Math 419W/519: (Intro to) Stochastic Math Modeling

Winter Semester 2021

Eastern Michigan University Creed

Land Acknowledgement

Official Course Catalog Entry

419: Models of randomness in a variety of fields: actuarial studies, economics, biology, engineering, and others as appropriate for student population. Discrete time Markov chains, Poisson processes and generalizations, time series, Brownian motion, and dynamic programming. An important part of the course is an opportunity for a student to become involved in an actual modeling problem.

519: Models of randomness in a variety of applications. Discrete and Continuous Time Markov chains, Renewal processes and generalizations, queueing theory, time series, Brownian motion, and dynamic programming. Completion of basic linear algebra and probability is assumed.

Prerequisites

419: Math 122 and at least one of Math/Stat 223, 311, 319, 360, 370

519: Linear algebra at the level of Math 122 and probability at the level of Math 360 is assumed.

Some experience using Excel, R, or Python will also be VERY helpful (or Matlab, or to a lesser extent Maple or Mathematica), but it is not strictly a prerequisite.

419 Follow-up courses: Math 436 Numerical Analysis, various statistics classes

519 Follow-up courses: various statistics classes

Class Format and Meetings

Format: Synchronous Online, with recordings available if needed.
Meetings: Tue, Thu 2:00pm-3:15pm in Zoom
"Final Exam" (actually, presentations) schedule: Tue Apr 27, 1:30-3:00 A HALF HOUR EARLY
Math 419W: CRN 24290, 3 credit hours.
Math 519: CRN 24288, 3 credit hours.

Class meetings will be mostly interactive lectures, with some time to discuss homework.

Instructor information

Professor Andrew Ross
Pray-Harrold 515m
andrew.ross@emich.edu
My homepage
(734) 487-1658, but I strongly prefer e-mail instead of phone contact.
Math department main office: Pray-Harrold 515 (734) 487-1444

Office Hours and other help

 
Mon/Wed: various meetings for research and with students
Tue/Thu:
	1:30	2:00	office hours	
	2:00	3:15	Math 419W/519
	3:15	4:00	office hours	
Fri
	11:00-12:00 once/month: department Colloquium (students invited!)
	12:30- 2:30 once/month: department meeting

I am also happy to make appointments if you cannot come to the general office hours. Please send me e-mail to arrange an appointment.

The Mathematics Student Services Center (or "Math Lab") is also here to help you, in Pray-Harrold 411. Their hours are posted here. Please give them a call at 734-487-0983 or just drop by. However, very few tutors have taken Math 419W/519, so while it's a good place to work (meet with classmates, etc.) the tutoring might not be as good as it usually is for 100, 200, and 300-level classes.

Many assignments in this course will be in the form of papers, which I want to be well written. I will be providing you with as much discipline-specific writing help as I can. You may also find it helpful to consult with the Academic Project Center for help in tuning up your writing.

Required materials

Heavy use of a computer is required in this class. We will mostly use Excel and R/Rstudio, but Python is also acceptable. Many days, it will be best to have a laptop in class, but if you can't bring one, you can probably buddy up with someone.

Recommended materials

Our recommended (not required) textbook is "Introduction to Probability Models", by Sheldon Ross (no relation to your instructor), published by Academic Press, any edition

Course Web Page

We will use the Canvas system. You are expected to keep an eye on your scores using the system, and get extra help if your scores indicate the need.

Supplementary Materials

Here is a list of books that I have found interesting and related to math modeling. Perhaps some of them will strike your fancy, too. I own the ones that are starred (*) and can lend them to you. Others you will have to find at the library or on the usual Internet booksellers. Links are given to Amazon, but I do not specifically endorse them or any particular bookseller. Of course, if you like a book you can see what similar books the online bookseller recommends.

• *Sheldon Ross, Stochastic Processes (2nd ed.)
• The Flaw of Averages by Sam Savage/Probability Management (in EMU Halle Library, electronically)
• Stochastic simulation [electronic resource] : algorithms and analysis / Søren Asmussen, Peter W. Glynn
• Informal Introduction to Stochastic Processes with Maple, Jan Vrbik, Paul Vrbik
• Probability and Stochastic Processes with Applications, free online!, Oliver Knill (more advanced than our class)
• Applied Stochastic Processes in science and engineering, free online!, M. Scott (more advanced than our class)
• Stochastic process in physics and chemistry, N. G. van Kampen
• *Bhat and Miller, Elements of Applied Stochastic Processes (3rd ed.)
• *Yates and Goodman, Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers (2nd ed.)
• James Solberg, Modeling Random Processes for Engineers and Managers
• *Marcel Neuts, Probability
• *Sidney Resnick, Adventures in Stochastic Processes
• *Linda Allen, An Introduction to Stochastic Processes with applications to Biology
• *Edward Kao, An Introduction to Stochastic Processes
• The Nature of Code
• Guide to Excel statistical functions
• Simulation (uses ExtendSim)
• An Elementary Introduction to Queueing Systems by Wah Chun Chan
• Performance Modeling and Design of Computer Systems: Queueing Theory in Action by Mor Harchol-Balter
• http://www.math.vu.nl/~koole/obp/obp.pdf http://www.win.tue.nl/~iadan/queueing.pdf
• Probabilistic Search for Tracking Targets: Theory and Modern Applications
• Stationary Stochastic Processes for Scientists and Engineers; Georg Lindgren, Holger Rootzen, and Maria Sandsten
• An Introduction to Statistical Learning, with Applications in R by James, Witten, Hastie and Tibshirani (Springer, 2013). As of January 5, 2014, the pdf for this book will be available for free, with the consent of the publisher, on the book website.
• It's sort of like the NIST data analysis handbook: http://www.epa.gov/caddis/index.html CADDIS: The Causal Analysis/Diagnosis Decision Information System Especially volume 4.
• Systems Biology: Mathematical Modeling and Model Analysis

Course Content

Course Goals

Our primary goal is to teach you to be a good (or great!) stochastic math modeler. To be a good modeler, you need:

• Good habits and procedures, just like a scientist,
• Knowledge of common math models, and
• Communication skills (writing and presenting), to publicize and get feedback on your models.

We have a few secondary goals, which may be more or less applicable to your personal situation:

• Get enough people together to form a few teams for the Math Contest in Modeling (MCM), usually in Jan or Feb. I participated in this 3 times as an undergrad and had a lot of fun. It is also a good resume-booster.
• Give future teachers some great ideas to show your kids how high-power math is used in the real world. You may enjoy reading Meaningful Math.
• Give computer-science students lots of interesting things to program. You may like reading this blog entry about math for programmers.
• Get everyone using the appropriate software for each problem: Excel, Matlab/Octave, or perhaps R, Sage, Python, Mathematica or Maple.

Outline/schedule

We will start by reviewing basic probability ideas. We will also learn how to simulate a variety of random variables using Excel or Matlab (your choice)--doing little simulations will help understand a fair amount of the theory we will learn.

Time Series are used for a variety of things in economics and the various sciences. This will be the most statistics-oriented part of the class.

Dynamic Programming is a method of optimizing one's decisions as they unfold in time. It often includes some model of randomness, because we don't know what the future will hold. It is also used in some pattern recognition problems, such as speech recognition and genomic searches/ DNA alignment.

After that, we will talk about Discrete Time Markov Chains (DTMCs), which are used to model a wide variety of phenomena, from people moving between socio-economic classes to babies learning where one word ends and the next begins. Then, we will talk about Poisson Processes, which are useful for modeling the arrival of demands (like phone calls or customers) or other time-based phenomena (radiation particles, asteroids, etc.)

We will also study Renewal Theory, in which many of the results are completely intuitive, but there is one important result (called the Inspection Paradox) that takes some getting used to.

Queueing Theory is the study of how long people (or items) have to wait to be served.

Reliability Theory is in the book, but we will not cover it in this course unless there is a demand for it and some extra time.

Brownian Motion is the basis of a lot of stock market models. It is essentially a random walk. We will also look at some generalizations.

Schedule

Detailed Learning Outcomes

By the end of the course, students will be able to:

• (General modeling skills):
• evaluate models by constructing simple test cases
• select the most important variables to start modeling with
• Sketch sensitivity-analysis graphs before commencing modeling
• (Communications skills):
• Write a technical report
• Differentiate between literature of varying quality, e.g. peer-reviewed vs. working paper vs. white paper vs. web site,
• Design appropriate figures to communicate models and results
• (Applied Time Series Analysis):
• Extract a trend from a data set
• Extract seasonality from a data set
• Examine residuals for autocorrelation
• (Discrete Time Markov Chains):
• Formulate a DTMC from the description of a situation.
• Find the steady-state distribution of a DTMC.
• Apply the steady-state distribution to compute performance measures.
• Compute transient distributions.
• 519 (Continuous Time Markov Chains): similar to DTMCs, above.
• (Queueing):
• Apply basic queueing formulas (Little's Law, VUT) when approximating real-life situations
• Describe queueing situations in standard notation, so they can look up appropriate solution methods
• Describe the difference between single- and multi-server systems
• (Poisson Processes):
• Graphically test a data set to determine if it is a Poisson process
• Use the memoryless property when appropriate in analyzing Poisson processes and exponential random variables.
• (Renewal Processes):
• Graphically test a data set to determine if it is a renewal process
• Describe the effects of the Inspection Paradox
• (Dynamic Programming):
• Formulate deterministic DPs (value function and end conditions) according to a described situation.
• Formulate stochastic DPs (value function and end conditions) according to a described situation.
• (Brownian Motion):
• Simulate Brownian motion using a computer.
• Simulate some variants of Brownian Motion using a computer.

Grading Policies

Attendance

Regular attendance is strongly recommended. There will be material presented in class that is not in the textbook, yet will be very useful. Similarly, there are things in the textbook that are might not be covered in class, but are still very useful. If you must miss a class, arrange to get a copy of the notes from someone, and arrange for someone to ask your questions for you. If you are stuck on occasion without your usual child care, you may bring your child to class, and need not even get advanced permission (this is my personal policy--I don't know if EMU has a policy). Please be considerate to your classmates if your child becomes disruptive.

My lectures and discussions mostly use the document camera, along with demonstrations in Excel and other mathematical software. I sometimes have PowerPoint-like presentations, and I distribute electronic copies.

Homework

Homework will be assigned about once or twice a week. It will sometimes be a small problem set designed to help you understand the behavior of math models. Other times, it will involve writing up a little paper on an assigned topic. All homework should be typed.

Homework papers should be submitted on-line via Canvas, where they may be checked by TurnItIn.com or a similar service. This is partly to help keep you honest, and partly to help you learn acceptable ways to cite the work of others. A side benefit is that sometimes TurnItIn finds papers relevant to your work that you would not have found otherwise!

Exams

There will be no exams unless the class has trouble being otherwise motivated. If you would like an interesting project, you could create a final exam for this course, along with a writeup justifying why each question is appropriate, and of course a solution key along with rubric for grading incorrect answers.

Projects

Instead of a mid-term and a final exam, you will do a mid-term and a final project. Your results will be reported in a paper and a presentation to the class. The grade for each project is split into:

• 10 percent for the project proposal (due 3 weeks before the project paper),
• 10 percent for an annotated bibliography (due 2 weeks before the project paper)
• 30 percent for a mandatory complete draft (due 1 week before the project paper)
• 10 percent for doing a peer review of someone else's complete draft (due 2 days after complete draft)
• 30 percent for final version of project paper
• 10 percent for PowerPoint-style slides (and presentating them, for midterm project)

Final presentations will be made during the time slot reserved for the final exam. If there will not be enough time to do all final presentations, then posters, random selection, or point-auction may be used.

Overall Grades

No scores will be dropped, unless a valid medical excuse with evidence is given. In the unfortunate event of a medical need, the appropriate grade or grades may be dropped entirely, rather than giving a make-up, at the instructor's discretion. You are highly encouraged to still complete the relevant assignments and consult with me during office hours to ensure you know the material.

Your final score will be computed as follows:

• 50 percent for all the homework together,
• 20 percent for the mid-term project, and
• 30 percent for the final project.

Once final scores are computed, we will use the following grading scale:

	92.0 and above : A
	88.0 to 92.0: A-
	84.0 to 88.0: B+
	80.0 to 84.0: B
	76.0 to 80.0: B-
	72.0 to 76.0: C+, etc.
	

Writing-Intensive Rationale

Whether you go into industry or academics, you will need to be able to write reports on the mathematical work you have done. Math 419 is designed to enable students to apply math modeling techniques to formulate and solve problems in applied mathematics/operations research. In this class, students learn how to present their findings in the format of a peer-reviewed scientific journal or technical report, and how to present their findings in the format of PowerPoint-type presentations. Of the final grade in Math 419, over 50 percent is based on the writing assignments. Students are provided with the tools to enable them to communicate successfully their modeling findings. They receive written and oral feedback on smaller, staged writing assignments, as well as opportunities for revision, providing them with the skills to improve their writing and excel at writing complete papers. Students will individually write two full-length math modeling papers and presentations (those with an interest in secondary education may substitute one lesson plan for a modeling paper). Students who successfully complete Math 419 have the ability to read critically and evaluate peer-reviewed journal articles and present their own research in the same format. As such, Math 419 meets the requirements of a Writing Intensive Course in the Major of the General Education program. Unofficial additional rationale thinking: On the other hand, who says the current system of writing expectations in academia or industry is the best possible system? The paragraph above is about learning to "play the game". If you are more interested in "changing the game" instead, let me know (well in advance of due dates) and we can work together to try new things.

Side note:

Math 419W is distinct from Math 311W, Mathematical Problem Solving, because 419W projects focus more on applied work where a substantial part of the difficulty is figuring out what problem we want to solve-do we optimize today's operations, or our tactics for the next few months, or our long-term corporate strategy? Also, Math 419W projects often start with real-world data that students obtain from their workplaces. Formal mathematical proofs are only rarely a part of Math 419W, whereas they are a mainstay of 311W. Computer simulations, computations, and sensitivity analysis are important parts of most 419W projects, while they are not usually important in 311W. Math 419W tends to consider stochastic (random) phenomena, while 311W considers deterministic formulas.

Advice from Other Math Modelling Students

In the last few semesters, I've asked my math modeling students to give advice to you, future math modeling students, based on their experiences in my course. Here are some of the highlights:

• * work in groups * start the first day assignment is given * don't take too many credits w/ this class * ask a lot of questions * utilize Dr. Ross
• Do go to his office hours more than you normally would; if you have a question ask don't wait.
• See Prof. Ross in office hours and don't be afraid to email him. He is usually very helpful and approachable.
• Plan on visiting Prof. Ross during office hours in order to do well in the class. You will learn a lot in the end, but be ready to work.
• [prof ross:] add a note to the syllabus stating something to the effect of, "This class will not be like other math classes. Instead of straight-up problems or proofs, the biggest amount of work will be setting up the models, exercises, etc. and in analysing what your results mean. It will not be the mathematical work done to obtain the results that is the tricky part." But word the note better.
• attend the office hours Prof Ross is really good at explaining & helping out with the homework
• WORK TOGETHER!
• Go to class. Go to office hours and pick project that you're energized about and interested in even if they're harder. It will make this math class the best one you've ever taken.
• If your project isn't working out, contact him ASAP and he'll work with you to find a new project. Don't drop the class!

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