Math Modeling, Fall 2009

(This page was created after the course concluded to archive some of the materials)

Reflections on course content

The chapter references are to Mooney and Swift. For many of the modules, I either modified modules from friends (like the traffic modeling module), or I designed my own materials (like the hidden Markov module).

  • Difference equations and compartment modeling, including analyzing fixed points and stability of fixed points. See chapter 1. I used NetLogo for this at the start of the unit, but in the end, NetLogo's limitations for a language made me not want to use it again. I'd rather use something like BM also looks like it might be interesting for this, but it costs quite a bit for our university.
  • Random variables to do discrete stochastic modeling and measure the effect of various random factors. See chapter 2. I used scipy to help analyze these things. See, for example:
  • State transition models using differential equations to model the transitions, including using a numeric differential equation solver to solve for state values. See the epidemic modeling unit. This was by far the most popular unit, and I felt like the students really grasped the epidemic unit. They particularly enjoyed it because it was easy for them to tweak the model and explore various other scenarios, like vaccines, etc.. See also
  • State transition models (non-Markov), including using eigenvectors and eigenvalues of the transition matrix to analyze the situation. See chapter 3.
  • Markov Processes, including using the eigenvectors and eigenvalues of the transition matrix to analyze the situation. See chapter 3.
  • Hidden Markov Models, including using the ``smart'' algorithms for determining likely outcomes or likely states (e.g., the Viterbi algorithm; not the naive algorithm). People also seemed to like the markov and hidden markov models. One person analyzed a situation using hidden markov models for their final project.
  • Partial differential equations, including using the method of characteristics to solve for a function. See the traffic unit. This unit was too technical. The PDEs lost people. However, one student said that they were thinking about the concepts of waves through traffic one time when they stopped in traffic, so it was at least somewhat useful.
  • Empirical modeling, including using the least squares technique to fit a line or other curve to some data and analyzing the fit using $R^2$. See chapter 4. This unit was okay. We delved into things more deeply than Mooney/Swift, though.

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