Seasonal component
The seasonal component provided in JD+ is based on Proietti[2000]
We consider that it is composed of latent variables corresponding to each period, which follow a multivariate random walk. We also impose that the sum of the latent variables is 0 at each point time.
For a periodicity of $s$, we define the state space form of the seasonal component as follows (West-Harrison form):
State vector
We consider first the state vector $\tilde \gamma_t$ such that $\tilde \gamma_{it}$ is the (unobserved) seasonal component of season $i$ observed at time $t$.
Using that definition would imply a time varying measurement error. To avoid it, we use the state vector $\gamma_t$ which is a rotation of $\tilde \gamma_t$ such that the first component corresponds to the current period. More precisely, $\gamma_{it} =\tilde\gamma_{\left( \left(t-i\right) \mod s \right), t}$.
Moreover, the constraint that the sum of the latent variables is 0 allows us to consider only $s-1$ periods (the missing one being the opposite of the sum of all the other periods).
Dynamics
\[T_t = \begin{pmatrix} 0 & 1 & \cdots & 0 \\ \vdots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & 1 \\ -1 & -1 & \cdots & -1\end{pmatrix}\]The covariance matrix of the innovations cantake different form. Using the terminology of Proietti, we consider:
| Model | V | S |
|---|---|---|
| Dummy | $\begin{pmatrix} 0 & \cdots & 0 \ \vdots & & \vdots\ 0 &\cdots &\sigma_{seas}^2\end{pmatrix}$ | $\begin{pmatrix}0 \ \vdots \ \sigma_{seas}\end{pmatrix}$ |
| Crude | $\sigma_{seas}^2\begin{pmatrix} 1 & \cdots & 1 \ \vdots & & \vdots\ 1 &\cdots & 1\end{pmatrix}$ | $\sigma_{seas}\begin{pmatrix}1 \ \vdots \ 1\end{pmatrix}$ |
| HarrisonStevens | ||
| Trigonometric |
Measurement
\[Z_t = \begin{pmatrix} 1 & 0 & \cdots & 0\end{pmatrix}\]Bibliography
Proietti T. (2000), “Comparing seasonal components for structural time series models”, International Journal of Forecasting, 16, 2, 247-260.