Empirical Modeling: Day 10

1. Major/Minor lunch: " Please drop-by for lunch between 11:30 and 1:30 in the center atrium. On the menu this week, we have chicken tenders, mac & cheese, chips, and dessert."

2. Friday Colloquim: Dr. Pamela Pyzza, Ohio Wesleyan University, will be joining us on Friday at 1pm. She will be giving a talk titled Models of Insect Olfaction in MEP 461.

In the original problem I said that you should be able to reproduce any mark on the stick -- now's your chance to show that you can.

Again, the main advantage of this approach is that it can be easily generalized to models containing many parameters.

Then we'll check the Keeling results using Mathematica and some linear algebra, to see that we get the same fit to Stewart's Keeling data as we did using Mathematica's LinearModelFit.

How would you adjust things to get the quadratic fit? Let's try!

There are several things I always consider when creating a regression model:

1. Graph the data first, and look for any "outliers" -- really unusual values. These could be due to mistakes -- coding errors, etc. -- so we want to catch those first (and fix -- or eliminate -- or live with them, depending on what we think is going on).

   If I see no pattern in the data, then I quit; otherwise, I choose my model and fit it.

2. Graphical check of Fit and Residuals - is there a pattern to the residuals? If so, we go back to the model-building step....
   • residual "errors" are supposed to be "iid" -- independent and identically distributed (and we hope for normally distributed).
   • Hopefully the residuals are not clustered, nor wildly different in size, nor different in appearance along the model.
   • if you graph the residuals as a histogram, you get something resembling a normal distribution (bell-shaped curve).

3. $R^2$: how much variation does my model explain?
   • We break variance down into two parts: a fraction due to the model, and a fraction due to the residual.
   • Let's derive this measure from squared "errors" of different types.

4. Confidence intervals for parameters (essentially equivalent to p-values, but perhaps more informative).

   The standard errors of the parameters pop out of the inverse matrix we compute, multiplied by the mean SSE. Once we have those, we have everything we need for confidence intervals.

   We're often interested to know whether we can exclude a certain value from a confidence interval -- e.g., can we conclude that the slope parameter $b$ in a linear regression $y(x)=a+bx$ is not 0? If so, we conclude that there is a non-zero slope, and the model suggests that $x$ drives values up or down, depending on the sign of $b$.

5. Eventually we may compare models, using statistics such as anova, AIC, BIC, etc. For now, we're going to focus on diagnostics pertaining to a single model.