- Roll
- Reminder: please keep the phones out of sight and sound.
- Your homework today was to read my final report. I'll pass
around a copy, and you can all sign it, or not -- if
you wish to dissent.
- Silas found an error in my Figure 3 report --
it's fixed now on-line, but let me just point it
out....
- Reminder: You have a homework assignment to
hand in on Monday.
- There's a new assignment as well, a group assignment, due
next Friday. I'll assign groups in a moment.
- No office hour after class: sorry, I have to dash to an appointment.
- I put my final report on-line. Discussion?
What modeling did I do in my approach to the problem which made use of champions of each era?
We'll talk more, later, about the appendix (discussion of the meaning of $a$).
- we talked Empirical versus Mechanical Models, and about Riegel's
formula in particular:
\[
t=2.299d^{1.07732}
\]
which he obtained by linear regression. He said
There are numerous ways in which equations may be fitted to data, and my choice of a power function is based on the desire to have an expression that represents a good compromise between accuracy and simplicity.
I disagree: I think that
- the power model seems reasonable (if you double
the race length, your time doesn't double --
representing linear change -- but rather it's somewhat
more -- super-linear).
- Riegel looked at the data, especially log-log plots;
- Riegel saw a straight line on the log-log plots:
Straight lines in log-log plots suggest power models.
- the power model seems reasonable (if you double
the race length, your time doesn't double --
representing linear change -- but rather it's somewhat
more -- super-linear).
- As a further example of empirical models -- but maybe
mechanical or structural as well, eventually! -- let's
consider my friend Dave's wooden stick, which I have cleverly scanned (much to the
amusement of our chair -- she's never seen anyone scan a stick
before. Sorry about the big files....). Feel free to look over
the stick itself, but I believe that I've captured all the important
info as well as one can with a scan.
- My friend Dave gave me this stick, but didn't know what it was
for. I'm a little surprised, because Dave is getting to be a pretty
good woodsman (he wasn't born in the north, but rather in the larger
southern cities of Ontario -- he's only been up north for about 20 years).
Once I knew that I would be teaching math modeling, I also knew that we'd be trying to decipher this stick.
I have since figured it out, through some detective work of my own; but now I'd like you to figure out a formula that gives rise to the markings -- and ultimately what it's used for (and you have one week).
- By plotting the data, we can attempt to figure out what
transformation might render it linear -- just as Riegel might have
looked at log-log plots to "see" a power model. Then we could build a
model of the numbers.
To get you started: For the $x$-axis, you have the inch measurements along each side of the stick.
- In any case, plotting is always a good idea. Use any plotting
software you have, and any subset (particular run) of the numbers to
work with, and attempt to find a formula that fits them well.
- And, while linear regression is not the only way to figure out the formula (it's hitting swatting a fly with a cannon, perhaps!), I did use it myself to ultimately discover the secret of Dave's stick.
- No particular links, but I'll add them here as you find interesting ones to share!