Modeling and Analysis of Time Series Data

University of Michigan STATS 531/631 Winter 2026. Instructor: Edward L. Ionides

Course description

This course gives an introduction to time series analysis using time domain methods and frequency domain methods. The goal is to acquire the theoretical and computational skills required to investigate data collected as a time series. The first half of the course will develop classical time series methodology, including auto-regressive moving average (ARMA) models, regression with ARMA errors, and estimation of the spectral density. The second half of the course will focus on state space model techniques for fitting structured dynamic models to time series data. We will progress from fitting linear, Gaussian dynamic models to fitting nonlinear models for which Monte Carlo methods are required. Examples will be drawn from ecology, economics, epidemiology, finance and elsewhere.

Additional information is in the syllabus. A provisional schedule for the topics is posted but deviations from this schedule may occur. Please follow the issues for the course GitHub repository at https://github.com/ionides/531w26. You are more than welcome to contribute issues and/or pull requests to the course repo. The repository is public, so your contributions should be polite and respectful.

631 includes a reading group where we discuss a research paper each week. Students registering for 631 are expected to have taken at least one core PhD-level class such as STATS 600.

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Class notes

1. Introduction

2. Estimating trend and autocovariance

3. Stationarity, white noise, and some basic time series models

4. Linear time series models and the algebra of ARMA models

5. Parameter estimation and model identification for ARMA models

6. Extending the ARMA model: Seasonality, integration and trend

7. Introduction to time series analysis in the frequency domain

8. Smoothing in the time and frequency domains

9. Case study: An association between unemployment and mortality?

10. Forecasting

11. Introduction to partially observed Markov process models

12. Introduction to simulation-based inference for epidemiological dynamics via the pypomp package.

    • Getting started with pypomp

13. Simulation of stochastic dynamic models

14. Likelihood for POMP models: Theory and practice

15. Likelihood maximization for POMP models

16. A case study of measles: Dynamics revealed in long time series

Other examples of POMP data analysis, covering diverse topics, are provided by the final projects for previous 531 courses: 2025, 2024, 2022, 2021, 2020, 2018, 2016.

Homework and participation assignments

• Homework 0. Some course preparation.

• Homework 1, due Sun Jan 18, 11:59pm. Solution.

• Homework 2, due Tue Jan 27, 11:59pm. Solution.

• Homework 3, due Sun Feb 8, 11:59pm. Solution.

• Homework 4, due Sun Feb 15, 11:59pm. Solution.

• Homework 5, due Sun Mar 15, 11:59pm. Solution.

• Homework 6, due Sun Mar 22, 11:59pm. Solution.

• Homework 7, due Sun Apr 5

Midterm exams.

These are drawn from the same problem bank as the daily quizzes. They are done without electronic devices and aim to reinforce the foundational course material.

• Exam 1. In class on Monday 2/16. Sample questions. With solutions. Sample midterm randomly drawn from the test generator; the actual test will use a different seed.

• Exam 2. In class on Wednesday 4/15. Sample questions. With solutions. Sample midterm randomly drawn from the test generator; the actual test will use a different seed. This is preliminary; more questions may be added before the test.

Midterm project

• Information.

• Midterm peer review report instructions

• 2026 midterm projects

• You are welcome to browse previous midterm projects. The course websites from 2021, 2022, 2024 and 2025 have a posted summary of peer review comments. Earlier projects are also available, from 2016, 2018 and 2020.

Final project

• Information

• You're welcome to browse previous final projects. The 2025 2024, 2022 and 2021 final projects have a posted summary of peer review comments. Earlier projects from 2016, 2018, 2020 may also be useful.

Using the Great Lakes cluster

• Great Lakes access will be set up after the midterm project and used for the second half of the course.

• Introductory notes for using our class account on the greatlakes cluster. This is optional but may be helpful for your final project.

• If you are already familiar with using R on Great Lakes, all you need to know is the class account: stats531w26s001_class.

• You are expected to use our class account only for computations related to STATS 531.

• Please share knowledge about cluster computing between group members, and/or on piazza, to help everyone who wants to learn these skills.

• Cluster-related questions can also be emailed to the U-M Information and Technology Services helpdesk, help@umich.edu

Acknowledgements and License

This course and the code involved are made available with a Creative Commons license. A list of acknowledgments is available.