Uncertainty ranking: SRRC

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 monotonic relationships that exist between Y^j and the different components X^i.

The basic method of hierarchical ordering using Spearman’s coefficients deals with the case where the variable Y^j monotonically depends on n_X variables \left\{ X^1,\ldots,X^{n_X} \right\}.

In such a situation, the standard rank correlation coefficients can be more useful in ordering the uncertainty hierarchically: the correlation coefficients \textrm{SRCC}_{X^i,Y^j} between the variables Y^j and X^i attempts to measure the linear influence of rX^i has on rY^j where rX^i (respectively rY^j) is the ranked i-th input variable (respectively the ranked output variable). The coefficients are measured using a linear regression model that links the variable rY^j to the n_x variables rX^1,\ldots,rX^{n_X}:

rY^j = a_0^j + \sum_{i=1}^{n_X} a_i^j rX^i + \varepsilon^j

\varepsilon^j describes a random variable with zero mean and standard deviation \sigma_{\varepsilon}^j independent of the input variables rX^i. If the random variables rX^1,\ldots,rX^{n_X} are independent and with finite variance \Var{rX^k} = (\sigma_k)^2, the variance of rY^j can be estimated as follows:

\Var{rY^j} = \sum_{i=1}^n (a_i^j)^2 \Var{rX^i} + (\sigma_{\varepsilon}^j)^2 = (\sigma^j)^2

The estimators for the regression coefficients a_0^j,\ldots,a_{n_X}^j, and the standard deviation \sigma^j are obtained from a sample of (Y^j,X^1,\ldots,X^{n_X}). Uncertainty ranking by linear regression ranks the n_X variables rX^1,\ldots, rX^{n_X} in terms of the estimated contribution of each rX^k to the variance of Y^j:

C^j_k = \frac{\displaystyle (a_k^j)^2 \Var{rX^k}}{\Var{Y^j}}

which is estimated by:

\widehat{C}^j_k = \frac{\displaystyle (\widehat{a}_k^j)^2 \widehat{\sigma}_k^2}{\displaystyle (\widehat{\sigma^j})^2}

where \widehat{\sigma}_i describes the empirical standard deviation of the sample of the input variables. This estimated contribution is by definition between 0 and 1. The closer it is to 1, the greater the impact the variable X^i has on the dispersion of Y^j.

The contribution to the variance C_i is sometimes described in the literature as the “importance factor”, because of the similarity between this approach to linear regression and the method of cumulative variance quadratic which uses the term importance factor.

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