identify those sources that contribute the largest amount of uncertainty in a given outcome of interest.

Three types of sensitivity analysis can be distinguished:

  • Screening, which is basically a general investigation of the effects of variation in the inputs but not a quantitative method giving the exact percentage of the total amount of variation that each factor accounts for. The main purpose of screening methods is to identify in an efficient way a short list of the most important sensitive factors, so that in a follow-up uncertainty analysis the limited resources can be used in the most efficient way.
  • Local SA, the effect of the variation in each input factor when the others are kept at some constant level. The result is typically a series of partial derivatives - or an approximation thereof-, one for each factor, that defines the rate of change of the output relative to the rate of change of the input.
  • Global SA, the effects on the outcomes of interest of variation in the inputs, as all inputs are allowed to vary over their ranges. This can be extended to take into account the shape of their probability density functions. This usually requires some procedure for sampling the parameters, perhaps in a Monte Carlo form, and the result is more complex than for local SA. In their book, Saltelli et al. (2000) describe a range of different statistics describing how this type of information can be summarized. Global SA is a variance-analysis based method, using indices expressing the contribution of parameters to the variance in the output (e.g. standardized rank correlation coefficients and partial rank correlation coefficients) (cf. Saltelli et al. 2000).

There is one particular (global) screening method for sensitivity analysis that we consider state of the art and recommend for its computational efficiency: the Morris algorithm (Morris, 1991). The typical case to apply this tool is if there are many parameters and available resources do not allow to specify probability density functions for a full Monte Carlo analysis.

The description of Morris given here is taken from Potting et al., (2001): "The Morris method for global sensitivity analysis is a so-called one step-at-a-time method, meaning that in each run only one input parameter is given a new value. It facilitates a global sensitivity analysis by making a number r of local changes at different points x(1r) of the possible range of input values. The method starts by sampling a set of start values within the defined ranges of possible values for all input variables and calculating the subsequent model outcome. The second step changes the values for one variable (all other inputs remaining at their start values) and calculates the resulting change in model outcome compared to the first run. Next, the values for another variable are changed (the previous variable is kept at its changed value and all other ones kept at their start values) and the resulting change in model outcome compared to the second run is calculated. This goes on until all input variables are changed. This procedure is repeated r times (where r is usually taken between 5 and 15), each time with a different set of start values, which leads to a number of r*(k+1) runs, where k is the number of input variables. Such number is very efficient compared to more demanding methods for sensitivity analysis (Campolongo et al. 1999).

The Morris method thus results in a number of r changes in model outcome from r times changing the input value of a given variable. This information is expressed in so-called elementary effects. These elementary effects are approximations of the gradient δyx of the model output y with respect to a specific value for input variable x. The resulting set of r elementary effects is used to calculate the average elementary effect (to lose dependence of the specific point at which each measure was taken) and the standard deviation. The average elementary effect is indicated by m, and the standard deviation by s. The s expresses whether the relation between input variable and model outcome has a linear (s = 0) or a curvi-linear (s > 0) character. (Campolongo et al. 1999) Curvi-linearity will be caused by curvi-linear (main) effects and interaction effects from the analysed input variable with other ones."

In summary, the Morris method applies a sophisticated algorithm for global SA where parameters are varied one step at a time in such a way that if sensitivity of one parameter is contingent on the values that other parameters may take, the Morris method is likely to capture such dependencies.