Fourier Amplitude Sensitivity Test for Efficient Variance Decomposition

Why use Fourier amplitudes for global sensitivity?

A QSP model can be expensive to evaluate, and a variance-based analysis may require many parameter combinations. The Fourier Amplitude Sensitivity Test (FAST) uses a structured periodic design: each uncertain parameter is assigned a frequency, the model is evaluated along a one-dimensional path through parameter space, and output variance is estimated from the resulting Fourier amplitudes. The design can estimate variance effects efficiently, but only when frequency assignment and sampling adequately separate those effects.

FAST is a variance-based method. Its main-effect target is the fraction of output variance attributable to variation in one parameter alone; its total-effect target includes that parameter's contributions through interactions as well. Interpret both relative to a specified output, input distributions, and parameter ranges. Vanilla FAST is designed for independent inputs. For dependent QSP parameters, its ordinary variance decomposition is generally not the appropriate interpretation.

From a periodic path to a Fourier estimate

FAST introduces an angle \(s\) and assigns an integer frequency \(\omega_i\) to each parameter. A convenient mapping for a continuous input with cumulative distribution function \(F_i\) is

Here \(\phi_i\) is an optional phase shift. As \(s\) traverses a full cycle, the transformed input has the specified marginal distribution. The frequencies and phases are chosen so the periodic design approximates the desired joint input distribution and separates spectral contributions. In practice, the model is evaluated at \(s_k=2\pi k/N\), for \(k=0,\ldots,N-1\). These are deterministic evaluations along a structured path, not independent random draws.

For the sampled outputs \(y_k=f(X_1(s_k),\ldots,X_d(s_k))\), center the values by subtracting their grid mean \(\bar y=N^{-1}\sum_k y_k\). For a non-Nyquist positive frequency \(q\), define the cosine and sine coefficients and their power as follows:

The power \(P_q\) is the sampled output variance represented by that Fourier frequency. If the assigned frequency of parameter \(X_i\) is \(\omega_i\), a basic FAST estimate of its main-effect variance is the sum of powers at its first \(M\) harmonics, \(\widehat V_i=\sum_{p=1}^{M}P_{p\omega_i}\). The interference factor \(M\) sets how many harmonics are included. The corresponding estimate is \(\widehat S_i=\widehat V_i/\widehat V\), where \(\widehat V=N^{-1}\sum_k(y_k-\bar y)^2\) is the grid variance. The estimate is meaningful only if these harmonics are distinguishable from other assigned frequencies and their combinations.

Frequency assignment and total effects

A main effect is associated with harmonics of a parameter's assigned frequency. Interaction terms can instead produce combination frequencies. For example, the product of a response component at \(\omega_1\) and one at \(\omega_2\) can generate frequencies \(\omega_1+\omega_2\) and \(|\omega_1-\omega_2|\). A useful assignment must therefore do more than give each input a different frequency: it must keep the harmonics used for main-effect estimates from colliding with other effects, including combinations that matter for the model.

A common FAST strategy assigns a target parameter a high frequency and the other parameters lower frequencies chosen so their own contributions lie in a separable low-frequency band. The low-frequency power then estimates the variance attributable to the other parameters without the target. The target's total effect is estimated by the complement:

Here \(\widehat V_{\sim i}\) is the sum of the resolved powers assigned to the complementary inputs in that design. The high-frequency target and its interactions are left out of that complement, so their variance is included by subtraction. This is the logic of extended FAST (eFAST): typically, a separate design is run for each target parameter. The estimate depends on the complementary band capturing the other inputs' variance without capturing the target's contribution. It is not valid simply to subtract the target's main-effect estimate from one; that would omit interactions.

Worked examples

Diagnostics, alternatives, and limitations

A frequency assignment is a design assumption, not a guarantee that the model has no aliasing. Aliasing occurs when distinct signal components produce the same sampled frequency, or when components above the resolvable range fold onto lower frequencies. For a grid of \(N\) points, non-Nyquist frequencies are at most \(\lfloor(N-1)/2\rfloor\). The maximum harmonic \(M\omega_i\) used for a main effect must remain within the resolvable range, and its harmonics must not overlap other assigned contributions. For even \(N\), the Nyquist component requires its separate calculation.

Useful checks include comparing the sum of all Fourier powers with the grid variance, inspecting power at unexpected frequencies, and repeating the analysis with a longer grid or alternative admissible frequency assignments. A material change in the sensitivity estimates across such designs suggests unresolved harmonics, aliasing, or inadequate quadrature. Low-frequency power can also fail to represent the complementary variance if the chosen complementary band excludes important components. Report the design and these checks, not just the final indices.

FAST is attractive when model evaluations are limited and a structured, periodic design is practical. Saltelli sampling estimates Sobol indices using paired Monte Carlo designs and is often easier to adapt or diagnose in that framework, though it commonly requires more model evaluations. FAST instead relies strongly on spectral separation; neither method removes the need to specify input distributions and verify numerical convergence. Derivative-based measures answer a different local-gradient question and should not be interpreted as variance fractions.

The method also has limits. A one-dimensional periodic path approximates integration over a multidimensional input distribution through frequency design; it does not independently sample every combination. Strong nonlinearities, discontinuities, narrow thresholds, or a limited grid can require more harmonics and samples than the design resolves. Ordinary FAST assumes independent inputs, and total-effect estimates rely on the complementary spectral band being correctly isolated. For correlated parameters, use a method and interpretation designed for dependence rather than applying these indices unchanged.

Common mistake