Day 32 in math modeling

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"To achieve great things, two things are needed; a plan, and not quite enough time.": Leonard Bernstein

Announcements:
- Stereopticon of the day...
- How did the visit with Professors Krug, Wilkinson, and Agard go?
- We'll get to the Markovian Galla (and sports!) on Wednesday.
Today: let's talk about rainfall.
- First of all, sorry about not making it clearer that you were to examine the rainfall reports from the cities you were working on before. Some of you went back and compared your old reports to what you did in rainfall...
- I must say that it is wonderful to read your thoughts about other reports, especially as I see you reflect on what you could do better next time (once you see something good in someone else's work).
  Many of you are doing a great job of self-reflection: how can I do better next time?
  It's also wonderful to hear you compliment each other's work. Some of you have identified individuals who seem to consistently do good work. It never hurts to compliment them directly, you know!:)
  I've sworn myself to secrecy, but you don't have to be secret in your compliments!
- Let's talk about your rainfall reports, and what's next.
  - We wanted to establish a relationship between rainfall and temperature, if we could. Some of you attempted to model rainfall based on temperature, and some of you didn't (preferring to attempt to model rainfall on time, with time as a surrogate for increasing temperature, perhaps).
  - When you use oscillatory terms, $\cos(\frac{2\pi t}{T})$ and $\sin(\frac{2\pi t}{T})$, you use them together or throw them out together -- they're really there to model the non-linear term \[ A\cos(\frac{2\pi (t-\phi)}{T}) \] and we need both of them to estimate $A$ and $\phi$.
    But my inclination is that, if one of them is significant, I'll keep both (not throw them both out if one of them is not significant).
    Those are sort of anti-symmetric policies, do you agree?
  - Express your oscillations in terms of periods. That's the fundamental, interpretable quantity. And you'll end up asking yourself "Hmmm.... what has a period of 365.25 days?" when you find it in your data....
  - Most of you failed to find any significant change in rainfall on linear temperature; those of you who went looking for it found that there was a significant effect temperature on rainfall.
- There are basically two approaches one can take to the rainfall: by month, or continuously and periodically. I like periodically, because one has an interpolator between months. But I have considered both types of models. Allow me to share my thoughts so far.
  - By Month:
    - What is the Gamma distribution?
    - Gamma fitting, by month
  - Continuously:
    - Matthew and Maria M. on pages 9 and 10 of their Mango report share a two-dimensional representation with the 3-d version which follows. They do a great job of relating the two.
    - A simple model, readily accessible and hopefully replicable over Togo.
    - A picture of the dynamic temperature/rainfall model, over two years. The model works into the future!

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