What differential equations topics are important for statistics?

[Algeh]

I'm considering taking a 400-level series of mathematical statistics courses next year, and they have a prerequisite of differential equations. (Math department stats as opposed to applied stats, which I've taken in multiple social science departments over the decades at both undergrad and grad levels to meet various graduation requirements for non-math degrees.)

So...technically, I have taken differential equations. In 2002. And I got a C in it. (The only C I've gotten in a college math course.) My memories are pretty much "there are vector fields that you generate with a proprietary computer program that comes with the textbook" and "lots of predator-prey models as examples", with a side of "kept falling asleep because it was early in the morning" and "this is all applied stuff with no theory and I need the underlying proofs before anything will actually make sense to me, so I'm just going to try and survive the semester as best I can", so assuming it's a "will need the content" prerequisite as opposed to "must be this devoted to taking math classes to ride" prerequisite I should probably spend some quality time re-visiting differential equations before I actually sign up for this stats series.

Any suggestions on what I might want to specifically work through from differential equations to be able to make sense of an upper-division undergrad math stats class?

If it matters, I want to take these stats classes because I'm teaching a high school statistics class, and I'd like to understand the topics more deeply so I can do a better job of teaching that class in future years. (Right now, I'm mostly just reading the class textbook and then occasionally looking at OER college stats textbooks on the internet when I'd like to understand something in more depth.) I've been teaching the Algebra 1/Geometry/Algebra 2 high school course sequence for a very long time and definitely have not thought about stats beyond the little bit included in those courses since grad school.

[warty goblin]

I'd email the professor, they'll have a better idea than internet randos about what their course requires.

That said, I have a PhD in statistics, and work as a statistician, and I have needed actual differential equations not at all. Stochastic differentials do show up in statistical models for population dynamics (I just read a pile of covid papers that use them) but directly using them isn't necessary. For any application I've encountered, you can just use a discrete process on a fine time grid. But if you want to review diff EQ for statistics, I'd concentrate on SIR and SEIR models.

More important to understanding statistics is calculus, particularly multivariate calculus. You can't really understand least squares without it, and all of maximum likelihood is in practice an exercise in multivariate calculus. Gradients, Jacobians, and Hessians are your friends here.

The other chunk of math you need is linear algebra, particularly matrix decompositions and properties of non-negative definite matrices.

[NichG]

I suppose there are things like Ito calculus for stochastic differential equations, Langevin dynamics, and Fokker-Planck formulations that could use differential equations, but I don't think they'd usually be in an undergraduate class...

Maybe for Kalman filtering? Is that even part of a stats class?

[warty goblin]

I suppose there are things like Ito calculus for stochastic differential equations, Langevin dynamics, and Fokker-Planck formulations that could use differential equations, but I don't think they'd usually be in an undergraduate class...

Maybe for Kalman filtering? Is that even part of a stats class?

I've taken stat classes that include the Kalman filter, but they were all upper Ph.D. level courses, and they were all derived using lots of clever conditional normal distribution tricks.

[Algeh]

The course description is here: http://pdx.smartcatalogiq.com/2020-2...s/400/Stat-461 (and extremely vague, as you can see).

It's been a while since I've last thought about calculus, although not as long as since I've thought about differential equations. (I took a "calculus refresher" summer grad course for high school math teachers about a decade ago, mostly because my old undergrad advisor was teaching it and I thought it'd be fun to take a class from him again. That went fine, and I suspect I could easily brush up on basic undergrad 200-level calculus again as needed.) Teaching Algebra 2 over and over until everything high-school-level about polynomials is pretty much automatic rather than tenuous is a big advantage for most (but not all) calculus topics, but it's certainly true that I could stand to spend some time brushing up on the multivariate stuff. (The Algebra 2 curriculum where I teach is pretty rigorous - finding roots for polynomials using things like the rational root theorem and Descartes' Rule of Signs, rational functions, logarithms, sequences and series, radian trig, and so on. Not just "Algebra 1 again, but this time you won't pass unless you can deal with quadratics" as seems to be the case in some other situations.)

Weirdly, it appears this stats course only requires the first two of three calc classes, which pretty much means not the stuff about multivariate/series. (Differential equations course, the only prereq for stats, requires the 2nd of 3 calc courses plus linear algebra, which I also haven't taken since 2002 but which actually made sense at the time so I figure I can review more quickly. Matrices are old friends from programming in C, so linear algebra felt more comfortable to start with even though we were doing pretty different things with them.)

...yeah, I should probably just find out who is teaching the stats class in the fall and email them. I just prefer not to bother people teaching classes I'm wildly unqualified to take, so I wanted to find out how ridiculous I was being first. (I really want to take this one: https://docs.google.com/document/d/e...LAUyDKiX6Y/pub but figure they don't need random ex-CS students who bailed partway through grad school and pivoted to teaching HS math instead crashing their Topics courses.)

[Eldan]

I mean, I'm not a statistician, but I am a scientist and use various statistics daily, and I have some postgrad statistics education and... we used differential equations to build models sometimes, but that's not the statistical part of it at al. I've never heard anyone ever even mention differential equations in stats.

[DavidSh]

Do you have anybody serving as an official Advisor? If you were a normal undergraduate, that's the kind of question an Advisor should be able to answer, or to relay.

[warty goblin]

So at a guess that's gonna be pretty standard mathstat, for which you really only need basic calculus - it says they don't go above bivariate distributions so hardly even any multivariate calculus. I have no idea why they specifically require diff EQ, but I'd be shocked if there was anything in there that required actual differential equations. I'd expect the material you actually need is being good at integration and differentiation.

At the end of the day, you just don't need that much heavy math for statistics outside of measure theory. But that's more a probability theory thing anyway, only really covered at the graduate level in theory intensive programs, and is more of less useless in actual practice.

[JeenLeen]

...yeah, I should probably just find out who is teaching the stats class in the fall and email them. I just prefer not to bother people teaching classes I'm wildly unqualified to take, so I wanted to find out how ridiculous I was being first. (I really want to take this one: https://docs.google.com/document/d/e...LAUyDKiX6Y/pub but figure they don't need random ex-CS students who bailed partway through grad school and pivoted to teaching HS math instead crashing their Topics courses.)

For both the class you plan on taking and the one you want to take, I recommend looking at who taught it the past few years. If it's the same professor almost all the time, call or e-mail them now to get an idea for this Fall. I've found most professors are more open to answering questions than I would have thought. There might also be a general advisor who could help; while formally they might only be the advisor for folk doing a Master's in math or stat, they might be willing to answer your question.
Also, I noticed that it said "Math 256 or equivalent". Definitely ask what counts as equivalent and see if your knowledge counts.

For what it's worth, I did a Master's in Applied Stat, ten years after I did my bachelor's. I did Calculus AP classes in high school, and one Stats class in college. All I really remembered was the basics of what integrating and taking the derivative was, and that was sufficient.
I did wind up looking at some math tutorials on Youtube for stuff I should have known (like u-substitutions) but had forgotten -- patrickJMT is great. Search Youtube for "patrickJMT differential" and see if it makes any sense, if you want to refresh your memory of differentials.
But the point being that, although it required two classes of Calc, my little bit I remembered from high school was enough to get me started.

On the other hand, I admit there was a class that was a survey of using different computer languages to program statistics. I love programming, so it looked ideal for me. But the professor clearly told me I would probably fail since I didn't have the mathematical understanding for the type of things the class was going to program. It was very good I asked ahead of time instead of assuming it was mostly a programming class, like the name implied.

[MrStabby]

I have never used DEs in any of the statistics work I have done. That said there are some fields where evolution of systems under uncertainty/noise are common, SDEs in quantitative finance for example.

Calculus is needed for proofs, certainly. The Fourier Transform and characteristic functions for the Central Limit Theorem for example, a good grounding in this kind of stuff is important. But DEs? Not so sure.

I guess I have benefitted from the intuition they can help with. Thinking of things like the normal distribution as the solution to a DE for the rate of change of probability with distance from the mean helped as a different perspective on what is going on.

[danzibr]

Kinda off topic...

I have a PhD in math (and was a teacher for 10 years), yet somehow dodged Diff Eq all together. Recently got The Great Courses Plus and there’s a pretty decent Diff Eq course on there. I’m really enjoying it, filling in what I consider a critical gap in my education.