Fourier amplitude sensitivity testing
Encyclopedia
Fourier amplitude sensitivity testing (FAST) is a variance-based global sensitivity analysis
method. The sensitivity value is defined based on conditional variance
s 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 referring to “main effect” or “total effect”.
computer program capable of analyzing either algebraic or differential equation systems was published in 1982. 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 and an extended FAST method able to calculate sensitivity indices referring to “total effect” was developed.
where 
. The ANOVA-like decomposition is
provided that
is a constant and the integral of each term in the sums is zero, i.e.
The conditional variance which characterizes the contribution of each term to the total variance of
is
The total variance is the sum of all conditional variances
The sensitivity index is defined as the normalized conditional variance as
especially the first order sensitivity
which indicates the main effect of the input
.
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
and 
are the real and imaginary part of 
respectively
as
where
is a set of incommensurate frequencies, i.e.
for an integer set of
if and only if 
for every 
.
Then the Fourier coefficients can be calculated by a one-dimensional integral according to the ergodic theorem
can be rational with all others being irrational. 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. A set of integers 
is approximately incommensurate to the order of 
if
for
where
is an integer. The exact incommensurate condition is a extreme case when 
.
Using the integer frequencies the function in the transformed one-dimensional integral is periodic so only the integration over a period of
is required. The Fourier coefficients can be approximately calculated as
The approximation of the incommensurate frequencies for a finite
results in a discrepancy error between the true Fourier coefficients 
, 
and their estimates 
, 
. The larger the order 
is the smaller the error is but the more computational efforts are required to calculate the estimates in the following procedure. In practice 
is frequently set to 4 and a table of resulted frequency sets which have up to 50 frequencies is available. (McRae et al., 1982)
, defines a search curve in the input space. If the frequencies, 
, are incommensurate, the search curve can pass through every point in the input space as 
varies from 0 to 
so the multi-dimensional integral over the input space can be accurately transformed to a one-dimensional integral along the search curve. However, if the frequencies are approximately incommensurate integers, the search curve cannot pass through every point in the input space. If fact the search is repeated since the transform function is periodic, with a period of 
. The one-dimensional integratal can be evaluated over a single period instead of the infinite interval for incommensurate frequencies; However, a compuatational error arises due to the approximation of the incommensuracy.
and
The non-zero integrals can be calculated from sampling points
where the uniform sampling point in
is
The total number of sampling points is
which should satisfy the Nyquist sampling criterion, i.e.
where
is the largest frequency in 
and 
is the maximum order of the calculated Fourier coefficients.
where only the partial sum of the first two terms is calculated and
for determining the number of sampling points. Using the partial sum can usually return an adequately good approximation of the total sum since the terms corresponding to the fundamental frequency and low order frequencies usually contribute most to the total sum. 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. Sine the integer frequencies are not exactly incommensurate there are two integers 
and 
such that 
Interference between the two frequencies can occur if higher order terms are included in the summation.
Similarly the total variance of
can be calculated as
where
denotes the estimated Fourier coefficient of the function of 
inside the bracket and 
is the squared Fourier coefficient of the function 
. Finally the sensitivity referring to the main effect of an input can be calculated by dividing the conditional variance by the total variance.
Sensitivity analysis
Sensitivity analysis is the study of how the variation in the output of a statistical model can be attributed to different variations in the inputs of the model. Put another way, it is a technique for systematically changing variables in a model to determine the effects of such changes.In any...
method. The sensitivity value is defined based on conditional variance
Conditional variance
In probability theory and statistics, a conditional variance is the variance of a conditional probability distribution. Particularly in econometrics, the conditional variance is also known as the scedastic function or skedastic function...
s 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
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
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
Nyquist–Shannon sampling theorem
The Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence...
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 referring to “main effect” or “total effect”.
History
The FAST method originated in study of coupled chemical reaction systems in 1973 and the detailed analysis of the computational error was presented latter in 1975. Only the first order sensitivity indices referring to “main effect” were calculated in the original method. A FORTRANFortran
Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...
computer program capable of analyzing either algebraic or differential equation systems was published in 1982. 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 and an extended FAST method able to calculate sensitivity indices referring to “total effect” was developed.
Variance-based sensitivity
Sensitivity indices of a variance-based method are calculated via ANOVA-like decomposition of the function for analysis. Suppose the function is where . The ANOVA-like decomposition isprovided that
is a constant and the integral of each term in the sums is zero, i.e.The conditional variance which characterizes the contribution of each term to the total variance of
isThe total variance is the sum of all conditional variances
The sensitivity index is defined as the normalized conditional variance as
especially the first order sensitivity
which indicates the main effect of the input
.Multiple Fourier series
One way to calculate the ANOVA-like decomposition is based on multiple Fourier series. The function in the unit hyper-cube can be extended to a multiply periodic function and the multiple Fourier series expansion iswhere the Fourier coefficient is
The ANOVA-like decomposition is
The first order conditional variance is
where
and are the real and imaginary part of respectivelyErgodic theorem
A multi-dimensional integral is required to evaluate for calculating the Fourier coefficients. One way is to transform the multi-dimensional integral into a one-dimensional integral by expressing every input as a function of a new independent variable aswhere
is a set of incommensurate frequencies, i.e.for an integer set of
if and only if for every .Then the Fourier coefficients can be calculated by a one-dimensional integral according to the ergodic theorem
Integer frequencies
At most one of the incommensurate frequencies can be rational with all others being irrational. 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. A set of integers is approximately incommensurate to the order of iffor
where
is an integer. The exact incommensurate condition is a extreme case when .Using the integer frequencies the function in the transformed one-dimensional integral is periodic so only the integration over a period of
is required. The Fourier coefficients can be approximately calculated asThe approximation of the incommensurate frequencies for a finite
results in a discrepancy error between the true Fourier coefficients , and their estimates , . The larger the order is the smaller the error is but the more computational efforts are required to calculate the estimates in the following procedure. In practice is frequently set to 4 and a table of resulted frequency sets which have up to 50 frequencies is available. (McRae et al., 1982)Search curve
The transform,, defines a search curve in the input space. If the frequencies, , are incommensurate, the search curve can pass through every point in the input space as varies from 0 to so the multi-dimensional integral over the input space can be accurately transformed to a one-dimensional integral along the search curve. However, if the frequencies are approximately incommensurate integers, the search curve cannot pass through every point in the input space. If fact the search is repeated since the transform function is periodic, with a period of . The one-dimensional integratal can be evaluated over a single period instead of the infinite interval for incommensurate frequencies; However, a compuatational error arises due to the approximation of the incommensuracy.Sampling
The approximated Fourier can be further expressed asand
The non-zero integrals can be calculated from sampling points
where the uniform sampling point in
isThe total number of sampling points is
which should satisfy the Nyquist sampling criterion, i.e.where
is the largest frequency in and is the maximum order of the calculated Fourier coefficients.Partial sum
After calculating the estimated Fourier coefficients, the first order conditional variance can be approximated bywhere only the partial sum of the first two terms is calculated and
for determining the number of sampling points. Using the partial sum can usually return an adequately good approximation of the total sum since the terms corresponding to the fundamental frequency and low order frequencies usually contribute most to the total sum. 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. Sine the integer frequencies are not exactly incommensurate there are two integers and such that Interference between the two frequencies can occur if higher order terms are included in the summation.Similarly the total variance of
can be calculated aswhere
denotes the estimated Fourier coefficient of the function of inside the bracket and is the squared Fourier coefficient of the function . Finally the sensitivity referring to the main effect of an input can be calculated by dividing the conditional variance by the total variance.