逆高斯過程之完整貝氏衰變分析

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姓名

古立丞(Li-Chen Ku) 查詢紙本館藏

畢業系所

統計研究所

論文名稱

逆高斯過程之完整貝氏衰變分析
(A Complete Bayesian Degradation Analysis under Inverse Gaussian Processes)

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摘要(中)

衰變分析包含兩個階段，首先對收集到的與產品失效時間相關之品質特徵值的衰變資料建構統計模型，而由於不同產品間可能存在差異性，衍伸出具隨機效應的衰變模型。其次根據選取的衰變模型推估衰變路徑首次達失效門檻值之時間，進而得失效時間分布之統計推論。另一方面，當樣本數少時，工程師在製程或試驗中對產品特徵與失效機制有深入的專業知識和經驗，因而融合先驗資訊與觀測資料的貝氏方法是很自然的選擇。文獻上關於衰變試驗的貝氏可靠度推論，多止於衰變路徑之建模與分析，在資料配適和模型選擇上鮮少有完備的貝氏論述;關於後續之失效時間的推論則多由衰變模型模擬而得，遑論探討失效時間模型之適合度檢定等重要議題。本文針對具單調性的衰變路徑資料，以具共軛先驗分布之逆高斯過程進行完整的貝氏可靠度分析。對衰變資料模型，除貝氏選模外，亦以貝氏適合度檢定，確認衰變資料的適切模型;繼而推得產品失效時間之貝氏預測分布及相關推論，同時再就貝氏適合度檢定驗證失效時間模型之合理性，提供完整的貝氏分析。最後以四筆實際資料驗證貝氏方法的可行性並與古典統計方法進行比較。

摘要(英)

Degradation analysis consists of two parts. First, a statistical model is constructed for the observations of a quality characteristic related to the failure time of the products. Possible unit to unit variation reflected in the decay rate and/or variability of degradation paths leads to random effects in the degradation models. Secondly, the lifetime inference of the product is derived based on the first passage time to a failure threshold of the selected degradation model. On the other hand, in particular for small sample sizes, Bayesian analysis incorporated with valuable prior information collected from expert opinion or experience, is a useful approach. However, most Bayesian research focuses more on analyzing the degradation data than deriving the lifetime inference. This thesis aims on a complete Bayesian approach not only to formulate the degradation model but also to deduce the consequent lifetime inference by using the Bayesian predictive analysis. Conjugate structure and hierarchical modelling are considered based on the inverse Gaussian processes for monotonic data. Bayesian model selection criteria are used to select the best degradation model and Bayesian goodness of fit is employed for model checking of the lifetime distribution. Four different examples are analyzed via the proposed approach and compared to the classical method.

關鍵字(中)

★ 階層貝氏
★ 馬可夫鏈蒙地卡羅
★ 逆高斯過程
★ 後驗預測 p -值
★ 適合度檢定
★ DIC
★ 預測分布

關鍵字(英)

★ hierarchical Bayesian
★ MCMC
★ inverse Gaussian process
★ posterior predictive p-value
★ goodness of fit
★ DIC
★ predictive distribution

論文目次

摘要 i
Abstract ii
誌謝 iii
目錄 iv
圖目錄 vi
表目錄 viii
第一章緒論 1
1.1 背景與研究動機................................. 1
1.2 文獻探討 .................................... 3
1.3 研究方法 .................................... 5
1.4 本文架構 .................................... 5
第二章逆高斯過程之衰變模型 7
2.1 符號與模型................................... 7
2.2 貝氏架構 .................................... 11
2.2.1 固定效應模型M0 ........................... 11
2.2.2 隨機效應模型M1 ........................... 13
2.2.3 隨機效應模型M2 ........................... 15
2.2.4 隨機效應模型M3 ........................... 16
2.3 MCMC演算法與貝氏推論........................... 17
2.4 貝氏模型選擇.................................. 19
2.4.1 DIC法則................................ 19
2.4.2 貝氏因子................................ 20
第三章壽命分布之貝氏推論與適合度檢定 22
3.1 壽命分布 .................................... 22
3.2 貝氏壽命推論.................................. 24
3.3 貝氏適合度檢定................................. 26
3.4 虛擬失效時間.................................. 28
第四章統計模擬與實例分析 30
4.1 分析步驟 .................................... 30
4.2 統計模擬 .................................... 31
4.3 雷射資料 .................................... 31
4.4 裂紋增長資料.................................. 39
4.5 火車車輪資料.................................. 44
4.6 接觸電阻資料.................................. 49
第五章結論與展望 56
參考文獻 57
附錄 62
A.1定理2.1之證明................................. 62
A.2定理2.2之證明................................. 64

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指導教授

樊采虹(Tsai-Hung Fan)

審核日期

2021-8-10

推文