Cast for two

Saturday, March 22, 2014

Comparing power models for cycling

On January 23, 1984, Francesco Moser set a new Hour Record of 51.151km/h at altitude in Mexico City. What probably contributed to that success was the theory developed by Prof. Conconi. He theorized that heart rate could be correlated with perceived exertion in order to allow Moser to cycle at the absolute maximum of his capability. Ever since, training with a heart rate monitor became more and more popular.

Training with a heart rate monitor has its limitations. Suppose, last week you drove your favorite time trail lap for 20 minutes at an average heartbeat of 155 beats per minute (bpm). This week you did the same test, riding the same distance but your heart rate was 5 beats higher on average. Does that mean that your condition lowered ? The answer is that you can't be sure. Maybe you had more headwind and had to push harder to ride at the same speed. Maybe you ate something before the test that still had to be digested. Maybe you didn't sleep well ? Maybe you were stressed. Heart rate is only an indication what is going on in the black box of your body but is influenced by a lot of external parameters that can't be controlled during a ride.

Enter the power meters. They measure exactly what power you're pushing instantaneously. In case of headwind, you'll drive slower but the power readings will be higher. It allows to assess if your body is performing better or not without guessing. Therefore power meters allow to train more scientifically. Power meters give objective values about the performance and hence can be better trusted to evaluate your training efforts.
A central curve helping to gauge your performance is the power duration curve. How many power can you generate for how long ? A typical power curve looks like this:

The horizontal axis is the duration in seconds (usually on a logarithmic scale) and the vertical axis is the delivered power in Watt. The maximum power one can generate for 1 second during a sprint is called the Peak Power. The power one can deliver for one hour is called the Functional Threshold Power (FTP). The limit of the power one can generate forever is called the Critical Power (CP).

But how do we obtain such a curve ? If you record your rides with a computer, software can derive this from the measured values. Since curves of different riders show up similarly, researcher started to believe a model could predict the curve based on some measurements. For example, record how much power you can delivers for 1 minute, 3 minutes and 20 minutes and the whole curve can be reconstructed. In the paper "Rationale and resources for teaching the mathematical modeling of athletic training and performance" by Clarke DC, Skiba PF contains a good overview of the state of the art. The formula to derive power in function of the duration is as follows: $P = AWC (1/t) + CP$ where AWC is the Anaerobic Work Capacity in Joule, $t$ is the duration in seconds and CP is the Critical Power in Watt. The AWC (called W' nowadays) represents the finite amount of energy that is available above the critical power. CP is power that can be sustained without the fatigue for very long time (longer than 10 hours). See also the paper by Charles Dauwe, "Critical Power and Anaerobic Capacity of Grand Cycling Tour Winners".  Recently, @acoggan, @veloclinic and @djconnel are working on more sophisticated models.

@veloclinic proposed in "Cycling Research Study Pre Plan":
$P(t) = \frac{W’_1}{t+\tau_1} + \frac{W’_2}{t+\tau_2}$
and since $W’ = P \times \tau $
$P(t) = \frac{P_1 \tau_1}{t+\tau_1} + \frac{P_2 \tau_2}{t+\tau_2}$
@veloclinic guesses that the new Trainings Peak model to be included in WKO4 is:
$P(t) = \frac{FRC}{t} (1-e^{-\frac{t}{\frac{FRC}{P_{max}-FTP}}}) + FTP + \alpha (t-3600)$
For more information about WKO's new model, watch the youtube video.
And Dan Connelly arrived at:
$P(t) = P_1 \frac{\tau_1}{t}( 1 - e^{-\frac{t}{\tau_1}} ) + \frac{P_2}{(1 + \frac{t}{\tau_2})^ {\alpha_2}}$

Fitting those models to my data, gives the following result (source code):

With some extra code, we can get the confidence interval of the models. For the veloclinic model, the 95% confidence interval is between the dotted green lines (source code):
The estimated veloclinic parameters and their 95% confidence interval are:
  • $P_1$ = 291.910642398 Watt [252.385602829  331.435681968]
  • $\tau_1$ = 28.6068269179 seconds [17.5113005392  39.7023532965]
  • $P_2$ = 800.776804574 Watt [726.249462145  875.304147003]
  • $\tau_2$ = 0.312268982434 seconds [-0.16239984039  0.786937805257]
The supposed WKO4 model gives this (source code):
The estimated WKO4 parameters and their 95% confidence interval are:
  • FRC = 20463.525586 Joule [13518.0176693  27409.0335028]
  • $P_{max}$ = 1011.29658965 Watt [937.344413201  1085.2487661]
  • FTP = 257.543501763 Watt [226.388909604  288.698093922]
  • $\alpha$ = -0.00512616028401 [-0.00777859140802  -0.00247372916]
And finally, Dan Connelly's model (source code):
The estimated djconnel parameters and their 95% confidence interval are:
  • $P_1$ = 730.972437292 Watt, [638.464114912 823.480759673]
  • $\tau_1$ = 19.9194404736 seconds [9.71207387339 30.1268070738]
  • $P_2$ = 324.66235674 Watt [246.999942142  402.324771339]
  • $\tau_2$ = 0.312268982434 seconds [-0.16239984039  0.786937805257]
  • $\alpha_2$ = 0.312268982434 [-0.16239984039  0.786937805257]
From those graphs one could derive that the WKO4 model fits best. Remark that this is only the case for my data and that is a too narrow basis to draw such a conclusion. Also my power data itself has to be taken by a grain of salt because I have no power meter and all power data is estimated from other measurements. The dots on the graph are derived from virtual power derived from the speed measurements on my Tacx trainer or by power estimated by Strava on climbs (in that case estimated power by Strava is fairly accurate).

Update: Apparently, the formula in the end of the blogpost from Dan Connely which I used in this blogpost was not correct and must be:
$P(t) = P_1 \frac{\tau_1}{t}( 1 - e^{\frac{-t}{\tau_1}} ) + \frac{P_2}{( 1 + \frac{t}{\alpha_2\tau_2} )^{\alpha_2}}$
Remark the extra $\alpha_2$ in the denomintator under $t$ in the last term of the formula.
Strangely, the confidence interval behaves strange. I have to look into it. The estimated djconnel parameters of the updated model and their 95% confidence interval are:
  • $P_1$ = 730.972437292 Watt, [638.464114912 823.480759673]
  • $\tau_1$ = 19.9194404736 seconds [9.71207387339 30.1268070738]
  • $P_2$ = 324.66235674 Watt [246.999942142  402.324771339]
  • $\tau_2$ = 8793.18635004 [-12682.9310423 30269.3037424]
  • $\alpha_2$ = 0.312268982434 [-0.16239984039  0.786937805257]
Only the $\tau_2$ parameter changed value, but the confidence interval indicated it does not converge. When starting the optimization function with other initial values, I get overflow errors. To be continued. As Dan remarked we can approximate $\frac{1}{( 1 + \frac{t}{\alpha_2\tau_2} )^{\alpha_2}} \approx (1-\frac{t}{\tau_2 \alpha_2})^{\alpha_2} \approx 1-\frac{t}{\tau_2}$ in the first order. In that case, we get:
The parameters of this simplified model are:
  • $P_1$ = 753.75304433 Watt [676.513606926 830.992481733]
  • $\tau_1$ = 27.1488626906 seconds [17.4753205281 36.822404853]
  • $P_2$ = 275.99764045 Watt [238.468417448 313.526863451]
  • $\tau_2$ = 53841.0314734 seconds [30850.1607097 76831.9022372]
Remark that in order to fit the curve the $\tau_2$ parameter must be very big to avoid to get negative power. Hence using the first order approximation is not a good idea.

Another idea I tested is to use a capacity for the second term of Dan Connely's function:
$P(t) = P_1 \frac{\tau_1}{t}( 1 - e^{\frac{-t}{\tau_1}} ) + \frac{P_2 \tau_2}{t+\tau_2}$
This gives the following fitting:
The parameters of this mixed djconel/veloclinic model are:
  • $P_1$ = 743.251464989 Watt [666.223087684  820.279842294]
  • $\tau_1$ = 23.648384644 seconds [14.5239744624  32.7727948257]
  • $P_2$ = 297.690593484 Watt [253.858306164  341.522880804]
  • $\tau_2$ = 24294.3750137 seconds [8540.40600978  40048.3440176]
The only thing I hoped to achieve is to offer some software to evaluate power models like those proposed by veloclinic, djconel and acogan. What we now need is a lot of power data so that we can ran those models quickly over that data to draw more solid/general conclusions.

Friday, January 31, 2014

Adding virtual power to TCX for the Tacx Blue Motion Cycling Trainer

I recently bought a cycle trainer for indoor training: Tacx Blue Motion T2600 for 185€ at, a local cycling store. Using my Garmin 800, i could record my heartrate, cadence and speed while riding a workout. Since I have no power meter on my bike, I was barred from a feature that higher and more expensive trainers offer. But in the documentation of the trainer, I found a graph showing a linear relation between speed and power. So if I just could add this to the recorded file before submitting it to Strava, I would have trainer for less than 200€ with power measurement. This is the graph:

From that graph we can derive that riding at 60 km/h in position 5 (the middle blue line) a power of 500 Watt is developed. Hence power = speed / 60 * 500 = speed / 6 * 50 . Remark that speed must be in Km/h dimensions to obtain the power in Watt. There's even an interactive graph at website of Tacx. The Garmin 800 measures speed by counting the number of revolution of the wheel in a time unit and multiplying this by the circumference of the wheels. I converted the .fit file, I obtained from the Garmin device, in TCX (a XML file) using Garmin Training Center (Free software available for Windows and Mac). This file basically contains a list of trackpoints (1 trackpoint every second). One trackpoint looks like this:
            <HeartRateBpm xsi:type="HeartRateInBeatsPerMinute_t">
              <TPX xmlns="" CadenceSensor="Bike">
At this time, the speed was 8.4530001 meter per second. To convert this to km/h, we have to divide by thousand and multiply by 3600 (the number of seconds in an hour). So speed_in_kmperh = speed /1000.0 * 60 *60 = 30.43080036 km/h. The power developed at that moment was : 30.43080036/6.0*50.0 = 253.590003 Watts. We convert to integer : 253 Watt. To add this to the TCX file, we add a line <Watts>253</Watts> as follows:
            <HeartRateBpm xsi:type="HeartRateInBeatsPerMinute_t">
              <TPX xmlns="" CadenceSensor="Bike">
The next step was to automate the calculation of the power and adding it to the TCX file. I wrote the following Python script to do that:
prompt> python > vpower_29-01-14\ 20-53-27.tcx
I uploaded the resulting TCX file to Strava and obtained this:
Of course, the power in this workout is based on the fact that I left the lever om my trainer on position 5 during the whole workout. If you change the position of the lever during the workout, this approach will give wrong results.

Is this approach of "virtual power" accurate ? Not as accurate as power meters on the bike but usable I would argue. The concept of "virtual power" is also supported by Trainer Road Software. Later, I found an interactive graph of the speed power relation for Tacx Blue Motion on the website of Tacx. From that graph, I could obtain more precise datapoints : at 60 km/h power is 407 Watt. So next time I use my script, I will use power = speed_in_kmperh / 60.0 * 407.0 as power formula.

Wednesday, January 15, 2014

Euler : great talk by William Dunham

Having a Google Chrome Cast at home increased my longform consumption on Youtube dramatically. For instance, I discovered this talk about the great mathematician Euler by William Dunham:

I learned from it that there's a simple polynomial discovered by Euler that generates 40 consecutive primes: $ \forall n \in [0,40] : x^2+ x+ 41$ is a prime number.
You can check it by running following Python code:
You may wonder if there are other polynomials that generate primes. The article "Prime Generating Polynomials" claims that the polynomial $(x^5 - 133x^4 + 6729x^3 - 158379x^2 + 1720294x - 6823316)/4$ generates 57 consecutive primes for $x \in [0,56]$. Also the Wolfram article " Prime-Generating Polynomial indicates that polynomial as a winner. There are of course formula's to generate prime numbers but i think the Euler polynomial is the fastest.
The solution to Euler problem 27 is also interesting. A second degree polynomial generates 71 primes (but they are not consecutive). The polynomial $x^2 - 61x + 971$ generates 71 primes for $ x \in [0,70]$. This is the longest solution when the absolute values of the coefficients are restricted to thousand. If we drop that restriction, an even stronger polynomial is found : $x^2 -79 x + 1601$ generates 80 primes. Remark that the generated primes are not unique. For example the numbers 1601, 41, 197, 797, 1373, 1523 are generated twice. From the 80 primes generated, 40 are unique. I think Euler would not have been impressed.

Tuesday, December 11, 2012

Random Forests are the new kid in machine learning town

Is was reading "Specialist Knowledge Is Useless and Unhelpful
When data prediction is a game, the experts lose out." I learned about the new algorithm that seems like a silver bullet for data mining problems: random forests. An explanation in layman's terms can be found on Quora. It still don't fully grasp the idea but I think its worth exploring further to add to a toolbox to solve problems. There exist a R package for random forests for some quick exploration on your datas. Happy hacking.

Monday, October 10, 2011

Design inspiration

This weekend I was browsing with zite app on Ipad. Especially in the "webdesign and user experience" section I encountered some interesting links:

Monday, August 22, 2011

Back from holiday

This video sums up how the holiday was. Two of the boys that are jumping are my sons. It was shot with a simple Canon Ixus 300HS. The video video resolution is only 320x240 pixels and was shot at 240 frames per second. Enjoy it.

Saturday, June 25, 2011

It's not the story, the people, the technology, it's the team

I was directed to the blog of Amir Salihefendic, and i bounced into this gem:

Finally, these are the operating principals of Pixar as quoted by Catmull:
  • constant review
  • it must be safe for people to tell the truth
  • communication should not mirror the organizational structure
  • people and how they function is more important than ideas
  • do not let success mask problems, do a deep assessment
  • mix up creative and technical people
So it's not the story, the people, the technology, it's the team, but remember: 
"Once one can articulate an important idea into a concise statement, then one can use the statement and not have to have a fear of changing behavior." (Ed Catmull, pixar).

Maybe I should read the pixar book that's on my reading list for some time now.