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 y = f(\vect{X}), the latter being represented by its Fourier expansion. \vect{X}=\{X^1,\dots,X^{n_X}\} is an input random vector of n_X independent components.

The extended FAST method consists in computing alternately the first order and the total-effect indices of each input. This approach relies upon a Fourier decomposition of the model response. Its key idea is to recast this representation as a function of a \mathit{scalar} parameter s, by defining parametric curves s \mapsto x_i(s), i=1, \dots, n_X exploring the support of the input random vector \vect{X}.
For each input, the same procedure is realized in three steps:
  • Sampling:

    Deterministic space-filling paths with random starting points are defined, i.e. each input X^i is transformed as follows:

    x^i_j = \frac{1}{2} + \frac{1}{\pi} \arcsin(\sin(\omega_i s_j + \phi_i)),
    \quad i=1, \dots, n_X, \, j=1, \dots, N

    where n_X is the number of input variables. N is the length of the discretization of the s-space, with s varying in (-\pi, \pi) by step of 2\pi/N. \phi_i is a random phase-shift chosen uniformly in [0, 2\pi] which enables to make the curves start anywhere within the unit hypercube K^{n_X}=(\vect{X}|0\leq x_i\leq1; i=1, \dots, n_X). The selection of the set \{\phi_1, \dots, \phi_{n_X}\} induces a part of randomness in the procedure. So it can be asked to realize the procedure Nr times and then to calculate the arithmetic means of the results over the Nr estimates. This operation is called \mathit{resampling}.

    \{\omega_i\}, \forall i=1, \dots, n_X is a set of integer frequencies assigned to each input X^i. 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):

    \omega_i = \frac{N - 1}{2M}

    with M 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 16 \leq \omega_i/N_r \leq 64.

    And the maximum frequency of the complementary set of frequencies is:

    \max(\omega_{-i}) = \frac{\omega_i}{2M} = \frac{N - 1}{4M^2}

    with the index ’-i’ which meaning ’all but i’.

    The other frequencies are distributed uniformly between 1 and \max(\omega_{-i}). The set of frequencies is the same whatever the number of resamplings is.

    Let us make an example with eight input factors, N=513 and M=4 i.e. \omega_i = \frac{N - 1}{2M} = 64 and \max(\omega_{-i}) = \frac{N - 1}{4M^2} = 8 with i the index of the input of interest.

    When computing the sensitivity indices for the first input, the considered set of frequencies is : \{64, 1, 2, 3, 4, 5, 6, 8\}.
    When computing the sensitivity indices for the second input, the considered set of frequencies is : \{1, 64, 2, 3, 4, 5, 6, 8\}.

    The transformation defined above provides a uniformly distributed sample for the x_i, \forall i=1, \dots, n_X oscillating between 0 and 1. In order to take into account the real distributions of the inputs, we apply an isoprobabilistic transformation on each x_i before the next step of the procedure.

  • Simulations:

    Output is computed such as: y = f(s) = f(x_1(s), \dots, x_{n_X}(s))

    Then f(s) is expanded onto a Fourier series:

    f(s) = \sum_{k \in \Zset^N} A_k \cos(ks) + B_k \sin(ks)

    where A_k and B_k are Fourier coefficients defined as follows:

    A_k &=& \frac{1}{2\pi}\int_{-\pi}^{\pi}f(s) \cos(ks) \, ds \\
    B_k &=& \frac{1}{2\pi}\int_{-\pi}^{\pi}f(s) \sin(ks) \, ds

    These coefficients are estimated thanks to the following discrete formulations:

    \hat{A}_k &=& \frac{1}{N} \sum_{j=1}^N f(x_j^1,\dots,x_j^{N_X}) cos \left( \frac{2k\pi (j-1)}{N} \right) \quad , \quad -\frac{N}{2} \leq k \leq \frac{N}{2}\\
    \hat{B}_k &=& \frac{1}{N} \sum_{j=1}^N f(x_j^1,\dots,x_j^{N_X}) sin \left( \frac{2k\pi (j-1)}{N} \right) \quad , \quad -\frac{N}{2} \leq k \leq \frac{N}{2}

  • Estimations by frequency analysis:

    The first order indices are estimated as follows:

    \hat{S}_i = \frac{\hat{D}_i}{\hat{D}} \\
              = \frac{\sum_{p=1}^M(\hat{A}_{p\omega_i}^2 + \hat{B}_{p\omega_i}^2)^2}
                     {\sum_{n=1}^{(N-1)/2}(\hat{A}_n^2 + \hat{B}_n^2)^2}

    where \hat{D} is the total variance and \hat{D}_i the portion of D arising from the uncertainty of the i^{th} input. N the size of the sample using to compute the Fourier series and M is the interference factor. Saltelli et al. (1999) recommended to set M to a value in the range [4, 6].

    The total order indices are estimated as follows:

    \hat{T}_i = 1 - \frac{\hat{D}_{-i}}{\hat{D}} \\
              = 1 - \frac{\sum_{k=1}^{\omega_i/2}(\hat{A}_k^2 + \hat{B}_k^2)^2}
                         {\sum_{n=1}^{(N-1)/2}(\hat{A}_n^2 + \hat{B}_n^2)^2}

    where \hat{D}_{-i} is the part of the variance due to all the inputs except the i^{th} input.