# 10 min. NP higher than 5 min. NP?

#### jstock

##### New Member
Hi,
I'm to lazy to do the math required to check wether it is possible for 10 minute NP to be higher than 5 minute NP, but it does look pretty strange to me. Any comments? HTML:
``````Peak 5min (256 watts):
Duration:          5:00
Work:              77 kJ
TSS:               8.7 (intensity factor 1.02)
Norm Power:        280
VI:                1.1
Distance:          1.992 km
Elevation Gain:            17 m
Elevation Loss:           12 m
Min        Max        Avg
Power:               0        461        256         watts
Heart rate:          144        169        161         bpm
Speed:               14.8        31.7        23.9         kph
Pace                 1:54        4:03        2:31         min/km
Altitude:            50        59        54         m

Peak 10min (248 watts):
Duration:          10:00
Work:              149 kJ
TSS:               19 (intensity factor 1.067)
Norm Power:        293
VI:                1.18
Distance:          3.85 km
Elevation Gain:            37 m
Elevation Loss:           20 m
Min        Max        Avg
Power:               0        544        248         watts
Heart rate:          143        171        160         bpm
Speed:               8.9        31.7        23.1         kph
Pace                 1:54        6:44        2:36         min/km
Altitude:            50        69        56         m``````

#### acoggan

##### Member
The reason that the 10 min normalized power is higher than the 5 min normalized power is because neither is (necessarily) your highest normalized power for that duration. That is, WKO+ picks out the "peaks" based on the average power, and then shows you the metrics for them (including the normalized power for any segment >5 min in duration).

If you want to know your highest normalized power across various durations, you need to generate a custom mean maximal power chart in which the "channel" is set to normalized power instead of plain ol' power.

#### jstock

##### New Member
acoggan said:
The reason that the 10 min normalized power is higher than the 5 min normalized power is because neither is (necessarily) your highest normalized power for that duration. That is, WKO+ picks out the "peaks" based on the average power, and then shows you all that metrics for them.
Ok, thanks. I've read that before, but somehow it slipped my mind.
/J

#### frenchyge

##### New Member
jstock said:
I'm to lazy to do the math required to check wether it is possible for 10 minute NP to be higher than 5 minute NP, but it does look pretty strange to me. Any comments? Not strange at all. Go as hard as you can for 3.5 minutes, soft pedal rest for 3 minutes, then hard again for 3.5 minutes and voila!

I'm too lazy to do the math to prove it to you, so you'll just have to believe it. #### GIH

##### New Member
Its definitely not possible.

#### frenchyge

##### New Member
GIH said:
Its definitely not possible.
Are you serious? Plug some numbers into the scenario I gave above and see if you still feel that way. #### daveryanwyoming

##### Well-Known Member
frenchyge said:
Are you serious? Plug some numbers into the scenario I gave above and see if you still feel that way. Actually in the scenario you just gave (3.5 max, 3 easy, 3.5 max) the 5 minute NP and 10 minute NP are equal assuming the max efforts are equal and the easy stretch is steady and equal.

Both 5 minute blocks have 7x30" at max and 3x30" at "easy", the 10 minute block has 14x30" at max and 6x30" at "easy" the average of the 30 second blocks taken to the 4th power are identical for the 5 and 10 minute blocks as are the fourth roots of those averages.

I suspect you're right frenchyge, but that particular test case doesn't support your argument -Dave

#### Alex Simmons

##### Member
daveryanwyoming said:
Actually in the scenario you just gave (3.5 max, 3 easy, 3.5 max) the 5 minute NP and 10 minute NP are equal assuming the max efforts are equal and the easy stretch is steady and equal.

Both 5 minute blocks have 7x30" at max and 3x30" at "easy", the 10 minute block has 14x30" at max and 6x30" at "easy" the average of the 30 second blocks taken to the 4th power are identical for the 5 and 10 minute blocks as are the fourth roots of those averages.

I suspect you're right frenchyge, but that particular test case doesn't support your argument -Dave
Well I just ran the numbers with:
350W for 3.5 min
100W for 3 min
350W for 3.5 min

10 min AP = 275W
10 min NP = 316W
5min max NP = 312W

But the OP's issue is more likely as Andy pointed out that the stats (including NP) shown are for the max Avg power segment but that does not necessarily coincide with max NP segment for any given duration.

#### Alex Simmons

##### Member
Of course NPs for durations this short are just an academic exercise anyway.... #### daveryanwyoming

##### Well-Known Member
Alex Simmons said:
Of course NPs for durations this short are just an academic exercise anyway.... Tru 'nuff

Still as an academic exercise I get exactly 320.37 watts NP for both the 10 minute and peak 5 minute based on your 350w, 100w test.

The more I think about this the more I realize a 10 minute NP that exceeds a 5 minute NP from the same data set is impossible. By definition the 5 minute peak NP in a given data set is a peak, if there was a 10 minute set that had a higher average (fourth power average) then the 5 minute peak would also be in that subset of the data and either exceed or equal the 10 minute average. It's not a physiology question it's a math question based on the definition of block averaging and what constitutes a peak.

It's pretty simple to build a spreadsheet with the first column representing 30 second power averages and the second representing those values to the fourth power. Run a short and long sliding block average through that second column to find the short and long peak averages. For any arbitrary data set there can't be a long peak that exceeds the short peak or by definition you didn't find the short peak.

Play with a spreadsheet like that for a while and you'll see you can't actually get a 10 minute NP that exceeds the 5 minute NP within the same workout.

-Dave

#### GIH

##### New Member
Just so ya'll know how much of a nerd I am. The arguments above suffice to show that it is impossible to have a 10 min NP higher than a 5 minute NP. But for those who want a very mathematical argument, here goes. Let p(t) be the power produced at time t. Let N(t) be the average of p(t) over the interval (t-15,t+15)seconds. Then NP is defined as the (sum (30*N(i*30)^4)/T)^(1/4) where T is the total number of seconds, and i ranges from 1 to T/30. I assume that this is the way its done. This is a norm on the space of bounded continuous functions on the interval (0,T). As such it satisfies the triangle inequality. Denote the normalized power of N(t) by NP(N(t)). This means that for any two functions P(t) and Q(t) NP(P(t)+Q(t))<=NP(P(t))+NP(Q(t)). Now divide T into two equal intervals, and define P(t)=N(t) for one of the intervals and P(t)=0 for the other. Likewise define Q(t)=N(t)-P(t). Then NP(N(t))<=NP(Q(t))+NP(P(t)). But P(t) and Q(t) represent an interval of length T/2 with power and an interval of length T/2 without power. So in this special case NPhalf(Q(t))=NP(Q(t))*2 etc. So this means that 2NP(N(t))<=NPhalf(P(t))+NPhalf(Q(t)). This means that either NPhalf(P(t)) or NPHalf(Q(t)) is greater than NP(N(t)).

In terms of our problem NP(N(t)) is the NP10, and NPhalf(Q(t)) and NPhalf(P(t)) are the NP5s of different parts of the ride.

So while in this case I'm correct, in the end you have the upper hand since this is a bicycling forum and I have know more about cycling than me. But if you have any math questions I likely could answer them.

#### frenchyge

##### New Member
daveryanwyoming said:
By definition the 5 minute peak NP in a given data set is a peak, if there was a 10 minute set that had a higher average (fourth power average) then the 5 minute peak would also be in that subset of the data and either exceed or equal the 10 minute average.
It is quite common to have 3min peak NP's that are higher than 2min peak NP's, as in the case of 1min at 400w, 1min at 100w, and 1min at 400w. The fact that the 2min peak is a subset of the 3min peak doesn't necessarily mean that it must be higher.

In this case, the problem is that 10min is twice as long as 5min (thus there's no overlap between 5min subsets). I hadn't realized that that would make the difference and last night I was about to agree with you and GIH that it wasn't possible for that specific combination, until Alex saved my bacon by running it through his spreadsheet.

GIH said:
Just so ya'll know how much of a nerd I am.
Join the club. Arguing about the math of cycling is just one of many ways to pass that annoying 'recovery' time. GIH said:
Let p(t) be the power produced at time t. Let N(t) be the average of p(t) over the interval (t-15,t+15)seconds. Then NP is defined as the (sum (30*N(i*30)^4)/T)^(1/4) where T is the total number of seconds, and i ranges from 1 to T/30. I assume that this is the way its done.
The 30sec smoothing in the WKO+ software actually looks at the average power from t to t+30, instead of t-15 to t+15. I'm not positive that is how it calculates NP, but I suspect that's why Alex's spreadsheet supported my otherwise erroneous assertion.

It is possible to have NP's for longer durations that are higher than the peak NP for periods contained therein, however. #### acoggan

##### Member
frenchyge said:
It is quite common to have 3min peak NP's that are higher than 2min peak NP's

Not in WKO+. #### acoggan

##### Member
frenchyge said:
The 30sec smoothing in the WKO+ software actually looks at the average power from t to t+30, instead of t-15 to t+15. I'm not positive that is how it calculates NP

It does, simply because there is no other alternative, i.e., you can't calculate a 30 s rolling average until you have at least 30 s worth of data.

#### frenchyge

##### New Member
acoggan said:
Not in WKO+. LOL. Okay, lets say 6min NPs that are higher than 5min peak NPs. #### daveryanwyoming

##### Well-Known Member
frenchyge said:
.....I was about to agree with you and GIH that it wasn't possible for that specific combination, until Alex saved my bacon by running it through his spreadsheet.....
I ran it through a NP spreadsheet as well and got very different results than Alex. Take a look at the attached spreadsheet. NP aside I have a hard time understanding how you can get different AP for the 5 and 10 minute segments given the 30 second numbers (350, 100) and the durations.

-Dave

#### frenchyge

##### New Member
I'm guessing Alex's computes the rolling average power on a second-by-second basis, which makes a difference (and is more correct).

#### GIH

##### New Member
Even if 1 second intervals of the 30 second rolling power are used I still certain that the max 5min NP will always be higher than the max 10minute NP. The same argument goes through. That said, the 5 minute 6 minute case works just as you described. I'm guessing there is a slight problem witht the way the spreadsheet deals with the initial points and the final points or something like that.

#### daveryanwyoming

##### Well-Known Member
frenchyge said:
I'm guessing Alex's computes the rolling average power on a second-by-second basis, which makes a difference (and is more correct).
Yeah, I suspect that the difference (four watts if I remember Alex's post correctly) is due to the rolling averaging and some assumptions about the starting conditions. If so it's more of an artifact of the NP definition than anything else but it would explain how you can get a higher average for a longer duration out of the same data set.

None of this discussion makes us faster, but it's fun nonetheless -Dave

#### jetnjeff

##### New Member
Couldn't be as simple as the first 5 minutes were at a steadier Iso-power and the second 5 minutes were more sprint and recover?

Thus increasing the NP for the 10 minutes but droping the AP, due to the rests?

Maybe being generally good at math, but now where near some in this forum lets me see the forest for the trees or maybe I am just seeing green 