DSpace collection: 博碩士論文

加速通用趨勢更新過程下的貝氏分析及其在鋰電池資料之應用;Bayesian Analysis of Accelerated Generic Trend Renewal Processes with an Application to Lithium-ion Battery Data

title: 加速通用趨勢更新過程下的貝氏分析及其在鋰電池資料之應用;Bayesian Analysis of Accelerated Generic Trend Renewal Processes with an Application to Lithium-ion Battery Data abstract: 針對經由充放電過程可重複使用但性能遞減的鋰電池衰變資料，通常可利用與時間相關的趨勢函數經變數變換轉換為衰變增量為獨立同分布的更新過程 (renewal process)，即趨勢更新過程 (trend renewal process) 來配適。囿於參數的不可辨別性，傳統趨勢更新過程限制更新分布之期望值為一，卻因此使模型中之參數失去原有的共軛 (conjugate) 結構，導致難以建構隨機效應模型描述具個別差異性的資料。為保留更新分布參數的共軛性，通用趨勢更新過程 (generic trend renewal process) 以趨勢函數參數之限制取代期望值為一的假設，俾能更具彈性地發展隨機效應模型。本文將通用趨勢更新過程推廣至加速通用趨勢更新過程 (accelerate trend renewal process; AGTRP)，在參數與加速應力呈對數線性關係下，分別考慮伽瑪、韋伯與逆高斯更新分布之 AGTRP 模型，利用參數之共軛性質，建立貝氏 AGTRP 模型，用以分析不同放電電流下的電池電容量比資料；在隨機效應模型中以階層貝氏 (hierarchical Bayes) 方法，引入隱藏變數捕捉電池間的個別差異。依據偏差訊息準則 (deviance information criterion; DIC) 以及對數邊際概似 (log marginal likelihood; LML) 值兩種常用的貝氏選模準則作為綜合依據，並由後驗預測 p-值 (posterior predictive p-value) 確認模型與資料之適切性。最後，根據加速資料配適之最適 AGTRP 模型，外插至正常應力水準下推估可修復產品的使用壽命。應用於電池充放電資料實例分析中，亦將 AGTRP 模型下所得電池的壽命指標與正常放電電流下衰變資料所得結果進行比較，經兩樣本 Kolmogorov-Smirnov 貝氏檢定，顯示兩組資料所得之預測壽命分布並無顯著不同，驗證由加速模型外插至正常應力水準下之壽命推論的合理性。;For the cyclic degradation data of lithium-ion batteries, which can be repeatedly charged and discharged but exhibit gradual performance decline, the trend renewal process (TRP) is commonly adopted. In this approach, time-dependent trend functions transform correlated data into independent and identically distributed increments that follow a renewal process. However, due to the issue of parameters identifiability, TRP models assume the expectation of the renewal distribution to be one. Such restriction destroys the conjugate structure of the parameters. The generic trend renewal process (GTRP), making the restriction on the trend-function parameter instead of expectation, preserves conjugacy and thus provides greater flexibility for modeling random effects. Based on this framework, this study develops Bayesian accelerated generic trend renewal processes (AGTRP) with gamma, Weibull, and inverse Gaussian renewal distributions respectively, by assuming a log-linear relationship between the parameters and the accelerated stress variable. Leveraging the conjugate structure, random-effects AGTRP models are further constructed by introducing latent variables within a hierarchical Bayesian model to capture unit-to-unit heterogeneity. Model performance is assessed using the deviance information criterion (DIC) and log marginal likelihood (LML), while posterior predictive p-value evaluates the corresponding model adequacy. The selected AGTRP model is extrapolated to draw the predictive life inference under normal use condition. A Bayesian two-sample Kolmogorov-Smirnov test comparing the extrapolated distribution with normal-stress experimental data shows insignificant differences between the resulting life distributions in the all data analysis, supporting the validity of AGTRP for extrapolative lifetime prediction.

群試迴歸模型中D-最適設計之理論極限;Theoretical Limits of $D$-Optimal Design in Group Testing Regression Models

title: 群試迴歸模型中D-最適設計之理論極限;Theoretical Limits of $D$-Optimal Design in Group Testing Regression Models abstract: 在公共衛生及流行病學中，相較於傳統個別試驗，群組試驗透過合併樣本檢測，有助於針對低盛行率疾病之研究能降低成本與誤差風險。在群組試驗中，群組大小與群組中個體自變數的分佈，會顯著影響群試迴歸模型參數估計的有效性。在實務中，研究者僅能將個體樣本分組，無法完全控制自變數於組間及組內的分佈。鑑於此，本論文探討群試迴歸模型在「理想情況」下的最適設計問題，藉由指定自變數之分佈，以達到參數估計變異的理論下界。當群組大小固定且只有一個自變數時，本研究刻劃了使用互補雙對數 (complementary log-log, cloglog) 鏈結函數的群試迴歸模型之D-最適設計的理論特徵。對於多變數的情況，我們利用隨機交換演算法來尋找最適設計。最後，我們探討群組試驗結構對 cloglog 模型下最適設計的影響，並與傳統個別試驗情境進行比較。;In public health and epidemiology, compared to traditional individual testing, group testing offers a cost-effective and error-reducing approach for screening low-prevalence diseases. In group testing, the group size and the distribution of individual covariates both within and between groups can greatly influence the efficiency of parameter estimation in group-testing regression models. In practical applications, researchers are able to assign individual samples into groups but have limited control over the covariate distributions within or across these groups. In light of this, we explore the optimal design problem for group-testing regression models in this thesis, under an idealized framework where covariate distributions are specified to achieve the theoretical lower bound of estimator variance. When the group size is fixed and only a single covariate is involved, we derive the theoretical properties of the D-optimal design for a group-testing regression model using a complementary log-log (cloglog) link function. For models involving multiple covariates, the randomized-exchange algorithm is employed to obtain optimal designs. Finally, we analyze how various group structures impact the optimal designs under the cloglog model and compare these findings with those obtained under the conventional individual-testing setting.

典型相關分析中維度檢定方法之比較;Comparison of Dimensionality Testing Methods in Canonical Correlation Analysis

title: 典型相關分析中維度檢定方法之比較;Comparison of Dimensionality Testing Methods in Canonical Correlation Analysis abstract: 典型相關分析(CCA)是一種用於衡量兩組變數之間線性關係的統計方法。其典型相關係數是透過變數集合的共變異數矩陣與交叉共變異數矩陣所進行的廣義特徵值分解而來，典型變數對則對基於對應的特徵向量建立。為了檢定典型相關是否顯著，目前最常使用的兩種檢定方法皆基於特徵值，分別為傳統的卡方檢定(假設維度固定且樣本數趨近無窮)與適用於高維條件下的 Tracy-Widom 檢定（假設維度與樣本數同時趨近無窮）。近年來，一種基於特徵向量的替代方法──變數填充下的維度推論方法（DIVA），被發展出來，原先用於充分維度縮減框架下的維度檢定。文中我們說明該方法也可應用於檢定 CCA 中的典型相關顯著性。為了評估這些方法在有限樣本下的表現，我們在不同的維度設定、樣本大小及相關強度條件下進行了綜合模擬研究。我們發現，當檢定兩組變數是否相關時，若相關性較低，卡方檢定表現較佳；若相關性較高，Tracy-Widom 檢定更為合適。而在估計顯著的典型變數對的個數時，當相關性較低時，卡方檢定效果較好，當相關性較高時，則以 s-DIVA 方法表現較佳。;Canonical Correlation Analysis (CCA) assesses linear relationships between two sets of variables. Canonical correlations are obtained via generalized eigen-decomposition of covariance and cross-covariance matrices, and canonical pairs are based on the corresponding eigenvectors. Two common eigenvalue-based significance tests are the traditional chi-square test (assuming fixed p and n → ∞) and the Tracy-Widom test (for high-dimensional settings where both p and n → ∞). In this work, the eigenvector-based “dimension inference using variable augmentation” (DIVA), originally developed for dimension testing in sufficient dimension reduction framework, is applied to CCA. We evaluates these methods via simulation studies with varying dimensions, sample sizes, and correlation strengths. Our numerical results show that the chi-square test performs better under weak correlations, while Tracy-Widom excels with strong correlations. For selecting number of significant canonical pairs, chi-square test is recommended for weak correlations, whereas DIVA is preferable for strong correlations.

錯標邏輯斯迴歸之D-最適設計;D-Optimal Designs for Mislabelled Logistic Regression

title: 錯標邏輯斯迴歸之D-最適設計;D-Optimal Designs for Mislabelled Logistic Regression abstract: 在實務應用中，二元反應資料常帶有一些錯誤標記。例如醫學檢驗的偽陰及偽陽，或是敏感問卷中隨機作答技術導致作答反應值的錯誤分類。在這些應用中，使用邏輯斯迴歸模型並加入錯誤標記之考量將有助於建構更準確的統計推論。在本論文我們將研究錯標邏輯斯迴歸模型之 D-最適設計問題。當只有單一解釋變數時，我們發現其D-最適設計與一般邏輯斯迴歸之D-最適設計同為等權重兩點設計，但這兩個支撐點不具對稱性。在多個解釋變數的情況下，我們則推廣隨機交換演算法，以求得錯標邏輯斯迴歸模型的D-最適設計，並討論其與一般邏輯斯迴歸之結果的異同之處。;In practical applications, binary response data often contain some misclassificationerrors, such as false positives and false negatives in medicaltesting, or response misclassification caused by random response techniquein sensitive questionnaires. Therefore, incorporating misclassification intologistic regression models for such problems can lead to more suitable statisticalinferences. In this thesis, we study D-optimal designs for mislabelledlogistic regression. When there is only one explanatory variable, wefind that there exists a D-optimal design having two support points withequal weights, as that in standard logistic regression, but these two pointsare not symmetric. In the case of multiple explanatory variables, we adaptthe randomized exchange algorithm to obtain a D-optimal design for themislabelled logistic regression and discuss the similarities and differencescompared to the results from the standard logistic regression.