Michael Redman - 2012 08 13

My original post to Usenet:

2012 August 13 1545 UTC i sought out to quantify the idea that there is an "equivalence" between either running the same distance a certain amount faster, or else running a certain amount farther at the same speed. same with swimming. by the same idea all the world record performances in, say, backstroke, across all distances, ought to be "equivalent" in the sense they all represent the best people can do. after some investigation i found this formula performance = ln(1 + distance*scale) * (distance*scale / time) i was able to find scale parameters that tightly focused my samples of scores for some current world records. for running, with scale=3.05, which physically is like using distance units of 1/3.05 m, or about a foot (anatomical coincidence?), the average was about 19x the standard deviation for the 11 current world record runs i sampled from 100m-100km (approximately doubling in distance each time), and about 29x for the middle 9. for swimming, with scale=116, the scores focused to average about 39x the standard deviation. i got my world records from wikipedia men's running meters seconds score, scale=3.05 100 9.58 182.2226989661 200 19.19 203.9191842155 400 43.18 200.811928634 800 101.01 188.4209275493 1500 206 187.188135625 3000 440.67 189.3997974528 5000 757.35 193.9580186086 10000 1604 196.33927671 20000 3321 202.3900565264 42195 7418 204.1143559653 100000 22413 171.8449762171 men's freestyle meters seconds score, scale=116 50 20.91 2403.7090604354 100 46.91 2314.2745753825 200 102 2286.3260325101 400 220.07 2265.5134268295 800 452.12 2347.7531143433 1500 871.02 2410.5371180953 the physical interpretation of the log term, is that at a given speed you have to go an amount of extra distance proportional to what you are already doing, in order to be equivalent to improving your average speed on the same distance by "1 unit". if you improve your speed for the race by "1 unit", since it is an average speed that is something you sort of did over the whole race. so it is not equivalent just to run 1 extra unit of distance at the end of the race. the log makes incremental distance proportional, just like exp models how the money in your bank account grows in proportion to what you have. so the basic form of the formula is really performance = ln(distance) * speed unfortunately the way that formula orders a given sample of races of different distances, is not invariant to your choice of units for distance. moreover with long distance units like kilometers, some races would be for less than 1 unit of distance, so in the basic formula the log term would turn negative and the formula would be physically meaningless so the 1+ term in the full formula reflects the physical reality that all races are for at least some positive distance. and the "scale" term is your choice of distance units. the log term is close to linear near 0 so when the distance units are long (fewer units of distance traveled) the formula approximates distance^2/time, which gives an edge to longer races. and the log term gets flatter towards infinity so when the distance units are short and more are traveled the formula approximates distance/time, which gives an edge to the sprints. however as the data above demonstrates, with a suitable choice of distance units the formula behaves well across the 3 or so decimal places of distance scale that the different races cover. does any one know if there is (or even can be?) another formula, which does the same thing, except that its ordering of performances is naturally invariant to a change of distance units?

Copyright 2015 Michael Redman.

IN GOD WE TRVST.