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My choir director shared a quote from Leonard Bernstein shortly before our last performance: "To achieve great things, two things are needed; a plan, and not quite enough time."
That's our paper!:) So we haven't much time. Basically we've got about four weeks... and it will go fast.
Here are some specific goals that I've thought of for each group. You will hopefully have some goals of your own! Think of mine as a starter set, at any rate.
Assignment for all: please have a member of each city's group email Jacob with a summary of identified issues in the three reports prepared so far (and any other issues identified since).
I might suggest that two separate pairs of eyes examine each issue.
\[ M(t)= a+b(t-1961)+\sum_{i=1}^{n}\left(\alpha_i \cos\left(\frac{2 \pi (t-1961)}{T_i}\right)+\beta_i \sin\left(\frac{2 \pi (t-1961)}{T_i}\right)\right) \]
to include elevation, latitude, and longitude.
Including them linearly would be a good start. That makes the parameters easy to interpret.
\[ M(t)= a+b(t-1961)+ \alpha_i \cos\left(\frac{2 \pi (t-1961)}{T_0+s}\right) + \beta \sin\left(\frac{2 \pi (t-1961)}{T_0+s}\right) \]
where $T_0$ is the smallest reasonable period (e.g. Nyquist limit).
(again, incorporating elevation, latitude, and longitude as well).
First of all, this course is called "Applied mathematical models". This is not "Theoretical mathematical models". So I hope that, by the time this course is over, you will have seen a variety of models applied to real problems, as we are now doing with the Togolese Meteorological data modeling.
This is from the syllabus for the course:
GOALS OF THE COURSE:
This is from the course description in the NKU catalog:
"Basic mathematical models arising in biology, psychology, sociology, political science, and decision science; exponential growth, predator-prey, Markov chain, learning theory, linear and nonlinear programming, waiting time, and simulation models."
Mathematicians seek to identify and understand patterns. We do that through mathematical models, which we can use to "summarize" the data, fill in the data, extend the data, predict the future, etc.
So we were obliged to talk about non-linear regression, and non-linear modeling in general.
We identified a pattern, and used linear regression to fit it: \[ CO_2(t)=a+bt+ct^2+d\sin(2 \pi t)+e\cos(2 \pi t) \] In fact, we want to think about that as a function of the form \[ CO_2(t)=a+bt+ct^2+ A \sin(2 \pi t + \phi) \] That's a quadratic trend, with a periodic oscillation representing the breathing of Earth's forests (which, as you might have noticed above, are under threat).
That non-linear model linearizes with a simple trig identity, however, so that we can use linear or non-linear regression to fit it. But that's because we know the period.
From this we were able to figure out something about how the Earth's forests breathe, which is pretty cool. We interpreted the parameters of the model, to provide some illumination or prediction.
We were also able to extend the diagnostics from linear regression to non-linear regression.
And we used some interesting software (InsightMaker) to allow us to work with these models.
Speaking of which, I never showed you my "final" model of this, and I want to now.
This model, a straight Lotka-Volterra model, can be fit using Newton's method, essentially, and I obtain a pretty good fit to the wolf and moose populations (especially if you compare it to our fits from those "more realistic" models, incorporating logistic growth for moose, and a reasonable kill rate -- which was fit in various ways via regression).
One of the interesting things is that the Lotka-Volterra system produces a large cycle (not as large as the theoretical 38-year cycle), but it's interestingly long, nontheless.
We kind of ran out of time to consider if further, because I felt obliged to get to some of the other parts of the catalog description (notably Markov Chains).
Before we get there, we'll look at a few models for an anthropological model of the Galla "cast" system in Ethiopia, and a learning model from psychology.
We'll probably compose a little Markov music. So let's end there today.
I've assigned you a little reading (sorry about the erratic quality of the scan -- I'll redo it when I get a chance). Ironically, the author of the Galla studies has written on mathematical anthropology, and I read one of his articles and was amused to discover this:
In short, elegant mathematical solutions to nonlinear problems are rare or nonexistent (Nering 1963:1). Hence the behavior of almost all anthropological components must be modeled by linear systems if they are to be modeled at all.
Nonlinear systems are equally common, and equally intractable, in pure mathematics. Mathematicians cope with them in the same way that physicists do: they approximate them with linear systems.