Documentation / Guide méthodologique :

The Sobol method is a Monte-Carlo based estimation that provides the first and total order sensitivity indices at the cost of requiring a total of code assessments. In order to produce these results, a design-of-experiments must be produced, whom size () has to be precised by the user. Instead of generating one design-of-experiments with the requested properties, the program generates actually twice as many data points, split in two different matrices that one could call and (both matrices are different and independent random samplings). The method is detailed in the following paragraph, separately for the first and total order indices, but a schematic view can be found at the end in Equation V.5.

The first step is to compute the first order sensitivity index, whose definition has been given in Equation V.2. This estimation is based on the measurement of the numerator, , which can be written from the definition of variance. Since the second part of previous formula is equivalent to the output expectation, the needed inputs to get the first order indices are: , and . The method discussed in the following paragraphs and illustrated by Equation V.5 is called the pick-and-freeze method. The matrix is passed to the code and assessment are done to get a vector of outputs (this is shown by the first line of Equation V.5). From this, the output properties can easily be estimated, leaving the first term only to be measured. In order to estimate it for the i-Th input variable, one needs to have two different random sampling with the i-Th component (to satisfy the conditional character of the expectation to be squared). This is where the second matrix is used: its i-Th column is replaced by the 's one (peek), creating an new matrix. The last term to be estimated can be computed as

(the mathematical development leading to this formula can be found in many references). This step, whose total cost is code assessments, is shown by the second line and the right-part of the third line in Equation V.5 (following the arrows).

Finally the total order indices are computed starting from the second part of Equation V.3. This second part can also be written as

which looks very much alike Equation V.2 used to compute the first order but instead of a condition on having known (frozen), it is the exact opposite: the condition is to freeze everything but . This is easily doable as this is the only difference between the and matrices. Following the same recipe as for the first order, only the has to be computed with the following formula:

The compulsory step is so to pass the matrix to the code, leading to code assessments, as shown in Equation V.5 by the left part of the third line.

Equation V.5.  Schematic view of the Sobol method

This method is, in Uranie, said to be "� la Saltelli" in contrast with the other implementation (previously used as default) which is said to be "� la Sobol". The difference between the two being the number of assessment used to get a certain precision: for the same results, the implementation "� la Sobol" was requesting and was offering more numerical results as five algorithms were used. The new implementation "� la Saltelli" requests only estimation but only three algorithms are run. This is summarised as follow where the bold name is the default stored in --first-- and --total--: