To be honest, this is what I'm getting at:

People are saying "NO" there's no way, and "YES" they're related. I'm just here to say the truth is somewhere in the middle. They're related, but there's a good deal of noise in the signal, and whether it's too much or not depends on the level of accuracy that's needed.

That reminder is the only meaningful contribution I have on this topic.

The truth is not in the middle; HR and power are related. You keep talking about accuracy, presumably in the reporting of watts produced, but we don't know if that's what the OP is looking to do. They merely said they were studying the relationship of HR to Power. Whether the PowerCal is a good alternative to a direct force powermeter, which is what I think you're on about, is a matter of user requirements (i.e. required accuracy), but why put the cart before the horse?

I gave up using a HRM years ago so I had to use a power and hr data file from a friend.

r(hr and watts): 0.72
r(30-second avg hr and 30-second avg watts): 0.83

That's interesting ...

Just for giggles, can you also derive the correlation between power and speed, preferably for a ride with varied terrain?

I accept this.

I don't see how Powercal can be much better than the Strava Guesstimator, as for instance, 130bpm up a hill and 130bpm down a hill will have drastically different power numbers. Say I were to coast (not pedaling) down a hill for 10 miles with a HR sitting around 110bpm, how would Powercal differentiate that from riding with my wife, where I will average 100-110bpm and still do 25-30kJ per mile worth of work?

Go back and see post #18; it was a reply to you, and answers that question.

Sure I can -- but it's not very interesting unless you expand the model. For example, for a ride I did this weekend that climbed 1500 feet and then descended back home the zero order correlation between power and speed was -0.3 because I was going slow uphill at high power (or what passes for high power for me) and was zooming back downhill at high speed and very low power.

If you expand the model to include total mass, air density, air speed, and maybe something for drive train losses, the model fit becomes much more interesting.

Just a hunch, but using a 30 sec moving average of power, but a CURRENT value for HR, as it is effectively already a moving average, one implemented very efficiently by biology, will probably get an even higher r.

And then, I'd try t from 20 to 60 seconds in 1 second intervals to see which lag factor was most effective (at least for that individual).

(and I've been meaning to do this for a while, but the stupid data files are all proprietary formats rather than the nice old CSV files I used to get from my power tap, and I hate XML with an absolute passion)

Actually, I checked that already. The r is 0.82.

I often see folks compare HR between athletes and then power between athletes and say see how much more reliable power is... Paraphrased of course. But yea.
On the edge numbers I see smoother lines for HR than power. I have not looked into sampling rates, but just about any Strava TT for a leader that post both power and HR - the HR plot is smoother.

I thought HR and power were fairly closely related and was one of the reference points for determining your fitness?
It obviously varies between individuals but if you maintain 200w at 140bpm and then 6 months later maintain 200w at 135bpm is this not a fair indication of improvement?
Also as the ride progresses the HR/Power becomes uncoupled and HR climbs for the same power. The longer this is delayed must be another indicator of improved fitness/stamina?

Although SQL Server isn't ideal for the type of mathematical analysis you're talking about, it makes shredding XML a breeze, and for your purposes, is a free download from Microsoft.

I've used it to do rudimentary analysis and updating of TCX/GPX files.

I just realized there was an error in my calculation, so I re-did the entire thing again. Here's the summary plot, showing his data at 1, 5, 10, 15, etc. second averages. I've suppressed the scales on the x and y-axes to hide his actual values but the x-axis is power and the y-axis is hr. The correlation coeff is in the lower right of each panel.

Sorry for my error; thanks for making me think about it again.

Isn't that a bit obvious? Output power has a large dynamic range and can change very quickly, e.g. it can go from maximum to zero in one pedal stroke. Heart rate, however, changes relatively slowly over a more limited range.

*I guess it's possible for your HR to go from maximum to zero in one pedal stroke, but you better hope it doesn't!

The HR plot is way less smooth in the mid ranges.

Geez, what are you guys on about this for? What is the point?

I don't know about anyone else but I hurt my toe, can't ride for a couple of days, and classes don't begin until Monday.

I live in Seattle. Forecast for January is 41 F and heavy rain, with occasional minor flooding.

But I still love bikes, and enjoy talking about cycling.

Caching! r=.96 @55sec moving average.

If that's not correlated, nothing is.