Variance-based sensitivity analysis

This incurs no loss of generality because any input space can be transformed onto this unit hypercube.

Now, further assuming that the f(X) is square-integrable, the functional decomposition may be squared and integrated to give, Notice that the left hand side is equal to the variance of Y, and the terms of the right hand side are variance terms, now decomposed with respect to sets of the Xi.

Higher-order interaction indices Sij, Sijk and so on can be formed by dividing other terms in the variance decomposition by Var(Y).

Note that this has the implication that, Using the Si, Sij and higher-order indices given above, one can build a picture of the importance of each variable in determining the output variance.

However, when the number of variables is large, this requires the evaluation of 2d-1 indices, which can be too computationally demanding.

It is given as, Note that unlike the Si, due to the fact that the interaction effect between e.g. Xi and Xj is counted in both STi and STj.

The Monte Carlo approach involves generating a sequence of randomly distributed points inside the unit hypercube (strictly speaking these will be pseudorandom).

To calculate the indices using the (quasi) Monte Carlo method, the following steps are used:[1][2] The accuracy of the estimators is of course dependent on N. The value of N can be chosen by sequentially adding points and calculating the indices until the estimated values reach some acceptable convergence.

For the estimation of the Si and the STi for all input variables, N(d+2) model runs are required.

In such cases, there are a number of techniques available to reduce the computational cost of estimating sensitivity indices, such as emulators, HDMR and FAST.