逆高斯過程之貝氏加速衰變試驗分析與序列預測

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童義軒(Yi-Hsuan Tung) 查詢紙本館藏

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統計研究所

論文名稱

逆高斯過程之貝氏加速衰變試驗分析與序列預測
(Bayesian Accelerated Degradation Analysis and Sequential Prediction based on Inverse Gaussian Processes)

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至系統瀏覽論文 (2026-7-11以後開放)

摘要(中)

對高可靠度產品，常用加速衰變試驗 (accelerated degradation test) 評估產品的壽命資訊。試驗中將產品置於較正常使用環境應力更嚴苛的高應力下，隨時間紀錄和產品壽命相關的品質特徵值 (quality characteristic) 之變化，建構與應力有關的加速衰變模型。衰變試驗中，定義產品壽命為衰變路徑首次抵達給定門檻值 (threshold) 的時間，在加速衰變試驗中經由衰變模型與應力的關係，可經外插至正常應力推估產品的壽命分布，因此應力與模型的關係也會對壽命推論產生影響。本文針對單調加速衰變資料，分別考慮三種應力與參數的關係，配適逆高斯 (inverse Gaussian) 過程，同時考量產品間不同的個別差異性，建構三種具隨機效應 (random-effects) 的貝氏階層模型 (Bayesian hierarchical model)，利用共軛先驗分布 (conjugate prior) 以吉布斯抽樣(Gibbs sampler) 得參數的近似後驗樣本。經貝氏準則及適合度檢定選取最佳的衰變模型提供完整的貝氏壽命預測推論。此外，對類似產品之可靠度推論，本文提出利用過去加速衰變試驗資料的後驗分布作為先驗資訊，配合逐次收集資料與先驗分布的更新，探討序列預測可靠度分析及試驗終止之準則，以縮短試驗時間。最後，並以三筆實例和模擬資料驗證所提方法之可行性。

摘要(英)

Accelerated degradation tests (ADTs) are widely used to assess the lifetime information of highly reliable products. In an ADT, products are placed in harsher environmental stress levels than the normal use condition and values of a quality characteristic (QC) related to the lifetime are observed over time. The ADT model, including a link function between the model parameter(s) and the stress levels, is formulated based on the degradation paths of values of QC with respect to time. The product lifetime can be inferred under normal use condition through extrapolation based on the link function. Therefore, the relationship between the parameter and the stress level is essential to deduce the lifetime inference in an ADT. In this thesis, three types of parameter-stress relationship for monotonic accelerated degradation data based on the inverse Gaussian processes are considered. In addition, to describe the unit-to-unit variation, three different random-effects models are used under ING {color{black}(inverse normal-gamma)} mixture distributions. Individual heterogeneity among products is taken into account by using Bayesian hierarchical models with latent variables. Taking advantage of the ING conjugate structure, the Markov chain Monte-Carlo procedure can be speeded up. A comprehensive Bayesian inference for lifetime prediction is provided through model selection and model checking. Furthermore, we propose a predictive inference which is constructed by the prior distribution sequentially to determine the optimal experimental time under a pre-specified accuracy for predicting the lifetime information. The feasibility of the proposed methods is demonstrated through three examples and the simulation study.

關鍵字(中)

★ 逆高斯過程
★ DIC準則
★ 邊際密度函數
★ 後驗預測p-值
★ 貝氏預測

關鍵字(英)

★ inverse Gaussian process
★ DIC criterion
★ marginal likelihood function
★ posterior p-value
★ Bayesian prediction

論文目次

摘要 v
Abstract vi
誌謝 vii
目錄 viii
圖目錄 x
表目錄 xi
第一章緒論 1
1.1 研究背景與動機 1
1.2 文獻探討 2
1.3 研究方法 4
1.4 本文架構 5
第二章逆高斯過程之貝氏加速衰變模型 6
2.1 逆高斯加速衰變模型 6
2.2 概似函數 8
2.2.1 固定效應模型 9
2.2.2 隨機效應模型 9
2.3 貝氏架構 12
2.3.1 固定效應貝氏模型 12
2.3.2 隨機效應貝氏模型 13
2.4 貝氏選模與適合度檢定 17
2.4.1 偏差訊息法則 (DIC) 17
2.4.2 對數邊際概似函數法則 (LML) 18
2.4.3 適合度檢定 19
第三章貝氏可靠度推論與序列預測分析 21
3.1 壽命分布 21
3.2 平均失效時間及 q-分位數 23
3.3 偽失效時間 24
3.4 序列預測分析 25
3.4.1 先驗分布之更新 26
3.4.2 序列預測 27
第四章實例分析與模擬結果 29
4.1 分析步驟 9
4.2 Device-B 資料 29
4.3 接觸電阻資料 34
4.4 金屬磨損資料 40
4.5 模擬分析 45
4.5.1 參數估計 45
4.5.2 模型選擇 46
4.5.3 序列分析 47
第五章結論與未來研究 52
參考文獻 53
附錄 Gelman-Rubin 統計量收斂圖 57

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指導教授

樊采虹(Tsai-Hung Fan)

審核日期

2023-7-18

推文