# Uncertainty ranking: PCC and PRCC¶

Partial Correlation Coefficients deal with analyzing the influence the random vector has on a random variable which is being studied for uncertainty. Here we attempt to measure linear relationships that exist between and the different components .

The basic method of hierarchical ordering using Pearson’s coefficients deals with the case where the variable linearly depends on variables but this can be misleading when statistical dependencies or interactions between the variables (e.g. a crossed term ) exist. In such a situation, the partial correlation coefficients can be more useful in ordering the uncertainty hierarchically: the partial correlation coefficients between the variables and attempts to measure the residual influence of on once influences from all other variables have been eliminated.

The estimation for each partial correlation coefficient uses a sample of size denoted by of the vector . This requires the following three steps to be carried out:

Determine the effect of other variables on by linear regression; when the values of the variables are known, the average forecast for the value of is then available in the form of the equation:

Determine the effect of other variables on by linear regression; when the values of the variables are known, the average forecast for the value of is then available in the form of the equation:

is then equal to the Pearson correlation coefficient estimated for the variables and on the -sample of simulations.

One can then class the variables according to the absolute value of the partial correlation coefficients: the higher the value of , the greater the impact the variable has on .

Partial *Rank* Correlation Coefficients (PRCC) are PRC coefficients
computed on the ranked input variables
and the ranked output variable .