Functionalities of performance modeling using power laws.
Preliminary results show that the power law not only models long efforts better but also allows us to estimate our marathon time with great precision when high-quality data is used.
Irati Ortiz, Frederic Sabater (substack) and I are working on an article that models the work-duration relationship through critical speed and power laws. Preliminary results show that the power law not only models long efforts better but also allows us to estimate our marathon time with great precision when high-quality data is used.
In this article, I will explain the functionalities this method offers. If you want personalized training zones and time estimates, you can participate in our study.
To participate, you just need to complete a 1200m and a 3600m test and connect with us on intervals.icu. (Instructions here). We need more participants to achieve statistical significance for our work.
Quality vs quantity of data
The key to making the model useful lies in carefully selecting high-quality data by manually choosing three tests that truly represent our current condition.
It is important to avoid mixing hot and cold weather days or tests conducted in competitions where motivation was maximal with less significant training efforts.
Although most people believe that having more data is better for the model, in reality, the opposite is true. In our study, we found that the more we ensured consistent test conditions, the better the coefficient of determination (R²) and the more accurate our long-duration performance estimates became.
Considering the work-duration relationship as a power law has strong theoretical implications for our understanding of physiology, but it also provides functionalities that the hyperbolic critical speed or power models could not offer, such as:
Performance modeling in long efforts.
We now have the tools to accurately estimate our performance in self-paced competitions lasting longer than one hour, whereas previously, heuristic rules had to be applied.
When applying these estimates to long-competition pacing, we observed that athletes either closely match the estimate or fall short, but they very rarely exceed it.
If traditional hyperbolic models of critical power or speed suggest that fatigue slows down below the critical threshold, while the power law indicates that performance continues to decline at the same rate, we often observe that the decline in performance becomes even steeper as duration increases.
In other words, the opposite of what hyperbolic critical speed models predict.
This happens because the power law models the ideal conditions in which the tests are performed. However, in long competitions, many new factors come into play, and most of them negatively impact performance.
On race day, it might be hot, you could become dehydrated, experience digestive issues, cramps, or pacing inconsistencies, among other things. Any of these factors, which are not accounted for in the tests used to determine the power law, will result in performance falling below the estimated value.
Therefore, we should consider these time or speed predictions as a theoretical maximum achievable under perfect conditions.
In cycling, since pacing is dynamic and depends on many factors, practical experience suggests that the power law can be used to predict the maximum normalized power for a long event, such as a 4-hour race, provided it does not include short, explosive efforts that can distort the normalized power calculation.
For example, using the power law, I could estimate that a cyclist will be able to generate a maximum of 280 normalized watts in 4 hours. If they exceed this early on, they will pay for it with greater fatigue later, and vice versa. Of course, variable pacing reduces accuracy, but it works very well for efforts such as long mountain climbs, mountain marathons, etc.
Calculating low-intensity training paces
When training at low intensities, the usual approach is to apply arbitrary percentages from short tests, such as a 20-minute test. The power law allows us to estimate with high precision the pace we could maintain in a marathon or a 3-hour effort in cycling or another sport.
This enables much more precise marathon pace training or for any other distance. It also helps us set anchors for reliable low-intensity training, such as determining the power we could sustain for 3 hours as the boundary between the yellow and green zones.
Athlete profile
Measuring the slope of the power law at different times of the season allows us to determine whether the athlete is becoming more explosive or more efficient with training.
Typically, in endurance sports, we observe that the exponent of the power law decreases: the fitter we are, the less energy-generating capacity we lose over time.
Furthermore, this exponent allows us to compare the profiles of different athletes and detect whether our performance declines faster or slower over time, which can help in selecting the ideal discipline or understanding which strategies suit us best.
For cyclist:
If you're a cyclist and you're interested, I've created an online calculator where you can determine the power law function that best models your performance to estimate your output over long durations (in watts).
About the scientific study
Who can participate?
Asphalt runners who, since June 2023, have completed or will complete within a year at least three of the following races: 5km, 10km, 15km, half marathon, and marathon.
Athletes who have times under 3h30′ for a marathon and under 45′ for 10k in men, and under 4h for a marathon and under 50′ for 10k in women.
What do you have to do?
Just three steps.
Fill out the questionnaire (link)
Perform two tests, 1200m and 3600m (instructions)
Sync your device with us on Intervals.Icu (tutorial)
What do you gain?
Helping scientific progress.
Getting to know yourself better.
An analysis of the pace at which you could run long races.
For any questions, write to us at modelovcrit@gmail.com