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研究生:陳峰彪
研究生(外文):Feng-Piao Chen
論文名稱:LCG係數的選擇與多重比較
論文名稱(外文):The multiple comparisons and the selection of coefficients for Linear Congruential Generators(LCG)
指導教授:蔡桂宏
指導教授(外文):Guei-hung Tsai
學位類別:碩士
校院名稱:銘傳大學
系所名稱:應用統計資訊學系碩士班
學門:數學及統計學門
學類:統計學類
畢業學年度:94
語文別:中文
論文頁數:77
中文關鍵詞:二維格子圖線性同餘產生器電腦模擬起始值統計檢定質元素
外文關鍵詞:Computer SimulationLinear Congruential GeneratorTwo Dimensional Lattice PlotSeedPrimitive ElementStatistical Tests
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由於電腦的普及及快速發展,在釵h的科學研究裡,電腦經被當作輔助工具。在統計的領域裡,也常會參考電腦模擬(Computer Simulation)的結果,來處理較為複雜的問題;而在數學方面,也可利用電腦模擬的方法,來解決數值分析上的問題。所以本研究選擇了目前常被使用的套裝軟體中,所預設的線性同餘產生器(Linear Congruential Generator;LCG)隨機數產生器。雖然LCG隨機數產生器已經被Lehmer發表有五十多年的歷史,不過目前還是最被廣泛使用。因此本研究探討LCG隨機數產生器在能產生最長週期(Maximal Period)的各質元素(Primitive Element)情況下之二維格子圖(Two Dimensional Lattice Plot)及統計檢定(Statistical Tests)的結果和評比。並提出了在避免重複模擬的情況之下,使用LCG隨機數產生器時,本研究提供了一些起始值選擇用表給使用者選用。
Due to the well-developed technology of computer, it is used as an assistant in plenty of researches. We would refer to the result of computer simulation to deal with more complicated statistical problems. In the aspect of mathematics, computer simulation can also explain the numeral analysis. Therefore, I select the nowadays most commonly used random number generator, LCG, as the main body. Although the LCG was introduced by Lehmer for more than fifty years, it is still used commonly among people as a random number generator. This thesis presents the performances of the LCG generator in the situation with different primitive elements by using Two Dimensional Lattice Plot and a variety of statistical tests. In order to avoid some kinds of reiterative simulations, we suggest the users of the LCG generator how to select the better seeds and give a few tables of seeds with different multipliers for reference.
中文摘要-----------------------------------------------------------------I
英文摘要-----------------------------------------------------------------II
致謝---------------------------------------------------------------------III
圖目錄-------------------------------------------------------------------VI
表目錄-------------------------------------------------------------------VII
程式目錄-----------------------------------------------------------------IX

第一章 緒論-------------------------------------------------------1
第一節 研究背景與動機--------------------------------------------1
第二節 研究目的---------------------------------------------------3
第三節 論文架構---------------------------------------------------4
第二章 文獻探討---------------------------------------------------5
第一節 實體隨機數產生器------------------------------------------5
第二節 常見的電腦隨機數產生器------------------------------------6
第三節 統計檢定理論概述------------------------------------------16
第三章 研究方法--------------------------------------------------18
第一節 線性同餘法係數的選擇-------------------------------------18
第二節 選擇係數的方法和流程-------------------------------------19
第三節 產生器的選取和比較----------------------------------------24
第四章 實際驗證及分析--------------------------------------------26
第一節 乘數B對於產生器的影響------------------------------------26
第二節 乘數B對於格子圖的影響------------------------------------31
第三節 乘數B對於統計檢定的影響----------------------------------44
第四節 起始值對於產生器的影響------------------------------------48
第五章 結論與建議-------------------------------------------------50
第一節 各係數對於產生器的影響------------------------------------50
第二節 未來方向與建議--------------------------------------------52
中英文參考文獻-----------------------------------------------------53
附錄1:程式集------------------------------------------------------56
附錄2:TESTU01 SmallCrush,B=16807程式輸出報表--------------61
附錄3:其他起始值統計檢定結果-------------------------------------71
附錄4:起始值選擇用表---------------------------------------------74
英文參考文獻

Deng, L.Y. and Lin, D. K. J., Random number generation for the new century, The American Statistician, 54(2):145–150, 2000.

Deng, L.Y. and Xu H., Design, search, and implementation of high-dimensional, efficient, long-cycle, and portable uniform random variants generator, Technical report, Department of Statistics, University of California at Los Angeles, 2002.

Eichenauer, J. and Grothe, H. and Lehn, J., Marsaglia’s lattice test and non-linear congruential pseudo random generators, Metrika, 35, 241-250, 1988.

Hull T. E., Dobell A. R. Mixed congruential random number generators for binary machines, J. ACM 11(1): 31-40, 1964.

Grothe H., Matrix generators for pseudo-random vector generation, Statist. 28, 233-238, 1987.

Knuth D. E., The art of computer programming Vol 2: Semi-numerical Algorithms, 2nd ed., Addison-Wesley, Reading MA., 1981.

Knuth, D. E. The art of computer programming Vol 2: Semi-numerical Algorithms, 3rd. Addison-Wesley, Reading, MA., 1998.

L’Ecuyer P., Efficient and portable combined random number generators, Communications of the ACM, 31(6):742–749 and 774, 1988.

L’Ecuyer P., Uniform random number generation. Annals of Operations Research, 53:77–120, 1994.

L’Ecuyer P., TestU01-1.0: Empirical testing of random number generators, (2006).

Lehmer D. H., On large-scale digital calculating machinery, Cambridge: Harvard University Press, 141-146, 1951.

Lidl, R. and Niederreiter H., Introduction to finite fields and their applications, Cambridge University Press, 1986.

Marsaglia, G., Random number fall mainly in the planes, Nat. Acad. Sci. Press, 60, 25-28,1968.

Marsaglia, G. and Tsay, L. H., Matrices and the structure of random number sequences, Linear Algebra and its Applications, 67:147–156, 1985.

Marsaglia G., DIEHARD: A battery of tests of randomness, http://stat.fsu.edu/~geo/diehard.html, (Last Update).

Niederreiter H., Random number generation and Quasi-Monte Carlo methods, vol.63 of SIAM CBMS-NSF Regional Conference Series in Applied Mathematics SIAM Philadelphia, (Last Update).

NIST, Specification for the Advanced Encryption Standard (aes), NIST special publication197 (FIPS-197), National Institute of Standards and Technology (NIST), http://csrc.nist.gov/encryption/aes/, (Last Update).

Ripley B. D., Thoughts on pseudorandom number generators. Journal of Computational and Applied Mathematics, 31:153–163, 1990.

Wichmann B. A. and Hill I. D., An efficient and portable pseudo-random number generator, Applied Statistics, 31(2):188-190, 1982.

Wu P. C., Multiplicative congruential random number generators with multiplier ±2k1 ± 2k2 and modulus 2p − 1, ACM Transactions on Mathematical Software, 23(2):255–265, 1997.

Zierler N., Linear recurring sequences J. SIAM, 7, 31-48, 1959.


中文參考文獻:

孟祥仁,隨機數的統計檢定與比較,銘傳大學應用統計研究所碩士論文,2004。
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