"There's only one hard and fast rule in running: sometimes you have to run one hard and fast."

Sunday, December 2, 2007

Comparing trail races

Training since last post

11/29. 4 miles in 35.5 Stiff from yesterday. (sub-zero windchill)
11/30. 4 miles in 35 (windchill of -10)
12/1. 10 in 90. This was supposed to be a long run before the snow came, but the storm hit three hours before expected. I was glad the RunnerBrewer and I decided not to run Afton. Just like I need to drive a few miles in the snow to remember how to turn and stop in bad conditions, the first bad storm makes me relearn how to dress and run. I didn't have a water- and wind-proof layer on my head and neck, so the ice pellets stuck to my clothes, and when they melted, refroze when I went into the wind. I wondered why I felt so cold so early and realized I couldn't feel my ears; as I put my hands over my ears to warm them, I knew I was close to frostnip (not frostbite, anyway). I headed straight home, immersed myself in warm water and spent the rest of the day shivering, trying to recover from what might've been the start of hypothermia.

I still won't run on a treadmill. You can't make me!

12/2. 10 in 88 First day of using my winter paths, which are a little less accurate than my summer ones (a bit long, rather than obsessively accurate). Shoveled 6 inches of snow for 2-3 hours before running. Made me think of my grandfather, who when he first saw a jogger and was informed he wasn't running away from anything, just exercizing, said, "Why doesn't he just get a job?" I've been reading others blogs where others are doing strength training (200 push-ups?!) and I think shoveling is just as good a workout. You with snowblowers and teenagers, you're all Ivan Drago to my Rocky (watching that wretched Rocky IV finally paid off).

Comparing Races

This is the geekiest thing I'll ever write, I hope. My last post said that one can use one's last races to predict one's next one, but that's not easy with trail races. The courses all have different terrain, the weather's variable, some races are more competitive than others, each runner is better at one ditance than another and some (most) runners don't run as hard as possible in every race.

I thought I could compare races mathematically. My background in population dynamics suggested a simple logistic curve would work, but it doesn't - one can force the data, but the results are meaningless. My background in statistical mechanics led me to see that in an ideal race, the number of runners plotted against the logarithm of finishing times led to a normal distribution for the leading edge (the trailing edge doesn't work because good runners are more likely to race than people who've never run - except in races like Race For the Cure, which has thousands of walkers messing things up). Using this means integrating gamma functions over logarithmic axes and that's where one chases the rabbit into Neverland.

My engineering background (are you wondering what I do for a living yet?) led me to the simple equation n= a+bT+cT(squared)+dT(cubed). In a smooth curve, a and c will be negative. In any race, one can find four data points that will give one four equations with four unknowns to solve, where -3a/b = b/c = -c/3d. This, in turn, should equal To, the theoretical time that no one could run (slightly less than the course record, presumably).

Last year's Moose Mountain Marathon gives a=-2048.4, b= 21.019, c= -.071469, d=.0000811.
Solving the cubic equation for n=0 gives To=238.5 instead of b/c=294.1 This is a typical result. The Sawtooth 100 mile gives a=-451.67, b=.83334, c=-.0005112, d=.000000105; b/c=1630 and To=1351.

Using the winners' times as fractions of the characteristic times of the equations suggest John Horn's finish was better than Wynn Davis'. I know both of these runners; John couldn't finish 100 miles and Wynn could run the marathon faster than John's winning time. The problem is that the populations differ; the 100 mile pulls in a more competitive group.

Any questions?


aharmer said...

Where does Chris' 50M time fit?

keith said...

Oh, man. You lost me at n= a+bT+cT(squared)+dT(cubed).

Your checkbook must be immaculate; and Adam Harmer is probably ecstatic with glee at reading ALL. THAT. DATA.

Good post.

Kel said...

So...what do you do for a living? ;)

SteveQ said...

Adam, I know what you really want to know is how that second place finisher in the 50 mile did. You can do the math...

Keith, my checks still have "19__" in the date line.

Kel, isn't my occupation obvious? I'm an undercover physical trainer inspector.

aharmer said...

I was hoping you'd do the math for me, but I'll give it a shot. My interest actually was in Chris, I've been curious about he versus Wynn for when they match up this year in the 100.

SteveQ said...

Adam, I'll show how to do easier comparisons in the next post. I think Chris was ahead of Wynn at Ice Age last year, but I don't know how good Chris is at 100 miles.