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相关变量随机数序列产生方法

马续波 刘佳艺 徐佳意 鲁凡 陈义学

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相关变量随机数序列产生方法

马续波, 刘佳艺, 徐佳意, 鲁凡, 陈义学

Generation of correlated pseudorandom varibales

Ma Xu-Bo, Liu Jia-Yi, Xu Jia-Yi, Lu Fan, Chen Yi-Xue
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  • 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.
    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.
      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.
      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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    [2]

    Xu S Y 2006 Monte Carlo Method and its Application in Nuclear Physics Experiment (2nd Ed.) (Beijing:Atomic Energy Press) p1(in Chinese)[许淑艳2006蒙特卡罗方法在实验核物理中的应用(第二版) (北京:原子能出版社)第1页]

    [3]

    Zhu Y S 2016 Statistic Analysis in High Energy Physics (Beijing:Science Press) p1(in Chinese)[朱永生2016高能物理实验统计分析(北京:科学出版社责任有限公司)第1页]

    [4]

    Landau D P, Binder K 2000 A Guide to Monte Carlo Simulations in Statistical Physics (2nd Ed.) (New York:Cambridge University Press) p1

    [5]

    Ferguson D M, Siepmann J I, Truhlar D G 1999 Monte Carlo Methods in Chemical Physics (New York:John Wiley & Sons, Inc.) p1

    [6]

    Matthew R B 2011 Ph. D. Dissertation (Hamilton:Mcmaster university)

    [7]

    Hu Z H, Ye T, Liu X G, Wang J 2017 Acta Phys. Sin. 66 012801(in Chinese)[胡泽华, 叶涛, 刘雄国, 王佳2017物理学报66 012801]

    [8]

    Wen D Z, Zhuo R H, Ding D J, Zheng H, Cheng J, Li Z H 2012 Acta Phys. Sin. 61 220204 (in Chinese)[文德智, 卓仁鸿, 丁大杰, 郑慧, 成晶, 李正宏2012物理学报61 220204]

    [9]

    Guo Q 2011 M. S. Dissertation (Jiangsu:Soochow University) (in Chinese)[郭强2001硕士学位论文(江苏:苏州大学)]

    [10]

    An F P, Balantekin A B, Band H R, et al. 2017 Chin. Phys. C 41 013002

    [11]

    An F P, Balantekin A B, Band H R, et al. 2016 Phys. Rev. Lett. 116 061801

    [12]

    Ivanov K, Avramova M, Kamerow S, Kodeli I, Sartori E, Ivanov E, Cabellos O 2013 Benchmarks for Uncertainty Analysis in Modelling (UAM) for the Design, Operation and Safety Analysis of LWRs, Volume I:Specification and Support Data for Neutronics Cases (Phase I), NEA/NSC/DOC (2013) 7 https://www.oecd-nea.org/science/docs/2013/nsc-doc2013-7pdf

    [13]

    Yamamoto A, Kinoshita K, Watanabe T, Endo T, Kodama Y, Ohoka Y, Ushino T, Nagano H 2015 Nucl. Sci. Engineer. 181 160

    [14]

    Park H J, Shim H J, Kim C H 2012 Sci. Technol. Nucl. Install. 2012 247

    [15]

    Curtis E M 2013 M. S. Dissertation (Hamilton:Mcmaster University)

    [16]

    Dan G C, Mihaela I B 2004 Nucl. Sci. Engineer. 147 204

    [17]

    Zwermann W, Aures A, Gallner L, et al. 2014 Nucl. Engineer. Technol. 46 3

    [18]

    Wan C H, Cao L Z, Wu H C, Zu T J, Shen W 2015 Atom. Energy Sci. Technol. 49 11(in Chinese)[万承辉, 曹良志, 吴宏春, 祖铁军, 沈伟2015原子能科学技术49 11]

    [19]

    Wang X Z, Yu H, Wang W M, Hu Y, Yang X Y 2014 Atom. Energy Sci. Technol. 48 9(in Chinese)[王新哲, 喻宏, 王文明, 胡赟, 杨晓燕2014原子能科学技术48 9]

    [20]

    Macfarlanef R, Kahler A 2010 Nuclear Data Sheets 111 2739

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  • 收稿日期:  2017-04-17
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  • 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.

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.

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