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 set made up of
values
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
.