Documentation / Guide méthodologique :

The estimation is done by estimating the correlation matrix of the output understudy with the different inputs, leading to a matrix which can be based on values (for SRC and PCC) or on ranks (for SRRC and PRCC). Once this matrix is estimated, it is inverted and the requested coefficients are estimated using the following relations (established in Ref [Iman85] and given here only for illustration purpose):

where and being respectively the number of the input and output under study in the correlation matrix. An important factor for the quality of the regression coefficients is the quality of the model which can be asserted with the value of the and the factors defined in Equation IV.1 and whose computation is performed as followed:

It can be considered that and must be superior to 0.7-0.8 in order to use the regression coefficients. However, these values are not guaranteed-threshold, one should be careful not to only rely on them to state that the underlying hypothesis is correct or not.

When considering SRC, one can rely on the equality introduced in Section V.1.1.1: , where is the Pearson coefficient between the output and the i-Th input. This is interesting, as uncertainty on the estimation of a correlation coefficient can be computed from Fisher's z-transformation [Fisher1921] under certain hypothesis. Given a certain sample of size N, the empirical estimation of a true correlation between two normal-distributed variables (independent and identically distributed) can be transformed into following this recipe:

The nice property of this newly-defined variable is that it follows asymptotically a normal distribution of mean and standard deviation . It is particularly appealing to notice that the standard deviation is independent of the correlation value itself, and only depends on the number of points provided in the initial sample. From there, in order to get a 95% confidence level on the correlation coefficient, one can start from this equation

and invert it to get the 95% confidence interval on the correlation coefficient itself, defined as

Equation V.4. 

This procedure would be fine if the quantity of interest was the correlation coefficient, but in our case, we're interested in the square value of this coefficient. To extrapolate the confidence interval on the correlation coefficient into a proper confidence interval on its squared value, we draw a large sample (of size ) of , following the expected asymptotic behaviour of the Fisher's z-transformed variable but using the estimated value instead of the true one : . This set of points is transformed into new coefficients which are squared in order to get a set of squared-correlation coefficients. From this set, the 2.5% and 97.5% quantile are estimated leading to a resulting 95% confidence interval on the squared-value of the estimated correlation coefficient (and thus on its Sobol interpretation in the linear case).

This procedure has been tested using an linear analytic model (for which it is possible to estimate the expected SRC coefficients) with normal-distributed independent and identically distributed inputs variables. Ten thousand design-of-experiments were generated and the theoretical Sobol indices were included in the estimated confidence interval in 95% of the cases. Running the same protocole with uniform distributions instead of normal ones, the theoretical Sobol indices were in the estimated confidence interval between 95% and 98% of the cases. This illustrate the fact that the resulting confidence interval can be considered exact only if the hypothesis stated above are respected. If not, it anyways provides an interesting insight on the way the estimation converges, without being a quantifiable range.

Finally, the procedure described above relies on the Pearson coefficient, to get an estimation of a confidence interval for Sobol indices in linear model. The exact same procedure can be followed using the Spearman correlation coefficient, which leads then to an estimation of a confidence interval for Sobol indices in monotonic model.