Day 16 in math modeling

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Announcements
- Your Ad Expenditures homework is returned.
  1. If Jacob E. is willing, I'll have him go over his homework solution with everyone. He'll be making many of the points that I'd planned to make (see below).
  2. Note that the parameters of the "shifted" exponential $ae^{bt}$ (for $t \in \{1970,...,1989\}$) can be obtained from the parameters of the original regression (call them $\alpha$ and $\beta$):
    - $b=\beta$ (remains unchanged)
    - $a = e^{ln(\alpha) - \beta*1900}$
    In actuality, this is the same exponential model! The estimates, residuals, etc. don't change.
    Shifts in the $x$ values don't change the model.
  3. Shifts in the $x$ values do change the power models, however. They have to, because the zero of the power model occurs at the zero of the $x$ values. Shift those, and the zero comes sooner or later, depending on the shift.
  4. The shifted power model (with power $\approx 209$) and the exponential model agree remarkably well.
  5. The power of the power models don't have any particularly sensible interpretation.
    
    Higher powers are needed to keep the curve "flat" as long as possible; so when we start at 1900, we can get by with an 8th degree (or so) power. But when we start at 1 A.D. or so, we've got to really up the power. The constant can shift up or down, but it does change the slope of the "flattish" part.
    We can relate the powers of the two models, using some basic mathematical principles: Flatness, etc.
  6. One thing that you might do is compare the models in their predictions. How do they compare if we use them to predict the ad expenditures 10 years down the road?
  7. Alyssa did a nice job of explaining various facets; she even considered the logistic model.
    She observed something that we noted last time about exponential functions: in $y(t)=e^{bt}$, an increase in $t$ of one unit means ad expenditures increase by a multiplicative factor of $e^b$ -- in "either model" (they're the same -- they just look different because of the shifted time scale).
  8. Important, big picture point: we log-transform to go over to "linearland"; but we've got to come back and talk about things in "non-linear land". That's where the scary things live....
- The second part of Mini-project 2 is due today, and then I'll assign you your third and final city and partner(s) for Mini-project 3. $Act by midnight tonight!$
Last time:
- We looked at how differential equations can arise from compartmental modeling.
- We derived a differential equation for the change in the infected population in an SIR model for infectious disease.
Today:
- We'll 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 $I(t)=I_0e^{-\alpha t}$).
- Then we'll move back to InsightMaker, to play 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.
Links:
- 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.

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