Power law for analysing and predicting best rowing times

[Nomath]

Sorry, if I duplicate earlier topics. I searched this forum for "power laws" but it looks like the search engine can't handle word combinations. The individual words throw up so many hits that it felt like typing "book" in a library search.

Power laws are used a lot in engineering to analyse data. You assume a logarithmic behaviour between a certain variable and the response for that variable. For example, the variable is Distance and the result you want to know is your expected PR for that distance. Of course, you have to submit some data that can be related to. For example, you have set a personal record for rowing 2K recently, resulting in a time T(2K), and now want to know what best time to expect on 5K.
The power law relationship is of the type T(x)/T(x0) = (x/x0)^a .
T(x0) is the known time for the known distance x0, T(x) is the predicted time for a distance x.
The quantity a is called the exponent.


That such a power law relationship is useful for describing rowing data is shown in the graph below. I searched the Concept2 Indoor Rowing rankings of 2020 for a person who entered his best seasonal records for both 500m, 1K, 2K, 5K and 10K and was happy to find one. Here are his results plotted in a graph with logarithmic scales for both Distance and Time. You can see quickly that his data lie almost perfectly on a straight line. This line has a slope of 1.070, which equals the exponent a in the above equation.



Now, what to think of an exponent 1.070? What does it mean? It means that if you double the distance, the time will increase by a factor 2^1.070 = 2.10. I.e. the time doubles plus 5% more.

Is this typical? Well, I analysed a lot of data, both from world class athletes and recreational sporters of various ages, from rowers, cyclists, skiers and speedskaters. I found that the power law holds up very well in all these sports. The exponent is rarely below 1.03 and rarely above 1.10. This seems to be the range of values that describes how a skilled athlete's body performs when the distance is shortened or lengthened.

Rowing. It is rare to find person who listed personal records in the 2020 season for 5 distances on the Concept2 ranking site. I analysed the data from 5 persons who entered PRs for 3 or 4 distances. All data fitted very well with an exponent in the range between 1.05 and 1.10.

Cycling The C2 bike ergometer is less popular than the rower, so there is less data available. I looked for persons who listed PRs in the 2020 season over the distances 1K, 4K and 40K, i.e. events lasting roughly between 2 and 90 minutes. I found 5 persons. The exponents ranged bewteen 1.043 and 1.064, most near 1.06.

Ski-erg The C2 ski ergometer engages heavily the muscles in arms and upper body. Legs are less involved. I looked at 5 persons who listed PRs for at least 4 distances from the range of 0.5K, 1K, 2K, 5K and 10K. Events typically lasted between 1 minute and 50 minutes. I found exponents ranging between 1.045 and 1.080.

Speed skating Speed skating championships are held over one weekend in which 4 distances are programmed : 500m, 1500m, 5000m and 10000m for men and 500m, 1500m, 3000m and 5000m for women. The men's races take between 35 seconds (500m) and 13 minutes (10K). The women's races take between 37 seconds and 7 minutes (5K). The winner has the shortest total time weighted over these 4 distances. The data I analysed are for the world championships 2019 held in Calgary. These are obviously world class atlethes in their prime. The top-4 men data fitted with exponents between 1.025 and 1.030. The top-4 women data showed exponents between 1.009 and 1.032. The exceptional woman with the low exponent of 1.009 is well known as a slow-starter and a diesel.
Surprisingly, despite the observation that the 500m race is very short and depends heavily on the start, which involves very different movements and takes a lot of skill, the 500m times fitted very well in the total range of distances.
This championship obviously forces competitors to distribute their effort equally over the 4 distances. They all have plenty of experience on how to pace each race. This probably explains the low exponent compared to the other sports.

How to use this?
If you have a PR for one distance and want to predict your best time at another distance, use an exponent of 1.07 in the power law equation.
If you are very experienced on how to pace a run for a much different distance, you may perhaps target a time using an exponent of 1.06.

Questions
1. Did somebody use a power law to analyse his personal data?
2. Do you want me to explain how to calculate your exponent if you have PRs on multiple distances?

For more explanation, see https://en.wikipedia.org/wiki/Power_law

[Ombrax]

Neat.

Could you show the plot with the erg data points for all the folks? (with a different symbol for each person so we could differentiate between them)

It would be interesting to see the scatter both within one person's data and between individuals.

[MartinSH4321]

Very interesting, thanks for sharing!

I checked my PBs for 500m-HM but I'm far away from a straight line. I got these exponents (rounded):
500m-1k: 1,2
1k-2k: 1,12
2k-5k: 1,08
5k-10k: 1,04
10k-HM: 1,05

I would interpret this results so:
- I'm power-dominant, so the exponents are higher for shorter distances
- my 2k and 5k are weak compared to my other results

[Citroen]

Nothing new here. This is just a re-hash of "Paul's Law" (which isn't really a law, more a conjecture).

http://danwork.blogspot.com/p/pauls-law.html

https://www.reddit.com/r/Rowing/comment ... pauls_law/
http://www.machars.net/

https://www.nonathlon.com/ has an extremely large dataset of scores from a wide age group of rowers.

[Nomath]

Could you show the plot with the erg data points for all the folks? (with a different symbol for each person so we could differentiate between them)

Thanks for your interest!
If I would make the same plot as in my first post you wouldn't see a lot, because the lines and data points would strongly clutter in a narrow band. That is because the dominant trend is that rowing times get longer as the distance increases. I had to do some mathematical juggling to get more resolution. The trick is to plot lap times instead of total times!
It can be shown that if the distance x consists of N laps and the distance x0 consists of N0 laps, then the power law formula becomes : tlap(N)/tlap(N0) = (N/N0)^(a-1) .
Here tlap(N) is the average lap time with doing N laps.
The nice thing is that because the exponent a is in the range 1.03-1.10, the new exponent (a-1) is a small and probably meaningful number. Remember that a value a=1 means that the time would increase exactly proportional to the distance. It's the difference with 1 that tells us about the struggle of the body to keep the time proportional to the distance. By taking 500m as one lap, we get an even simpler formula: tlap(N)/tlap(1) = N^(a-1).
The graph below shows how the average 500 m times increase as the distance increases from 0.5K (N=1) to 10K (N=20).
Sorry for this technical talk, but otherwise you could possibly misunderstand what the plot shows.



I collected data from 10 rowers in the 2020 season rankings. The choice was not random. I had to search for persons who ranked 4-5 best times over the range 0.5K-10K. The easiest way to find them was to look for recurring names in small groups. That's why I looked at small countries, like my own NED or NOR or FIN or CAN, or GBR in narrow age groups. I also observed that the persons who rank their performances over the full range are mostly no slugs. Most of them are in the top-25% of their group.
Persons were numbered 1-10 in an arbitrary way
Note that both the x-axis and the y-axis have a logarithmic scale.

The slope of the lines (a-1) varies between 0.056 to 0.130. The average value is 0.081 (sd =0.020).
Most lines show a convex shape, which means that the power law does not hold exactly.

[Ombrax]

That is a convenient way to see the differences between individuals, and see that although in general some are clearly faster than others, the effect of longer rows on the body (and the mind, since that too plays a role) is quite similar.

Enjoy your number crunching

[Nomath]

Nothing new here. This is just a re-hash of "Paul's Law" (which isn't really a law, more a conjecture).

I have never heard of "Paul's Law". Searching on the internet I saw that the term is mainly used in the rowing community, but also made it to the Urban Dictionary with some very creative definitions.
I read that Paul's Law implies that in rowing when you double the distance, you should add 5 seconds to the 500m time. It this is what it says , it definitely differs from the Power Law because the Power Law by nature leads to an increase that is proportional, not a fixed number. The increase for doubling the distance is a factor (2)^a, where the exponent a can be a range of numbers. Only if a is near 1.07, and t500=120 sec, will the increase in 500m time be near 5 sec.

The Power Law is not even a conjecture, it is just a mathematical formula that may or may not fit the empirical data. In engineering it often does and also in many human-powered sports that measure speed. What to my knowledge is not well known and possibly new, is that very different sports show a very similar value for the exponent.

[Nomath]

Reading further in the scientific literature about athletic records, I found a free-access article that explains that there is a transition from anaerobic to aerobic energy production of the body between 150-170 seconds from the start. Typical 500 m rowing times between 1:20 and 2:00 are clearly on the anaerobic side and a part of the 1000 meter run is in the aerobic range. This probably explains why most 500 m points do not fit well on a straight line in a log-log plot.
The exponent found for 'aerobic' world running records is 0.07-0.08 (i.e. 1.07-1.08 for total time), which agrees very well with the rowing data.
See https://www.academia.edu/30157986/Scali ... ld_records

[gregsmith01748]

Here's another model, calibrated with the C2 data. A key insight is that the different people have different power profiles.

https://analytics.rowsandall.com/2018/0 ... t-are-you/

[Nomath]

Greg, thanks for the link. The article seems a very thorough analysis. I will take more time to read it.

However, I am personally less interested in world records, because they are a mix of different individuals, different years, sometimes different equipment, training levels, altitudes, etc. I prefer to analyse the best results of individuals, preferably from a short time span, preferably under the same conditions. The ideal, for me, are the results from speed skating on ice competitions , because they are held on one weekend and involve 4 distances from 0.5K to 10K, with trained athletes who know how to pace each distance.

In the cited article I found a formula for Paul's Law. This formula differs from the Power Law. Let me show the difference in a graph where I took the average 500m time, 90 sec, from the group of 10 rowers that I analysed above, as a start time for the calculations. For the Power Law, I took the average exponent 0.081 that I calculated for this group.



The differences are small at 1K, but at 10K there is a difference of 3.1 sec per lap, amounting to 1 minute for the total time.

[gregsmith01748]

Individual variation is a main point of that article. It introduces the idea of two kinds of athletes: sprinters and stayers. The sprinters have a greater fall off in power with duration and the stayers have less. There’s a fun variation of Paul’s law for this. Paul’s law is basically double the distance, add 5 to the pace. A lot of lightweight rowers aspire to “double the d, add three”.

[Allan Olesen]

In running, this is known as the Riegel Formula. Typical exponents are 1.05-1.10. It works best for longer durations, where you are staying below the anaerobic threshold.

It is actually strange, that the exponents come so close in running and rowing. In running, the necessary power input is generally considered more or less proportional to velocity. In Concept2 rowing, the necessary power input is per C2 definition proportional to velocity^3. I would have expected this to result in an exponent closer to 1.02-1.03 for rowing.

For shorter, anaerobic efforts, you may want to look into the critical power model as defined by Monod many years ago. It basically says that you have a power output, CP, which you can hold for ever (where "for ever" in reality is less than 1 hour for most of us). And then you have a given amount of work, W', which you can put on top of this power. This means that your maximal power output for any duration, t, is CP + W' / t. This model holds up quite well for medium long efforts, but not for super short efforts, where your actual max. power output will be the limit.

[Nomath]

I am not familiar with power analysis in running. However, I would expect that for running the air drag is a substantial part of the total resistance too. You can feel if you run in a headwind or in a tailwind! The power to overcome air resistance is proportional to v³, or more accurately to v*(v-vw)², where v is the ground speed and vw is the component of the wind speed in the direction you are heading.

I am studying the article on Rowing Analytics, but I already have some doubts about the critical output power CP, at least where it is said that a useful CP model should go to 0 as duration goes to infinity. I challenge that condition! I know many cyclists, including myself, who can go on for more than 5 hours at a 100W output level if it were not for non-energy-related stops.

[Allan Olesen]

I am well aware of the relationship between velocity and the power needed to overcome wind resistance. But a runner uses the majority of his effort for other purposes than overcoming wind resistance, which can be easily seen by comparing running speeds to cycling speeds - even if we assume that the runner has twice the wind drag of a cyclist at the same velocity, this doesn't even come close to explaining the lower velocity of the runner.

Regarding what you read on a rowing website about CP: I haven't read that article. The CP model does not originate from rowing. But I would like to clear up one apparent misunderstanding. Monod's CP model does not say that CP goes toward 0. It says that CP is constant, until it isn't anymore. That is one known weakness of the model, and that is why I said that you can't use it for long duration efforts.

I would like to add that the CP model has some physiologic backing. CP is just around the intensity, where you start going anaerobic. When you go anaerobic, you start building up lactate in your blood. The higher you go above CP, the faster you build up lactate. If we make some assumptions...:

• Your CP is constant over time

• Your growth in lactate concentration is proportional to the amount of power in excess of CP

• Your "pain limit" for resulting lactate concentration is independent from duration

...then the CP model can actually be used to describe anaerobic efforts in the duration range where lactate tolerance will be setting the limit for your intensity.

[Allan Olesen]

By the way: If you use power in cycling, then you are probably familiar with FTP and FRC (or AWC). Those are just other terms for CP and W', coined by dr. Andrew Coggan who introduced the concept to cycling.