Empirical Modeling: Day 14

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

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

1. if one does a log-log transformation of both $x$ and $y$, and the data look linear, then you build a power model: $y=\alpha x^b$.

   (e.g. Riegel's models)

2. If you log-transform only the $y$, and get a linear looking cloud, then you build an exponential model: $y=\alpha e^{b x}$.

   (I've asked you for an exponential model for CO₂. You log-transform the trend data, and plot. Then linear regress, if appropriate.)

3. Those oscillations in Keeling's model? We typically model them with sinusoidal functions, as \[ A \sin[2\pi\frac{t-t_0}{T}] \] where
   1. $A$ is the amplitude
   2. $T$ is the period of the oscillation (frequency $f=1/T$)
   3. $t_0$ is the phase shift

   This is possible with linear regression, because we can use a trig identity

   \[ \sin[\theta + \phi] = \sin[\theta]\cos[\phi]+\sin[\phi]\cos[\theta] \]

   to write \[ A \sin[2\pi(t-t_0)] = A \left( \sin[2\pi t]\cos[2\pi(-t_0)] + \sin[2\pi(-t_0)]\cos[2\pi t] \right) \equiv c\cos[2\pi t] + d\sin[2\pi t] \]

   That is, we can write the given sinusoidal function as a sum of a sine and cosine, each a function of time. Then we can deduce both $A$ and $t_0$ from the coefficients $c$ and $d$ obtained by linear regression.

4. We had a look at an example (Ad Expenditures), where these linear techniques aren't sufficient to help us build a linear model. It also illustrated how one might load the Keeling data into Mathematica (in case you hadn't already begun).

   In general, it's better if we can build the model with non-linear regression, which we'll begin next week, after the exam.

In general, for your models you should

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

• and confidence intervals for the parameters.

  As we discussed last time, the confidence intervals of the parameters of a linearizable model are not those of the linear regression parameters. Let's consider the examples of the three models we've linearized, and come 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?

[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....]