Simple closed-form solution for alpha in detect_clearsky

[agodbehere]

The algorithm in clearsky.detect_clearsky optimizes for a scaling parameter alpha by finding a value that minimizes the root mean-squared error (RMSE) between measurements and a scaled clear-sky model. Currently, the implementation computes the RMSE and uses scipy.optimize to iteratively minimize the function. However, a simple closed-form solution exists, bypassing the need to call scipy.optimize.

The objective function is $f(\alpha) = \sqrt{\frac{\sum_i (G_i - \alpha C_i)^2}{n}}$, where $G$ represents measurements of GHI and $C$ represents the clear-sky model.

Since $f(\alpha)$ is nonnegative, $\arg\min_\alpha f(\alpha) = \arg\min_\alpha f^2(\alpha)$. The value $n$ is a constant and similarly does not affect the solution.

Thus, we end up with a modified objective function $\hat{f}(\alpha) = \sum_i (G_i-\alpha C_i)^2$. Solving for $\frac{d\hat{f}}{d\alpha} = 0$ yields the optimal value: $\hat{\alpha} = \frac{\sum_i G_i^2}{\sum_i G_iC_i}$. The result is nonnegative, so the constraint $\alpha > 0$ is satisfied implicitly.

[cwhanse]

Thanks for pointing this out. The implementation went through several iterations and this slipped by.

Could you check that formula for the minimum? I get $\sum GC / \sum C^2$

[agodbehere]

You are correct!

In "Identification of Periods of Clear Sky Irradiance in Time Series of GHI Measurements" (Reno & Hansen), they have $\alpha$ scaling $G$ (the measurements) instead of $C$ (the model), while the code scales $C$ instead. I mixed things up when converting to the code's perspective.

[cwhanse]

I had forgotten about that typo in the 2016 paper. The intent is to scale the clear-sky model by $\alpha$, not to scale the GHI measurements.

Are you interested to submit a pull request to replace the call to scipy.optimize?

[agodbehere]

I would be happy to do so. Thank you!

…

On Wed, Aug 21, 2024 at 09:49 Cliff Hansen ***@***.***> wrote: I had forgotten about that typo in the 1996 paper. The intent is to scale the clear-sky model by \alpha, not to scale the GHI measurements. Are you interested to submit a pull request to replace the call to scipy.optimize? — Reply to this email directly, view it on GitHub <#2171 (reply in thread)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AANUIXR5H2XOPWNNTIZGYELZSSLE3AVCNFSM6AAAAABM2QO4ZSVHI2DSMVQWIX3LMV43URDJONRXK43TNFXW4Q3PNVWWK3TUHMYTANBQHAYDAMI> . You are receiving this because you authored the thread.Message ID: ***@***.***>