Choice of class as preparation for (potential) PhD in Statistics

[LostLamb]

Hi all, I was hoping to get some advice on which course I should take in the coming semester. I only need one more class to complete my undergrad and I am currently faced with three choices:

1) Functional Analysis

2) Stochastic Processes II (i.e. continuous-time Markov chains, martingales)

3) Nonparametric Statistics

To provide some background, here's the list of math/stats classes (with grades) I've completed

Math

Linear Algebra (A), Calculus I,II (B+) -- Used pass/fail option here, Calculus III (i.e. Multivariable) (B+), Probability (A), Mathematical Stats (A-), Real Analysis I (B+), Real Analysis II (A-), Real Analysis III (Metric Spaces) (B+), Stochastic Processes I (A-), Ordinary Differential Equations (A-), Advanced (Measure-theoretic) Probaility (grades not released yet but probably A- and above)

Statistics

Intro to Computing (B+) -- Used pass/fail option here, Regression Analysis (A+), Simulation (A+), Categorical Data Analysis (A+), Data Mining (A), Computer-Intensive Statistics (A+), Linear Models (A-), Bayesian Statistics (A), Multivariate Statistics (A), Structural Equation Modeling (A), Psychometrics (A)

As you can probably tell, my math grades are less than stellar, especially in analysis. Considering how real analysis is so important, I was wondering if getting a good grade in functional analysis would kind of make up for it. However, the main drawback of taking functional analysis is that the professor teaching it next semester is known for making math modules a lot more difficult than they usually are. So taking functional analysis is kind of like a gamble: if I do well then great, but if not I may end up getting a really ugly grade. On the other hand, I know for a fact that it would be easier to score a decent grade for the other 2 classes. Any thoughts?

[bayessays]

Obviously an A in a tough math class will help you some, but it sounds far from guaranteed. Your math grades aren't awful, so I don't think taking one more class and getting an A would be such a huge boost that you need to do it. I'd say take whichever class you want to take - you already have more of a math and stats background than most applicants.

[LostLamb]

19 hours ago, bayessays said:

Obviously an A in a tough math class will help you some, but it sounds far from guaranteed. Your math grades aren't awful, so I don't think taking one more class and getting an A would be such a huge boost that you need to do it. I'd say take whichever class you want to take - you already have more of a math and stats background than most applicants.

Thanks so much for your input! If you don't mind, I have another quick question: Out of the three classes, which one do you think is the most useful (ignoring the difficulty of the class) for statistics research (especially in the more applied areas)? 

The main issue i have with pure-ish math classes (like functional analysis) is that they tend to be a little too detached from statistics and oftentimes i find it quite hard to maintain sufficient motivation to power through the deluge of theorems, lemmas and proofs (e.g. while studying metric spaces, most of the time i was like "who the heck cares if a set is compact!". Of course, now I see why people care after studying some advanced probability and optimisation theory). This motivation issue is probably made worse by fact that my main background is in psychology and I ventured into statistics by accident. So it would be great to get some insight into how each topic is relevant to actual statistics/methodology research!

 

 

Edited December 8, 2018 by LostLamb
typo

[orchidnora]

Typically, for top stats Ph.D. programs, it’s best to take as many theoretical math courses as possible. Generically, functional analysis might look best if you can get a good grade. 

However, if you want something directly applicable to applied statistical research, I would recommend taking nonparametric stats. An at least basic understanding of these methods is necessary imo for many data analysis projects. Of course, if you don’t take it now, you’ll probably end up seeing it in grad school anyway. 

On the other hand, if you really enjoyed Stochastic Process I and might wanted to study it in grad school, then take Stochastic Process II. 

Really, it’s up to you: how you want to spend your semester and what you intend to do in grad school. When applying you’ll have the SOP to emphasize any strengths in coursework, so think about what you want your coursework to represent about you. 

[bayessays]

I would look at it this way - if you don't enjoy the classes where you have to prove things like this, you won't enjoy the type of research that requires proving things like that all the time, so take the non-parametric stats class. Unless you go to Stanford or really want to research in those areas, you can be a PhD statistician and know nothing from those two classes. Some PhD stats programs might require you to take a similar stochastic processes class though.  If you're into applied research and don't really love the super mathy parts, I'd consider looking into biostatistics programs where your profile would be attractive to very good programs. 

[LostLamb]

Thanks for all the advice! As for proofs, it's not that I hate it, but it's not like I love it either. But I certainly think it's important to be able to read and digest the proofs though. Otherwise I don't think one can get a good grasp of the literature in whatever subfield one is working in. 

I'm really more into applied research (but not too applied to the point where you are just using tools developed by others to work solely on empirical data). Will look more into biostats programs!

[bayessays]

For sure, I get it - I like math too and understanding some proofs is important.  But you'll probably never have to use anything from those first two classes in research if you don't go out of your way to do so.  There is a huge range of research in between using already-made tools to do empirical research and functional analysis (almost the entire field of statistics!), and in a few years you will think of this as a very inconsequential choice in your career. 

[theduckster]

Like everyone here has suggested, do whichever one interests you more.

I will break from everyone else and say that a class in functional analysis can actually be quite useful in serious statistical research, as functional analysis is very relevant in the areas of statistical learning theory as well as - surprise surprise - functional methods in general (nonparametric regression, various kernel methods, etc). Do take a gander at those subfields of statistics and see if they interest you enough to the point where a class on functional analysis class seems useful. However, even then it probably won't be necessary. You'll probably be served just fine with coursework in (graduate) real analysis if you want to pursue serious research in these areas.

 

Edited December 9, 2018 by theduckster

[LostLamb]

 

50 minutes ago, theduckster said:

Like everyone here has suggested, do whichever one interests you more.

I will break from everyone else and say that a class in functional analysis can actually be quite useful in serious statistical research, as functional analysis is very relevant in the areas of statistical learning theory as well as - surprise surprise - functional methods in general (nonparametric regression, various kernel methods, etc). Do take a gander at those subfields of statistics and see if they interest you enough to the point where a class on functional analysis class seems useful. However, even then it probably won't be necessary. You'll probably be served just fine with coursework in (graduate) real analysis if you want to pursue serious research in these areas.

 

Yeah one reason why I was looking at functional analysis is that quite a lot of people have mentioned its relevance to things like kernels, Bayesian nonparametrics and functional data analysis. For now I'm browsing through kreyszig's Introductory functional analysis with applications to see if I would be somewhat interested in the material before committing an entire semester to it. 

Thanks again for everyone who replied! Wasn't really expecting this much input!

[bayessays]

Functional analysis is used in some areas of statistics, but unless you are doing theoretical research in that area, what I said was that taking a class in it is probably overkill - most PhD statisticians probably couldn't tell you what subjects a class in functional analysis would even cover. Differential geometry is used in shape analysis and some people design experiments using algebraic geometry, but you probably shouldn't take those classes either. 

[LostLamb]

Got it! The only reason I'm sparing some thought for functional analysis is because it's related to the applied area my honors thesis is in. I think at my stage it's not necessary to actually use the results from functional analysis. But since I have to pick a class anyway I'm just toying with the idea of getting more background.

[Stat Assistant Professor]

7 hours ago, LostLamb said:

 

Yeah one reason why I was looking at functional analysis is that quite a lot of people have mentioned its relevance to things like kernels, Bayesian nonparametrics and functional data analysis. For now I'm browsing through kreyszig's Introductory functional analysis with applications to see if I would be somewhat interested in the material before committing an entire semester to it. 

Thanks again for everyone who replied! Wasn't really expecting this much input!

I am doing some research on Bayesian nonparametrics and kernel-based regression right now and have been mostly learning it on my own "on the fly." 

Agreed with bayessays to take the class that interests you most. A PhD and postdoc teaches you to learn things "on the fly," which means: a) learning just enough about tangentially related areas to answer your research question (like the basics of functional analysis and reproducing kernel Hilbert spaces -- I do not need to become an expert in these areas), and b) you will probably learn "just enough" just by having a well-versed person explain it to you and by skimming a lot of papers, one section of a textbook, or lecture slides (the latter is especially helpful). When I do look at theorems and proofs, I only carefully read the parts that are needed for understanding the "general" technique and that I think would be useful for solving my own problem (not every lemma or technical detail is going to be relevant/useful).

[LostLamb]

On 12/9/2018 at 11:01 PM, Stat PhD Now Postdoc said:

I am doing some research on Bayesian nonparametrics and kernel-based regression right now and have been mostly learning it on my own "on the fly." 

Agreed with bayessays to take the class that interests you most. A PhD and postdoc teaches you to learn things "on the fly," which means: a) learning just enough about tangentially related areas to answer your research question (like the basics of functional analysis and reproducing kernel Hilbert spaces -- I do not need to become an expert in these areas), and b) you will probably learn "just enough" just by having a well-versed person explain it to you and by skimming a lot of papers, one section of a textbook, or lecture slides (the latter is especially helpful). When I do look at theorems and proofs, I only carefully read the parts that are needed for understanding the "general" technique and that I think would be useful for solving my own problem (not every lemma or technical detail is going to be relevant/useful).

Thanks very much for sharing your insights! Having read all the responses here (and after consulting friends who majored in pure math), I think I will be taking nonparametrics for a grade and will only be auditing functional analysis out of interest. 

Really appreciate everyone's views on this issue!