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 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. Sampling 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 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 by 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. Since 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. References Sensitivity analysis Fourier series