#### hedgie

##### New Member
I am looking to do a PhD in Stats. I could take one or both: stochastic processes, and/or numerical linear algebra. Any advice whether these would be of value, or could one just self study the two?

I have already sought some very useful opinions. I just prefer polling

I do not know what research areas I'll wish to do.

#### Dason

Do you have access to a syllabus or a course overview for the courses? There is quite a bit of variability in what courses covering those materials could actually attempt to cover.

#### hedgie

##### New Member
Course descriptions:

Stochastic Processes:

Introduction to the basic concepts and applications of stochastic processes. Markov chains, continuous-time Markov processes, Poisson and renewal processes, and Brownian motion. Applications of stochastic processes including queueing theory and probabilistic analysis of computational algorithms.

Numerical Linear Algebra:

Further study of matrix theory, emphasizing computational aspects. Topics include direct solution of linear systems, analysis of errors in numerical methods for solving linear systems, least-squares problems, orthogonal and unitary transformations, eigenvalues and eigenvectors, and singular value decomposition. Usually offered in the spring semester.

The syllabus I have access to are old so the material covered may have evolved more now.

Stochastic from syllabus:
DESCRIPTION: This course is an introduction of the basic concepts and applications of stochastic processes. Stochastic processes studied are Markov chains, continuous-time Markov processes, Poisson and renewal processes, and Brownian motion. Applications of stochastic processes include queueing theory, communication networks, finance, and others.

Book: Intro To Stochastic Modeling, Pinsky

OUTLINE
TOPICS # OF 75 MINUTE LECTURES
PRELIMINARIES 4
POISSON PROCESS 5
RENEWAL PROCESS 2
MARKOV CHAINS 5
CONTINUOUS-TIME MARKOV CHAINS 5
BROWNIAN MOTION 5

Numerical Linear Algebra from the Syllabus:

Book: Fundamentals Of Matrix Computation, Watkins

This course is a continuation of Intro to LA on a less abstract level than an "algebraic" graduate course in linear algebra. The course will be accessible to engineers and physical science students though duplication of introductory material given in intro to numerical analysis will be avoided. The motivation of the course is to emphasize some important computational aspects of matrix theory which are often neglected by linear algebra courses at all levels, and yet which (in the real world) comprise the essence of the subject. This course covers the solution of linear systems, least square problems, eigenvalue problems, and the singular value decomposition. Lectures in the text book will be followed, but Part IV, the interative methods, will not be covered. This topic is regarded as fundamental to applied math and hence furnishes required knowledge for ensuing professional careers. The outline of the topics on this course is the following:

1. Review Linear Algebra Basic Concepts.
a. Vector spaces, subspaces, and bases
b. Vectors and matrices and their norms
c. Linear transformations and their matrix representations.
2. Conditioning and Stability
a. Condition numbers
b. Floating point arithmetic
c. Stability of various algorithms
3. Linear Equation Solving
a. Gaussian Elimination
b. Pivoting
c. Stability of Gaussian Elimination
d. Cholesky Factorization
4. Least Square Problems
a. Orthogonal Matrices
b. QR Factorization
c. Gram-Schmidt Process
d. Householder Transformation
5. Eigenvalue Problems
a. Canonical Forms
b. Algorithms for eigen problems
c. Generalized eigen-problems
6. Singular Value Problems
a. Singular value decomposition (SVD)
b. Computing the SVD

#### Dason

Have you taken a linear algebra course before?

#### bryangoodrich

##### Probably A Mammal
I'd also wonder what your research interests are. I mean, the numerical linear algebra can be very handy in today's computational world. I assume it covers numerical methods for linear algebra. Of course, you can pick up a numerical analysis class (or text) just about anywhere to pick up those skills. I'd be partial to theory, and stochastic processes is far more appealing to me. With that said, if-when I do go for my stats/math PhD, my research interests would be in computational methods. I'd probably take both courses!

#### hedgie

##### New Member
Yes I've taken two linear algebra courses. The standard intro to linear algebra course and the proof based undergraduate course. My inclination is to take both but that will detract from some other obligations, and I have to pay for these courses as opposed to them being paid for in grad school.

Would it be advisable to pick up the topics on my own?

#### bryangoodrich

##### Probably A Mammal
From my experience, numerical programming isn't that difficult. It is less theory, more typing. Like, coding the bisection method to find the root of a polynomial is simple as hell. Every book begins by pointing out the intermediate-value theorem. Do I need to have taken a course in real analysis to appreciate that? Not really! Unless you want to justify your methods, you're basically doing cookbook stuff. It's good to know the linear algebra algorithms that an applied linear algebra course should teach, such as LU factorization, but if you're just wanting to be introduced to that stuff to learn how to program using it, then you'd want to take the course. Usually it's easy to pick up from a numerical analysis book if you're familiar with linear algebra. I literally read through the entire chapter of my book in a night (granted, the syllabus for your class covers 3 chapters from my book: LA, Least Squares, and Eigen-problems). Of course, programming it out is another story! The algorithms, though, aren't difficult. If you need a class to give you that structure, take the course. If you're good at teaching yourself programming, I'd save the money. You can always take a numerical analysis course in the future when it is relevant (or you have funding!). It'd make an easy course during your dissertation work or to fill an elective in your later years (which better have funding! lol)

#### hedgie

##### New Member
Thanks Bryan that was all very helpful! What about stochastic's is that easy to pickup on my own or doable I should say nothing ever winds up too easy!

Thanks again, ...any recommendations on a numerical analysis - numerical linear algebra book? And what language is best to learn in? The book at my University I uses matlab.

#### bryangoodrich

##### Probably A Mammal
Matlab is good, but I'm biased towards R because I know everything you're going to do can be done in it! Matlab is closer to more rigorous (low level) programming. Unless you know R, I'd probably recommend just gaining the Matlab skills. Frankly, you can do it in any programming language (C/C++, Java). I wouldn't be surprised you could get away with doing in it Python with the right libraries imported.

The book I used in my class, which isn't special, is Numerical Analysis by Sauer. My professor used, I believe, Numerical Analysis by Faires and Burden. I also modified one of my algorithms (double-integral solver) based on the one they use in their book, at least that's what the online reference cited. If you want to focus solely on numerical methods for linear algebra, I'd probably just check out what Amazon has for that and look at the reviews.

Stochastic processes you can do on your own if you like learning statistical theory on your own. I, personally, do not. I'm lazy and won't do homework lol I've never had the class, though. I heard it is good, and can be tough. At the graduate level, I would just assume it will be tough! haha

#### Dason

Personally I'd prefer to have a teacher for the stochastic processes especially if you've already had some linear algebra exposure.

#### hedgie

##### New Member
Thanks Bryan. I think I'll stick to R as that was one of my projects to learn this semester. Any other languages anyone recommends to pick up? I have access to all my uni software. I know matlab/octave somewhat just curious what else besides R and SAS I'll need for a PhD.

Dason what do you think the value of either or both courses is (given I do not know my research interests, though I am interested in all stats and probability subjects)? I may take the stochastic processes course in person then.

#### bryangoodrich

##### Probably A Mammal
While not particularly advantageous to statisticians, there's Mathematica and Maple out there. They're more oriented toward numerical programming in general. If you were a math major, I'd say check them out. If you can be familiar with Matlab or Octave, I think you'll be fine. If you can do that stuff in SAS (maybe SAS/IML), that would be impressive! If you can do some of that stuff in Python, that's your swiss army knife right there (seriously, Python connects to everything!)

#### Dason

Mathematica is actually a really nice program and I use it every now and then. It can be very powerful if you know how to use it. But if you were to learn something other than R or SAS I'd recommend Python or C.

#### hedgie

##### New Member
Thanks Bryan and Dason.

I know a little python a little. Is it advisable to pick up more python and potentially C or just stick with R. I guess I am asking what you guys are using in grad school and the real world on a regular basis.

Also Dason stochastic or the numerical LA course do you see a lot of value I taking them or waiting till I narrow down research interests ?

#### bryangoodrich

##### Probably A Mammal
Like I said, Python can be a swiss army knife. It is very, very powerful, especially with the scope of what it can do effectively and how it can interconnect with a lot of technology. I wouldn't go out of my way to learn C for numerical programming, but it can be very helpful, especially when you need to do a large computational thing, say, within R. You can create a C or Fortran module and call it from within R. Handy to know how to do, but not something you're gonna depend upon in general. When it comes to business, SAS dominates. R is pretty dominant in academia. The import thing is to be able to take the algorithms in theory and put them into code to be computed. How you get that done doesn't really matter, because that just requires learning the syntax of another language.

#### Dason

Also Dason stochastic or the numerical LA course do you see a lot of value I taking them or waiting till I narrow down research interests ?
I see value in both. I personally think the stochastic course would be a little harder to pick up on your own which is why if I were in your situation I would probably take that. Linear algebra is really important but it sounds like you have a good enough foundation there that for the time being you could do without the additional course. I'm not saying that it wouldn't be a good course to take - just that I think the stochastic processes course would be a little more difficult to do on your own. Plus learning theory is always beneficial in one way or another - if not just to help improve your mathematical thinking skills.

#### hedgie

##### New Member
Thanks Bryan and Dason!

Dason so in your position youd want to take both courses? I'd love to except for funding!!

Is stochastic too much to learn on ones own...I assumed because it was more an applies. course it wouldnt be as bad. Also I assumed for certain I'd be repeating it in grad school.

I was going to take the semester to work on SAS, R, and python but is it advisable to take courses beyond the first programming course? I assumed that was definitely self teachable.

Thanks so much to everyone!

#### Dason

Well I guess it sounds more applied than any stochastic course I've ever seen but stochastic processes typically involves quite a bit of theory. I guess it depends on how the professor handles it.

Programming is self teachable but you can definitely learn a lot from taking courses. There are certain things that you might not get from books when it comes to programming whereas if you have a teacher they'll correct you or tell you why it's a bad idea to do the seemingly not-so-bad thing in your code.

#### ANDS!

##### New Member
Thanks Bryan that was all very helpful! What about stochastic's is that easy to pickup on my own or doable I should say nothing ever winds up too easy!

Thanks again, ...any recommendations on a numerical analysis - numerical linear algebra book? And what language is best to learn in? The book at my University I uses matlab.
When I did NA, I used R; it was quite easy and the instructor appreciated having an alternative to what everyone else was using. As others have said though, it is a breeze.

#### noetsi

##### No cake for spunky
I think the best advice, given above, is to take things that are going to be useful in your career (and disertation). The one thing you don't want to do as you go through a doctoral program is lose sight of your disertation. Nothing is quite so painful as finishing your course work and realizing you need several more courses to do what you need to do in that (speaking from experience here as I did exactly that).