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We should put some examples of this in the documentation.
In short, a higher order means we have higher frequency terms and so will be able to fit more quickly-changing and complex seasonality patterns.
This figure from Wikipedia shows a Fourier series approximation to a square wave, with 1, 2, 3, and 4 components: https://en.wikipedia.org/wiki/Fourier_series#/media/File:Fourier_Series.svg . You can see that as the order is increased, the Fourier series is able to better represent the quick change of the square wave. The downside of using a very high order is just that it is more terms in the model and so overfitting becomes a risk.
The default of 10 typically is appropriate for seasonality on the scale of a year. If it looks like your seasonality effect changes very quickly and the Prophet seaosnality estimate is lagging and not able to capture it all, then try increasing the order.
Thanks for making the package!
As a beginner, i'm not sure what the fourier_order parameter can help with. Could someone shed some light on it in the docs?
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