Uncertainty ranking: Pearson’s correlation

This method deals with analyzing the influence the random vector \vect{X} = \left( X^1,\ldots,X^{n_X} \right) has on a random variable Y^j which is being studied for uncertainty. Here we attempt to measure linear relationships that exist between Y^j and the different components X^i.

Pearson’s correlation coefficient \rho_{Y^j,X^i}, defined in , measures the strength of a linear relation between two random variables Y^j and X^i. If we have a sample made up of N pairs (y^j_1,x^i_1), (y^j_2,x^i_2), …, (y^j_N,x^i_N), we can obtain \widehat{\rho}_{Y^j,X^i} an estimation of Pearson’s coefficient. The hierarchical ordering of Pearson’s coefficients is of interest in the case where the relationship between Y^j and n_X variables \left\{ X^1,\ldots,X^{n_X} \right\} is close to being a linear relation:

\begin{aligned} Y^j \simeq a_0 + \sum_{i=1}^{n_X} a_i X^i \end{aligned}

To obtain an indication of the role played by each X^i in the dispersion of Y^j, the idea is to estimate Pearson’s correlation coefficient \widehat{\rho}_{X^i,Y^j} for each i. One can then order the n_X variables X^1,\ldots, X^{n_X} taking absolute values of the correlation coefficients: the higher the value of \left| \widehat{\rho}_{X^i,Y^j} \right| the greater the impact the variable X^i has on the dispersion of Y^j.

(Source code, png, hires.png, pdf)

../../_images/ranking_pearson-1.png

API:

Examples:

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