- A reminder to put your phones away, please.
- We had a better look at the Keeling data, and saw that it looks like anything BUT a straight line! Nonetheless, it turns out that we can use linear regression to find the parameters of a best-fitting model.
- Bolker, in his "Beastiary" of functions, calls
$f(x)=a+bx+cx^2$ the "simplest nonlinear model"; but in another
sense, it is linear: It is linear in the parameters, $a$,
$b$, and $c$.
Any time that the model is linear in the parameters, then we can use simple linear regression to find a "best-fit" (using the criterion of least sum of squared residuals).
So it turns out that we can use linear regression to find the best parameters of this quadratic "nonlinear model" to the Keeling data, and we'll do that today.
- Furthermore, for the Keeling data,
we can 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 greek parameters determine the periodicity of the trig functions, which we have already decided is one year -- so we know those! Hence we can use linear regression to fit this model: \[ CO_2(t)=a+bt+ct^2+d\sin(2 \pi t)+e\cos(2 \pi t) \] to this "refined" Keeling data. In fact, this is a graph of monthly averages. We can obtain the daily (and even hourly!) Keeling data, if we wish.
- By the way, these regression methods are going to serve us well
even when we begin fitting structural models (rather than just
empirical) models.
And we will be able to fit some models non-linear in the parameters using these methods as well, via either
- data transformations, or
- iterative procedures (that is, we'll make successively better approximations, but using linear regression over and over).
- We're going to see how to use Mathematica to find linear models that fit Stewart's Keeling data;
- We're going to "do the math", to see how to find the parameters $a$ and $b$ of the model $y=a+bx$ that minimize the sum of the squared residuals, using (multivariate) calculus;
- We'll check the results using Mathematica, to see that we get the same fit to Stewart's Keeling data.
- "The IPCC stands by its equation of 450 ppm = 2 degrees C [of warming]...."
- An extract from Curve Fitting via the
Criterion of Least Squares, by John Alexander. A
bit old-fashioned, but it's got all the major ideas,
and some nice examples.
Alexander also illustrates that one cannot simply invert the regression equation $y=a+bx$ to get the regression equation $x=\frac{y-a}{b}$. So it really matters in linear regression which variable is considered "independent" and which is considered "dependent".
- My multivariate derivation.