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## Generation of correlated pseudorandom varibales

Ma Xu-Bo, Liu Jia-Yi, Xu Jia-Yi, Lu Fan, Chen Yi-Xue
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• #### 摘要

当采用蒙特卡罗方法对很多问题进行研究时，有时需要对多维相关随机变量进行抽样.之前的研究表明：在协方差矩阵满足正定条件时，可以采用Cholesky分解方法产生多维相关随机变量.本文首先对产生多维相关随机变量的理论公式进行了推导，发现采用Cholesky分解并不是产生多维相关随机变量的唯一方法，其他的矩阵分解方法只要能满足协方差矩阵的分解条件，同样可以用来产生多维相关随机变量.同时给出了采用协方差矩阵、相对协方差矩阵和相关系数矩阵产生多维随机变量的公式，以方便以后使用.在此基础上，利用一个简单测试题和Jacobi矩阵分解方法对上述理论进行了验证.通过对大亚湾中微子能谱进行抽样分析，Jacobi矩阵分解和Cholesky矩阵分解结果一致.针对核工程中的不确定性分析常用的238U辐射俘获截面协方差矩阵进行分解时，由于协方差矩阵的矩阵本征值有负值，导致很多矩阵分解方法无法使用，在引入置零修正以后发现，与Cholesky对角线置零修正相比，Jacobi负本征值置零修正的误差更小.

#### 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.

#### 作者及机构信息

###### 通信作者: 马续波, maxb@ncepu.edu.cn
• 基金项目: 国家自然科学基金（批准号：11390383）和中央高校基本科研业务费（批准号：2015ZZD12）资助的课题.

#### Authors and contacts

###### Corresponding author: Ma Xu-Bo, maxb@ncepu.edu.cn
• Funds: Project supported by National Natural Science Foundation of China (Grant No. 11390383) and Fundamental Research Funds for the Central Universities of China (Grant No. 2015ZZD12).

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#### 施引文献

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##### 出版历程
• 收稿日期:  2017-04-17
• 修回日期:  2017-05-16
• 刊出日期:  2017-08-05

## 相关变量随机数序列产生方法

• 1. 华北电力大学核科学与工程学院, 北京 102206
• ###### 通信作者: 马续波, maxb@ncepu.edu.cn
基金项目: 国家自然科学基金（批准号：11390383）和中央高校基本科研业务费（批准号：2015ZZD12）资助的课题.

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