# FAST

Fourier amplitude sensitivity testing (FAST) is a variance-based global sensitivity analysis method. 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. Then the ergodic theorem is applied to transform the multi-dimensional integral to a one-dimensional integral in evaluation of the Fourier coefficients. A set of incommensurate frequencies is required to perform the transform and most frequencies are irrational. To facilitate computation a set of integer frequencies is selected instead of the irrational frequencies. The integer frequencies are not strictly incommensurate, resulting in an error between the multi-dimensional integral and the transformed one-dimensional integral. 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. Using integer frequencies in the integral transform, the resulted function in the one-dimensional integral is periodic and the integral only needs to evaluate in a single period. 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. However the calculation by FAST is usually limited to sensitivities referred to as “main effects” or “first-order effects” due to the computational complexity in computing higher-order effects.

## Reference

- R. I. Cukier, C. M. Fortuin, Kurt E. Shuler, A. G. Petschek, and J. H. Schaibly. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. I theory.
*The Journal of Chemical Physics*, 59(8):3873–3878, October 1973. - Gregory J. McRae, James W. Tilden, and John H. Seinfeld. Global sensitivity analysis - a computational implementation of the Fourier amplitude sensitivity test (FAST).
*Computers and Chemical Engineering*, 6(1):15–25, 1982.