3 Basics
3.3 Sensitivity Analysis Methods
3.3.2 Global approach: Fourier Amplitude Sensitivity Test
On the other hand, if the changes in ∆ and ∆ are taken in opposite directions, the boundary is lower ∆ ∆ 0.701. Thus, the principal component Δ is very sensitive to changes and Δ is less sensitive.
Figure 3.12: The approximated region for the matrix with
The main idea of the FAST method is to convert this -dimensional integral into an equivalent one-dimensional integral. Therefore, the parameters are transformed by a set of known transformation functions , so-called search curves
, 1, 2, … , , Eq. 3-54
whereby is a set of frequencies and is a scalar variable, called search variable, which varies over the range ∞ ∞.
According to the ergodic theorem, Weyl shows that the expression for the expected objective function value in Eq. 3-53 is identical to the following one-dimensional integral (Weyl 1938)
→
1
2 , , … , . Eq. 3-55
One demand on the search curve is that it is space-filling, i.e. it passes arbitrarily close to any point in the -dimensional parameter space. To ensure this, the frequencies have to be selected properly so that they are incommensurate, i.e.
0, . Eq. 3-56
However, the space-filling curve is only an ideal which cannot be realized numerically due to the fact that the frequencies cannot be truly incommensurate; therefore, at most one of the incommensurate frequencies can be rational and all others have to be irrational. The numerical value of these irrational numbers cannot be stored exactly in a computer and, hence, an approximation of the incommensurate frequencies by a rational number is required.
To calculate these frequencies, Schaibly and Shuler propose an appropriate set of integer frequencies which implies that the parameters , 1, 2, … , are periodic in on the finite interval , (Schaibly and Shuler 1973). Hence, Eq. 3-55 can be modified to
1
2 , , … , . Eq. 3-57
and the variance of the objective function is then given by
1
2 , , … , . Eq. 3-58
Hereafter , , … , is denoted by .
FREQUENCY SELECTION
The integer frequencies are selected according to Schaibly and Shuler (Schaibly and Shuler 1973) such that they are approximately incommensurate to order , i.e.
0,
| | 1,
Eq. 3-59
whereby the order can be determined by the investigator. The larger the chosen value of , the better the coverage of the space. If → ∞, then the frequencies are completely incommensurate.
The frequency set can be calculated by a trial and error procedure (Cukier et al. 1973) (Cukier, et al., 1973). Schaibly et al. present a set of frequencies which are free of interferences to the fourth order for systems with 5 to 19 considered parameters (Schaibly and Shuler 1973). The listed frequencies have the smallest that still satisfies the conditions from Eq. 3-59 and are so-called minimal sets.
FOURIER EXPANSION
The objective function is expanded to Fourier series
. Eq. 3-60
The Fourier coefficients and are given by 1
2 ∙
1
2 ∙ ,
Eq. 3-61
whereby and .
From Parseval’s theorem, it can be obtained under certain conditions that 1
2 . Eq. 3-62
The squared mean value of can be simplified by using Eq. 3-60
1 2 1
4 .
Eq. 3-63
Inserting Eq. 3-62 and Eq. 3-63 into Eq. 3-58 leads to an expression of the variance in terms of the Fourier coefficients
2 . Eq. 3-64
The part of the variance which corresponds to the uncertainty in the th parameter, the partial variance from Eq. 3-33, can be determined by evaluating the effect of the frequency on the total variance. Therefore, the Fourier coefficients for the frequency and its higher harmonics are summed up
2 . Eq. 3-65
The Fourier amplitudes decreases as increases so that can be approximated by
2 , Eq. 3-66
whereby is the maximum harmonic that is considered and equals the order chosen for the integer frequencies in Eq. 3-59.
The first-order sensitivity indices of Eq. 3-34 are then given by the ratio
∑
∑ . Eq. 3-67
The FAST method requires one model evaluation for each parameter combination which is the main component of the computational cost. Thus, it is desired to minimize the required number of model evaluations. This can be achieved by utilizing the symmetry properties of the search curve. If the frequency set only comprises odd integers, the search curves sin , 1, 2, … , in Eq. 3-54 become symmetric about so that the following symmetry properties hold:
Eq. 3-68
2 2
2 2 .
The range of the search variable s can then be restricted to and the Fourier coefficients in Eq. 3-61 are given by
0
1
0 1
Eq. 3-69
Applying these symmetry properties reduces the required model evaluations by one half.
FAST SAMPLING
The search curve given in equation Eq. 3-54 cannot be utilized in real problems because the number of points on the search curve is infinite. Hence, a finite subset of these points has to be selected. From the Nyquist criteria follows that the minimum number of points has to be taken corresponding to the maximum frequency of the frequency set (Cukier et al.
1975)
2 1. Eq. 3-70
In this manner, harmonics up to an order of can be considered by the analysis. The larger the chosen value of the order , the greater the likelihood that the Fourier amplitude of each input frequency reflects solely the uncertainty of the corresponding parameter. On the other hand, the larger the chosen , the larger the maximum value of the frequency set which still satisfies Eq. 3-59 and the larger the number of sample points required for the evaluation of the Fourier amplitudes. The order is mostly chosen to be 4 or higher. A symmetric and uniformly spaced sample of the search variable in the interval , , including 0, is achieved by (Cukier et al. 1978)
2
2 1
, 1, 2, … , Eq. 3-71
SEARCH CURVE
The search curve in Eq. 3-54 should provide a uniformly distributed sample of the model parameter . Therefore, several transformation functions have been proposed (see Cukier et al. 1973, Koda et al. 1979, Saltelli et al. 1999). Within this work, the search curve suggested by Saltelli et al. is used (Saltelli et al. 1999)
1 2
1 , 1,2, … , Eq. 3-72
This is a set of straight lines which oscillate between 0 and 1 and the empirical distribution can be regarded as more or less uniform.
Saltelli et al. proposed, additionally, a modification of Eq. 3-72 to obtain more flexible sampling schemes. The disadvantage of all these transformations is that they always return the same points. This can be avoided by a random phase-shift chosen uniformly from the interval 0, 2
1 2
1 , 1,2, … , Eq. 3-73
However, the random shift causes that the symmetry properties of in Eq. 3-68 no longer hold. Hence, the search curve has to be sampled over the interval , . By selecting various sets , different search curves can be realized. This procedure is called resampling and denotes the number of curves used per parameter. The minimum sample size from Eq. 3-70 has to be redefined by
WORKING EQUATIONS
For a numerical calculation of the Fourier coefficients, the integrals in Eq. 3-69 have to be approximated by sums to realize their computation. The computational procedure of the FAST method is performed by steps outlined in Figure 3.13 as suggested in (Koda et al.
1979).
2 1 . Eq. 3-74
Figure 3.13: Implementation of the FAST-method without random phase-shift