Day 12 in math modeling

• Sally said to me "I'm sorry, I had written in my planner that the linear model (including oscillation) to the more detailed Keeling data is due next Friday."

  But it was my fault, because I reviewed my "agendas" for the two days prior, and I had written "Due next Friday" on both of them (a holdover from the previous week's reminder). If anyone else was deceived, and didn't turn in this homework, then please do submit it.

• There is an interesting talk coming up Thursday night, by the author of Code Girls

• There's a new assignment for Friday (in addition to your evaluations of mini-project 1s). I'll describe it below.

• We discovered how to linearize an annual periodic oscillation with a phase shift, $\sin(2 \pi t + \theta)$, by including matching sine/cosine terms to the Keeling data (with a quadratic fit to the trend).

  We could fit other periods of oscillation by adding in pairs of the form $\sin(2 \pi \frac{t}{T})$ and $\cos(2 \pi \frac{t}{T})$, where $T$ is the desired period. You might consider this if you see other periods (or suspect other periods) in your temperature data on your mini-project 2.

  What about the Keeling data -- might you suspect other periods?

• I've turned this example into a homework, due Friday: 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, which can be computed as \[ R^2=1-\frac{SSE}{SS_{Tot}}=\frac{SS_{Reg}}{SS_{Tot}} \] (but the second equality only holds in the linear regression case!)

    The second form actually allowed for values greater than 1 in the non-linear case! That's not good...:) Ironically, the improved version above could actually be negative, but, in that case, you'd certainly say that your model is very, very bad!

• 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!

The key to non-linear regression is linear regression! That may seem odd, but I will mention again that frequently in mathematics we often cast a problem from the non-linear world over to where we obtain a linear problem, then bring that solution back (and hope for the best).

In the case of the linearizations we have done, we only did this once. In the case of non-linear regression, we will do it multiple times (iteratively), hoping for constant improvement.

After I describe Newton's method, I'll

• go through an actual example with you: my non-linear regression derivation; and then
• give you a chance to play with it a bit, using this code.

• 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.