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## Monte Carlo analysis

**In the following is explained how a Monte Carlo analysis can be used in cause-effect chain modeling. The descirption of other applications of the Monte Carlo analysis can be found** ** here ** **.**

After defining probability density functions for all relevant stochastic input variables, see 'How to use', the Monte Carlo analysis can be carried out. From all probability density functions (pdf's), values of the stochastic variables will be chosen randomly.

*Figure 1: example schematization of a Monte-Carlo analysis*

### Dependencies between different input-variables

An important question for the set-up of the Monte Carlo analysis is whether or not dependencies exist between the different stochastic input variables. If dependencies or correlations exist (or if two variables depend on the same, third variable), these have to be incorporated in the analysis. It wouldn't be correct to take random samples from the pdf's of these input variables as if they were independent from each other. However, in case of a weak correlation between variables, the effect on the model results of this correlation might be negligible. In that case using correlated samples in the Monte Carlo analysis is unnecessary.

### Numerical models

The application of a Monte Carlo analysis might be difficult if numerical models form part of the quantification of the cause-effect chain. In case of a long computation time of the numerical model, doing a large number of simulations within the Monte Carlo analysis might take a very long time or is even virtually impossible (dependent on computation capacity). A well-considered simplification of the numerical model might be necessary. An example can be found in Van Kruchten (2008). Smaller sample sizes might also be a solution using specific sampling techniques (e.g., Latin hypercube). Using these techniques one can cover most of the parametric space with less samples.

## Results

The probabilistic analysis can be carried out in two different ways, leading to a different presentation of the results.

### Option 1: change of the population size due to dredging

Within each single simulation of the Monte Carlo analysis, the cause-effect chain has been modeled for the reference as well as the dredging scenario, using the same sample of input variables. In this way, for example a population size has been modeled twice per simulation (using the same sample): one time including the effect of dredging and one time without the effect of dredging (reference). Per simulation the modeled population size for the dredging scenario is compared with the modeled population size for the reference scenario. This results in a change of the population size due to the dredging. In case of a Monte Carlo analysis with 10,000 simulations, the change of the population size is computed 10,000 times, for 10,000 different samples of input variables. From these 10,000 results the probability distribution can be constructed for the change of the population size due to dredging.

Figure 2 shows an example of such probability distribution function. The horizontal axis shows the possible relative change of the population size due to dredging. The vertical axis shows the probability that the impact that will occur is smaller (value more positive) than the accompanying value at the horizontal axis.

*Figure 2: Example of probability distributions, option 1*

### Option 2: population development in a reference and dredging scenario

A second option is to carry out the Monte Carlo analysis twice (2*10,000 simulations): once for the reference scenario and once for the sand mining scenario. Within each simulation, the population size at *t=0* can be compared with the population size at *t=x*. The ratio between these population sizes shows whether a population grows or declines. A probability distribution of this ratio can be constructed for the reference as well as the dredging scenario.

An example is shown in Figure 3. In the reference scenario the probability is about 50% that the population decreases over 35 years. For some dredging scenarios the probability on a decline of the population size increases.

*Figure 3: Example of probability distributions, option 2*