Sensitivity analysis by Fourier decomposition¶
FAST is a sensitivity analysis method which is based upon the ANOVA decomposition of the variance of the model response , the latter being represented by its Fourier expansion. is an input random vector of independent components.
Deterministic space-filling paths with random starting points are defined, i.e. each input is transformed as follows:
where is the number of input variables. is the length of the discretization of the s-space, with varying in by step of . is a random phase-shift chosen uniformly in which enables to make the curves start anywhere within the unit hypercube . The selection of the set induces a part of randomness in the procedure. So it can be asked to realize the procedure times and then to calculate the arithmetic means of the results over the estimates. This operation is called .
is a set of integer frequencies assigned to each input . The frequency associated with the input of interest is set to the maximum admissible frequency satisfying the Nyquist criterion (which ensures to avoid aliasing effects):
with the interference factor usually equal to 4 or higher. It corresponds to the truncation level of the Fourier series, i.e. the number of harmonics that are retained in the decomposition realized in the third step of the procedure.
In the paper [saltelli1999], for high sample size, it is suggested that .
And the maximum frequency of the complementary set of frequencies is:
with the index ’’ which meaning ’all but ’.
The other frequencies are distributed uniformly between and . The set of frequencies is the same whatever the number of resamplings is.
Let us make an example with eight input factors, and i.e. and with the index of the input of interest.When computing the sensitivity indices for the first input, the considered set of frequencies is : .When computing the sensitivity indices for the second input, the considered set of frequencies is : .etc.
The transformation defined above provides a uniformly distributed sample for the oscillating between and . In order to take into account the real distributions of the inputs, we apply an isoprobabilistic transformation on each before the next step of the procedure.
Output is computed such as:
Then is expanded onto a Fourier series:
where and are Fourier coefficients defined as follows:
These coefficients are estimated thanks to the following discrete formulations:
Estimations by frequency analysis:
The first order indices are estimated as follows:
where is the total variance and the portion of arising from the uncertainty of the input. the size of the sample using to compute the Fourier series and is the interference factor. Saltelli et al. (1999) recommended to set to a value in the range .
The total order indices are estimated as follows:
where is the part of the variance due to all the inputs except the input.