Day 11 in math modeling

• Your linear model (including oscillation) to the more detailed monthly Keeling data is due.

• I have assigned groups and towns for Mini-Project 2. Let's talk about that a bit.

• One more comment from your mini-project 01 that I wanted to share (and that I didn't get to last time):
  1. Evaluate: Whatever model you choose, it's important that you be able to interpret the parameters. What do they mean? The Atakpame team was the only one I've noticed (yet) that took a crack at this....

We were looking at some examples, including some of the functions from the "Bestiary", which can be linearized -- meaning that we can use linear regression to estimate their parameters even though the models are non-linear in the parameters.

• Before we get back to the linearization exercises we were from last time, let's find out when during the year that the Keeling data hits its peak. This is back to the problem of evaluating the results of our model, and interpreting parameters.

  To do that, we use the results of our Keeling model's trig terms (thanks Connor), and one of those important trig identities: either

  • The cosine of a sum: $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$
  • The sine of a sum: $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$

• Advertising expenditures. This example is from Mooney and Swift's A Course in Mathematical Modeling (p. 169). They suggest two models for the data:
  • Power model: $y(t)=at^b$
  • Exponential model: $y(t)=ae^{bt}$
  • How do they compare?
  • What problem does the power model pose, relative to starting value of time $t$? Does it change if we represent years as 1970 rather than 70?

    How do we interpret the power? (Remember my remarks above about interpreting the parameters of your model!)

    Does the exponential model suffer the same problem? What is the impact of a shift in time scale?

  • Note that I have changed the definition of $R^2$ on you: \[ R^2=1-\frac{SSE}{SS_{Tot}}=1-\frac{\sum_{i=1}^n(y_i-\hat{y}_i)^2}{\sum_{i=1}^n(y_i-\overline{y})^2} \] I was abusing a property of the linear model to attempt to compute $R^2$. This version is essentially comparing SSE across models, only, and reduces to the $R^2$ for the linear model.

    My previous version matched the linear version in the linear case, but actually allowed for values greater than 1! That's not good...:) Ironically, this version could actually be negative, but, in that case, you'd certainly say that your model is very, very bad!

    By the way, some people don't like to use $R^2$ at all for non-linear models.

• Sea Anemone reaction time (from Lamm's UMAP model, and in this book)
  • Duration of Response is a function of size (in "conventional units"!). How should it vary, do you think? What do you think from the scatterplot?

  • This model is a move away from an empirical model, and towards a structural model!

• Using Mathematica's LinearModelFit to find linear models that fit Stewart's Keeling data

• 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 linear algebra derivation.