Wednesday, March 16, 2011

The Power of Mathematics

Life is the art of drawing sufficient conclusions from insufficient premises. –Samuel Butler

At one point in my life I thought I might want to be an actuary. I am a firm believer that mathematics can be used to explain (and predict) events. Take for instance the heart rate as it correlates to fitness and ultimately race performance.

Theoretically one could collect enough data in order to accurately predict a marathon time. Gather a history of the following training data: pace, course conditions (hills, wind, surface), distance, human conditions (health, fatigue, diet, weight, hydration, sleep, resting heart rate, running heart rate). Develop an equation or some artificial intelligence algorithms based on those factors, and then predict your race given the environment.

One key is to note that most models do much better "interpolating" results as opposed to "extrapolating" results. In other words, if all of the data input distances range between 6 miles and 13 miles then my model will best estimate times between those distances. If I use the model to predict a 20 mile race then I am extrapolating and it's likely to be less accurate. This applies not only to distance but also to heart rate and every other factor for that matter.

There's the rub. Since we so seldom work out at race pace, and never for race pace at race distances, the model will almost always be forced to extrapolate. Of course there are elegant mathematical methods to reduce this limitation, but they can only be so good. I think perhaps the biggest flaw in this approach is probably the general lack of data. If you ran every day for 100 years you probably wouldn't generate enough data to generate a comprehensive set that produced high r squared values. Which brings us to Mr. Butler's quote above. We all go through life making decisions aimed at returning the highest payout (however you choose to measure "payout"), with data sets far too small to be able to know definitively that our choice is correct. In short, we take risks based on our best knowledge. It's the same thing with our race predictions -- I'm going to pick a race goal pace at the beginning of the OKC marathon and hope I can hold on to it for 26 miles. It's risky, and I've almost always crashed in marathons, but I'm hoping this one will be different.

Of all those factors I mentioned above I tend to believe that the heart rate (both resting and running) is the most significant variable that drives predicted performance. To that extent I'm fairly happy with results like I recorded today.

8.29 miles, 1:04:48, 7:48 pace, 130 HR avg

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