軟體是現今社會的基礎，人們在生活各層面大量地仰賴軟體。在軟體品質的屬性中，軟體可靠度通常被認為是最重要軟體品質的因素之一，並且是影響系統可靠度的重要因素。在軟體開發及測試的過程中，造成軟體故障的錯誤會被分析及修復。實務上，錯誤發生的各自時間及造成的原因會被蒐集與紀錄。許多軟體可靠度成長模型發表於文獻中，並致力於從蒐集的軟體錯誤資料估計軟體可靠度。三十多年來，非齊次卜瓦松過程（Non-Homogeneous Poisson Process, NHPP）模型在軟體可靠度工程中是個相當成功的工具，這些模型提供了創新的概念及分析架構描述軟體測試中錯誤的發生。
　　然而，須注意的是，當資料的特性（資料型態）不符合模型的特性時，在實務上盲目地使用軟體可靠度模型可能無法得到有意義的結論。並且，並不存在專案管理者或開發者可遵循的任何準則或策略，為特定的軟體錯誤資料集選擇最適當的軟體可靠度模型。從實務應用的觀點而言，討論與發展一套建議以篩選與平滑蒐集的錯誤資料是有需要的。另一方面，於配適軟體可靠度模型前等待蒐集大量的錯誤資料，在許多情況下未必可行。
　　在本論文裡，我們建議使用統計資料分析的方法探究軟體錯誤資料的特性，以改善軟體可靠度建模的精確性。這裡我們將軟體可靠度模型概括性地分成兩個類別：屬於NHPP模型及非屬NHPP模型。同樣地，軟體錯誤資料可以被歸為兩個類別：屬於NHPP型態資料及非屬NHPP型態資料。我們提出一個兩階段方法，以檢驗錯誤資料遵循NHPP之假設。提出的方法中，第一階段將整筆資料視為一個區間檢驗。此外，拉普拉斯趨勢檢驗亦在本論文裡使用。在提出的方法中，若資料未通過第一階段測試，資料將在第二階段以區間分割及（或）移除離群值之方式進一步調整，並再次檢驗。
　　透過我們提出的兩階段檢驗，我們可以推論一筆資料是否屬於NHPP型態，以及該資料是否適當地用於基於NHPP的軟體可靠度模型。如此一來，在我們能正確判斷資料是否可視為NHPP型態後，便能做最終決定選擇最適合的NHPP模型為軟體錯誤過程建立模型，並且軟體可靠度預測結果是可信且有幫助的。我們執行了基於25筆真實軟體錯誤資料的實驗，並詳細地討論我們提出的兩階段檢驗的表現。最後，我們用了一種資料平滑方法改善未通過兩階段檢驗的資料。我們以一個案例呈現，探討以平滑資料用於典型參數型軟體可靠度模型、原始資料用於典型參數型軟體可靠度模型，以及原始資料用於非參數軟體可靠度模型的結果。使用我們所提出的兩階段檢驗方法，專案管理者將能分析與使用蒐集的錯誤資料來改善規劃的精確性及軟體品質。

Software is a part of the foundation of our society nowadays. People heavily rely on software in many aspects of their lives. Among the software quality attributes, software reliability is generally accepted as one of the most important factors of software quality and it is also an important factor that impacts system reliability. During the process of software development and testing, the faults that cause the software failures will be analyzed and fixed. Practically, the individual times at which failure occurs, and the root causes of failure, will be collected and recorded. There are a lot of Software Reliability Growth Models (SRGMs) published in the literature and many efforts have been made to estimate the software reliability from collected software failure data. Over the past three decades, the Non-Homogeneous Poisson Process (NHPP) models have been quite successful tools in software reliability engineering and they have provided a novel conceptual and analytic framework for describing the failure occurrence during software testing.
But it should be noted that, in practice, blindly applying SRGMs might not lead to meaningful results when the characteristics of data (i.e., data type) don’t fit the characteristics of the model. Also, there does not exist any guidelines or policies that project managers or developers can follow to select the most appropriate SRGMs for a particular failure data set. From the viewpoint of practical application, there is a need to discuss and develop a set of recommendations for filtering and smoothing the collected failure data. On the other hand, waiting to collect a substantial amount of failure data before being able to fit SRGM(s) might not be feasible in many cases.
In this thesis, we propose to use the methods of statistical data analysis to investigate the characteristics of software failure data and to improve the accuracy of software reliability modeling. Here SRGMs will be broadly partitioned into two categories: the “NHPP models” and the “not belonging to the class of NHPP models”. Similarly, software failure data can be classified into two categories, the “NHPP-type data” and the “not NHPP-type data”. We will propose a two-phase method to verify the hypothesis that failure data follows an NHPP. The first phase of our proposed method is to examine the whole dataset as an interval. Additionally, the Laplace trend test is also used in this thesis. If a dataset doesn’t pass the first phase examination, the dataset will be further adjusted by region partitioning and/or outliers removing, and then it will be re-examined in the second phase of the proposed method. In general, trend tests include graphical tests and analytical tests. The Laplace test is generally used for identifying trends in grouped data or time-series and it is one of the most commonly used methods, because it is often found to be the most appropriate one when failures follow the NHPP.
Through the proposed two-phase test and the Laplace trend test, we will be able to infer whether a dataset is the NHPP-type and whether it is properly used by NHPP-based SRGMs. In this case, the final decision of choosing the most suitable NHPP model(s) to model the software failure process would be made, and the software reliability prediction result is trustworthy and helpful after we can correctly judge whether the dataset is said to be absolutely NHPP-type or not. Experiments based on 25 real software failure data are performed and discussed in much detail in order to assess the performance of our proposed two-phase methods. Finally, a data smoothing method is also used to improve and reassess this failure data when it failed the two-phase test. A case study is presented and discussed for the result of typical parametric SRGMs with the smoothed data and with the original data as well as some non-parametric approaches. Using our proposed method, project managers and developers will be able to analyze and use collected failure data to help improve both planning accuracy and software quality.

Abstract i
Abstract in Chinese iv
Contents vi
List of Tables viii
List of Figures ix
List of Symbols x
Acronyms and Abbreviations x
Notation xi
Chapter 1 Introduction 1
Chapter 2 Background and Related Work 9
Chapter 3 Two Phase Test for NHPP 21
3.1 Investigating the assumption of NHPP 21
3.2 Proposed Two-Phase Test Procedure 26
The Chi-Squared Test 27
The Kolmogorov-Smirnov Test 28
The Cramér-von Mises Test 28
The Autocorrelation 29
The Box-Ljung Test 29
The BDS Test 29
Chapter 4 Failure Data Analysis 34
4.1 Failure Data 34
4.2 Test Results in the First Phase 36
4.2.1 Poisson Distribution Test 37
4.2.2 Serial Independence Test 39
4.2.3 Summary of Main Results in the First Phase 46
4.3 Test Results in the Second Phase 49
Chapter 5 Further Discussions and Analysis on the Data Failed Two-Phase Test 57
5.1 Data Smoothing 58
5.2 A Case Study for the Failed Poisson Distribution Test – DS20 59
5.2.1 Smoothed Data 59
5.2.2 Curve Fitting with Smoothed Data 61
5.2.3 Result Discussion 64
5.3 Thread to Validity 67
Chapter 6 Conclusion 69
References 73
Appendix 83
Appendix A. Raw Data (Cumulative Faults) 83
Appendix B. The BDS Test Result (R Report Output) 116
Appendix C. Test Result for Regions of DS3 125

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