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研究生:蔡侑霖
研究生(外文):Yu-Lin Tsai
論文名稱:多重聚集在圓周上的分配
論文名稱(外文):The Distribution of Multiple Clusters on the Circle
指導教授:林千代林千代引用關係
指導教授(外文):Chien-Tai Lin
學位類別:碩士
校院名稱:淡江大學
系所名稱:數學學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2003
畢業學年度:91
語文別:中文
論文頁數:53
中文關鍵詞:間距變數史德林數
外文關鍵詞:spacingstirling number
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多重聚集( Multiple Clusters )常被流行病學家用來作為檢定病情
的隨機均勻性(uniformity)或群集性(clustering)。 Glaz (1993)曾提出多重聚集可做為檢定資料分布是否均勻的合理統計量。
在這篇論文中,我們首先推導出圓周上聚集個數動差的一般式,
再利用Huffer及Lin (1995) 所寫的程式把 值計算出來,最後便可
得到圓周上聚集個數的分配。因為聚集個數的機率分配非常複雜且
不易計算,我們將利用Huffer及Lin(1995) 所提出的三個以動差法
(method of moments )為原則的逼近法來求取聚集個數之機率分配的
近似值。前兩種逼近法是具有兩種不同參數值的複合卜瓦松逼近法
( Compound Poisson ),第三種逼近法是牽涉一個簡單的二階段馬可
夫鏈( Markov Chain )模式。三種方法所求得的近似值與我們計算出
來的實際機率值比較之後,發現第三種逼近法表現較為準確。
Multiple clusters are usually used by epidemiologist to test the uniformity or clustering of data. Glaz (1993) has proposed that multiple clusters can be a reasonable statistic for testing uniformity.
In this thesis, we obtain a general expression for the moments of multiple clusters on the circle, and then apply the programs proposed by Huffer and Lin (1995) to evaluate the distribution of multiple clusters on the circle. Because the computation of the distribution of multiple clusters on the circle is very complicate and hard to calculate, we adopt Huffer and Lin (1995)’s three different methods approximations by moments to find the distribution of multiple clusters. The first two methods are related to the compound poisson distribution, and the third approximation involes a simple two-state Markov Chain. From the exact probabilities, we obtaind from our procedure discussed in this thesis, we find the third approximation has a better performance than the other two methods.
目 錄
第一章 前言 1
第二章 間距變數
第一節 定義及特性 3
第二節 基本遞迴 (recurrence) 定理 5
第三節 矩陣符號 6
第四節 實例說明 7
第三章 圓上聚集個數的動差
第一節 定義 13
第二節 一般式的推導 17
第三節 實例說明 20
第四章 圓上聚集個數的分配
第一節 圓上聚集個數的機率分配 28
第二節 圓上聚集個數分配之逼近法的比較 36
第五章 結論 46
附錄A 環繞型 ( i = 1 ~ 7) 的推導展開式 47
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