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題名 以特徵向量法解條件分配相容性問題
Solving compatibility issues of conditional distributions by eigenvector approach
作者 顧仲航
Ku, Chung Hang
貢獻者 宋傳欽
Sung, Chuan Chin
顧仲航
Ku, Chung Hang
關鍵詞 條件機率矩陣
相容性
不可約化
可約化
不可約化對角塊狀矩陣
特徵向量法
最近似聯合分配
兩人零和有限賽局
conditional probability matrix
compatibility
irreducible
reducible
IBD matrix
eigenvector approach
most nearly compatible joint distributions
2-player finite zero-sum game
日期 2010
上傳時間 4-Sep-2013 15:14:23 (UTC+8)
摘要 給定兩個隨機變數的條件機率矩陣A和B，相容性問題的主要課題包
含：(一)如何判斷他們是否相容？若相容，則如何檢驗聯合分配的唯一性
或找出所有的聯合分配；(二)若不相容，則如何訂定測量不相容程度的方
法並找出最近似聯合分配。目前的文獻資料有幾種解決問題的途徑，例
如Arnold and Press (1989)的比值矩陣法、Song et al. (2010)的不可約
化對角塊狀矩陣法及Arnold et al. (2002)的數學規劃法等，經由這些方法
的啟發，本文發展出創新的特徵向量法來處理前述的相容性課題。

當A和B相容時，我們觀察到邊際分配分別是AB′和B′A對應特徵值1的
特徵向量。因此，在以邊際分配檢驗相容性時，特徵向量法僅需檢驗滿足
特徵向量條件的邊際分配，大幅度減少了檢驗的工作量。利用線性代數中
的Perron定理和不可約化對角塊狀矩陣的概念，特徵向量法可圓滿處理相
容性問題(一)的部份。

當A和B不相容時，特徵向量法也可衍生出一個測量不相容程度的簡單
方法。由於不同的測量方法可得到不同的最近似聯合分配，為了比較其優
劣，本文中提出了以條件分配的偏差加上邊際分配的偏差作為評量最近似
聯合分配的標準。特徵向量法除了可推導出最近似聯合分配的公式解外，
經過例子的驗證，在此評量標準下特徵向量法也獲得比其他測量法更佳的
最近似聯合分配。由是，特徵向量法也可用在處理相容性問題(二)的部份。

最後，將特徵向量法實際應用在兩人零和有限賽局問題上。作業研究的
解法是將雙方採取何種策略視為獨立，但是我們認為雙方可利用償付值表
所提供的資訊作為決策的依據，並將雙方的策略寫成兩個條件機率矩陣，
則賽局問題被轉換為相容性問題。我們可用廣義相容的概念對賽局的解進
行分析，並在各種測度下討論賽局的解及雙方的最佳策略。
Given two conditional probability matrices A and B of two random
variables, the issues of the compatibility include: (a) how to determine
whether they are compatible? If compatible, how to check the uniqueness
of the joint distribution or find all possible joint distributions; (b)
if incompatible, how to measure how far they are from compatibility
and find the most nearly compatible joint distribution. There are
several approaches to solve these problems, such as the ratio matrix
method(Arnold and Press, 1989), the IBD matrix method(Song et
al., 2010) and the mathematical programming method(Arnold et al.,
2002). Inspired by these methods, the thesis develops the eigenvector
approach to deal with the compatibility issues.

When A and B are compatible, it is observed that the marginal distributions
are eigenvectors of AB′ and B′A corresponding to 1, respectively.
While checking compatibility by the marginal distributions, the
eigenvector approach only checks the marginal distributions which are
eigenvectors of AB′ and B′A. It significantly reduces the workload.
By using Perron theorem and the concept of the IBD matrix, the part
(a) of compatibility issues can be dealt with the eigenvector approach.

When A and B are incompatible, a simple way to measure the degree
of incompatibility can be derived from the eigenvector approach.
In order to compare the most nearly compatible joint distributions
given by different measures, the thesis proposes the deviation of the
conditional distributions plus the deviation of the marginal distributions
as the most nearly compatible joint distribution assessment standard.
The eigenvector approach not only derives formula for the most
nearly compatible distribution, but also provides better joint distribution
than those given by the other measures through the validations
under this standard. The part (b) of compatibility issues can also be
dealt with the eigenvector approach.

Finally, the eigenvector approach is used in solving game problems.
In operations research, strategies adopted by both players are assumed
to be independent. However, this independent assumption may not
be appropriate, since both players can make decisions through the
information provided by the payoffs for the game. Let strategies of
both players form two conditional probability matrices, then the game
problems can be converted into compatibility issues. We can use the
concept of generalized compatibility to analyze game solutions and
discuss the best strategies for both players in a variety of measurements.
參考文獻 [1] Arnold, B. C. and Press, S. J. (1989), Compatible conditional distributions.
Journal of the American Statistical Association, 84, 152-156.
[2] Arnold, B. C., Castillo, E., and Sarabia, J. M. (2002), Exact and near
compatibility of discrete conditional distributions. Computational Statistics
& Data Analysis, 40, 231-252.
[3] Arnold, B. C., Castillo, E., and Sarabia, J. M. (2004), Compatibility
of partial or complete conditional probability specifications. Journal of
Statistical Planning and Inference, 123, 133-159.
[4] Song, C. C., Li, L. A., Chen, C. H., Jiang, T. J., and Kuo, K. L.
(2010), Compatibility of finite descrete conditional distributions. Statistical
Sinica, 20, 423-440.
[5] Mayer, C. D. (2000), Matrix analysis and applied linear algebra. Society
for Industrial and Applied Mathematics.
[6] 姚景星、劉睦雄(1994)，作業研究應用篇，台灣東華出版社，台北市。
描述 碩士
國立政治大學
應用數學研究所
96751008
99
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0096751008
資料類型 thesis
dc.contributor.advisor 宋傳欽zh_TW
dc.contributor.advisor Sung, Chuan Chinen_US
dc.contributor.author (Authors) 顧仲航zh_TW
dc.contributor.author (Authors) Ku, Chung Hangen_US
dc.creator (作者) 顧仲航zh_TW
dc.creator (作者) Ku, Chung Hangen_US
dc.date (日期) 2010en_US
dc.date.accessioned 4-Sep-2013 15:14:23 (UTC+8)-
dc.date.available 4-Sep-2013 15:14:23 (UTC+8)-
dc.date.issued (上傳時間) 4-Sep-2013 15:14:23 (UTC+8)-
dc.identifier (Other Identifiers) G0096751008en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/60076-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學研究所zh_TW
dc.description (描述) 96751008zh_TW
dc.description (描述) 99zh_TW
dc.description.abstract (摘要) 給定兩個隨機變數的條件機率矩陣A和B，相容性問題的主要課題包
含：(一)如何判斷他們是否相容？若相容，則如何檢驗聯合分配的唯一性
或找出所有的聯合分配；(二)若不相容，則如何訂定測量不相容程度的方
法並找出最近似聯合分配。目前的文獻資料有幾種解決問題的途徑，例
如Arnold and Press (1989)的比值矩陣法、Song et al. (2010)的不可約
化對角塊狀矩陣法及Arnold et al. (2002)的數學規劃法等，經由這些方法
的啟發，本文發展出創新的特徵向量法來處理前述的相容性課題。

當A和B相容時，我們觀察到邊際分配分別是AB′和B′A對應特徵值1的
特徵向量。因此，在以邊際分配檢驗相容性時，特徵向量法僅需檢驗滿足
特徵向量條件的邊際分配，大幅度減少了檢驗的工作量。利用線性代數中
的Perron定理和不可約化對角塊狀矩陣的概念，特徵向量法可圓滿處理相
容性問題(一)的部份。

當A和B不相容時，特徵向量法也可衍生出一個測量不相容程度的簡單
方法。由於不同的測量方法可得到不同的最近似聯合分配，為了比較其優
劣，本文中提出了以條件分配的偏差加上邊際分配的偏差作為評量最近似
聯合分配的標準。特徵向量法除了可推導出最近似聯合分配的公式解外，
經過例子的驗證，在此評量標準下特徵向量法也獲得比其他測量法更佳的
最近似聯合分配。由是，特徵向量法也可用在處理相容性問題(二)的部份。

最後，將特徵向量法實際應用在兩人零和有限賽局問題上。作業研究的
解法是將雙方採取何種策略視為獨立，但是我們認為雙方可利用償付值表
所提供的資訊作為決策的依據，並將雙方的策略寫成兩個條件機率矩陣，
則賽局問題被轉換為相容性問題。我們可用廣義相容的概念對賽局的解進
行分析，並在各種測度下討論賽局的解及雙方的最佳策略。		zh_TW
dc.description.abstract (摘要) Given two conditional probability matrices A and B of two random
variables, the issues of the compatibility include: (a) how to determine
whether they are compatible? If compatible, how to check the uniqueness
of the joint distribution or find all possible joint distributions; (b)
if incompatible, how to measure how far they are from compatibility
and find the most nearly compatible joint distribution. There are
several approaches to solve these problems, such as the ratio matrix
method(Arnold and Press, 1989), the IBD matrix method(Song et
al., 2010) and the mathematical programming method(Arnold et al.,
2002). Inspired by these methods, the thesis develops the eigenvector
approach to deal with the compatibility issues.

When A and B are compatible, it is observed that the marginal distributions
are eigenvectors of AB′ and B′A corresponding to 1, respectively.
While checking compatibility by the marginal distributions, the
eigenvector approach only checks the marginal distributions which are
eigenvectors of AB′ and B′A. It significantly reduces the workload.
By using Perron theorem and the concept of the IBD matrix, the part
(a) of compatibility issues can be dealt with the eigenvector approach.

When A and B are incompatible, a simple way to measure the degree
of incompatibility can be derived from the eigenvector approach.
In order to compare the most nearly compatible joint distributions
given by different measures, the thesis proposes the deviation of the
conditional distributions plus the deviation of the marginal distributions
as the most nearly compatible joint distribution assessment standard.
The eigenvector approach not only derives formula for the most
nearly compatible distribution, but also provides better joint distribution
than those given by the other measures through the validations
under this standard. The part (b) of compatibility issues can also be
dealt with the eigenvector approach.

Finally, the eigenvector approach is used in solving game problems.
In operations research, strategies adopted by both players are assumed
to be independent. However, this independent assumption may not
be appropriate, since both players can make decisions through the
information provided by the payoffs for the game. Let strategies of
both players form two conditional probability matrices, then the game
problems can be converted into compatibility issues. We can use the
concept of generalized compatibility to analyze game solutions and
discuss the best strategies for both players in a variety of measurements.		en_US
dc.description.tableofcontents 目次i
中文摘要iii
Abstract v

1 緒論1
1.1 研究動機與目的. . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 研究架構. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 文獻探討3
2.1 比值矩陣法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 IBD矩陣法. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 不相容程度的測量. . . . . . . . . . . . . . . . . . . . . . . . . 8

3 特徵向量法12
3.1 特徵向量法的簡介. . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 不可約化條件機率矩陣的相容性檢驗. . . . . . . . . . . . . . . . 15
3.3 可約化條件機率矩陣的相容性檢驗. . . . . . . . . . . . . . . . . 21
3.4 各種不相容程度測量方法的比較. . . . . . . . . . . . . . . . . . 28

4 求最近似的聯合機率分配36
4.1 評量近似聯合分配的標準. . . . . . . . . . . . . . . . . . . . . . 36
4.2 各種不相容測度下之最近似聯合分配. . . . . . . . . . . . . . . . 37
4.3 實例說明與比較. . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 在賽局問題上的應用43
5.1 賽局的介紹. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 條件矩陣的廣義相容性. . . . . . . . . . . . . . . . . . . . . . . 48
5.3 廣義相容與賽局解的關係. . . . . . . . . . . . . . . . . . . . . . 51
5.4 以特徵向量法求賽局混合解. . . . . . . . . . . . . . . . . . . . 55

6 結論60

參考文獻61		zh_TW
dc.format.extent 695059 bytes-
dc.format.mimetype application/pdf-
dc.language.iso en_US-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0096751008en_US
dc.subject (關鍵詞) 條件機率矩陣zh_TW
dc.subject (關鍵詞) 相容性zh_TW
dc.subject (關鍵詞) 不可約化zh_TW
dc.subject (關鍵詞) 可約化zh_TW
dc.subject (關鍵詞) 不可約化對角塊狀矩陣zh_TW
dc.subject (關鍵詞) 特徵向量法zh_TW
dc.subject (關鍵詞) 最近似聯合分配zh_TW
dc.subject (關鍵詞) 兩人零和有限賽局zh_TW
dc.subject (關鍵詞) conditional probability matrixen_US
dc.subject (關鍵詞) compatibilityen_US
dc.subject (關鍵詞) irreducibleen_US
dc.subject (關鍵詞) reducibleen_US
dc.subject (關鍵詞) IBD matrixen_US
dc.subject (關鍵詞) eigenvector approachen_US
dc.subject (關鍵詞) most nearly compatible joint distributionsen_US
dc.subject (關鍵詞) 2-player finite zero-sum gameen_US
dc.title (題名) 以特徵向量法解條件分配相容性問題zh_TW
dc.title (題名) Solving compatibility issues of conditional distributions by eigenvector approachen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [1] Arnold, B. C. and Press, S. J. (1989), Compatible conditional distributions.
Journal of the American Statistical Association, 84, 152-156.
[2] Arnold, B. C., Castillo, E., and Sarabia, J. M. (2002), Exact and near
compatibility of discrete conditional distributions. Computational Statistics
& Data Analysis, 40, 231-252.
[3] Arnold, B. C., Castillo, E., and Sarabia, J. M. (2004), Compatibility
of partial or complete conditional probability specifications. Journal of
Statistical Planning and Inference, 123, 133-159.
[4] Song, C. C., Li, L. A., Chen, C. H., Jiang, T. J., and Kuo, K. L.
(2010), Compatibility of finite descrete conditional distributions. Statistical
Sinica, 20, 423-440.
[5] Mayer, C. D. (2000), Matrix analysis and applied linear algebra. Society
for Industrial and Applied Mathematics.
[6] 姚景星、劉睦雄(1994)，作業研究應用篇，台灣東華出版社，台北市。		zh_TW