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# STAT 687 - Math of Reliability

## Welcome!

Welcome to STAT 687 - Math of Reliability! This course presents the theory and application of statistical methods used to analyze reliability & survivability data. To accomplish this we'll integrate real-world data, advanced statistical concepts, and data analytics software tools.

## Course Learning Objectives

After completing this course, students will:

1. Thoroughly understand the basic concepts on which statistical methods for life-data analyses are based, specifically:

• How life-data analysis methods differ from other statistical methods
• The importance of statistical censoring
• Differences between right, left, and interval censoring
• Why all data is interval censored data
• How to use nonparametric methods for fitting models to data
2. Be able to implement maximum likelihood (ML) estimation, specifically:

• How to construct likelihood functions for various types of data
• How to fit data to a parametric model by maximizing likelihood functions analytically, graphically, and computationally
• The importance of ML estimation in the frequentist and Bayesian domains
• The purpose and importance of the relative likelihood and profile likelihood functions
3. Understand various point and interval estimation techniques, specifically:

• How compute estimates for parameters and functions of parameters from data
• How to compute confidence intervals and credible intervals for parameters and functions of parameters from data
• How to apply the Delta method to compute uncertainty propagation for an arbitrary function of the parameters
• Advantages and disadvantages between various procedures used to compute confidence intervals
4. Use software to analyze real-world reliability data and communicate the results of their analyses, specifically:

• Students will use R & RStudio to compute results from real-world life data
• Students will communicate analysis techinques and results together using rmarkdown
• Students will build interactive plots and tables using shiny

## Pre-requisites & Co-requisites

Unless otherwise waived by the course instructor, students are required to have completed the following courses prior to enrolling in STAT 687 - Math of Reliability.

• STAT 602

Requests to waive these required courses must be made in accordance with AFIT EN policy guidelines and must be approved by the course instructor. Waivers my be approved for students that have has successfully completed a similar courses or if special arrangements are made to meet AFIT requirements.

## Course Resources (Required & Optional)

The required text for this course is Statistical Methods for Reliability Data by Bill Meeker and Luis Escobar. Although the first edition of this text was written in 1998, it is still regarded by many as the gold standard reference for advanced statistical methods for life-data analyses.

The Meeker and Escobar text references a well executed software, called SPLIDA, that was originally written in the S-Plus language (SPLIDA stands for S Plus Life Data Anaysis). While the S-Plus language is still in use, it has largely been replaced by R. Thus, Dr. Meeker started work on an alpha version of the SPLIDA package, modified to run in R, called RSplida. By Dr. Meeker's own admission, the effort to port SPLIDA to the R language was rushed and incomplete. In 2015, I began working with Dr. Meeker to finish updating RSplida to an R package. The package has since been renamed SMRD and we hope to publish the it to the CRAN in 2018. The SMRD package is based on the textbook and will be used throughout this course.

Some additional reliability textbooks are listed below. While these texts aren't required for the course, they contain important material that complements the Meeker and Escobar text in many ways.

• Charles E. Ebeling
An Introduction to Reliability and Maintainability Engineering, 2nd ed., Waveland Press, Long Grove, IL 2010

• Marvin. Rausand and Arlnot Hoyland
System Reliability Theory: Models, Statistical Methods & Applications 2nd ed., Wiley-Interscience, Hoboken, NJ 2004

In addition to the textbooks listed above, the following list contains references to more reliability resources that can be accessed online.

## Grading Policy & Course Deliverables

In this course, grades will be assigned according to the values shown in the table below. The following sections detail the requirements of each graded deliverable.

(1.00 - 0.93] A
(0.93 - 0.90] A-
(0.90 - 0.87] B+
(0.87 - 0.83] B
(0.83 - 0.80] B-
(0.80 - 0.77] C

### Homework (33% of the overall course grade)

Homework will be assigned throughout the course to help you learn the material. If you don't complete the assignments, you won't do well in the course. You're encouraged to work together on the homework assignments, but you won't learn much from copying someone else's work, so don't do it. You may use any available resource to complete the assignments, however you must cite them. Homework will be graded on completeness, (i.e. full credit will be given when a "complete" attempt to each problem is made) with one caveat, see Exams. Solutions will be posted after the assignments are turned in. Questions to the instructor, both in class and during office hours, are welcomed and encouraged.

Homework assignments will be distributed and collected using GitHub Classroom. Thus, to complete the assignments each student will first need to create an account on GitHub, if they don't already have one.

Tip: Don't use your awesome.student@afit.edu email address to create your account. Use an email address that you'll have access to after you graduate. If desired, you can add your AFIT address as a secondary email.

The homework process in this class will be as follows

1. Prior to the first assignment, each student will be assigned to a two or three-person team on GitHub.

2. For each assignment and each team I'll create a private repository containing an edit-able shell of the assignment.

3. Teams will complete the assignment by entering their work into the shell, along with any supplementary files.

4. On the assigned turn-in date, the each team's completed homework assignments will be 'collected' on GitHub.

5. Once all of the assignments have been collected, teams will no longer have write access to the assignment repository (you'll still have read access).

6. Assignment solutions will be posted to the repository.

### Exams (33% of the overall course grade)

I've chosen to modify the standard exam process in a way that I believe is (1) fair to you and (2) easy for me to grade. After receiving your completed homework assignments, I'll choose 3-4 exercises from each homework set to serve as exam questions. These selected exercises will be evaluated more closely than the others and the resulting grade on these exercises will serve as your exam score. A comprehensive final exam (take-home) will be given after the final class meeting (due date TBD).

### Final Project (33% of the overall course grade)

The final project is intended to develop your skills in applying the reliability concepts learned in this course. The goal of the project is to work with your teams to perform a reliability analysis using a dataset that you find out in the wild. Your grade on the project will be based on the quality of the analysis and final outbrief -- taking project difficulty into consideration. So, don't make the project too hard (duh) or too easy. Each team should select an unclassified, non-FOUO dataset, good sources of such data are:

• Current or prior thesis projects
• Industrial studies (e.g. Samsung S7 failures, GM ignition switch failures)
• System performance test data (e.g. Tesla battery degradation)
• Analytics competition datasets (e.g. from Kaggle)

The final project has the following milestones:

1. Deliverable: data description
Due: End of Week 3
Purpose: This deliverable serves to ensure that each team has identified a dataset that's appropriate for the project
Format: Two informal paragraphs describing your chosen dataset (i.e. Why was the dataset chosen? Where did you find it? What do you want to learn from your analysis?)

2. Deliverable: Preliminary analysis
Due: End of Week 7
Purpose: This deliverable serves as a sanity check to ensure your team is on track.
Format: By this point you should have performed a preliminary analysis of the data. Write two informal paragrahs describing your initial results and any problems encountered.

3. Deliverable: Final outbrief
Due: During the scheduled Finals time for this class
Purpose: Present the results of each team's analyses to the rest of the class
Format: An rmarkdown presentation containing no more than six content slides and at least one shiny application

## Software/Computer Programming

A key component of this course involves developing the skills and knowledge to create reproducible & dynamic data products to present results from your research. In previous offerings of this course, I allowed students to use any software package to complete their assignments. This became difficult, for the students to complete their work and for me to grade them. So, I've decided to require you to use the R programming language to complete and submit your assignments.

I realize that some of you may be new to coding or may have never coded before. Don't worry, you don't need an extensive background in R or to be successful in this course. Each week I'll provide you with LOTS of code examples that you can copy/paste to help you get more familiar with the software. Also, many created several demo presentations to get you up to speed and I'm always willing to help out when needed. The first link provided for Week 1 in the schedule below walks you through the process of getting the R/RStudio tool-chain installed and ready for the course.

Each of these tools listed below will be used this quarter:

• R Project for Statistical Computing
• RStudio IDE
• Git/Github
• LaTeX/Mathjax
• Pandoc Markdown
• HTML5, CSS3, and JavaScript (don't need to know these - already built in!)

## Course Schedule

The course schedule below is TENTATIVE and is therefore subject to change. Any changes made to this schedule will be clearly communicated by the instructor.

Week Topics.Presented
1 Introduction to R [1], [2]
Managing your R Workflow [1], [2]
Chapter 1 - Reliability Concepts and Reliability Data
2 Chapter 2 - Models, Censoring, and Likelihood for Failure-Time Data
Chapter 3 - Nonparametric Estimation
3 Chapter 4 - Location-Scale-Based Parametric Distributions
Chapter 5 - Other Parametric Distributions
4 Project Review 1
Chapter 6 - Probability Plotting
5 Chapter 7 - Parametric Likelihood Fitting Concepts: Exponential Distribution
Chapter 8 - Maximum Likelihood for Log-Location-Scale Distributions
Chapter 11 - Parametric Maximum Likelihood: Other Models
6 Chapter 9 - Bootstrap Confidence Intervals
Chapter 10 - Planning Life Tests
7 Chapter 12 - Prediction of Future Random Quantities
Chapter 14 - Introduction to the Use of Bayesian Methods for Reliability Data
8 Project Review 2
Introduction to Shiny
Chapter 15 - System Reliability Concepts and Methods
9 Chapter 17 - Failure-Time Regression Analysis
Chapter 18 - Accelerated Test Models
10 Chapter 19 - Accelerated Life Tests
Chapter 20 - Planning Accelerated Life Tests
Course Review
11 Project Out-Briefs
Final Exams Due

### Other Noteworthy Dates

In addition to the course schedule, you should take note of the following dates.

• 04 Jul - Indenpendence Day (No Classes)
• 05 Jul - AETC Family Day (No Classes)
• 06 Jul - Last Day to Drop Without Record
• 17 Aug - Last Day to Withdraw Without a Grade
• 30 Aug - Summer Quarter Classes End
• 31 Aug - AETC Family Day (No Classes)
• 03 Sep - Labor Day (No Classes)
• 04 Sep - Final Exam Week Begins
• 07 Sep - Final Exam Week Ends

## Important Policy Statements

All students must adhere to the highest standards of academic integrity. Students are prohibited from engaging in plagiarism, cheating, misrepresentation, or any other act constituting a lack of academic integrity. Failure on the part of any individual to practice academic integrity is not condoned and will not be tolerated. Individuals who violate this policy are subject to adverse administrative action including disenrollment from school and disciplinary action. Individuals subject to the Uniform Code of Military Justice may be prosecuted under the UCMJ. Violations by government civilian employees may result in administrative disciplinary action without regard to otherwise applicable criminal or civil sanctions for violations of related laws. (References: Student Handbook, ENOI 36-107, Academic Integrity)

### Attendance Policy Statement

Attendance at all class sessions and exams is mandatory for military and civilians assigned to AFIT as full-time students except for extenuating circumstances. Part-time students are expected to attend scheduled classes, and absences should be explained to the instructor. The student should provide advance notice, if possible. Scheduled classes and exams are defined by the instructor and they are documented in the course schedule. (References: Student Handbook, Graduate School Catalog)

AFIT and the Graduate School of Engineering and Management affirm the right of each student to resolve grievances with the Institution. Students are guaranteed the right of fair hearing and appeal in all matters of judgment of academic performance. Procedures are detailed in ENOI 36-138, Student Academic Performance Appeals.

### My Teaching Philosophy

As AFIT graduates, you should be expected to know how to approach and solve real-world problems AND present your results in a meaningful way so that decision makers can make defensible decisions.

As AFIT instructors, we do a disservice to our students by not teaching new and improved ways to produce and share your results. Further, we do a disservice by teaching you to solve problems using tools that you won't have access to after leaving AFIT. Therefore, I re-built this course using the R/RStudio tool-chain to help you produce better results...faster.