Instructor: Tim Chumley
Office: Clapp 423
Phone: 413-538-2525
e-mail: tchumley
Office Hours: Mondays & Wednesdays 4:00-5:00, Thursdays 11:00-12:00; additional availability by appointment
Textbook: Introduction to Stochastic Processes with R by Robert P. Dobrow, ISBN: 9781118740651;
on library reserve under QC20.7.S8 D63 2016;
available as a free e-text
Announcements
Announcements will be posted on the Course Announcements Moodle forum throughout the semester, but essentially all other materials will be posted on this page.
Syllabus
Check the syllabus for all the important class policies (grades, attendance, etc.).
Homework
There will be weekly homework assignments throughout the semester to be turned in. Please read these guidelines for writing in our class.
- General information. A selection of problems will be assigned to be written up individually and turned in each week.
- These problems will be due Thursdays at 5 pm.
- You may work with others but the writing should be done on your own.
- Gradescope. Homework will be turned in through Gradescope.
- You should be enrolled automatically. Please let me know if you have any issues logging in.
- Gradescope has made a short tutorial on submitting homework.
- Collaboration. I want you to work together on the homework! The process of explaining your ideas to one another really helps in learning math. However, you must write up the assignments on your own and avoid copying others’ work directly. Also, please only write what you understand so that I know where to help, and please avoid using online forums like Math StackExchange, solutions manuals, or similar resources. A huge part of learning in this class comes from figuring out how to get unstuck by spending time thinking on your own or talking to me and classmates; these other resources can end up being counter-productive in the long term.
- Rewrites. Homework is for practice, and you are not expected to write perfect answers from the start!
- You will be allowed to submit revisions of most problems for full credit each week.
- Your revisions will be due on Thursdays at 5 pm.
- Please resubmit (only the problems you’re revising) on Gradescope.
| Assignment | Due |
|---|---|
| Homework 0 | Sep 4 |
| Homework 1 | Sep 11 |
| Homework 2 | Sep 18 |
| Homework 3 | Sep 25 |
| Homework 4 | Oct 2 |
| Homework 5 | Oct 9 |
| Homework 6 | Oct 23 |
| Homework 7 | Oct 30 |
| Homework 8 | Nov 6 |
| Homework 9 | Nov 13 |
| Homework 10 | Dec 4 |
Exams
There will be three exams. The dates for the exams are subject to change slightly.
| Exam | Due Date | Format | Material |
|---|---|---|---|
| Exam 1 | Oct 17 | take-home | TBA |
| Exam 2 | Nov 21 | take-home | TBA |
| Exam 3 | Dec 16 | take-home | TBA |
Course plan
Our plan is to cover most of chapters 1-3 and 6-7 in the textbook, and possibly some of 4, 5, or 8, time permitting. Below is a rough outline of what is to be covered week by week through the semester. Please check back regularly for precise details on what is covered, as well as postings of class materials like lecture notes. A note about files: LaTeX and LyX source code is available for nearly all typeset documents. Simply change the file extension from .pdf to .tex or .lyx in the URL.
Introduction
Wednesday
- Topic: Sections 1.1, 1.2: Introduction. We start to get familiar with the notion of a stochastic process and discuss the terms state space, index set, Markov chain, transition state diagram, and transition matrix.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 0.
Chapter 2
Monday
- Topic: Sections 2.1, 2.2: Introduction to Markov Chains. We give a formal definition of a discrete time, discrete state Markov chain and begin to discuss how the transition matrix of a Markov chain can be used to understand multi-step transition probabilities using the conditional law of total probability.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 1 and make sure you can run RStudio in class next time. Read Section 2.3 up through page 56.
Wednesday
- Topic: Section 2.3: Basic computations. We discuss how to use matrix powers to understand \(n\)-step transition probabilities, the distribution of the \(n\)th step of a Markov chain, and the joint distribution of a Markov chain at multiple time steps.
- Class materials: Lecture notes, worksheet
- After class: Read about a random walk on a cycle in Section 2.4 and about simulation in Example 2.19 of Section 2.5.
Chapter 3
Monday
- Topic: Section 2.4, 3.1: Long term behavior and limiting distributions. We discuss the possible long-term behavior of Markov chains through numerical computation. This discussion will serve as motivation for the analytical study of limiting distributions we will take on in Chapter 3.
- Class materials: Lecture notes, worksheet, RMarkdown file
- After class: Read the Proportion of Time in Each State portion of Section 3.1. Make sure to bring a laptop to class next time and download the rref.Rmd file in advance.
Wednesday
- Topic: Section 3.2: Stationary distributions. We begin our discussion of how to turn the problem of understanding limiting distributions from a question about limits to a question about algebra. This entails introducing the idea of a stationary distribution and solving linear systems of equations.
- Class materials: Lecture notes, worksheet
- After class: Finish Homework 2 and today’s worksheet.
Chapter 3
Monday
- Topic: TBA
- Class materials: Lecture notes, worksheet, worksheet solutions
- After class: Read Examples 3.8 and 3.11 and the definition of communication class.
Wednesday
- Topic: Section 3.3: Communication classes. We discuss what it means for states to communicate and get introduced to the ideas of recurrence and transience. This discussion, while separate from limiting and stationary distributions, is the first piece to developing an understanding of the Limit Theorem for Regular Matrices and more broadly the question of which Markov chains have a limiting distribution.
- Class materials: Lecture notes, worksheet
- After class: Read pages 96-98 on recurrence and transience.
Chapter 3
Monday
- Topic: Section 3.3: Recurrence and transience. Our aim is to deepen our understanding of recurrent and transient states. We discuss the related ideas of excursions, regeneration, and the geometric distribution in order to give a quantitative criterion for showing a state is recurrent or transient.
- Class materials: Lecture notes
- After class: Read Theorem 3.3, Corollary 3.4, and Lemma 3.5.
Wednesday
- Topic: Section 3.3: Canonical decomposition. We discuss how the recurrence and transience series criterion, briefly introduced last time, is derived, and then show how it can be used to show that states in a communication class are either all recurrent or all transient.
- Class materials: Lecture notes, worksheet
- After class: Read about first step analysis in Section 3.4, focusing on Example 3.17.
Chapter 3
Monday
- Topic: Section 3.4: Irreducible Markov chains. We discuss finite state irreducible Markov chains and a theorem that states that these Markov chains always have a unique stationary distribution whose components can be found using the expected length of an excursion. We discuss a computational method called first step analysis for finding the expected length of an excursion.
- Class materials: Lecture notes
- After class: Read about periodicity in Section 3.5. Put your focus on Lemma 3.7 and Example 3.18.
Wednesday
- Topic: Sections 3.5, 3.6: Periodicity and ergodicity. Through examples we have seen two main obstructions for the existence of a limiting distribution with positive components: multiple communication classes (which obstructs the limiting matrix from having equal rows) or states having a periodic structure (which prevents \(\lim_{n \to\infty} P^n\) from existing like in the random walk on the 6-cycle). We discuss periodicity in more depth and a new theorem, the Limit Theorem for Ergodic Markov chains, which tells us that once these obstructions are overcome we must have a limiting distribution: any finite state, irreducible Markov chain whose states have period 1 must have a limiting distribution with positive components.
- Class materials: Lecture notes, Ergodicity
- After class: Read the opening discussion in Section 3.8 up to Example 3.27.
TBA
Monday
- Topic: Fall break, no class.
Wednesday
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Wednesday
- Topic: Thanksgiving break, no class.
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TBA
Monday
- Topic: TBA.
- Class materials: Lecture notes, worksheet
- After class: Work on Exam 2. Enjoy your break! Keep in touch!
Getting help
Here are a few ways to get help:
- Office Hours: Mondays & Wednesdays 4:00-5:00, Thursdays 11:00-12:00; additional availability by appointment
- Study groups: Other students in the class are a wonderful resource. I want our class to feel like a community of people working together. Please get in touch if you’d like me to help you find study partners, and please reach out if you’d like others to join your group. You may work together on homework, explain and check answers, but make sure you write what you know on homework in order to get good feedback.
- Message board: I’ve set up a question and answer forum for asking and answering questions about class material, including homework problems. My plan is to check the forum regularly and answer questions as they come up, but my hope is everyone in the class will pitch in and answer others’ questions as another form of participation in the class.
Resources
- Our textbook also has a useful collection of R scripts available; contained there are all the R code snippets you’ll notice interspersed in the text.
- I’ve collected some resources to help you with some basics of RMarkdown.
- Here is a RMarkdown/LaTeX template file for writing nicely formatted documents, along with its pdf output.
- A LaTeX quick reference is available for commonly used symbols.
- RStudio Server is a cloud service that lets you edit and compile R and RMarkdown files through a web browser so that no local installation is needed. The server is hosted on the MHC network and you need to be on the VPN to access it if you’re away from campus.
- You can also install R and RStudio locally on your personal computer (you must install R before RStudio), or you can also use RStudio Cloud, which is a commercial RStudio cloud service with a free tier.
- Here are some useful RMarkdown documents that have code snippets for ideas that will be used through the class.