.. _ranking_pcc: Uncertainty ranking: PCC ------------------------ This method deals with analyzing the influence the random vector :math:\vect{X} = \left( X^1,\ldots,X^{n_X} \right) has on a random variable :math:Y^j which is being studied for uncertainty. Here we attempt to measure linear relationships that exist between :math:Y^j and the different components :math:X^i. The basic method of hierarchical ordering using Pearsonâ€™s coefficients deals with the case where the variable :math:Y^j linearly depends on :math:n_X variables :math:\left\{ X^1,\ldots,X^{n_X} \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^j} between the variables :math:Y^j and :math:X^i attempts to measure the residual influence of :math:X^i on :math:Y^j 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^j} uses a set made up of :math:N values :math:\left\{ (y^j_1,x_1^1,\ldots,x_1^{n_X}),\ldots,(y^j_N,x_N^1,\ldots,x_N^{n_X}) \right\} of the vector :math:(Y^j,X^1,\ldots,X^{n_X}). 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^j by linear regression; when the values of variable :math:\left\{ X^j,\ j\neq i \right\} are known, the average forecast for the value of :math:Y^j is then available in the form of the equation: .. math:: \begin{aligned} \widehat{Y^j} = \sum_{k \neq i,\ 1 \leq k \leq n_X} \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 variable :math:\left\{ X^j,\ j\neq i \right\} are known, the average forecast for the value of :math:Y^j is then available in the form of the equation: .. math:: \begin{aligned} \widehat{X}^i = \sum_{k \neq i,\ 1 \leq k \leq n_X} \widehat{b}_k X^k \end{aligned} #. :math:\textrm{PCC}_{X^i,Y^j} is then equal to the Pearsonâ€™s correlation coefficient :math:\widehat{\rho}_{Y^j-\widehat{Y^j},X^i-\widehat{X}^i} estimated for the variables :math:Y^j-\widehat{Y^j} and :math:X^i-\widehat{X}^i on the :math:N-sample of simulations. One can then class the :math:n_X variables :math:X^1,\ldots, X^{n_X} according to the absolute value of the partial correlation coefficients: the higher the value of :math:\left| \textrm{PCC}_{X^i,Y^j} \right|, the greater the impact the variable :math:X^i has on :math:Y^j. .. topic:: API: - See :py:func:~openturns.CorrelationAnalysis_PCC .. topic:: Examples: - See :doc:/auto_data_analysis/manage_data_and_samples/plot_sample_correlation .. topic:: References: - Saltelli, A., Chan, K., Scott, M. (2000). "Sensitivity Analysis", John Wiley \& Sons publishers, Probability and Statistics series - J.C. Helton, F.J. Davis (2003). "Latin Hypercube sampling and the propagation of uncertainty analyses of complex systems". Reliability Engineering and System Safety 81, p.23-69 - J.P.C. Kleijnen, J.C. Helton (1999). "Statistical analyses of scatterplots to identify factors in large-scale simulations, part 1 : review and comparison of techniques". Reliability Engineering and System Safety 65, p.147-185