Uncertainty ranking: SRC

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.

The principle of the multiple linear regression model consists in attempting to find the function that links the variable Y^j to the n_x variables X^1,\ldots,X^{n_X} by means of a linear model:

Y^j = a_0^j + \sum_{i=1}^{n_X} a_i^j X^i + \varepsilon^j

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

\Var{Y^j} = \sum_{i=1}^n (a_i^j)^2 \Var{X^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 X^1,\ldots, X^{n_X} in terms of the estimated contribution of each X^k to the variance of Y^j:

C^j_k = \frac{\displaystyle (a_k^j)^2 \Var{X^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:

  • See CorrelationAnalysis_SRC()

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