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Even if it is known, on a physical or ecological base, which type of quantitative relation (e.g. linear, sigmoid or exponential) holds between two factors in the cause-effect chain, the exact value of the relevant parameters might be uncertain. These parameters may be uncertain when no measurements are available or when measurements show some scatter.

Figure 1 shows a relation between impact y and cause x, from which measurement data are available. On the basis of these data as well as physical / ecological knowledge the relation is expected to be linear; y=f(x)=a*x. The value of parameter a can be derived from the measurement data. However, the measurements show a large scatter. In case of a deterministic approach it would be necessary to do a conservative assumption on the value of a. A conservative assumption of f(x) could for example be the red line in Figure 1.

Figure 1: Example of linear relation F(x)=a*x with uncertain parameter a

In a probabilistic approach it is possible to take into account the variation of a. The measurement data can be used to estimate a probability density function (pdf) of a. Parameter a is calculated for the line from the origin to each measurement point individually. From the resulting set of different values of a the pdf is estimated. Within the Monte Carlo analysis, each simulation a different value of a will be used, randomly sampled from the pdf.

Figure 2: Estimate of probability density function of a

NB: In this example the cause of the variation of factor a is not known. Before taking into account uncertainties in this way, it has to be excluded that the variation of the uncertain factor is caused by another variable, from which the variation is already taken into account separately. It should be avoided that some uncertainties are incorporated more than once in the probabilistic analysis.