Harmonic regression

Gist

You got some seasonal curves (aka 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.