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| Title: | 有限離散條件分配族相容性之研究
A study on the compatibility of the family of finite discrete conditional distributions. |
| Authors: | 李瑋珊
Li, Wei-Shan |
| Contributors: | 宋傳欽
Song, Chuan-Cin
李瑋珊
Li, Wei-Shan |
| Keywords: | 相容
比值矩陣
秩1正擴張矩陣
不可約化塊狀對角矩陣
二分圖
圖形法
修正比值矩陣法
廣義秩1正擴張矩陣
compatibility
ratio matrix
ROPE matrix
IBD matrix
bipartite graph
Graphical Method
Revised Ratio Matrix Method
GROPE matrix |
| Date: | 2008 |
| Issue Date: | 2009-09-19 12:07:30 (UTC+8) |
| Abstract: | 中文摘要
有限離散條件分配相容性問題可依相容性檢驗、唯一性檢驗以及找出所有的聯合機率分配三層次來討論。目前的文獻資料有幾種研究方法，本文僅分析、比較其中的比值矩陣法和圖形法。
二維中，我們發現簡化二分圖的分支與IBD矩陣中的對角塊狀矩陣有密切的對應關係。在檢驗相容性時，圖形法只需檢驗簡化二分圖中的每個分支，正如同比值矩陣法只需檢驗IBD矩陣中的每一個對角塊狀矩陣即可。在檢驗唯一性時，圖形法只需檢驗簡化二分圖中的分支數是否唯一，正如同比值矩陣法只需檢驗IBD矩陣中的對角塊狀數是否唯一即可。在求所有的聯合機率分配時，運用比值矩陣法可推算出所有的聯合機率分配，但是圖形法則無法求出。
三維中，本文提出了修正比值矩陣法，將比值數組按照某種索引方式在平面上有規則地呈現，可降低所需處理矩陣的大小。此外，我們也發現修正比值矩陣中的橫直縱迴路和簡化二分圖中的迴路有對應的關係，因此可觀察出兩種方法所獲致某些結論的關聯性。在檢驗唯一性時，圖形法是檢驗簡化二分圖中的分支數是否唯一，而修正比值矩陣法是檢驗兩個修正比值矩陣是否分別有唯一的GROPE矩陣。修正比值矩陣法可推算出所有的聯合機率分配。
圖形法可用於任何維度中，修正比值矩陣法也可推廣到任何維度中，但在應用上，修正比值矩陣法比圖形法較為可行。
The issue of the compatibility of finite discrete conditional distributions could be discussed hierarchically according to the compatibility, the uniqueness, and finding all possible joint probability distributions. There are several published methods, but only the Ratio Matrix Method and the Graphical Method are analyzed and compared in this thesis.
In bivariate case, a close correspondence between the components of the reduced bipartite graph and the diagonal block matrices of the IBD matrix can be found. When we examine the compatibility, just as simply each diagonal block matrix of the IBD matrix needs to be examined using the Ratio Matrix Method, so does each component of the reduced bipartite graph using the Graphical Method. When we examine the uniqueness, just as whether the number of the diagonal blocks of the IBD matrix is unique needs to be examined, so does the number of the components of the reduced bipartite graph. The Ratio Matrix Method can provide all possible joint probability distributions, but the Graphical Method cannot.
In trivariate case, this thesis proposes a Revised Ratio Matrix Method, in which we can present the ratio array regularly in the plane according to the index and reduce the corresponding matrix size. It is also found that each circuit in the revised ratio matrix corresponds to a circuit in the reduced bipartite graph. Therefore, the relation between the results of the two methods can be observed. When we examine the uniqueness with the Graphical Method, we examine whether the number of the components in the reduced bipartite graph is unique. But with the Revised Ratio Matrix Method, we examine whether each revised ratio matrix has a unique GROPE matrix. All possible joint probability distributions can be derived through the Revised Ratio Matrix Method.
The Graphical Method can be applied to the higher dimensional cases, so can the Revised Ratio Matrix Method. But the Revised Ratio Matrix Method is more feasible than the Graphical Method in application. |
| Reference: | \\bibitem{Arnold1989}Arnold, B. C. and Press, S. J. (1989), Compatible conditional distributions. \\textit{J. Amer. Statist. Assoc.}, \\textbf{84}, 152-156.
\\bibitem{Arnold2004}Arnold, B. C., Castillo,E., and Sarabia, J. M. (2004), Compatibility of partial or complete conditional probability specifications. \\textit{J. Statist. Plann. Inference}, \\textbf{123}, 133-159.
\\bibitem{Kuo2008}Kuo, K. L. (2008), \\textit{New tools for studying the Ferguson-Dirichlet process and compatibility of a family of conditionals.}，政治大學應用數學系博士論文。
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\\bibitem{Rossman1995}Rossman, Allan J. and Short, Thomas H. (1995), Conditional probability and education reform：Are they compatible？ \\textit{Journal of Statistics Education}, \\textbf{v.3, n.2}.
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\\bibitem{Song2009}Song, C. C., Li, L. A., Chen, C. H., Jiang, T. J., and Kuo, K. L. (2009), Compatibility of finite discrete conditional distributions.\\textit{Statistica Sinica.}(Accepted) To appear.
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\\bibitem{程代展2007}程代展、齊洪勝(2007)，矩陣的半張量積：理論與應用，科學出版社，北京市。
\\bibitem{林義雄1987}林義雄(1987)，初等線性代數(第二冊)，九章出版社，台北市。 |
| Description: | 碩士
國立政治大學
應用數學研究所
94751006
97 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G0094751006 |
| Data Type: | thesis |
| Appears in Collections: | [應用數學系] 學位論文
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