Harmonic regression

Gist

You got some seasonal curves (aka Disease- RSV), then harmonic regression could be one method. It’s also called dynamic harmonic regression.

Definition

We know from Fourier Transformation that any curve can be explained by a sum of sine and cosine waves. Therefore, the harmonic regression for the dependent variable $Y_{t}$ at time $t$ is:

Y_{t} = β_{0} + \sum_{j = 1}^{k} α_{j} c o s (f_{j} t) + \sum_{j = 1}^{k} γ_{j} s i n (f_{j} t) + ϵ_{t} .

And we can code this in R as:

lm(y ~ sin(2*pi*t/P) + cos(2*pi*t/P))

Advantages

It allows any length seasonality
For data with more than one seasonal period, Fourier terms of different frequencies can be included;
The smoothness of the seasonal pattern can be controlled by K, the number of Fourier sin and cos pairs – the seasonal pattern is smoother for smaller values of K;
The short-term dynamics are easily handled with a simple ARMA error.