臺灣博碩士論文加值系統

本文主要目的為介紹一個新的隨機取樣法(stochastic simulation)，本方法適用於貝氏模型更新(Bayesian model updating)、貝氏模型選定(Bayesian model class selection)及貝氏模型平均(Bayesian model averaging)，稱之為漸變型馬可夫鍊蒙地卡羅法(transitional Markov chain Monte Carlo approach, TMCMC)。本方法主要理念為避免直接從較困難的事後機率密度函數(posterior probability density functions, posterior PDF)中取樣，而是從一系列較容易取樣的機率密度函數中取樣，進而得到posterior PDF之樣本，而此一系列之PDF會逼近到posterior PDF。TMCMC基本架構為馬可夫鍊蒙地卡羅法(Markov chain Monte Carlo approach, MCMC)，卻較MCMC更加多功及強健(robust)。TMCMC可以從困難的PDF中取樣，例如：多峰形態(multi-modal)PDF及非常尖銳的PDF。TMCMC可以在給定資料的情況下估算出假定模型之可能性(證據值、evidence)，這對於貝氏模型選定非常重要。本文藉由四個範例，展示TMCMC於貝氏模型更新、貝氏模型選定及模型平均之應用。

This thesis presents a newly developed stochastic simulation for Bayesian model updating, model class selection and model averaging, named the transitional Markov chain Monte Carlo approach (TMCMC). The idea behind TMCMC is to avoid the problem of sampling from difficult posterior probability density functions (PDF) but sampling from a series of PDFs that converge to the posterior PDF and that are easier to sample. The TMCMC approach is based on Markov chain Monte Carlo (MCMC), while it is more versatile and robust than MCMC. It is shown that TMCMC is able to draw samples from some difficult PDFs, e.g. multi-modal and very peaked PDFs. The TMCMC approach can also estimate evidence of the chosen probabilistic model class conditioning on the measured data, a key component for Bayesian model class selection and model averaging. Three examples are used to demonstrate the effectiveness of the TMCMC approach in Bayesian model updating, model class selection and model averaging.

中文摘要 I
ABSTRACT II
致　　謝 III
目　　錄 IV
表 目 錄 VI
圖 目 錄 VII
符 號 表 VIII

第一章 緒論 1
1.1 研究動機及目的 1
1.1.1 模型更新 2
1.1.2 模型選定、模型平均 2
1.1.3 研究目標 3
1.1.4 傳統隨機取樣法的限制 4
1.2 研究方法 5
1.3 論文架構 5

第二章 文獻回顧 7
2.1 模型更新 7
2.1.1 一般模型更新 7
2.1.2 貝氏模型更新 8
2.2 模型更新之應用 9
2.3 模型選定 11
2.4 模型平均 12

第三章 問題定義及貝氏模型更新之困難 13
3.1 問題定義 13
3.2 隨機取樣法應用於貝氏模型更新所遭遇之困難 15
3.2.1 重要性取樣法 15
3.2.2 Metropolis-Hasting演算法 18
3.2.3 混合蒙地卡羅法 20
3.2.4 拉普拉斯漸近法 23
3.2.5 Adaptive Metropolis法 24
3.2.6 隨機取樣法小結 25
3.3 貝氏模型選定所遭遇之困難 26

第四章 漸變型馬可夫鍊蒙地卡羅法 28
4.1 基本TMCMC演算法 30
4.2 TMCMC演算法 34

第五章 估算值之機率特性 36

第六章 範例 39
6.1 混合常態分布 39
6.2 四樓線性結構 42
6.3 深開挖案例研究 46
6.4 土壤性質之高斯過程回歸研究 50
6.5 範例討論 55

第七章 結論討論與未來方向 57
7.1 結論討論 57
7.2 未來方向 58

參考文獻 59
表 61
圖 66

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