Mathematical Modeling
Math 350, Spring 2023

This course explores the process of building, analyzing, and interpreting mathematical descriptions of physical processes. Topics may include feature extraction, partial differential equations, neural networks, statistical models. The course will involve some computer programming, so previous programming experience is helpful, but not required.

There is no required text for this course. Class notes will be provided.

Beginning Material

Master version of the class notes. I'll be changing this as we go.
Videos for some topics (taped Spring 2021). These are optional, and they may be helpful.
Course Syllabus
List of Topics

Daily Links

Week 1:
- Intro To Matlab
- The N-Armed Bandit Code
Week 2: Finishing N-armed bandit, looking at some basic stats.
- Homework for the N-armed Bandit.
- Handed out Chapter 2 (Statistics) (See the full version of the class notes in the first link).
Week 3: Line of Best Fit and Linear Algebra
- Homework 1: Find the line of best fit to the Hanford data, which is a script file (meaning to load the data, just type "HanfordData", and the matrix X (which is 9 x 2) is loaded. We'll do another example of this in the lab next Monday.
- Homework 2 (4.1 in the notes), pg. 40: 1, 3, 5, 6, 7. You may do the computations on Matlab/Octave.
- Mean and Variance of the Projection Handout.
- Homework 3 (4.3, p. 43) 3, 4, 5.
- Homework due on Tuesday, Feb 7 at 11:59PM Turn in solutions to Homework 1 above, 5, 6, 7 from Homework 2, and problems 2, 3 to the Mean and Variance handout. (uploaded to Canvas).
Week 4: SVD (Feb 6-10)
- (Monday Lab) Lab Examples and Exercises
- Wed: Discussed eigenvalues/eigenvectors and the Spectral Theorem. HW: 1, 2 on page 47.
- Fri: The SVD.
  Homework: 1, 2, 8 on page 49.
- Homework to turn in for Week 4: From Monday's Lab: #2 (just give the command), 3, 4. From Wed's HW: Turn in #1, and from Fri's HW: 1, 2, 8. Due: Wed, Feb 15 at 11:59PM.
Week 5 (Feb 13-17)
- Mon: Finish chapter 4.
- Wed: Review
- Fri: Exam 1
Week 6 (Feb 22-24) Short week - The Best Basis (Chapter 5)
- Mon: No classes
- Wed: Start the "best basis". Remember that the take home portion of the exam is due at 11:59PM.
- Fri: Eigenfaces
Week 7 (Feb 28-Mar 3)
- Mon: Computer Lab- Best Basis
- Wed: Finish up PCA. See an interesting application here., which is summarized in PDF format here.
- Fri: Data clustering
Week 8 (Mar 6-10)
- Mon: Clustering in the Matlab computer lab
- Wed: Neural Gas training
- Wed: Kohonen's Self Organizing Map (SOM)
  Wikipedia page for SOM (Nice animations!)
  Sample training session
- Fri: DBSCAN (slides from Dr. Gao, SUNY Buffalo)
  A video showing DBSCAN in action!
  A video on training the DBSCAN algorithm
Week 9 (Mar 27-Mar 31) Finish clustering, then go to Optimization.
- Monday: Clustering Homework, Mar 27
- Wed: Start Chapter 9 (Optimization)
- Fri: Optimization
  - How to deal with Matlab functions. Also the Matlab code for the bisection algorithm, Newton's method, and multidimensional Newton.
  - (Matlab function file) bisect.m
  - (Matlab function file) NewtonMethod.m
  - (Matlab function file) MultiNewton.m
  - (Matlab function file) testfunc.m (A test function for the multivariate Newton).
Week 10 (Apr 3-7)
- Mon: Matlab in the computer lab Week 10 Lab is here.
- Wed: Finish up optimization. Start gradient descent and stochastic gradient descent.
  - (For Wed/Fri) Links for Gradient Descent and SGD
- Fri: Finish up the optimization and gradient descent. (Next Wed is Exam 2).
Week 11 (Apr 10-14)
- Mon: Review for the exam.
- Wed: Exam 2
- Fri: (PDF) Chapter 10: K-Nearest Neighbors Chapter
Week 12 (Apr 17-21)
- Mon: Finish kNN classifier, look at kNN regression.
  Here is a folder with the "helper" files and two example apps for the iris data.
- Wed: (PDF) Chapter 11: Linear Neural Networks
- Fri: Linear nets, continued. (Code to follow). If time, begin Radial Basis Functions.
Week 13 (Apr 24-28)
- Mon: Computer lab for Monday
- Wed: Neural Nets
- Fri: Neural Nets
Week 14 (May 1- May 5)
- Mon: Neural Nets- Backpropagation and building from scratch
  - (PDF) Full chapter on Neural Nets
  - (Video) Backpropagation of Error
  - (Video) Backprop part 2
  - (Video) Gradient Descent
  - (Video) Backprop is Gradient Descent
  - (Video - Matlab) Discussion of the code - Matlab
  - Folder with Code and Examples
  - Homework: Fill in the worksheet and look over the neural nets we built from scratch. Try running them in either Octave or Python.
- Wed: Autoencoders
- Fri: Deep Learning

Exam Links

Exam 1 Links:

UPDATED: You may prepare a half-sheet (8.5 x 5.5 inches, or equivalent area) of notes. Please turn in the notes with the exam.

Exam 1 Take Home Portion of the Exam

Youtube video on how to "publish" the M-file (so you can upload it). This is easier than uploading the finished m-file and screenshots, although if you get stuck, you can do that.
Take home portion of the exam, in PDF
M-file for Question 1
Data file for Question 1 (Exam1Q1.mat)
M-file for Question 2
Data file for Question 2 (Exam1Q2.mat)
M-file for Question 3

Exam 2 Links:

You may prepare one sheet (8.5 x 11 inches, or equivalent area) of notes. Please turn in the notes with the exam.

Take Home Portion of Exam 2

Here is the page with the exam and links

Final Exam Links:

Final Exam instructions (Final exam is due 11:59PM on Monday, May 15th. No exceptions!
(Matlab mat file) FinalExamData01.mat for Problem 1
(Matlab mat file) mushrooms.mat for Problem 3
(Matlab mat file) diabetes1.mat for Probelm 4

Matlab/Octave Notes

Octave is a free version of Matlab. Some things about it may differ from Matlab, but for the things we'll do in class, it shouldn't be a problem.
Octave online, so you don't have to install it
Example installation of Octave with Ubuntu
Example installation of Octave with Windows 10
"Cheat Sheet" for Matlab - Python - R. Useful to compare!
Numpy for Matlab users
Week 4 data

Mathematical Modeling Math 350, Spring 2023