Controlling seasonality with Fourier order

Seasonality is at the heart of how Prophet works, and Fourier series are how seasonality is modeled. To understand what a Fourier series is, and how the Fourier order relates to it, I'll use an analogy from linear regression.

You may know that increasing the order of a polynomial equation in linear regression will always improve your goodness-of-fit. For example, the simple linear regression equation is , with β1 being the slope of the line and β0 the y-intercept. Increasing the order of your equation to, say, will always improve your fit, at the risk of overfitting and capturing noise. You can always achieve an R2 value of 1 (perfect fit) by arbitrarily increasing the order of your polynomial equation higher and higher. The following figure illustrates how higher-order fits start to become quite unrealistic and overfit, though:

Figure 4.10 – Linear regression with higher-order polynomials

...