comparison of TSS models on "standard" rides

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I did a comparison of an "experimental" model for TSS with the conventional model.

The experimental model is based on the following assumptions:

1. there is an aerobic cost to generating power, proportional to total kJ of work done.
2. there is an anaerobic cost to generating power, proportional to (for each time segment) blood lactate levels over baseline, which Andy has modeled as proportional to P^4. (this is in the NP formula).
3. at FTP, the two costs are about equal. In other words, doing the same hill twice, as slow as you can go, has about the same cost as doing it once @ FTP. (I make no claims on this assumption except that it's probably incorrect :)).

Additionally, there's the following highly experimental enhancement:

1. A certain form of fatigue is added proportional to P^4
2. This fatigue decays away exponentially with a 1-hour time constant.
3. Doing work when thus fatigued has a higher cost than doing it when fresh, in proportion (1+F)^2 (why not?)
4. F=1 corresponds to doing steady work @ FTP, F=0 is well-rested for several hours.

This I'm SURE is wrong -- too many assumptions.

Anyway, I compared two experimental TSS models, one with the fatigue term, one without, to the conventional TSS model, for a variety of workouts. By definition they agree for 1 hour @ TSS (I normalize the models to give 100 in this case. The conventional model actually gives me 100.16, due to rounding extending the workout over 1 hour. to which it's sensitive).

I estimated FTP from histograms of power for each workout. Bogus, obviously. But I had to use something... I append these values at the end.

Interestingly, my formulas always give a bit less than the conventional model in these examples (except constant power). The reason is the conventional model is boosted by periods of relative inactivity, for example coasting, in excess of what those periods would score on their own, while the experimental formulas are not (the fatigue term decays during coasting, so the TSS is decreased a bit by it, while the no-fatigue model isn't affected). Not surprisingly, adding the fatigue term tends to boost the TSS of longer rides relative to shorter ones.

Dan

http://www-tcad.stanford.edu/%7Edjconnel/cycling/maximal_power_curve_simulation/TSS_comparison.png

Assumed FTP values:

g010102:
g010103:
ftp = 280

montrealwc:
ftp = 350

jiminy-ahm:
jiminy-jwh:
ftp = 375

jiminy-wrb:
bear-ahm:
sterling-ahm:
housatonic:
tt:
ftp = 400

riis-amstel97:
riis-lbl92:
ftp = 430

A couple comments/questions:

I did a comparison of an "experimental" model for TSS with the conventional model.

The experimental model is based on the following assumptions:

1. there is an aerobic cost to generating power, proportional to total kJ of work done.
2. there is an anaerobic cost to generating power, proportional to (for each time segment) blood lactate levels over baseline, which Andy has modeled as proportional to P^4. (this is in the NP formula).

Comment #1:

I think it is mistake to view the physiological "cost" of training to have independent aerobic and anaerobic components. In particular, blood lactate should not be viewed as a marker of the anaerobic cost, as regardless of O2 availability, lactate will be produced any time the rate of pyruvate production exceeds its rate of entry into the mitochondria. Indeed, to bring this back to TSS: the blood lactate data that I used to derive the 4th order weighting came from submaximal bouts of exercise lasting at least 10 min, during which the vast majority of energy was produced aerobically.

Comment #2:

The original, "raw" (i.e., not standardized across individuals) formula for TSS is:

"raw" TSS = duration (s) x normalized power (W) x (normalized power (W)/functional threshold power (W))

Obviously, the product of duration and power is work, i.e., the formula already takes into consideration the amount of work that is performed (as well as the relative intensity at which that work is performed).

Anyway, I compared two experimental TSS models, one with the fatigue term, one without, to the conventional TSS model, for a variety of workouts.

...which to my eye, actually seem to correspond fairly well (i.e., all the additional assumptions and mathematical manipulations didn't have a huge impact). Related to that, though, a couple of questions:

1) why use a log-log plot? and

2) why define the experimental TSS as the independent variable?

For the record, I reduced the FTP of the "g01010x" plots, as I realized they were too high, after Andrew responded. Those were virtually entirely above-threshold efforts. A video of the race in question from this year is here (way off topic): http://media.putfile.com/San-Bruno-Hill-Climb-Jan-1-2007

A couple comments/questions:
Comment #1:

I think it is mistake to view the physiological "cost" of training to have independent aerobic and anaerobic components. In particular, blood lactate should not be viewed as a marker of the anaerobic cost, as regardless of O2 availability, lactate will be produced any time the rate of pyruvate production exceeds its rate of entry into the mitochondria. Indeed, to bring this back to TSS: the blood lactate data that I used to derive the 4th order weighting came from submaximal bouts of exercise lasting at least 10 min, during which the vast majority of energy was produced aerobically. Okay -- I'll just leave (P/FTP) + (P/FTP)^4 as a heuristic formula that below a certain point, doing work has a fixed cost, while above a certain point, it costs more the faster you do it. I'd thought the fourth power in your formula was based on blood lactate levels.

Comment #2:

The original, "raw" (i.e., not standardized across individuals) formula for TSS is:

"raw" TSS = duration (s) x normalized power (W) x (normalized power (W)/functional threshold power (W))

Obviously, the product of duration and power is work, i.e., the formula already takes into consideration the amount of work that is performed (as well as the relative intensity at which that work is performed).
That's not a fixed cost of doing work, though, since I can climb arbitrarily much at arbitrarily small cost if I'm willing to do so arbitrarily slowly (with arbitrarily small gears, obviously, and arbitrarily good balance :)). My conjecture was at some point it no longer gets easier.

...which to my eye, actually seem to correspond fairly well (i.e., all the additional assumptions and mathematical manipulations didn't have a huge impact). I noticed the same thing! The ordering of workouts is little effected. The 1 hour TT is the notable difference.

Related to that, though, a couple of questions:

1) why use a log-log plot? and
The data span a broad range, and I wanted to highlight fractional differences in TSS, rather than absolute differences. The log plot naturally shows fractional differences as a given deviation on the plot.

2) why define the experimental TSS as the independent variable?That was a mistake. I was too lazy to fix it. The experimental TSS should really be on the y-axis.

Rather than change the power part of the TSS calculation, an alternative is to change the time part.

Rather than change the power part of the TSS calculation, an alternative is to change the time part.You could imagine some sort of pseudo-time which attempted to prune low-intensity powers from the mix. For example, limiting to formulas which for high sustained powers converge to actual duration:

duration' = integral { (P*/FTP)^4 / (1 + (P*/FTP)^4) dt )

and use this in place of exercise duration.

For P << FTP, then

duration' → integral { (P*/FTP)^4 dt }

remember:

TSS = (100/hr) × (integral { (P*/FTP)^4 dt } / duration)^(1/2) × duration

Plug in duration' for duration

TSS → (100/hr) × (integral { (P*/FTP)^4 dt } / integral { (P*/FTP)^4 dt } )^(1/2) × integral { (P*/FTP)^4 dt }
= (100/hr) × integral { (P*/FTP)^4 dt }

This linearizes the equation! No more problem with low power segments.

However, still a problem: Now TSS drops like a rock when NP << FTP, much faster than before.

And this still keeps the nonlinearity for larger powers. So better to simplify.

Dan

However, still a problem: Now TSS drops like a rock when NP << FTP, much faster than before.
There are other formulations of time that one could use. But the main point is that there is no particular reason why the "problem" (if problem there be) is with the power side of the equation, and Andy's use of "straight" time isn't sacred (except that currently TSS looks sorta kinda like a scaled version of kJ, which could be both a good thing and a bad thing). In addition, from a TSB point of view, I don't see anyone doing anything that depends on the scale values of TSS -- all that matters (it appears) are the ordinal values. So as long as any new transformation is monotonic you can still use it the way it was intended.

There are other formulations of time that one could use. But the main point is that there is no particular reason why the "problem" (if problem there be) is with the power side of the equation, and Andy's use of "straight" time isn't sacred (except that currently TSS looks sorta kinda like a scaled version of kJ, which could be both a good thing and a bad thing). In addition, from a TSB point of view, I don't see anyone doing anything that depends on the scale values of TSS -- all that matters (it appears) are the ordinal values. So as long as any new transformation is monotonic you can still use it the way it was intended.True. For example, a more generalized form of the time derivation:

duration' = integral { (K P* / FTP)^4 / (1 + (K P* / FTP)^4 }
where K controls at what point time matters. For example, for K = 4, time is deweighted by 2x @ FTP/4, instead of @ FTP for K=1.

But its important to keep the same P^4 dependence for low P, in order to cancel the term (integra P^4l to 1/4 power) which causes the nonlinearlity.

The core problem which isn't being addressed here, except in the limit of small P, is that ride segments are weighted P^4 but rides are weighted P^2, so where you separate rides matters. The solution is to weight them the same. Which is what I did by using a simple integral equation (and Rick did much earlier).

Dan

The core problem which isn't being addressed here, except in the limit of small P, is that ride segments are weighted P^4 but rides are weighted P^2, so where you separate rides matters. This is the issue that I try to ignore. Sometimes I'm better at that than other times.