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研究生:姚禹丞
研究生(外文):Yu-Cheng Yao
論文名稱:奈米溶膠保存期限之統計推論
論文名稱(外文):Shelf-life Prediction of Nano-Sol Products
指導教授:曾勝滄曾勝滄引用關係
指導教授(外文):Sheng-Tsaing Tseng
學位類別:博士
校院名稱:國立清華大學
系所名稱:統計學研究所
學門:數學及統計學門
學類:統計學類
論文種類:學術論文
論文出版年:2016
畢業學年度:104
語文別:中文
論文頁數:64
中文關鍵詞:奈米溶膠試驗雙常態混合分布保存期限估計pH值加速衰變模型期望條件最大演算法模型誤判分析最佳試驗計畫
外文關鍵詞:Nano-solmixture of two normal distributionsshelf-life predictionpH acceleration modelexpectation/conditional maximization (ECM) algorithmmodel mis-specificationoptimal test plan
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本論文針對奈米溶膠產品的保存期限進行相關壽命推論工作。有別於傳統在非液態高可靠度產品上,採用溫度、溼度或電壓來執行加速壽命/衰變實驗,本研究以酸鹼 (pH) 值作為加速因子來推估奈米溶膠一類的液態高可靠度產品之保存期限。
奈米溶膠中奈米顆粒之分布為一典型的直方圖值 (histogram-valued) 資料,本論文提出以雙常態混合模型來描述奈米顆粒分布,進而建構出pH值加速衰變模型來完整描述在不同pH值下,奈米溶膠內的奈米顆粒分布隨時間之動態變化情形;然後透過期望條件最大演算法 (Expectation/Conditional Maximization algorithm) 來求得模型中未知參數的最大概似估計值,進而預測產品於正常使用條件下之保存期限及其95%信賴區間。
本論文亦探討在試驗總成本不超過事先給定的預算下,尋找pH值加速衰變試驗之最佳試驗配置 (optimal testing plan),使產品保存期限估計值之近似變異數可極小化;亦即決定在每個pH值下的最佳量測次數、量測頻率及其樣本數。藉由敏感度分析 (sensitivity analysis) 可發現本研究所提的最佳試驗計畫之表現頗為穩健。
最後,本論文探討了奈米顆粒分布發生誤判時,對於產品保存期限估計值之準確度 (accuracy) 與精確度 (precision) 的影響。具體來說,本研究推導出當模型發生誤判時,產品保存期限之估計量的大樣本漸近分布,藉此結果可探討模型誤判對於產品保存期限之影響。由實證研究可發現當模型發生誤判實,其對於產品保存期限估計值之準確度及精確度的影響皆不容忽視。

Motivated by nano-sol data set, this thesis addresses the shelf-life prediction of a nano-sol product. Instead of using temperature, humidity or voltage to do accelerated life/degradation experiments in the past, this study used pH as accelerating factor to predict the shelf-life of nano-sol product.
The observed distribution data of nano-particles in the nano-sol are histogram-valued frequencies. A mixture of two normal distributions was used to describe the particle size distribution. The time evolution of the particle size distribution under different pH values was described by a pH accelerated degradation model. The maximum likelihood estimator (MLE) for unknown parameters in the pH accelerated degradation model can be solved by applying Expectation/Conditional Maximization (ECM) algorithm. The estimated shelf-life under normal-use-condition and corresponding 95% confidence intervals can then be obtained.
An optimal test plan for pH accelerated degradation model can be obtained by minimizing the approximate variance of the estimated shelf-life of the nano-sol product under the constraint that the total experimental cost not exceeding a pre-specified budget. The sensitivity analysis reveals that the optimal test plan is quite robust to moderate departures from the model parameters.
Finally, the effects of model mis-specification were discussed. Specifically the asymptotic large sample distribution of the shelf-life estimates was derived when the particle size distribution described in the first topic is wrongly fitted with inappropriate model. The result show that the effects on the accuracy and precision of the product’s shelf-life are critical.

目錄
第一章 緒論 1
1.1 前言 1
1.2 衰變模型之簡介 2
1.2.1 隨機效應之衰變模型 2
1.2.2 隨機過程之衰變模型 3
1.3 研究主題與動機 4
1.4 研究架構 5
第二章 文獻回顧及問題描述 7
2.1 相關文獻回顧 7
2.1.1 一般化加速衰變模型 7
2.1.2 模型誤判之理論 8
2.2 動機例子與問題描述 11
2.3 奈米顆粒之分布及本研究主題 13
第三章 pH值加速衰變模型之建構 15
3.1 問題描述 15
3.2 pH值型加速衰變模型架構 17
3.3 pH值加速衰變模型之期望條件最大演算法 21
3.4 奈米溶膠於正常使用條件下之保存期限估計及其信賴區間 23
3.5 動機例子回顧 24
3.6 有母數拔靴模擬分析 25
3.7 結論 27
附錄3.1: pH值雙常態混合模型之期望最大演算法 28
附錄3.2: 及 之表示式 30
附錄3.3: 之計算方式 32
第四章 pH值加速衰變試驗之最佳實驗配置 38
4.1 前言 38
4.2 衰變試驗之最佳化問題 39
4.2.1 成本函數 40
4.2.2 最佳化模式 40
4.2.3 最佳實驗配置之演算法 40
4.3 動機例子說明 41
4.3.1 最佳實驗配置 41
4.3.2 敏感度分析 42
4.4 結論 44
第五章 pH值加速衰變模型之誤判分析 47
5.1 資料導向無母數分析方法 47
5.2 模型誤判的問題描述 48
5.3 當pH值加速衰變模型被誤判為資料導向無母數模型時,對奈米溶膠保存期限估計值的影響 49
5.4 動機例子回顧 53
5.5 結論 55
第六章 總結及後續研究 58
參考文獻 61


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