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I have binned energy production data (in Wh) in 5-minute intervals. This means that, at the timestamp of t + 5 minutes, the value represents the amount of energy generated in the interval from t to t + 5 minutes. Here is a generated sample:

enter image description here

So between time 0/60 and time 5/60, 1 Wh was generated; between time 5/60 and time 10/60, 4 Wh were generated; etc.

I would like to estimate the continuous process that underlies these discrete values. To that end, I first generate the associated step-function for power (in W).

enter image description here

My goal is to have a curve such as the following to estimate the continuous process. This will allow me to integrate across any interval and have an estimate of energy generation at a finer grain.

enter image description here

This is fairly simple just using splines. However, my constraint is that the integrals of the continuous function must match the integrals of the step-function at the discrete intervals in which the raw data are provided. So, $\int_\frac{0}{60}^\frac{5}{60} f(t) dt = 1$, $\int_\frac{5}{60}^\frac{10}{60} f(t) dt = 4$, $\int_\frac{10}{60}^\frac{15}{60} f(t) dt = 2$, etc.

Does anyone know of a way to do this? I'm fine hard-coding the math, but if there are packages in either Python or R that can do this, that would be great.

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