Day 3 in math modeling

Here's another example: A modeling misadventure with corn seedlings.

I asked you to read this because straight lines are not enough: we need a variety of functions, for a variety of situations. I thought that Ben's comment on the first page was funny: "If you begin to feel bogged down you can skip ahead and use the section for reference as needed." (I can see why one could get bogged down!)

If you did get bogged down, then I hope that you at least had a look at the pictures, on pages 123, 124, 128, 132!

So yesterday, my January, 2018 edition of Science appeared, and I've flagged all the linear regressions in the Research Section (in 77 pages, there are ten examples!).

• To the question: "Are surface temperatures in Togo, West Africa, rising?"

  Our first answer: We're not sure! (And that's okay! But do we have enough to continue, or do we bail out?)

  It appears that there's more going on for the minima than for the maxima; but there does seem to be something going on. That is, we've identified a pattern, consistent with "leaning yes", at least in some of the cities.

  That means we can proceed to phase II: modeling.

  We suggested that we might need to consider other things, like

  1. whether there is variation across Togo (it should help to view a map!)
  2. where the towns are located relative to the coast,
  3. elevation.
  4. Are the towns bunched together, or do they provide a good coverage of Togo?

• What do we need to know from Togolese meteorologists?
  • What kind of averages are these? Based on what little Binao has said, I'm thinking that they're averages of daily minima or daily maxima for the entire year.

  • Why is there a zero value in the Kara data? (I received word from Binao that it was a mistake -- the value is 20.7705555555555.)

    (Permit me to explain why I would put it in that way....)

  • How were these measurements made? Were they all done in a consistent fashion, town to town, year to year?

• What's next? Impressions are nice, but models are better....
  • Some suggested straight line models. This is a very sensible idea, since we're looking to see if something is increasing -- and linear functions are the simplest functions that can increase.
  • Jacob suggested a model that increased for awhile, then tailed off. That can't be linear, but must be more interesting. We might look at the "hockey stick" model.

  • There were suggestions that we might incorporate elevation and distance from the ocean into our models, when we see the dramatic difference between sites.

• In our discussion of model-fit I used the term "residual" without definition-- I apologize for that. So I'll rectify that today!

• Received the data and the question: "Are surface temperatures in Togo, West Africa, rising?"
• Divied up the data, and had a look at ten different towns across Togo -- scatterplots of data \[ \{(t_i,y_i)\}_{i\ \in\ Years} \] where $t_i$ is a year, and $y_i$ is the corresponding minimum or maximum temperature (in $^oC$) for a particular town, of which which there ten.
• We've discovered rather wide variation between towns, but those that showed pattern appeared to show an increase in temperatures (rather than a decrease, say). This suggests that we test that.

• Did I overhear Patrick say that he performed regression on his data, and had a "significant" increase in temperature for at least one of his data sets?

  Let's suppose that there were no change in temperature across the years. Then we might expect temperatures to vary erratically, randomly over the course of time about the mean min or max. So we would expect our scatter to flit about the means, call them $Max$ and $min$, for a particular town.

  That model, which we might call the "null model", would be constant functions \[ y^M(t)=Max \] and \[ y^m(t)=min \] (where $y^M$ indicates we're talking about maxima, and $y^m$ indicates minima) and the data would include some random fluctions (which we might characterize as "noise") \[ y^M_i=Max+\epsilon_i \] or \[ y^m_i=min+\tau_i \]

This function is linear in $t$, and has as its graph a straight line.

Patrick has moved on from the Planning part of UPCE, to the "Carrying Out" of the plan, and even to the Evaluation part -- I heard him assert that the upward temperature trend was significant.

So he has carried out UPCE -- but "E" should not be thought of as "End"! We iterate -- do UPCE again -- using what we learned in the first phase.

In ordinary least squares regression, or objective is to compute $a$ and $b$ so as to minimize the sum of squared residuals: \[ S(a,b)=\sum_{i}(y_i-y(t_i))^2=\sum_{i}(y_i-(a+bt_i))^2 \]

But today I want to show how our notion of "linear models" can be extended to more interesting cases. Consider atmospheric CO₂ (the Keeling data from Mauna Loa) in more detail than Stewart showed (he was looking at yearly means):

• What kind of model might you suggest? (Does it look like a straight line?:)
• If you split it into a "trend", and a "seasonal variation" term, how would you characterize each?
• Ben Bolker makes another comment in the Bestiary of functions, that illustrates an important flaw in our mathematical vocabulary: we're lazy mathematicians, and we sometimes say things that aren't quite true!

  On page 120, he calls $f(x)=a+bx+cx^2$ the "simplest nonlinear model". In what sense is $f$ nonlinear? In another sense it is linear, however! It is linear in the parameters, $a$, $b$, and $c$!

  So it turns out that we can use linear regression to find the best parameters of this "nonlinear model". So we'll also be doing that next week.

• Consider a model \[ CO_2(t)=a+bt+ct^2+d\sin(\alpha t)+e\cos(\beta t) \] which is linear in $a$, $b$, $c$, $d$, $e$, but non-linear in $\alpha$ and $\beta$. But those last determine the periodicity of the trig functions, which we have already decided is one year -- so we don't need to find those! Hence we can use linear regression to fit this model: \[ CO_2(t)=a+bt+ct^2+d\sin(2 \pi t)+e\cos(2 \pi t) \]