The sensitivity value is defined based on conditional variances which indicate the individual or joint effects of the uncertain inputs on the output.
FAST first represents conditional variances via coefficients from the multiple Fourier series expansion of the output function.
However, the integer frequencies can be selected to be incommensurate to any order so that the error can be controlled meeting any precision requirement in theory.
Next, since the continuous integral function can be recovered from a set of finite sampling points if the Nyquist–Shannon sampling theorem is satisfied, the one-dimensional integral is evaluated from the summation of function values at the generated sampling points.
FAST is more efficient to calculate sensitivities than other variance-based global sensitivity analysis methods via Monte Carlo integration.
The FAST method originated in study of coupled chemical reaction systems in 1973[1][2] and the detailed analysis of the computational error was presented latter in 1975.
[3] Only the first order sensitivity indices referring to “main effect” were calculated in the original method.
A FORTRAN computer program capable of analyzing either algebraic or differential equation systems was published in 1982.
[4] In 1990s, the relationship between FAST sensitivity indices and Sobol’s ones calculated from Monte-Carlo simulation was revealed in the general framework of ANOVA-like decomposition [5] and an extended FAST method able to calculate sensitivity indices referring to “total effect” was developed.
[6] Sensitivity indices of a variance-based method are calculated via ANOVA-like decomposition of the function for analysis.
One way to calculate the ANOVA-like decomposition is based on multiple Fourier series.
in the unit hyper-cube can be extended to a multiply periodic function and the multiple Fourier series expansion is where the Fourier coefficient is The ANOVA-like decomposition is The first order conditional variance is where
A multi-dimensional integral must be evaluated in order to calculate the Fourier coefficients.
Then the Fourier coefficients can be calculated by a one-dimensional integral according to the ergodic theorem [7]
Since the numerical value of an irrational number cannot be stored exactly in a computer, an approximation of the incommensurate frequencies by all rational numbers is required in implementation.
Without loss of any generality the frequencies can be set as integers instead of any rational numbers.
The exact incommensurate condition is an extreme case when
results in a discrepancy error between the true Fourier coefficients
is the smaller the error is but the more computational efforts are required to calculate the estimates in the following procedure.
, defines a search curve in the input space.
, are incommensurate, the search curve can pass through every point in the input space as
However, if the frequencies are approximately incommensurate integers, the search curve cannot pass through every point in the input space.
The one-dimensional integral can be evaluated over a single period instead of the infinite interval for incommensurate frequencies; However, a computational error arises due to the approximation of the incommensuracy.
The approximated Fourier can be further expressed as and The non-zero integrals can be calculated from sampling points where the uniform sampling point in
is the maximum order of the calculated Fourier coefficients.
After calculating the estimated Fourier coefficients, the first order conditional variance can be approximated by where only the partial sum of the first two terms is calculated and
Additionally, the Fourier coefficient in the summation are just an estimate of the true value and adding more higher order terms will not help improve the computational accuracy significantly.
Interference between the two frequencies can occur if higher order terms are included in the summation.
denotes the estimated Fourier coefficient of the function of
Finally the sensitivity referring to the main effect of an input can be calculated by dividing the conditional variance by the total variance.