## Explaining the Performance Predictors

The calculator/performance predictor takes a distance and corresponding time as input, and uses them to compute several pieces of information and predict comparable performances for other events.

First, the Purdy points are reported, demonstrating what the performance is worth. The Purdy points tables provide a way of assigning a "worth" to performances of most any distance, with comparable performances across distances being worth an equal number of points. Also known as the Portuguese Scoring Tables, the Purdy calculations match the lookup charts in "Computerized Running Training Programs" by James B. Gardner and J. Gerry Purdy. The calculations are a PHP version of the C program kindly presented by Patrick Hoffman on Cross Country, Track, and Running Analysis.

The second item is your VO2 Max, with the calculations based on Jack Daniels' "Oxygen Power Performance Tables for Distance Runners." VO2 Max represents maximum oxygen consumption, or in simplest terms, aerobic capacity. Endurance athletes typically have higher VO2 Max readings. Although VO2 Max may not be a perfect predictor of who will win between any two athletes, it does do a good job of predicting comparable performances for distances over 1500m. Distances shorter than 1500m do not rely on aerobic capacity nearly as much, thus they are better predicted by other models. Also reported along with VO2 Max is the VO2 at the calculated time and pace, as well as the percentage of your VO2 Max.

Next come the performance predictors. They are, in order:

1. Purdy Points: Taking the Purdy points calculated above for the original distance and time, a time is calculated for each new distance that is worth the same number of points.

2. VO2 Max: This is the most complicated model to calculate, as the times for each distance must be found by narrowing down a time prediction until they are "close enough" through various combinations of Newton's Method and derivatives of quadratic equations. For the Run-Down calculator, times within a tenth of a second or 0.001% of the time prediction, which ever is less, were deemed reasonable. This is especially true since at most a tenth of a second in a distance race is negligible and the VO2 Max predictions are not very accurate in the first place for events that are short enough for 0.001% to be meaningful. The standard predictions for calculating VO2 Max are:

percent_max = 0.8 + 0.1894393 * e^(-0.012778 * time) + 0.2989558 * e^(-0.1932605 * time)
vo2 = -4.60 + 0.182258 * velocity + 0.000104 * velocity^2
vo2max = vo2 / percent_max

where time is in minutes and velocity is in meters per minute. These equations are also used for working backward to determine a time corresponding to a known VO2 Max and distance, although it requires approximating percent_max, combining equations, and treating vo2 as a quadratic equation to solve for velocity, which is in turn used to calculate time (time = distance / velocity) and check how close the initial time estimate was.

3. Dave Cameron's Model: This model is based on the top 10 times in the world at each distance, using them to compute comparable performances across distances. The speed vs. distance model works well for post-1945 records at 800m through 10k. From 1964 onward it also works well for the marathon. The calculations are as follows, with new_time and old_time being in seconds and new_dist and old_dist being in meters:

a = 13.49681 - (0.000030363 * old_dist) + (835.7114 / (old_dist^0.7905))
b = 13.49681 - (0.000030363 * new_dist) + (835.7114 / (new_dist^0.7905))
new_time = (old_time / old_dist) * (a / b) * new_dist

4. Pete Riegel's Model:   t2 = t1 * (d2 / d1)^1.06
where t1 equals the initial time, d1 equals the initial distance, d2 equals the new distance being calculated for, and t2 equals the predicted time for the new distance.

The formula was developed by Pete Riegel and published first in a slightly different form in Runner's World, August 1977, in an article in that issue entitled "Time Predicting." The formula was refined for other sports (swimming, bicycling, walking) in an article "Athletic Records and Human Endurance," also written by Pete Riegel, which appeared in American Scientist, May-June 1981.

Of the 4 prediction models, the Purdy points method does the best job of predicting comparable performances in events shorter than a mile, and it appears to be equally accurate for events that are longer in duration. Permission has been received from the cited sources to present their work here.