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 . 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 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 Ergodic theorem A multi-dimensional integral must be evaluated in order to calculate the Fourier coefficients. One way to evaluate this multi-dimensional integral is to transform it into a one-dimensional integral by expressing every input as a function of a new independent variable , as follows 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 Implementation 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 if for where is an integer. The exact incommensurate condition is an 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) Search