Empirical Modeling: Day 6

Said Thad: "Yeehah! Cue the Rocky theme song!"

"I'm justified in claiming my GOATness for all time, lucky underwear and nursing home choice notwithstanding.

Furthermore, "The data was fun, but the analysis comments were even better. Glad that a brief mention of time comparisons led to a real life application for your class. Please pass on my thanks.

"Too fun."

From Thomas: only silence....:(

Comments, questions, concerns?

• The Basic Problem

• Linear functions are arguably the most important in mathematics. At the beginning of a semester I have my calculus classes (e.g. mat129) take an exam that is (somewhat) predictive of their chance of success in calculus:

  $grade(K)=3.1K + 27.3$
  Final Course Grade (%)
  	KEMTP score $K$

• Let's talk about whether straight line models -- the simplest models -- are appropriate, and look at a couple of examples.

  In particular, I want to emphasize one of my favorite aphorisms: don't fit a banana with a straight line. That is, if your model is missing something important (e.g. concavity in a set of data), then fix the model!

  First of all, regarding straight line models: The Basic Problem document asks "Have you ever read an article in Scientific American, Physics Today, Science or other periodical, and come across a picture like this one...." (a linear regression).

  So my most recent edition of Science appeared, and I've flagged all the linear regressions in the Research Section that I found (there are several examples).

• Let's continue by looking at a few "misadventures in modeling":
  1. James Stewart's Calculus, p. 25: your introduction to the Keeling CO₂ data. What would you see in the residuals (that is, the difference between the data and the model)? Stewart describes interpolation, extrapolation, and the dangers of his model (p. 27).
     

     "We therefore predict that the CO₂ level will exceed 420 ppm by the year 2029. This prediction is risky because it involves a time quite remote from our observations. In fact, we see from Figure 6 that the trend has been for CO₂ levels to increase rather more rapidly in recent years, so the level might exceed 420 ppm well before 2029.

  2. Here's another example (Mooney and Swift, p. 171): A modeling misadventure with Advertising Expenditures.

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

• So we might consider a "modified" Basic Problem: to find a function that fits the data (whether a straight line or not). For that it is good to have a healthy stable of functions to choose from, many of which are illustrated in the Bestiary of functions, from Ben Bolker's Ecological Models and Data in R.

  I asked you to read this because straight lines are not enough: we need a variety of functions, for a variety of situations. For example, Riegel thought a power function would be nice. 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!

  

• Now we want to see how to use Mathematica to perform linear regression in the case of Stewart's version of the Keeling CO₂ data.

  We will use this Mathematica code to find "linear" models that fit Stewart's Keeling data (that included a quadratic model, which was linear in its parameters (more on that as we derive the equations next time).

  I'll walk you through this -- especially important if you're new to Mathematica.

• Then you might modify things a bit to do some linear regression on the data for Dave's stick....