- Thank you for your second reports: they all arrived on
time, with a flurry of activity around 11 p.m.....:)
That means that they're all available for you to begin
reviewing for Friday, here.
And I've started to go over them, and they look good! Interesting stuff. A few new models this time.
- You have a preliminary part due this Friday (your evaluation of mini-projects 2), and
- the third report will be due on the Friday just before break.
- Hot off the presses:
Alaska's Bering Sea Lost a Third of Its Ice in Just 8 Days:
"Globally, sea ice is at record lows as the polar regions warm
faster than the rest of the planet. Along the Alaska coast,
it's affecting people's lives."
- El Nino is another long period oscillation:
- Jacob talked through his
ad expenditure homework solution, which did a
really good job of comparing the linear model $a+bt$
with two other (non-linear) models, a power model
($at^b$) and an exponential model ($ae^{bt}$), as well as
summarizing the important features of a power model
under a simple shift in time.
The exponential model is insensitive to simple shifts in the time scale, which is an advantage to that model.
- We continued our examination of an InsightMaker SIR model
for infectious disease. We got through most of it, but
haven't added in the infection piece yet. So we'll
continue with that first today.
- Now we're in the business of doing more "structural"
modeling -- that is, modeling the actual dynamic relationships
-- causative relationships -- between variables in a system.
Our tool of choice for doing this is differential (or difference) equations, but we're using InsightMaker to allow us to focus on the relationships, rather than focusing on the equations. In the end, however, being mathematicians, we're going to want to look "under the hood" at the DEs....
- I want to start by reviewing De Vries, et al.,
and talking about a parallel "difference equation" for the
infected population (and we'll see how it converges to the
differential equation solution, which was $I(t)=I_0e^{-\alpha
t}$).
One of the reasons that this is important is because, when all is said and done, we generally solve differential equations numerically by approximating them by difference equations. So we're really studying the solutions of difference equations.
- Then we'll move back to InsightMaker, to finish with
the "compartmental model" for infectivity, and see how to
change it in order to turn it into an SIR model.
- Then we'll have a look at the same thing within Mathematica, only expressed in the language of differential equations.
- InsightMaker has a demo (here) of the SIR building
process, in case you struggled last time -- or would just like to go
over it again. They take you through it step-by-step.
- I have a Newton's method sidenote to show you, to illustrate how or why it works for non-linear regression: fitting a single parameter model to a single data point. I borrowed the strategy from the Mathematica demo file for Newton's Method.