University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School June 2020 Bayesian Reliability Analysis for Optical Media Using Accelerated Degradation Test Data Kun Bu University of South Florida Follow this and additional works at: https://scholarcommons.usf.edu/etd Part of the Statistics and Probability Commons Scholar Commons Citation Bu, Kun, "Bayesian Reliability Analysis for Optical Media Using Accelerated Degradation Test Data" (2020). Graduate Theses and Dissertations. https://scholarcommons.usf.edu/etd/8438 This Thesis is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. Bayesian Reliability Analysis for Optical Media Using Accelerated Degradation Test Data by Kun Bu A thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts in Statistics Department of Mathematics and Statistics College of Art and Sciences University of South Florida Major Professor: Lu, Lu, Ph.D. Ramachandran, Kandethody, Ph.D. Li, Mingyang, Ph.D. Date of Approval: June 19, 2020 Keywords: Statistics, Bayesian Data Analysis, Reliability Analysis, Median Lifetime Prediction Copyright © 2020, Kun Bu Acknowledgments First of all, I would like to sincerely thank my master thesis advisor Dr.Lu, Lu for the continuous support of my study and research, also for her patience, motivation, enthusiasm, and profound knowledge. Her guidance helped me in all the time of research and writing of this thesis; without her, I could not have imagined having a better advisor for my thesis study. At the same time, I would like to express my appreciation to the committee members Dr. Ramachandran, Kandethody, and Dr.Li, Mingyang, for their support and encouragement. My sincere thank also goes to my classmates and friends: Weiru Han, Oluwaseun Otunuga, Nam Phuong Nguyen, Chengcheng Mou, Heidi Harding, Bob Welch, Audrey Welch for the wisdom and spirit they bring to me, for the assistance and support they enlighten me, for the help and advice they provide me. Especially I thank the development team of Stan. With their professionalism and e↵ort, my work in parameters estimation can be realized in a quick, efficient, and robust language. IsincerelythankoneoftheStanteamdevelopersDr.BobCarpenter,whoisactivelyhelping Stan beginners at The Stan Forums. Last, I would like to thank my family: my parents Guangshuang Bu and Jianping Du, for their unconditional love and care, always support and encouragement. Table of Contents Table of Contents ...................................... i ListofFigures ....................................... iii List of Tables . iv 1Introduction......................................1 1.1 Background . 1 1.2 Goal and Motivation . 6 1.3 Overview . 10 2LiteratureReview...................................11 2.1 Previous Works . 11 3Methodology......................................14 3.1 Model for Degradation Data . 14 3.1.1 DegradationDataFromISO10995:2011. 14 3.1.2 Hierarchical Degradation Model . 16 3.2 Parameters Estimation . 17 3.2.1 Gibbs Sampler . 18 3.2.2 JAGSandSTAN .......................... 20 3.2.3 STAN Convergence . 21 3.2.4 STANOutput ............................ 26 4 ReliabilityAnalysis .................................. 28 4.1 Degradation Path Model . 29 4.2 Monte Carlo Simulation to Draw Degradation Paths . 30 4.3 Quantile Lifetime Prediction with Credible Interval . 35 5 SensitivityAnalysis .................................. 40 5.1 Priors Selection . 40 5.2 Comparison of Parameter Estimation . 41 5.3 Comparison of Predicted Reliability and Quantile Lifetime . 43 5.4 Overall Prediction Results . 45 6ConcludingRemarks.................................47 References ......................................... 49 i Appendix A Three basic blocks in RStan (data, parameters, and model) . 52 Appendix B Example of transformed data and transformed parameters . 53 Appendix C The R code for model function of F (t).................55 Appendix D R code for sensitive analysis using di↵erentpriors. 56 AppendixE OriginalDataTablefromISO10995:2011[1]. 62 ii List of Figures 1.1 DegradationPathsFromtheISO10995:2011ADTData . 3 1.2 A Part of ISO Data after Data Processing . 6 1.3 AFlowChartofthisStudy .............................. 9 3.1 Density Plots for Convergence Parameters . 23 3.2 TracePlotsforConvergenceParameters . 24 3.3 TracePlotsforNon-convergenceParameters . 25 3.4 ACF Plots for Convergence Parameters . 26 4.1 500 Simulated Degradation Paths Cross the Threshold . 32 4.2 Reliability Curve with 95% Credible Interval in Real Time Hours . 33 4.3 Reliability Curve with 95% Credible Interval in Log Scaled Time . 34 4.4 GraphicalExplanationofSimulatingReliabilityCurve . 35 4.5 Graphical Explanation of Simulating Quantile Lifetime with CI . 37 4.6 QuantileLifetimePredictionwith95%CredibleInterval . 38 5.1 Reliability Curve with 95% Credible Interval in Log Scaled Time . 46 iii List of Tables 3.1 ISO Testing Conditions . 15 3.2 Parameters Estimation From STAN output . 27 4.1 Quantile Lifetime Prediction with 95% Credible Interval . 39 5.1 Priors Selected in Stan . 42 5.2 Parameters Estimation Comparison using Di↵erent Priors . 43 5.3 Comparison of the Predicted Quantile Lifetime using Di↵erent Priors . 44 E.1 Original Degradation Data Test Condition A . 62 E.2 Original Degradation Data Test Condition B . 63 E.3 Original Degradation Data Test Condition C . 64 E.4 Original Degradation Data Test Condition D . 65 iv Abstract ISO (the International Organization for Standardization) 10995:2011 is the international standard providing guidelines for assessing the reliability and service life of optical media, which is designed to be highly reliable and possesses a long lifetime. A well-known chal- lenge of reliability analysis for highly reliable devices is that it is hard to obtain sufficient failure data under their normal use conditions. Accelerated degradation tests (ADTs) are commonly used to quickly obtain physical degradation data under elevated stress conditions, which are then extrapolated to predict reliability under the normal use condition. This stan- dard achieves the estimation of the lifetime of recordable media, such as Magneto-Optical media, via an accelerated degradation test for measuring the error rate of these hard devices under elevated temperature and relative humidity levels. The observed degradation mea- sures are modeled with regression analysis to predict the unobserved failure time, which is then used as observed failure time for estimating the lifetime distribution and predict the device quantile/median lifetime. However, the ISO 10995:2011 analysis fails to consider the uncertainty of the predicted failure times, as well as the heterogeneity of the test units, and hence could lead to imprecise and overconfident estimation. This thesis presents a Bayesian method to analyze the ISO degradation data, which (1) provides more accurate quantifica- tion of uncertainty through the use of a hierarchical degradation path model, (2) includes random e↵ects to capture the unit-to-unit variation for improving analysis of heterogene- ity, and (3) o↵ers more straightforward implementation for estimating reliability and its associated uncertainty based on general ADT data. v 1 Introduction There is a common consensus that predicting the archival life of optical media plays an increasingly important role in its market and industry. Existing standards have documented methods for predicting the lifetime of recordable media, which is high reliable product. The aim of the ISO 10995:2011[1] is to provide guidance on current practice based on conduct- ing accelerated degradation tests for understanding the underlying failure mechanism and predicting the archival life of optical media products. Optical media are designed to be long-life devices. It is often impossible to observe suffi- cient failure data under its normal use condition during a reasonable test period. Accelerated degradation tests (ADTs) become popular in testing highly reliable products to quickly in- quire the sufficient failure data. In ADTs, the test units are assigned to elevated stress levels or exposed to harsher use conditions to speed up the failing process, and measures on charac- teristics related to the physical degradation process are obtained, analyzed, and extrapolated to predict the product lifetime under normal operating conditions. In ISO 10995:2011, four di↵erent test conditions were used with di↵erent combinations of the temperature (T) and relative humidity (RH) levels. It was believed that elevated stress conditions would accel- erate the rate of chemical reactions, which will then lead to the degradation of the ideal material properties that ultimately causes disk failure. 1.1 Background The worldwide standardization system consists of both the ISO (International Or- ganization for Standardization) and the IEC (International Electrotechnical Commission). 1 ISO or IEC participates in the development of international standards through technical committees established by their respective organizations to deal with technical activities in specific areas. The purpose of ISO/IEC is to develop and maintain standards in the fields of information technology and communications technology. ISO 10995:2011 [1] uses the ADTs to estimate and predict the lifetime of optical media. Particularly, it uses the degradation data obtained