The structural time series model is a linear gaussian ssm with a specific structure:
is a composition of states of all latent components, , where is the state of latent component at time step . - Observation noise is the same for all
, so .
example for hierarchical or grouped time series
Consider an example where we are forecasting across multiple time series.
We want a shared (global) autoregressive effect and seasonalities, and then each site to have its own autoregressive effect and seasonalities.
In this case, we can make up the latent components such as
\mathbf{H}^T_t = \begin{pmatrix}
1 & 1 & 1 \
1 & 1 & 1 \
0 & 0 & 0 \
1 & 0 & 0 \
1 & 0 & 0 \
0 & 0 & 0 \
0 & 1 & 0 \
0 & 1 & 0 \
0 & 0 & 0 \
0 & 0 & 1 \
0 & 0 & 1 \
0 & 0 & 0 \
\end{pmatrix}