Monday, 23 January 2012

Fractal geometry of running

I'm almost done reading Nassim Taleb's The Black Swan (second edition). The main focus of the book is on "Black Swan" events, which are unexpected events that are both under-represented and therefore ignored in many models of the economy/finance/statistics, etc. It sounds even as I write this sentence silly that someone would need to pointing out extreme events are rare but important. But the point is more subtle having to do with how we perceive low odds of 1 in 10,000 (happens once a decade) the same as really low odds of 1 in 10,000,000,000 (happens once an eon). For instance although a human population's height and weight are Gaussian (the chances of meeting an 8ft tall person is closer to the latter value), if incomes were distributed in a Gaussian manner then the odds of a Warren Buffet existing would be the same as a 14 ft. person. Rather incomes are scaleable; as incomes increase the fraction of comparatively richer people remains the same. This leads to his point that we spend most of our lives in a Madelbrotian universe more than a Gaussian one.



I'm pretty much on board with these ideas already, but it was nice to have them clarified. I'm a fan of Mandelbrot and love playing with fractals, fractal dimensions and objects like the Sierpinski pyramid below.



So I'm almost done the book, but I wanted to write something I came across in his post-script essay regarding exercise. It's a brief point, about five pages, but I find it a significant one. Taleb advocates a large volume of exceedingly low intensity exercise followed by bursts of high intensity exercise. Now for him low intensity exercise is 15 hours of very easy walking (per week) and 15 minutes of weight lifting (twice per week). Now Taleb isn't an exercise scientist per se (he is a self-described intellectual flâneur) and I don't want to quibble with how he precisely keeps fit (these quotas would make any doctor satisfied with your health). But his point was that he noticed that a quality exercise regime also involves scalable values like wealth, only now it's regarding exercise intensities. This observation is significant because many people exercise too long at precisely one intensity. Taleb's aside led me to think more about the idea of training, but with a fractal twist.




One of the most significant numbers quoted by endurance coaches is that all serious runners (and other endurance athletes) do about 80% of their running at 'low' intensity and the remaining 20% at 'high' intensity. It's important to note the difficulty in pinning down any precise references to this number. Suffice to say that the exercise scientist Steve Magness corroborates the value for top Kenyan runners and Norwegian skiers (and I've seen similar quotes in other sources too, i.e. here). Anecdotal evidence is sufficient if it consistently matches the training structure of the highest level athletes, better justification than studying elite reading habits anyway.

Dividing all of your training into only two camps is, however, simplified to the point of incompleteness. For starters this rule of thumb does not account how one spends their resting hours, which are spent as relaxed as possible, and may not specify how fast is 'fast' or how slow is 'slow' (relatively speaking). Note that the 20/80 split is in terms of mileage run, not minutes run (if written in terms of minutes, the split is more like 15/85). Regardless, the 20/80 rule does not provide enough insight because there are too few scales (i.e. only one). What I want to do is expand this pyramid a bit  to show the scalable ratios for relative intensities


To take a 'typical' fast running, here's the training breakdown scheme as borrowed from Florence Kiplagat's running sessions for the month of December (2008):

Distance run (km) % of total
Regeneration  Mileage  (< 4’10”)           309 52.5%
Basic Aerobic Mileage  (4’10”-3’40”)                 212.5 36.1%
Aerobic Endurance Mileage (3’40-3’20”)           22.4 3.8%
Aerobic  Power  Mileage  (3’20”-3’)    40.7 6.9%
Specific Speed Endurance  (3’-2’45”)                        0.4 0.1%
Medium  Length  Hills  (200m >< 300m)                                  4 0.7%
Total  Mileage   589

It's not as if anyone obeys a precise set of laws, but using real training data gives us an anchor to reality. I'll normalize this data into an imaginary day of three intensities: 'slow', 'medium', and 'fast'. Obviously you don't run the same volume of each kind each day, or even every week. Also no sprinting was reported, but this is usually a very small number milage-wise. There were 31 days of running for December, so that means 589 km/31 days = 19 km/day. Assembling the data we have a clear difference in volume:

km run % of total km/day time (min/day)
Easy Mileage     (sub 3'40")                     521.5 88.54% 16.82 67.29
Tempo  (3’40” -3’00”)          63.1 10.71% 2.04 7.12
Speed  (3'00"- 2’45”) 4.4 0.75% 0.14 0.41
Total  Mileage   589 100.00% 14.37 74.83

Roughly speaking we have for our prototype day 17 km of easy, 2 km of medium, and 0.1 km of hard, or 67 min of easy, 7 min of medium, and 0.4 min of hard (assuming she ran, on average 4', 3'20", and 2'50" for these paces, respectively). If we include rest as part of the schedule -the remaining 22.75 hours of the day- the hierarchy is complete:

time(min/day) ratio
rest   1365  20
Easy (jogging)    67    9
Medium (tempo)      7 17
Fast(speedwork)   0.4

The first ratio is "total time spent resting"/"time spent jogging" = 1365/67 = 20.

Note there isn't a consistent value for the ratio. This is to be expected since tiny changes can easily alter the ratio, probably why top runners keep such careful logs of each intensity and not just the total time spent running. But note how large each ratio is, being 20, 9, and 17. Considering this was a light training month for Florence, there was a relatively low amount of mileage and low intensity, so these ratios would decrease to something less in a more intense month. For a month of higher intensity and more running in total, it is easy to re-arrange the totals to something else (maybe to 5? or even 2?), but the idea remains intact.


The ratios are the relative measure of time spent between adjacent effort levels and the principal reason I felt compelled to write this entry. Using these ratios, we see the relative amount of time spent in a comparatively easier state. The first obvious point is that a lot of your running time is spent running easy. But the interesting thing is your time spent running at tempo pace also overwhelms your speed volume, and speed is in turn way more than your stride volume. This might get overlooked if runners quickly round the mileage totals; perhaps they may think that a week of 20% quality means a 50/50 split between tempo and speed. So for a given week maybe I run one tempo session of 12 km and another bit of tempo during a long run via 20 minutes of pickup for a total of 18 km. In the same week the amount of speed could be only be 4 km.

There is no magic number, but the theory of the pyramid stacking scheme emphasizes just how little of your time is spent running fast. When studies emerge telling us that tempo is important, it is because without tempo there is no support for the speed (racing) pace, the most vital and dangerous training pace of all. Almost anyone can stride (because there is so little of it), do a few weights (done carefully) and run easy (if it does not stress your body). But to build up to a race pace at race distance is very, very hard. It requires the support of each layer above and below. This is the pace your are trying to maximize volume-wise.

     Running a 10-21.1 km race at race pace implies, due to the structure of the pyramid, that you are briefly inverting your pyramid (i.e. spending more time at race pace than tempo), which in turn causes stress on your body given the unsustainable stacking. You require a careful buildup to this inversion in your training, but inevitably as your pyramid approaches this inversion your training becomes more fragile, like nearing the end of Jenga game. This careful balance of speed versus time is why so few are successful at running and why even the best athletes require a regeneration period after endurance competition(s).

Note that changing the shape of your pyramid will also add a time dimension to your stacking, which is another fractal element I'll write about later. That's all for now. 

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