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.. _ranking_pcc:
Uncertainty ranking: PCC and PRCC
---------------------------------
Partial Correlation Coefficients
deal with analyzing the influence the random vector
:math:`\inputRV = \left( X_1,\ldots,X_{\inputDim} \right)` has on a random
variable :math:`Y` which is being studied for uncertainty. Here we
attempt to measure linear relationships that exist between :math:`Y`
and the different components :math:`X_i`.
The basic method of hierarchical ordering using Pearson’s coefficients
deals with the case where the variable :math:`Y` linearly
depends on :math:`\inputDim` variables
:math:`\left\{ X_1,\ldots,X_{\inputDim} \right\}` but this can be misleading
when statistical dependencies or interactions between the variables
:math:`X_i` (e.g. a crossed term :math:`X_i \times X_j`) exist. In such
a situation, the partial correlation coefficients can be more useful in
ordering the uncertainty hierarchically: the partial correlation
coefficients :math:`\textrm{PCC}_{X_i,Y}` between the variables
:math:`Y` and :math:`X_i` attempts to measure the residual influence
of :math:`X_i` on :math:`Y` once influences from all other variables
:math:`X_j` have been eliminated.
The estimation for each partial correlation coefficient
:math:`\textrm{PCC}_{X_i,Y}` uses a sample of size :math:`\sampleSize` denoted by
:math:`\left\{ \left(y^{(1)},x_1^{(1)},\ldots,x_{\inputDim}^{(1)} \right),\ldots, \left(y^{(\sampleSize)},x_1^{(\sampleSize)},\ldots,x_{\inputDim}^{(\sampleSize)} \right) \right\}`
of the vector :math:`(Y,X_1,\ldots,X_{\inputDim})`. This requires the
following three steps to be carried out:
#. Determine the effect of other variables
:math:`\left\{ X_j,\ j\neq i \right\}` on :math:`Y` by linear
regression; when the values of the variables
:math:`\left\{ X_j,\ j\neq i \right\}` are known, the average
forecast for the value of :math:`Y` is then available in the form
of the equation:
.. math::
\begin{aligned}
\widehat{Y} = \sum_{k \neq i,\ 1 \leq k \leq d} \widehat{a}_k X_k
\end{aligned}
#. Determine the effect of other variables
:math:`\left\{ X_j,\ j\neq i \right\}` on :math:`X_i` by linear
regression; when the values of the variables
:math:`\left\{ X_j,\ j\neq i \right\}` are known, the average
forecast for the value of :math:`X_i` is then available in the form
of the equation:
.. math::
\begin{aligned}
\widehat{X}_i = \sum_{k \neq i,\ 1 \leq k \leq d} \widehat{b}_k X_k
\end{aligned}
#. :math:`\textrm{PCC}_{X_i,Y}` is then equal to the Pearson
correlation coefficient
:math:`\widehat{\rho}_{Y-\widehat{Y},X_i-\widehat{X}_i}`
estimated for the variables :math:`Y-\widehat{Y}` and
:math:`X_i-\widehat{X}_i` on the :math:`\sampleSize`-sample of simulations.
One can then class the :math:`d` variables :math:`X_1,\ldots, X_{\inputDim}`
according to the absolute value of the partial correlation coefficients:
the higher the value of :math:`\left| \textrm{PCC}_{X_i,Y} \right|`,
the greater the impact the variable :math:`X_i` has on :math:`Y`.
Partial *Rank* Correlation Coefficients (PRCC) are PRC coefficients
computed on the ranked input variables
:math:`r\inputRV = \left( rX_1,\ldots,rX_{\inputDim} \right)`
and the ranked output variable :math:`rY`.
.. topic:: API:
- See :meth:`~openturns.CorrelationAnalysis.computePCC`
- See :meth:`~openturns.CorrelationAnalysis.computePRCC`
.. topic:: Examples:
- See :doc:`/auto_data_analysis/manage_data_and_samples/plot_sample_correlation`
.. topic:: References:
- [saltelli2000]_
- [helton2003]_
- [kleijnen1999]_
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