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研究生:林億雄
研究生(外文):Yi-Hsiung Lin
論文名稱:可維修系統之轉折點分析的研究
論文名稱(外文):A Study on Change Point Analysis for Repairable Systems
指導教授:任眉眉任眉眉引用關係
指導教授(外文):Mei Mei Zen
學位類別:博士
校院名稱:國立成功大學
系所名稱:統計學系碩博士班
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:2008
畢業學年度:97
語文別:英文
論文頁數:99
中文關鍵詞:有限制的最大概似估計方法不均勻布瓦松過程核密度估計方法S型軟體可靠度成長模型加權式核密度估計方法貝氏分析方法可維修系統
外文關鍵詞:Constrained MLEBayesian analysisS-shaped reliability growth modelNon-homogenous Poisson processRepairable systemsKernel methodWeighted kernel method
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本研究主要探討可維修系統可靠度的問題,一般而言我們稱該系統的可靠度衰退是指當系統兩相鄰錯誤發生的間隔時間持續縮短;反之,當系統兩相鄰錯誤發生的間隔時間持續延長,我們稱此系統可靠度成長。然而,一個系統執行中可能會受很多因素的影響,例如:工作環境的改善、重大策略的實行、資源的重新配置等使該系統的可靠度發生改變,其系統可靠度從成長至衰退(或衰退至成長)。上述的問題等同在不均勻布瓦松過程中探討可靠度狀態的改變,常見系統模型如:S型軟體可靠度成長模型。首先,我們從最簡單具有一個轉折點的布瓦松過程,其事件發生頻率在轉折點前後不同探討開始。對於具有一個轉折點的布瓦松過程,在其參數估計上我們使用貝氏分析方法。至於轉折點的先驗分布選擇,我們提出使用具有單峯性質的分布並以貝氏因子為比較選擇基準。研究發現具有單峯性質的轉折點先驗分布,其表現明顯優於均勻先驗分布;同時,在轉折點估計上我們提出使用有限制的最大概似估計方法,並且證明有限制的最大概似估計量具有一致性。接續,我們將系統模型推廣至具有單調性質的不均勻布瓦松過程中(例如:Power law process)。關於其錯誤再現率,我們提出使用無母數核密度估計方法。研究發現傳統無母數核密度估計方法無法保留系統可靠度的單調性質,因此我們提出使用無母數加權式核密度估計方法用於保留系統可靠度單調性質的限制。最後,我們亦以無母數加權式核密度估計方法推廣至估計S型軟體可靠度成長模型之具有單峯性質的錯誤再現率,該方法可以成功保留具有單峯限制的錯誤再現率,經模擬發現該方法有不錯的估計效果。
The main purpose of this study is to discuss the problem of reliability in repairable systems. Generally, we say that a repairable system is deteriorating (reliability decay), if the time between failures tend to get shorter with advancing age; if the time between failures tend to increase, then we say that the system is improving (reliability growth). In fact, the reliability from growth to decay (or vice versa) in a system may be affected by many factors such as improvement of the environment, strategy and resources. It is closely related to the problem of changing reliable state for a non-homogenous Poisson process. For example, the S-shaped software reliability growth model is a kind of these models. This study is focused on the detection of time at which point processes undergo abrupt changes. First, we started with the simplest model, assuming events occur according to a Poisson process, whose rate changes at an unknown change point. We used Bayesian analysis for Poisson process with a change-point. Regarding to the prior of change-point, we suggested using unimodal property. On the choices of prior, we chose Bayes factors for comparison. Our research found that it is more realistic to put a unimodal prior on it, while outlines the important feature of prior beliefs. Meanwhile, a maximum likelihood estimator for the change-point subject to a suitable constraint was proposed, which was shown to be consistent. For power law process, we used nonparametric kernel method to estimate the intensity function. Since its intensity function is monotonic, we proposed a weighted kernel estimate subject to the monotonic constraint. For an S-shaped software reliability growth model, its intensity function is unimodal. A weighted kernel estimate fitted the model appropriately, where the ordinary kernel estimate didn't guarantee the unimodality for an S-shaped growth model.
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Frame of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 A Change-point Poisson Process 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Parametric Inference for the Intensity . . . . . . . . . . . . . . . . 11
2.2.1 British Coal-mining Disaster Data . . . . . . . . . . . . . . 11
2.2.2 Parametric Bayes Estimation of the Intensity . . . . . . . . 12
2.2.3 Constrained Maximum Likelihood Estimators of the Intensity 30
2.2.4 Consistency of the Constrained MLE . . . . . . . . . . . . 33
2.2.5 Bootstrap Confidence Intervals for the Constrained MLE’s . 37
2.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Power Law Process with a Change-point 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Power Law Process . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.1 Parametric inference for the intensity . . . . . . . . . . . . 42
3.2.2 Nonparametric inference for the intensity . . . . . . . . . . 42
3.2.3 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Power Law Process with a change-point . . . . . . . . . . . . . . . 50
3.3.1 Parametric inference for the intensity . . . . . . . . . . . . 52
3.3.2 Nonparametric inference for the intensity . . . . . . . . . . 55
3.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 S-shaped Software Reliability Growth Models 57
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Parametric NHPP-gamma-k model for the intensity . . . . . . . . . 58
4.2.1 Parametric estimation of the intensity . . . . . . . . . . . . 58
4.2.2 Confidence region . . . . . . . . . . . . . . . . . . . . . . 59
4.2.3 Bayesian Inference for the NHPP-gamma-k Model . . . . . 61
4.3 Nonparametric inference for the intensity . . . . . . . . . . . . . . 62
4.3.1 The unimodality kernel estimation method . . . . . . . . . 62
4.3.2 The confidence region by Bootstrap resampling . . . . . . . 63
4.3.3 Boundary effects adjustment . . . . . . . . . . . . . . . . . 64
4.4 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4.1 Parametric estimation . . . . . . . . . . . . . . . . . . . . . 64
4.4.2 Nonparametric estimation . . . . . . . . . . . . . . . . . . 66
4.4.3 Comparison between parametric and nonparametric approaches 67
4.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 Conclusions 72
Bibliography 74
A Estimation Methods 82
A.1 Gibbs Sampler Method . . . . . . . . . . . . . . . . . . . . . . . . 82
A.2 Metropolis-Hastings Algorithm . . . . . . . . . . . . . . . . . . . . 83
A.3 Kernel Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
A.4 Kernel Estimation with Constraints . . . . . . . . . . . . . . . . . . 85
B Constructing Approximate Confidence Intervals 89
B.1 Normal-Approximation Confidence Intervals . . . . . . . . . . . . 89
B.2 Bayesian Credible Sets . . . . . . . . . . . . . . . . . . . . . . . . 90
B.3 Bootstrap Resampling Confidence Intervals . . . . . . . . . . . . . 91
C Estimation in the PLP-CP Model 94
D Estimation of the change-point in the PLP-CP Model 97
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