Correlation Between Inputs in the Uncertainty Tool

OSCAM allows correlation to be defined between pairs of inputs. Correlation means that the two distribution samples are not generated independently but instead there is some relationship between the values that are being generated. OSCAM uses rank correlation, which is defined as meaning that the ranks of values (in order of smallest to highest value) are similar to some specified degree. Rank correlations allow completely different probability distributions to be correlated effectively. Correlation is defined in the range -1 to 1.

Technical explanation
When defining correlation between two uncertainty inputs, one input will be Independent and the other input will be Dependent. The Independent input sample will be generated first and then the Dependent sample will be generated so that it displays the required degree of rank correlation. Note that the heuristic for generating the sample with the required correlation will not exactly generate the target rank correlation (unless the target is 1 or -1), but it will be close to the target value. The Cholesky decomposition with a Monte Carlo approach is used for samples from two simple distributions, with a simple distribution X sorted into the same rank order as the Independent distribution Y to generate another simple distribution, Y, with approximately the correct rank correlation (it is a sample from a population with the required correlation). The Dependent input sample is then generated using the distribution and parameters that are defined in the OSCAM and that sample is then sorted into the same rank order as distribution Y. This means that the Independent distribution and the Dependent distribution have the same rank correlation as the simple distributions X and Y. The approach is commonly used by statistical analysis packages and Excel add-ins and generates samples with close to the required degree of correlation (generally within +/- 0.05 of the required correlation) very quickly. OSCAM can generate two correlated samples of 10,000 values each in a fraction of a second.

Defining correlation between two uncertainty inputs

Any existing correlation relationships are shown on the Uncertainty Input form in the Uncertainty Input Selection List. The Correlation column shows the input identifier (sector code and input number) of the Independent uncertainty input for any correlation relationship, and the target correlation coefficient value.

Correlation between two uncertainty inputs can be defined via the Set Sampling Distribution dialog for the Dependent input (i.e. the input whose sample will be sorted to exhibit the required rank correlation). You can view the Set Sampling Distribution dialog by double-clicking on the Distribution cell, or clicking the ellipses button for the appropriate input.

Uncertainty input list showing correlation relationships

Uncertainty inputs table showing correlation coefficients

The related input to be correlated with is shown at the bottom of the Set Sample Distribution dialog. If no correlation relationship is defined then this will be shown as "<None>".

Set Sampling Distribution dialog with no correlation defined and select correlation button highlighted

Set Probability Distribution dialog with no correlation defined

Clicking on the ellipses button next to the Correlated Input box allows selection of an Independent input to be correlated with. The Correlation Input Selection form will be displayed. Note that only uncertainty inputs that have already been selected from the Uncertainty Input tree, as shown in the Uncertainty Input List, will be available for selection.

Correlation Selection dialog with no correlation selected

Correlation Selection dialog with None selected

The name of the Dependent Input will be shown in the dialog caption at the top. All uncertainty inputs are shown in the table. A filter option allows all uncertainty inputs to be viewed or only inputs from a specified sector. The <None> option will always be shown at the top of the list.

The current selection of the Independent input associated with the Dependent input will be shown by an arrow in the left-hand column of the table. The Dependent input is shown grayed in the text and cannot be selected (this would define correlation with itself). Any ineligible inputs for selection will be shown in red. These cannot be selected because they would cause circular correlation relationships (i.e. a chain of correlation relationships that leads back to the Dependent input). In the example above, MT5 I-L Consumables Cost is shown in red because MT4 O-L Consumables Cost (the input that we are defining the correlation relationship for) is already the Independent input for MT5. This can be seen in the "Correlation" and "Independent Correlation Variable For ..." columns. More information can bee seen by hovering the mouse cursor over any of the cells and viewing the expanded comments and explanations that are shown in the Long Description panel at the bottom of the main OSCAM application form. See below for more information on correlation chains and circular references.

A different Independent correlated input can be selected by clicking on the appropriate row in the table. The selected input will be shown as the highlighted row, but the selection will not be applied until the Apply or OK button is clicked. The Apply button updates the Current Setting (i.e. changes the row with the arrow) without closing the dialog, while the OK button which updates the Current Setting and closes the dialog.

Correlation selection dialog with input selected

Correlation Selection Dialog with input MT3 selected

After clicking the Apply button the arrow will be updated and the Current Setting name will also be updated. The Correlated Value input box allows the correlation coefficient to be edited with a value between -1.000 and 1.000. Clicking the Cancel button will exit the dialog with the Current Setting as the selected Independent input, as shown at the bottom of the dialog.

Correlation Selection dialog after correlation value entered and Apply is clicked

Correlation Selection dialog after Apply clicked

Clicking the OK or Cancel button will close the dialog and the Set Sampling Distribution dialog will be shown again with the updated settings. The Correlated Value in this dialog can be edited to change the change the correlation value.

Set Sampling Distribution dialog showing correlated input

Set Sampling Distribution dialog with correlated input specified

Correlation chains and circular references

An uncertainty input can be the Independent input for several Dependent inputs. For example the number of NFOs and enlisted aircrew crew could both reference the number of pilots as the Independent input in a correlation relationship. It is also possible to select an uncertainty input as the Independent input which itself is the Dependent input in another correlation relationship. For example, the number of NFOs may be correlated with the number of pilots as the Independent input, and number of enlisted aircrew correlated with the number of NFOs as the Independent input. In this example there is an indirect correlation between pilots and enlisted aircrew through correlation relationship chaining. This is perfectly acceptable provided that the same uncertainty input does not appear twice in the same chain. If the input did appear twice in the chain this would mean that it was indirectly correlated with itself, which is not permitted.

Example of permitted correlation chain and potential circular reference

Correlation chain and circular reference example

OSCAM prevents circular correlation chaining in the Correlated Input Selection Dialog by checking the chains and displaying any input that would create a chain in red, and preventing the selection of those inputs. If the mouse is hovered over a cell in the Correlation column the full correlation chain will be shown in the Long Description panel of the main OSCAM application.

Input MT5 is displayed in red and cannot be selected since this would create a circular reference

MT5 is displayed in red showing and cannot be selected to prevent circular reference