Fourier Amplitude Sensitivity Test

In the FAST method the m-dimensional integral in Eq. (5.24) can be transformed to a one-dimensional integral using the following function  

During the FAST analysis, N parameter sets are defined by selecting N points equidistantly in the interval n s x, then the corresponding F,- model results are calculated (once for each parameter set) and a Fourier analysis of the results is carried out. In this way the variance fractimi of the total variance of K,- can be obtained  

If the Fourier coefficients and their harmonics are calculated at the frequency coj, then the partial variance caused by parameter Xj is calculated from 

In this equation index r refers to the r-th harmonics of the base frequency coj. The partial variance Sy is the fraction of the total variance of the model result T, caused by parameter xf. 

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Examples of mathematical methods include nominal range sensitivity analysis (Cullen Frey, 1999) and differential sensitivity analysis (Hwang et al., 1997 Isukapalli et al., 2000). Examples of statistical sensitivity analysis methods include sample (Pearson) and rank (Spearman) correlation analysis (Edwards, 1976), sample and rank regression analysis (Iman Conover, 1979), analysis of variance (Neter et al., 1996), classification and regression tree (Breiman et al., 1984), response surface method (Khuri Cornell, 1987), Fourier amplitude sensitivity test (FAST) (Saltelli et al., 2000), mutual information index (Jelinek, 1970) and Sobol s indices (Sobol, 1993). Examples of graphical sensitivity analysis methods include scatter plots (Kleijnen Helton, 1999) and conditional sensitivity analysis (Frey et al., 2003). Further discussion of these methods is provided in Frey Patil (2002) and Frey et al. (2003, 2004). 

The first widely used global method was the Fourier Amplitude Sensitivity Test (FAST) (for a review see [83]). In the FAST method, all rate parameters were simultaneously perturbed by sine functions with incommensurate frequencies. Fourier analysis of the solution of the model provided the variance crf(t) of concentration i, and also the variance o- (t) of c, arising from the uncertainty in the /th parameter. Their ratio... 

McRae, G. J., Tilden, J. W., and Seinfeld, J. H. (1982) Global sensitivity analysis - a computational implementation of the Fourier Amplitude Sensitivity Test (FAST), Computers and Chem. Engineering. 6, 15-25. 

The algorithms for the (Extended) Fourier Amplitude Sensitivity Test (EFAST) and the Random Balance Design (RBD) depend on the ergodic (measure-preserving) map... 

Fourier-based techniques. For these methods, the input parameter realisations have to fulfil special frequency properties. Then a fi quency decomposition of the output maps different frequencies attributed to the input factors to different fractions of the variance of the output. Different frequency selection schemes have been developed, named Fourier Amplitude Sensitivity Test (FAST), Extended FAST (EFAST), and Random Balance Design (RBD). They can estimate first and/or total effects. 

Several algorithms have been proposed to compute Sobol s indices. Only the Fourier Amplitude Sensitivity Test (FAST), see Cukier et al. (1973), has been considered in this study. Itwas developed, before Sobol provided his variance decomposition, to compute first... 

Saltelli, A., Bolado, R. An alternative way to compute fourier amplitude sensitivity test (FAST). Comput. Stat. Data Anal. 26,445-460 (1998)... 

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