以中性模型分析動物的群聚行為

Abstract

動物群聚問題源自於生物數學。生物學家希望找到適當的模型來估計及預測動物群聚的狀況，在本篇報告中， 我們會專注在計算出該隨機變量序列的動差以及中央動差，首先我們會先算出隨機變量序列的期望值，接著我們會利用奇異點分析， 計算出該參數的動差以及中央動差，並且在中央動差的部分我們會使用兩種不同的方式證明，會發現使用奇異點分析的計算會相對一般證明方式簡潔， 凸顯出使用奇異點分析的優點。

當得到動差以及中央動差之後我們就嘗試利用動差法去計算該隨機變量序列是否有一個極限分佈，可惜的是，這個參數並沒有辦法經由動差法得到它的極限分佈。

本篇報告中使用的分析工具為奇異點分析，是解析組合中主要使用的一項工具，我們會透過分析生成函數的奇異點精確地計算出隨機變量序列的量， 甚至還可以知道我們算出來的值的誤差值有多大，是一個非常有用的分析方法。

以下是我的論文概述，在第一章我們會先介紹什麼是動物群聚問題，並且也會介紹前人所做的結果，以及本篇論文中所研究的模型，在第二章中， 我們會介紹解析組合的工具以及一些相關的機率定理，在第三章中我們會計算動物群具模型群數的動差以及中央動差， 並且還會證明該隨機變量序列會符合強大數法則及弱大數法則。第四章我們會計算關於二元搜尋樹去掉葉子之後的外點個數的中央動差以及它的極限分佈。 在第五章中我們會繪製出這兩個參數的統計圖表。最後我們會在第六章給一個結論。

Biologists have long been interested in finding appropriate models for the clustering behavior of social animals. Recently, they have proposed the so-called neutral model. In this thesis, we will study the group pattern problem for the neutral model. More precisely, we will find moments and central moments for the number of groups under the neutral model. In doing so, we will start with the mean and then discuss all moments. As for central moments, we will use two different approaches to find asymptotic expansions. These results will then be used to prove weak and strong laws of large numbers. Moreover, our results show that the limiting distribution cannot be found by the method of moments. This is rather surprising since another quantity, namely the number of terminal nodes in random binary search tree, satisfies almost the same recurrence and the limit law of this quantity can indeed be found with the method of moment. For the sake of comparison, this result will be shown in this thesis as well.

As for the methodology, we will use singularity analysis throughout this thesis. Singularity analysis is a major tool in analytic combinatorics. It can be used to find asymptotic expansions of sequence from the local behavior around singularities of its generating function. Moreover, the error of approximation can be controlled as well. We will see that singularity analysis is an efficient tool in studying the group pattern problem under the neutral model.

We conclude by giving a short outline of this thesis. In Chapter 1, we are going to introduce the group pattner problem in more details and state some known results. In Chapter 2, we will introduce singularity analysis and review some theorems from probability theory. In Chapter 3, we will compute moments and central moments of the number of groups under the neutral model. In addition, we will prove weak and strong laws of large numbers. In Chapter 5, we will calculate central moments of the number of terminal nodes in random binary search trees and show that these results lead to the limit law. In Chapter 5, we will do some numerical computations concerning the limit law of the two quantities analyzed in the previous chapters. Finally, we will give a conclusion in Chapter 6.