Modeling Class Notes

Math 350 Daily Links (Spring 2021): Week 1 to 11

Week 1: Jan 25-29
- Mon 1/25: Section 7.2 of Boyce/Diprima (See Canvas)
  - Video 1 of 2 (Review of Matrices)
  - Video 2 of 2 (Review of Matrices)
  - Homework: Section 7.2: 1, 4, 7, 8, 9, 11, 13*, 21*, 23*. Due on Thursday, Jan 28 (Upload to Canvas).
- Wed 1/27: Section 7.3 of Boyce/Diprima (See Canvas)
  - Video 1 of 3 (Four subspaces)
  - Video 2 of 3 (Numerical example)
  - Video 3 of 3 (Eigenvalues and Eigenvectors Review)
  - Homework: Section 7.3: 1, 3, 8*, 14*, 15, 16, 18*, 31. Due on Friday, Jan 29 (by midnight).
- Fri 1/29: Section 7.1 (Intro to Modeling Systems)
  - An overview of systems of DEs
  - Modeling interconnected tanks
  - Shifting coordinates using equilibrium solution
  - Homework: Section 7.1: 1, 5, 18*, 22*. Homework due on Tuesday next week.
  - Optional video- Shows you how the book got the equations for the two mass/spring system
  - Optional video- Shows you how to use Octave-Online to compute eigenvalues/eigenvectors and inverses. We'll have a more formal intro into this later.
Week 2: Working on solving x'=Ax.
- Mon, Feb 1
  - Video 1 of 3: Linear Systems
  - Video 2 of 3: Eigenvalues and Eigenvectors
  - Video 3 of 3: Numerical example
  - Quick example of code to plot your solutions. The online plotter is linked here.
  - Homework: Section 7.5: 1, 3, 4*, 12*, 15, 25. Due on Thursday, Feb 4.
- Wed, Feb 3: Live Session: Lots of links below, grouped into 4: The first grouping would have been the usual videos looking at eigenvalues (not live). The second group below is the population modeling- This is "group work", so feel free to work together on the questions given in the PDF below. The last link below is the live video in case you want to check something (optional, since the info you need is given in the previous videos). Sorry about all the links!
  - Continuing Ch 7: Eigenvalues.
    - Complex Eigenvalues (Part 1)
    - Complex Eigenvalues (Part 2), with Octave Online
    - An eigenvalue with algebraic multiplicity equal to 2 (2x2 systems only)
    - PDF notes on the 3d (and final) case
    - Homework: Solve 1(a,b), 2(a,b), 3(a,b) from the exercises in the PDF document above. Due Monday.
  - Group Work- due next Tuesday.
  - Video from the "live" session:
    - Live up until we broke into groups
    - Video after the groups to the end (Octave, similar to above)
- Fri, Feb 5: Numerical Methods (for solving ODEs)
  - (PDF) Summary of the next 4 videos (each about 10 minutes long)
  - (Video) Euler's Method
  - (Video) Euler's Method in Matlab/Octave
  - (Video) Backwards Euler's Method
  - (Video) Improved Euler's Method
  - Video on setting up Octave online for class.
  - PDF notes for geting Octave set up, and some introductory commands and exercises
  - Matlab/Octave "cheat sheet" (in PDF)
  - Homework note: There are two sources of homework for tonight- The textbook (sections 8.1 and 8.2), and also towards the end of the PDF file for getting Octave set up (linked above). For the Octave command problems, you can write your results on a piece of paper to turn in.
  - Homework:
    For each algorithm below, write the first solution by hand (you don't need to go farther than two steps in). All other tasks can be done on the computer.
    Section 8.1: 1(a,c)*, 5(b,d)*
    Section 8.2: 1(a,c)*, 5(a,c)*
    Octave PDF: Exercise set I: 1, 3, 4, 6*, 8. (Due: next Wed, Feb 10)
Week 3: Feb 8-12
- Mon, Feb 8
  - (PDF) Notes on 8.3, 8.5, 8.6. Also, homework and Octave code.
  - (Video) How to share files in Octave
  - (Octave Link) Takes you to the "bucket" of code discussed in the video.
  - (Video) Summary of Euler methods and discussion of numerical error.
  - (Video) The error in Euler's method, setting up Runge-Kutta.
  - (Video) The Runge-Kutta Method and Octave example
  - (Video) Conversions to use first order solvers.
  - Homework: Work through the exercises on the PDF in the first link. (Turn them all in- there's only 3 questions). (Due Thur, Feb 11)
- Wed, Feb 10: Graphical analysis of a system of DEs.
  - Online slope field plotter for systems
  - The Poincare Diagram
  - Graphical analysis, Part 1 (Ch 9.1)
  - Graphical analysis, Part 2 (Ch 9.1)
  - Graphical analysis, Part 3 (Ch 9.1)
  - Poincare analysis (Part 4 of 4)
  - Homework: Section 9.1. For each of these, try (a,b,c,d): 1, 3, 5, 7, 9. You should try them all, but I won't have you turn them in as I might not be able to get these back before the exam.
- Fri, Feb 12: Linearization and Modeling.
  - (PDF) Linearization
  - Video of Live Session
  - Octave Code Focus
  - Homework: Section 9.3: 1, 2, 5, 7.
Week 4: Feb 15-19
- Mon, Feb 15: President's Day- No classes.
- Wed, Feb 17: Modeling with Systems of Differential Equations (9.4-9.5)
  - Example from 9.3, Setting up 9.4. Some class announcements.
  - Example from 9.4
  - More from 9.4-9.5
  - Method of Nullclines, Finishing 9.5
  - For the homework, rather than finding eigenvalues and eigenvectors of the linearized system, we will just use the Poincare Diagram to classify the equilibria. For each of the following, first find the equilibria, then classify. Use the nullclines to analyze the direction field as well, as we did in the last video. The systems are from 9.4: 1, 3, 5 and 9.5: 1, 3.
- Fri, Feb 19: The S-I-R model.
  - (PDF) Hard copy notes of the SIR model, with homework questions.
  - Setting up the model (Video 1)
  - Analyzing the model (Video 2)
  - Numerical solution - Example in Octave
  - Homework: Work through the questions in the PDF file from today.
Week 5: Feb 22-26
- Mon: Review day. See "Exam 1 Links" below.
- Wed: Exam is available Wed-Thu.
- Fri Feb 26: Linear Algebra
  - PDF notes with exercises.
  - Video 1 of 3
  - Video 2 of 3
  - Video 3 of 3
  - Homework: Exercise 1-8, pg 43-44. Turn in solutions to 1, 3, 6, 8. Also, computer problem \#7 that was added later. Due on Tues, Mar 2d. (Refers to the updated notes with the computer code included just before the exercises).
Week 6: Mar 1 - 5
- Mon, Mar 1: Continuing with linear algebra- The 4 subspaces, Eigenvectors.
  - Updated PDF notes (previous link updated as well)
  - Video: Four fundamental subspaces.
  - Video: More linear algebra
  - Video: Matrix factorization
  - Homework: Be sure to add the computer problem (#7, p 44) to Friday's HW.
    Page 46: 1, 3, 4, 6. Turn in solutions on Thu, Mar 4th.
- Wed, Mar 3: The SVD (the notes are a continuation of Monday's notes)
  - Video 1: Worked Example after The Spectral Theorem
  - Video 2: The Spectral Decomposition
  - Video 3: The SVD and example by hand
  - Video 4: Worked out Exercise 5, 6 p. 53
  - Video 5: SVD and the 4 subspaces; the Reduced SVD
  - Homerwork: Programming exercise, pg. 51, Exercises 1-4 and 8 on p 53. Turn in #1, 2, and 8 on Tues, Mar 9.
- Fri, Mar 5: Applying the SVD- Generalized Inverses and Image processing
  - Update of notes (PDF). Start on p. 56
  - Homework PDF: The Computer Lab
  - Video files:
  - Download files (right-click the links below, "Save Target As..."
Week 7: Mar 8-12
- Mon, Mar 8: Spring Break Day. Nothing due today- Take the day off!
- Wed, Mar 10: Some statistics today! You may already be familiar with most of this, but the last part, the Covariance matrix, plays a key role on Friday.
  - (PDF) Notes for today ("Chapter 2", pg 17-22).
  - Video 1: The mean
  - Video 2: Centering, Double-Centering a matrix. Variance.
  - Video 3: Standard Deviation, Correlation Coefficient.
  - Video 4: The Covariance Matrix and starting some exercises.
  - Homework: Pg 22 of today's notes. Do #1, #2 on the computer, and the rest by hand: 1-7. Be sure to work 3-7 consecutively, but only turn in solutions to #2, 6, 7. Due: Mon, Mar 15.
- Fri, Mar 12 (Mid-Semester): The Best Basis (aka Pincipal Components Analysis)
  - (PDF) Notes for today
  - Video 1: Motivation and the Cov Matrix
  - Video 2: Cov matrix and SVD; Projection of the mean
  - Video 3: Proj and variance; Error for the Best Basis
  - Video 4: Minimizing the error is same as maximizing a nice quantity
  - Video 5: Maximize the nice quantity.
  - Video 6: The Best Basis
  - Video 7: Summary and connections to PCA
  - Homework: Last page of today's notes (p. 64), problems 1-7. Most of them are short computer computations to verify the formulas in the text. Due Wed, Mar 17.
Week 8: Mar 15-19
- Mon, Mar 15: Eigenfaces
  - (PDF) Full linear algebra chapter (finishes notes from Friday)
  - (Video) Matlab and Eigenfaces (abt 30 min- Sorry!)
  - (Video) Python and Eigenfaces (you don't need to watch both Matlab and Python versions) (About 20 minutes).
  - (Matlab MAT file used in the video and the text) allFaces2.mat (Right-click and "Save As...")
  - (Matlab M-file) EigenfaceExample.m: Script file of commands from video and text.
  - (Python script file) EigenfaceExample.py: Script file in Python.
  - (Matlab MAT file for homework) Math350Homework.mat: Data file for the homework/project.
  - (Video) Short video introducing the HW on Octave.
  - Homework: On the last page of today's notes, you'll see the Eigenfces project outline. You may work together, but in no more than 2 per group. Each person needs to upload their work- What to submit? A script file showing your commands, and the figures that the script file produces (screenshots are OK if you're not sure about how to work with them). Due by Thursday this week.
- Wed, Mar 17: Homework Catch Up
  - (Video) Homework help for Friday's stuff (#1-3). The first two problems were really designed to be done by hand- Some started with computer code for those, but I think its much easier just to do them by hand (again, #1, #2).
  - (Video) Homework help, #4 (Matlab/Octave)
  - (Video) Homework help, #5 (Matlab/Octave)
  - (Video) Homework help, #6-7 (Matlab/Octave)
  - (Video) Homework help, #3, 4 (Python)
  - (Video) Homework help, #5, 6 (Python)
  - (Video) Homework help, #7 (Python)
- Fri, Mar 19: Data clustering
  - Chapter on Data Clustering
Week 9: Data Clustering
- Mon, Mar 22: K-Means Clustering (Links moved here)
- Wed, Mar 24: The Neural Gas Algorithm
- Fri, Mar 26:
  - A rewrite of the clustering chapter, updating DBSCAN (Topic for today).
  - Videos:
    - Part 1
    - Part 2
    - A video showing DBSCAN in action! What you're seeing is how the data is collected over time into clusters using DBSCAN.
  - Homework is here. Problem 1 and 2 are done in the included videos, so just turn in solutions to Problem 3. Due next Wednesday
  - Review Homework (Due Tuesday): Turn in the Computer Lab, Mar 5th that didn't get a due date! You only need to turn in the second part (the part with the SVD of the image).
Week 10: Optimization
- Mon, Mar 29: Bisection, Newton's Method
  - (PDF) Optimization Chapter
  - (Video) Ch 9: Introduction/Motivation
  - (Video) Ch 9: Bisection
  - (Video) Ch 9: Newton's Method
  - Corrected: (Matlab) Bisection and Newton's Method Examples - (main.m). Run this as a function- from the command window, type: main. The original file is here if you want to compare.
  - (Python) Bisection and Newton's Method Examples - (ExScript1.py)
  - Homework: Page 105 of the notes 1-3. Turn in 1, 2. (Be sure to use the sample scripts above) DUE: Thursday, Apr 1.
- Wed, Mar 31: Optimization (linearization, multidimensional Newton's Method). No HW will be collected, but will be important for the upcoming exam!
  - (PDF) Updated notes, Ch 9
  - (Video) Review of Linearization
  - (Video) Finish linearization, start Newton's Method
  - (Video) Comments on Multivariate Newton
  - (Video) Comments on the Code (Sorry about the intermittent flashing on the video- I don't know why that happens (I'll have to try some different software).
  - Code:
  - Homework: p. 114: 1, 3, 5, 7. Use the provided code to answer #5.
- Fri, Apr 2: Optimization. No HW will be collected, but will be important for the upcoming exam!
  - Webpage with links for Gradient Descent and Stochastic Gradient Descent.
Week 11 (Exam 2 this week)
- Mon, Apr 5: Catch up/Review
- Wed, Apr 7: Spring Break Day
- The exam will be available any time on Friday or Saturday. More details to come.