具共變數韋能衰退隨機過程之經驗貝氏可靠度分析

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李宜馨(Yi-hsin Li) 查詢紙本館藏

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統計研究所

論文名稱

具共變數韋能衰退隨機過程之經驗貝氏可靠度分析
(An Empirical Bayesian Reliability Analysis of Degradation Test Based on Wiener Process with Covariates)

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摘要(中)

本文考慮具共變數韋能衰退隨機過程之貝氏可靠度分析, 其中漂移係數為共變數之線性函數。實務上同款產品的衰退特徵可能存在個別差異, 並不適合用同一模型來描述所有個體, 因此我們以三種不同的模型, 在共同的先驗分配下, 分別經由馬可夫鏈蒙地卡羅方法(MCMC) 進行貝氏可靠度推論。另一方面當主觀先驗資訊微弱或無法確認資料來源的真實模型時, 我們先經由觀測資料估計具個別差異之模型中參數的先驗分配建立經驗貝氏模型, 模擬結果驗証經驗貝氏方法可在模型不確定時取其折衷進而降低選模錯誤的風險。也就是說, 在先驗資訊不足或不確定產品間是否有差異, 經驗貝氏模型可提供較穩健的可靠度推論。

摘要(英)

For high reliability products, the degradation test can provide more information than accelerated life test to assess the lifetime distribution. In this thesis, we consider the degradation test based on Wiener processes in which the drift coefficient is linear in the covariates. In practice, there may have unit-to-unit variantion of the products with the same type. Therefore, it may not be appropriate using the same model to fit all products. Here, we aim on the Bayesian approach with three difference models. Empirical Bayes methods are used to determine the prior parameters using all the observed data. It not only provides a compromise in the model uncertainty with the advantage of ”borrowing the strength” among the data, but also solve the situation when the prior information is vague. Further, we are interest in reliability inference under given covariates. Simulation study shows that the empirical Baysian method can reduce the risk for fitting wrong models and yield robust reliability inference.

關鍵字(中)

★ 衰退試驗
★ 韋能隨機過程
★ 馬可夫鏈蒙地卡羅方法
★ 貝氏方法
★ 經驗貝氏

關鍵字(英)

★ degradation test
★ Wiener process
★ Markov chain Monte Carlo
★ Bayesian approach
★ empirical Bayes

論文目次

摘要i
Abstract ii
誌謝iii
目錄iv
圖目次vi
表目次vii
第一章緒論1
1.1 研究動機. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 文獻探討. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 研究方法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
第二章具共變數韋能衰退隨機過程之貝氏可靠度分析7
2.1 韋能衰退隨機過程與失效時間. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 具共變數之韋能隨機過程. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 具個別差異性之高維度模型. . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 具部分差異性之折衷模型. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.3 無個別差異之低維度模型. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 產品失效時間之貝氏推論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
第三章經驗貝氏模型17
3.1 經驗貝氏方法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 EM演算法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 共軛性估計法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
第四章模擬研究24
4.1 具個別差異性之高維度模型資料. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 具部分差異性之折衷模型資料. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 無個別差異之低維度模型資料. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.4 穩健性分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.5 案例分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
第五章結論與展望50
參考文獻51

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指導教授

樊采虹

審核日期

2015-7-24

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