What does the Centroid of area under Rate vs Time plot represent?

[geetar_king]

What does the "Centroid" of area under Rate vs Time plot represent?

Does the 'centroid' of the area under a rate vs time plot represent anything?

I have a bunch of rate vs time plots and was trying to think of a way to compare them, other than just cumulative volume which is area under the rate vs time plot. Would centroid somehow weigh in on the rate?

Wondering if this represents anything physical...

Cheers

 

[haruspex]

I think the rate co-ordinate of the centroid will be the mean square of the rate divided by the mean rate. So it gives a measure of variance. Nothing springs to mind for the time co-ordinate.

 

[geetar_king]

Thanks haruspex, if the rate coordinate of the centroid represents the variance, then I could use that for comparison.

I was hoping to find something for comparison to weigh in on the value of the rate. If higher rate = better, variance won't help me in comparison. I don't want average rate because you lose detail of amount of time spent at higher vs lower rate

I wonder if area moment of inertia of the centroid would give me this? or some sort of weighted average maybe.

 

[haruspex]

You need to decide what attribute it is you're looking for. You say say it's not the average rate, or the peak rate, or the variance... so what is it?

 

[geetar_king]

I want some attribute to distinguish higher rates in a shorter amount of time versus lower rates for a long amount of time. Eg tall skinny area under curve vs short and wide area.

I'm not sure how to do this without simply looking for shortest time, or highest avg rate.

 

[haruspex]

OK, but why do you not want to use highest average rate?

 

[Redbelly98]

If it's desirable to have a high average rate and a short amount of time, how about

(average rate) × (1/Δt)

$$= \frac{\left( \frac{\int R \ dt}{\Delta t} \right)}{\Delta t} = \frac{\int R \ dt}{(\Delta t)^2}$$

That would mean the same rate applied over a short time is "better" than the same rate applied over a long time. Not sure if that is desirable, without knowing what this is supposed to be used for.