Empirical Modeling: Day 15

1. Cross-Country
2. Log Ruler
3. Keeling

For the exam you will need a calculator and your notebook.

In general, for your models you should

• provide $R^2$ (of the linear models you obtain)

• and confidence intervals for the parameters.

  The confidence intervals of the parameters of a linearizable model are not those of the linear regression parameters. We considered the three models we've linearized, and came up with some confidence intervals for the non-linear model parameters.

  When it comes time for the sinusoidal model, it might be good to review tan and arctan.

• Provide an intepretation of the parameters, if you can. For example, in the sinusoidal model for the Keeling data, the parameter $A$ represents the seasonal variation in the amount of CO₂ (in parts per million) gained or lost by the forests because of gain in leaves or loss and decomposition of leaves. (Important question -- is $A$ varying in time?)

  From $t_0$ we can determine the dates of maximal drawdown and return of CO₂ to the atmosphere.

• Discuss residuals.

• Evaluate your models. Which seems best? Which worst? Etc. What would be the real-world consequences of a linear versus quadratic versus an exponential growth in CO₂?

We'll also consider a few other smaller mini-projects (e.g. cannabilistic beetles!:)

Last time we were discussing the meaning of parameters in a regression model, and interpreting $A$ (amplitude) and $t_0$ (phase) of the oscillatory part of a Keeling regression model. Let's look at some issues in the

• Cross Country

  The equation of interest is Riegel's "endurance equation", \[ t=ax^b \] where $t$ is in minutes, $x$ in km, and $a$ and $b$ "are constants unique for each activity."

  Riegel says that "exponent $b$ of the endurance equation may be referred to as the "fatigue factor," because its value determines the rate at which average speed decreases with distance and time required to finish the race...." He notes that $b$ is unit independent: if we express $x$ in different units, $b$ doesn't change (it will impact $a$, however).

  But about $a$, Riegel says that ``a is dependent on the units of measurement chosen and has no particular absolute significance, although it provides a measure of relative speed.''

  In our paper, I argue that it is better to break $a$ into two new parameters, and it comes into view a little better. Let's talk about that (p. 11 of our paper).

  In the end, I want to rewrite Riegel's \[ t=ax^b \] so that it becomes (for Thomas, say) \[ t_{Thomas}(x)=17.25\left(\frac{x}{5}\right)^{1.07732} \]

  For world-class runners ($a=2.299$ and $b=1.07732$) the equation is

  \[ t(x)=13.02 \left(\frac{x}{5}\right)^{1.07732} \] (I just looked up the world's record for men in a 5K: 12:38 or so.)

• Log Rule

  For this paper, I want to look at a structural model, which gives rise to the Ontario rule (with the proper choice of parameters).

  H. H. Chapman, who wrote the book on log measurement (Forest Mensuration), tells us that ``... American log rules were made by the practical mill men who were not versed in the use of graphic or formula methods of obtaining average values. The results were in some cases good, in other, very bad. The most sensible and in every respect the soundest and best method of making a log rule ever devised was that of plotting the dimensions of 1-inch boards on circles drawn to the size of logs of different diameters, and then computing the board foot contents from the diagram. In this way one of the earliest and best of the old log rules, the old Scribner rule, was made.''

  • Take a look at p. 4, for a structural interpretation of the derivation of the coefficients of the Ontario rule.

  • Let's look at p. 10 for one approach to dividing up a log.

  • A couple of points:

    • Our structural rule is comparatively difficult to calculate: but with linear regression we can approximate it with a quadratic quite well, and that's as easy as the other rules to use.

    • We can verify some claims using our model -- i.e. that the loss to kerf (the width of the saw blade) is as expected, and that the loss from slag (the bark sides) goes to 0 as the diameter goes to "infinity".

    • Our rule ignores taper, and defects - with which real lumber mills are very concerned.

[ael: by the way -- ecosystems worldwide are already suffering serious damage.... The world won't "suddenly heat up" if we reach 450 ppm -- it's heating up right now, and we're seeing bumblebees struggle with the heat, for example. They pollinate our food -- think about it....]