國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
30
第五章 結論
本篇論文介紹如何使用新的 R 套件"rBeta2009"來生成貝他分配及狄氏分配 亂數並評估此套件之有效性、準確性及隨機性。新套件"rBeta2009"無論在 32 位 元或 64 位元處理器的環境下其 CPU 時間比現有的 R 套件"stats"及"MCMCpack"
有顯著的減少,其增進比例約為 10%到 80%。利用 Kolmogorov-Smirnov 檢定與 Ljung-Box 檢定,本文也驗證了此套件的準確性及隨機性。以實際應用的角度來 說,套件"rBeta2009"可以完全取代其他的 R 套件來執行貝他分配及狄式分配的 亂數產生。
此外,我們也介紹了四個可以利用"rBeta2009"套件來產生的分配,分別是反 貝他分配、反狄式分配、Liouville 分配及在凸面區域上的均勻分配。這對於需要 大量運算及電腦模擬相關分配的研究工作者(或企業界)有很大的幫助。
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
31
參考文獻
[1] A.C. Atkinson and Whittaker (1976). A Switching Algorithm for the Generation of Beta Random Variables With at Least One Parameter Less Than One.
Proceedings of the Royal Society of London, Series A, 139, pp. 462-467.
[2] K. Alam, R. Abernathy and C.L.Williams (1993). Multivariate Goodness-of-Fit Tests Based on Statistically Equivalent Blocks. Communication in Statistics, Theory Methods 22, pp. 1515–1533.
[3] A.G. Ashraf and S. Tamás (2009). On Numerical Calculation of Probabilities According to Dirichlet Distribution. Annals of Operations Research, 177, pp.
185–200.
[4] T.W. Anderson (1966). Some Nonparametric Procedures Based on Statistically Equivalent Blocks. Proceedings of International Symposium on Multivariate Analysis. P.R. Krishnaiah ed., Academic Press Inc., New York, pp. 5–27.
[5] T. Bdiri and N. Bouguila (2011). Learning Inverted Dirichlet Mixtures for Positive Data Clustering. Neural Information Processing, ICONIP 2011, Part II, LNCS 7063, pp. 71–78. Springer, Heidelberg.
[6] T. Bdiri and N. Bouguila (2011). Learning Inverted Dirichlet Mixtures for Positive Data Clustering. Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, RSFDGrC 2011. LNCS, 6743, pp. 265–272. Springer, Heidelberg.
[7] R.C.H. Cheng (1978). Generating Beta Variates with Nonintegral Shape Parameters. Communications of the Association for Computing Machinery, 21, pp. 317-322.
[8] C.W. Cheng, Y.C. Hung and N. Balakrishnan (2012). Package "rBeta2009".
URL http://cran.r-project.org/package=rBeta2009.
[9] R.V. Foutz (1980). A test for goodness of fit based on empirical probability
‧
measure. The Annals of Statistics, 8, pp. 989–1001.
[10] K.T. Fang, G.L. Tian and M.Y. Xie. (1997). Uniform Distribution on Convex Polyhedron and Its Applications. Department of Mathematics, Hong Kong Batist University, No. 149.
[11] D.A.S. Fraser (1957). Nonparametric Methods in Statistics, JohnWiley & Sons, NewYork.
[12] T. Hahn (2005). CUBA—a library for multidimensional numerical integration.
Computer Physics Communications, 168, pp. 78–95.
[13] Y.C. Hung, N. Balakrishnan and Y.T. Lin (2009). Evaluation of Beta Generation Algorithms. Communications in Statistics - Simulation and Computation, 38, pp.
750-770.
[14] Y.C. Hung, N. Balakrishnan and C.W. Cheng (2011). Evaluation of Algorithms for Generating Dirichlet Random Vectors. Journal of Statistical Computation and Simulation, 81, pp. 445-459.
[15] J.R.M. Hosking (1981). Fractional Differencing. Biometrica, Vol. 68, No. 1981, pp. 165-176.
[16] J.R.M. Hosking (1980). The Multivariate Portmanteau Statistic. Journal of American Statistical Association, 75, pp. 602-608.
[17] B. Jarle and O.E. Terje (1991). An adaptive algorithm for the approximate calculation of multiple integrals. ACM Transactions on Mathematical Software, Vol. 17, No. 4, pp. 437-451.
[18] B. Jarle and O.E. Terje (1991). Algorithm 698: DCUHRE: An adaptive algorithm for the approximate calculation of multiple integrals. ACM Transactions on Mathematical Software, Vol. 17, No. 4, pp. 452-456.
[19] M.D. Jöhnk (1964). Erzeugung von betaverteilten und gammaverteilten zuffallszahlen. Metrika, 8, pp. 5-15.
‧
optimization. Advances in Applied Probability, 20, pp. 476-478.[21] K. Lange (2005). Applications of the Dirichlet distribution to forensic match probabilities, Genetica , 96, pp. 107–117.
[22] G. Laval, M. SanCristobal and C. Chevalet (2003). Maximum-likelihood and Markov chain Monte Carlo approaches to estimate inbreeding and effective size from allele frequency changes. Genetics, 164 (3), pp. 1189-1204.
[23] J. Liouville (1839). Note sur quelquess integrals définies. Journal de Mathématiques Pures et Appliquées, 4, pp. 225-235.
[24] G.M. Ljung and G.E.P. Box (1978). On a Measure of Lack of Fit in Time Series Models. Biometrika, 65, pp. 297-303.
[25] R.E. Madsen, D. Kauchak and C. Elkan (2005). Modeling Word Burstiness Using the Dirichlet Distribution. Proceeding of the 22nd International Conference on Machine Learning, pp. 545-552.
[26] A.J. McNeila and J. Nešlehová (2010).From Archimedean to Liouville copulas.
Journal of Multivariate Analysis, 101, 8, pp. 1772-1790.
[27] J.B. McDonald and R.J. Butler (1987). Some generalized mixture distributions with an application to unemployment duration. The Review of Economics and Statistics, 69, pp. 232–240.
[28] D.G. Rameshwar and St. P.R. Donald (1987). Multivariate Liouville distribution.
Journal of Multivariate Analysis, 23, pp. 233-256.
[29] R.J. Serfling (1980). Approximation Theorems for Mathematical Statistics.
JohnWiley & Sons, NewYork.
[30] B.W. Schmeiser and A.J.G. Babu (1980). Beta Variate Generation via Exponential Majorizing Functions. Operations Research, 28,pp. 917-926.
[31] K. Sjölander, K. Karplus, M. Brown, R. Hughey, A. Krogh, I. S. Mian and D.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
34
Haussler (1996). Dirichlet mixtures: A method for Improving Detection of Weak but Significant Protein Sequence Homology. The Computer Application in Bioscience, 12, pp. 327-345.
[32] H. Sakasegawa (1983). Stratified rejection and squeeze method for generating beta random numbers. Annals of the Institute Statistical Mathematics, 35,pp.
291-302.
[33] H. Sahai and R.L. Anderson (1973). Confidence regions for variance ratios of the random models for balanced data. Journal of the American Statistical Association, 68, 344, pp. 951-952.
[34] J.W. Tukey (1947). Non-parametric estimation II. Statistically Equivalent Blocks and tolerance regions – the continuous case. The Annals of Mathematical Statistics, 18, pp. 529–539.
[35] G.G. Tiao and I. Guttman (1965a). The inverted Dirichlet distribution with applications. Journal of the American Statistical Association, 60, pp. 793–805.
[36] G.G. Tiao and I. Guttman (1965b). The inverted Dirichlet distribution with applications. Journal of the American Statistical Association, 60, pp. 1251-1252.
[37] G.R. Warnes (2010). Various R programming tools. URL http://cran.r-project.org/web/packages/gtools/gtools.pdf.
[38] E.P. Xing, M.I. Jordan, R.M. Karp and S. Russell (2002). A Hierarchical Bayesian Markovian Model for Motifs in Biopolymer Sequences. Proceedings of Advances in National Information Processing Systems, pp. 1489-1496.
[39] H. Zechner and E. Stadlober (1993). Generating beta variates via patchwork rejection. Computing, 50,pp. 1-18.