Is TSS additive?



hmronnow

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Aug 12, 2006
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I'm a recreational cyclist who likes numbers and data-analysis. Having a powermeter and analysing rides is fun for me, and the 'performance manager' in WKO works nicely as a motivational tool - trying to work up the TSS.

But, I just wanted to point out a detail:
TSS (training stress score) is based on IF (intensity factor), which is based on NP (normalized power). Normalised power is the 4th root of the average of power to the 4th power (squared squared, or matematically put P^4). The reason for this is very smart - it takes into account that training feedback is more than linear in training load - NP rewards you e.g. for interval training.

However, there is a mathematical problem here. If we defined TSS as the sum over P^4, then all is fine - only draw back that the unit for TSS is not watts but watts^4.

The wish to keep simple units (more understandable for lay people and those cyclists who don't have a PhD in physics) lead to using the 4th root. However, in doing so TSS is no longer additive.

A simple example:
Suppose we have FTP of 200W and ride 1 hour at 200W, download to WKO, which rewards us 100TSS. We then take another 1 hour ride at 100W, which WKO rewards us with 50TSS. That would give total 150TSS.

Now, lets instead take one 2 hour ride, riding 200W first hour and 100W second hour. NP will be ((200^4+100^4)/2)^1/4=171W. This corresponds to an IF=0.854 for 2 hours and is rewarded by 171TSS.

We did exactly the same training, but in second example we were rewarded 21TSS more. In other words, I think the concept of TSS is mathematically ill founded.

But, there may not be reason to despair. I think what saves the concept is that the P^4 is anyway just a crude approximation (and generalisation across all athletes) to the physiological fact that intense training pays off. In other words, the mathematical error built into the TSS concept is probably inferior to the approximation in applying the simple expression P^4 to a physiological effect, and using this same formula for all different athletes.

In other words, TSS is an approximation - and as such it works.
 
hmronnow said:
A simple example:
Suppose we have FTP of 200W and ride 1 hour at 200W, download to WKO, which rewards us 100TSS. We then take another 1 hour ride at 100W, which WKO rewards us with 50TSS. That would give total 150TSS.

Now, lets instead take one 2 hour ride, riding 200W first hour and 100W second hour. NP will be ((200^4+100^4)/2)^1/4=171W. This corresponds to an IF=0.854 for 2 hours and is rewarded by 171TSS.

We did exactly the same training, but in second example we were rewarded 21TSS more. In other words, I think the concept of TSS is mathematically ill founded.

Not exactly. The first hour among others depletes your glycogen stores and continuing the ride in a fatigued state is more "consuming" than doing it fresh.

The question is how long could/should the break between two training impulses be so that the calculation is additive.
 
Yeah, this topic has been pounded to death on various forums. TSS isn't additive across segments and worse the overall CTL/ATL averaging is effectively an additive process. There are a couple frequent flyers on the Google wattage list that rarely pass up the opportunity to repeat their arguments that TSS/CTL/ATL is structurally flawed in a mathematical sense. Alternatives have been proposed that might be more mathematically elegant, but typically lack grounding in physiology so IOW nice math that doesn't map to first principles of exercise physiology.

Personally I'd split a workout with breaks over about 15 minutes into multiple shorter workouts, but other folks approach this differently.

I think your statement:
...In other words, TSS is an approximation - and as such it works...
sums it up nicely and of course it's an approximation as is the Monod model or any other simple relationship that attempts to describe physiological processes. The body and its responses are pretty damn complex and any of these modeling tools will have theoretical shortcomings, the trick is to work within the limits of the model.

So don't try to extrapolate Monod curves out to pace your 12 hour event and don't intentionally get off the bike and rest between your on bike efforts to rack up a lot of TSS.

-Dave
 
daveryanwyoming said:
I think your statement: sums it up nicely and of course it's an approximation as is the Monod model or any other simple relationship that attempts to describe physiological processes. The body and its responses are pretty damn complex and any of these modeling tools will have theoretical shortcomings, the trick is to work within the limits of the model.

FWIW, here is part of what I wrote when I foisted the idea on the world back in 2003:

"A greater limitation to the entire concept, though, is that the
basic premise – i.e., that you can adequately describe the training load
and the stress it imposes on an individual based on just one number
(TSS), completely ignoring how that “score” is achieved and other
factors (e.g., diet, rest) – is, on its face, ridiculous...Nonetheless, I believe
that TSS (and IF) should prove useful to coaches and athletes for
evaluating/managing training."

I would submit that the latter prediction has proven to be true...
 
hmronnow said:
I think what saves the concept is that the P^4 is anyway just a crude approximation (and generalisation across all athletes) to the physiological fact that intense training pays off. In other words, the mathematical error built into the TSS concept is probably inferior to the approximation in applying the simple expression P^4 to a physiological effect, and using this same formula for all different athletes.

IMO, the normalized power concept/algorithm is more sound/robust than the notion of TSS (or TRIMP, EPOC, Foster's session RPE, etc.).
 
hmronnow said:
A simple example:
Suppose we have FTP of 200W and ride 1 hour at 200W, download to WKO, which rewards us 100TSS. We then take another 1 hour ride at 100W, which WKO rewards us with 50TSS. That would give total 150TSS.

Now, lets instead take one 2 hour ride, riding 200W first hour and 100W second hour. NP will be ((200^4+100^4)/2)^1/4=171W. This corresponds to an IF=0.854 for 2 hours and is rewarded by 171TSS.

We did exactly the same training, but in second example we were rewarded 21TSS more.
If I am not mistaken, Tudor Bompa used to refer to this as training density. He'd use an algorithm to quantify training load (way back then) and to the best of my knowledge, he'd come to the same conclusion. Two workouts same duration same intensity would produce different scores if density isn't the same (IOW if training frequency would differ).

I am not expert, but they may have done that on purpose.
 
SolarEnergy said:
If I am not mistaken, Tudor Bompa used to refer to this as training density. He'd use an algorithm to quantify training load (way back then) and to the best of my knowledge, he'd come to the same conclusion. Two workouts same duration same intensity would produce different scores if density isn't the same (IOW if training frequency would differ).

I am not expert, but they may have done that on purpose.

:confused: I would venture into saying that if the math problem exposed by the OP was an attempt to replicate what you describe... the good doctor Coggan would have mentioned it...just a hunch. ;)
 
quick question

if i have an ftp of 200W and a ride at 100W for an hour is the TSS for that ride not 25 rather than 50?

thanks

p
 
pauln99 said:
if i have an ftp of 200W and a ride at 100W for an hour is the TSS for that ride not 25 rather than 50?

Yes.

Edit: but the math in the second example was similarly wrong, so it all essentially still works out as the OP intended. :)

2nd case: IF = .854, so TSS = (.854)^2 * 2 hrs *100 = 146 TSS
 
midlife said:
I would venture into saying that if the math problem exposed by the OP was an attempt to replicate what you describe... the good doctor Coggan would have mentioned it

I wish I could claim that the issue of RATSS ("recovery augmented TSS") was something that I had anticipated/deliberately intended, but it isn't. Intentional or not, though, I can make a very strong case (from a physiological perspective) that TSS should not be additive across segments. Moreover, the issue of how to properly weight duration, and especially the interaction between intensity and duration, is not unique to TSS, but applies to all similar scoring systems (e.g., TRIMP, EPOC, etc.) as well. Most importantly of all, however, is the fact that it is actually the modeling process itself that is amenable to improvement, rather than how the input function to such models (e.g., the impulse-response model) is calculated.
 
acoggan said:
Intentional or not, though, I can make a very strong case (from a physiological perspective) that TSS should not be additive across segments.

Help my weak brain please. This means if I go for a day long training ride and stop for lunch it should appear as two workouts in WKO not one ? right ? (cause right now i come back home and shove one big workout in there)
 
midlife said:
Help my weak brain please. This means if I go for a day long training ride and stop for lunch it should appear as two workouts in WKO not one ? right ? (cause right now i come back home and shove one big workout in there)
That is my understanding, which seems to have been confirmed by Dave a little earlier in the thread
daveryanwyoming said:
Personally I'd split a workout with breaks over about 15 minutes into multiple shorter workouts, but other folks approach this differently.

My understanding of this thread is that the OP (who is very good at maths) found out that he'd get two different scores for the same training dose (60minFTP / 60min50%FTP) depending on if this dose was achieve through one single workout vs two different workouts (segments). He suggested that TSS (while being fairly addictive) wasn't additive from segment to segment. Others have replied that it shouldn't be additive from segment to segment, quite the opposite. Some stated that in order to avoid segments to be additive, it may even be better to split a longish workout that has significant rest periods in it, into smaller chunks. Again, this would be done in order to avoid segments to behave in a way the OP suggested they should.

To tell you the truth though, for now, I personally don't worry about these things. There's so much room on the "sunny side" of this concept that I enjoy playing with it as is.

And besides, one should aim at gaining a decent level of mastery over the concept before becoming truly relevant in seeing windows of opportunities for improvement. In other words, every one should do his own 1n study and see to which extent the concept is holding the road given one's training/logging/racing habits. And that takes time. Every season wide test, well, takes a full season.
 
SolarEnergy said:
That is my understanding, which seems to have been confirmed by Dave a little earlier in the thread

You are right .. thanks

SolarEnergy said:
And besides, one should aim at gaining a minimal level of mastery over the concept before being fully relevant in attempting to criticize or improve it.

[x] slippery slope initiated