Random number X | Result before modulo m |
---|---|
0 | 17 |
17 | 204 |
4 | 61 |
61 | 688 |
88 | 985 |
85 | 952 |
52 | 589 |
89 | 996 |
96 | 1073 |
73 | 820 |
37 | 424 |
24 | 281 |
8 | 908 |
5 | 105 |
72 | 72 |
9 | 809 |
For the other distributions mentioned above we need the inverse cumulative distribution formula. This means that the random number represents a cumulative probability. This, inserted in the formula, generates the required random draw x from the distribution. For the details see the pages on the distributions mentioned above.
In the LHS case the resulting random draws are not coming in a random order, because the sampling interval are dealt with in sequence. For many follow-up calculations in the MC model it is required to "shake" the vector of random draws to get a random order by "sort and carry" routines.
For many purposes this method is good enough. However, when a graph is drawn of the correlated vectors, i.e. Y versus X, a quite unnatural picture is obtained: the "Saturn effect". The correlated elements are forming an almost perfect straight line, while the unsorted parts form a almost circular cloud of points around it. It is looking at the planet Saturn from its side.
For a negative correlation, the sorted parts of the vectors have to be put into reverse order by a "flip routine".
The Saturn effect can be avoided by using a more sophisticated method as follows.
Assume we wish to simulate a correlation of r = 0.5 between two variables, X and Y, represented by vectors of random draws from their respective distributions. The algorithm is basically a combination of statistics and a sorting procedure:
In the context of aggregation of reserve estimates we may have to deal with more than two vectors. In those cases the vectors that have the same dependence are treated in the above way. Other vectors (prospects) with a different interdependence are separately summed, after which the sums of groups are summed to the grand total. Within a group of 4 prospect with degree of dependence r the correlation matrix would look like follows:
If, for some reason, a correlation matrix with several different intercorrelations would be required, a more sophisticated statistical method would be required (Haugh, M., 2004).
When the prospects have full dependence, the POS of the sum (POS_{Dependent}) will be equal to the maximum POS in the prospect set. In case of complete independence of the prospects, the maximum POS (POS_{Independent}) is reached. In between these two extremes we can interpolate between these values in a linear manner, using the correlation coefficient r. Note that the POS_{Independent} >= POS_{Dependent} in all cases.
In formulas:
For 100% dependent prospects (r = 1.0) the "POS_{dep}" of the sum becomes:
For the independent prospects (r = 0.0) the "POS_{ind} of the sum becomes:
For any r the formula is:
That the difference between the dependent and independent case is proportional with r can be seen in the following Monte Carlo result, in which three prospects were summed with the individual POS's as indicated below the graph, of which 0.90 is the maximum:
Unrisked Sum. Although the above calculation can be done by hand, the of the "unrisked" sum of prospects requires a Monte Carlo addition, including the correlation, whereby in each cycle a prospect resource is added to the total, but only if the binomial simulation of its POS is 1. For the Mean Success Volume again a hand calculation might suffice, provided we have the means of the unrisked volumes of the prospects available. The each mean is multiplied by its POS and added to the sum.
Here n is the number of Monte Carlo cycles.
Similar rules are used for probabilities (POS), using the binomial distribution. If not too skewed, the binomial can be approximated by the normal distribution, especially in this context, with many MC cycles (= large samples).
For an estimate of POS derived from n Monte Carlo cycles, its standard deviation (s is given by the following formula.
Here is an example of confidence ranges from the prospect appraisal program
Gaeapas with 1000 MC cycles.
If the number of MC cycles is small in comparison with the uncertainties from the analysis, it may be that the calculated confidence limits at the lower end become less than zero. In that case the lower confidence must be set to zero. At the upper end, for POS, the 95% limit may exceed 1.00. Also there the POS limit has to be set to 1.00.