Generation of correlated pseudorandom varibales

Abstract

When Monte Carlo method is used to study many problems, it is sometimes necessary to sample correlated pseudorandom variables. Previous studies have shown that the Cholesky decomposition method can be used to generate correlated pseudorandom variables when the covariance matrix satisfies the positive eigenvalue condition. However, some covariance matrices do not satisfy the condition. In this study, the theoretical formula for generating correlated pseudorandom variables is deduced, and it is found that Cholesky decomposition is not the only way to generate multidimensional correlated pseudorandom variables. The other matrix decomposition methods can be used to generate multidimensional relevant random variables if the positive eigenvalue condition is satisfied. At the same time, we give the formula for generating the multidimensional random variable by using the covariance matrix, the relative covariance matrix and the correlation coefficient matrix to facilitate the later use. In order to verify the above theory, a simple test example with 33 relative covariance matrix is used, and it is found that the correlation coefficient results obtained by Jacobi method are consistent with those from the Cholesky method. The correlation coefficients are more close to the real values with increasing the sampling number. After that, the antineutrino energy spectra of Daya Bay are generated by using Jacobi matrix decomposition and Cholesky matrix decomposition method, and their relative errors of each energy bin are in good agreement, and the differences are less than 5.0% in almost all the energy bins. The above two tests demonstrate that the theoretical formula for generating correlated pseudorandom variables is corrected. Generating correlated pseudorandom variables is used in nuclear energy to analyze the uncertainty of nuclear data library in reactor simulation, and many codes have been developed, such as one-, two-and three-dimensional TSUNAMI, SCALE-SS, XSUSA, and SUACL. However, when the method of generating correlated pseudorandom variables is used to decompose the 238U radiation cross section covariance matrix, it is found that the negative eigenvalue appears and previous study method cannot be used. In order to deal with the 238U radiation cross section covariance matrix and other similar matrices, the zero correction is proposed. When the zero correction is used in Cholesky diagonal correction and Jacobi eigenvalue zero correction, it is found that Jacobi negative eigenvalue zero correction error is smaller than that with Cholesky diagonal correction. In future, the theory about zero correction will be studied and it will focus on ascertaining which correction method is better for the negative eigenvalue matrix.