structural time series models

The structural time series model is a linear gaussian ssm with a specific structure:

• ${\mathbf{z}}_t$ is a composition of states of all latent components, ${\mathbf{z}}_t=[c_{1,t},c_{2,t},...]$, where $c_{i,t}$ is the state of latent component $c_i$ at time step $t$.
• Observation noise is the same for all ${\mathbf{y}}_t$, so ${\mathbf{R}}_t\to {\sigma }_t$.

$$\begin{array}{rl}{\mathbf{z}}_{t+1}& ={\mathbf{F}}_t{\mathbf{z}}_t+{\mathbf{q}}_t,{\mathbf{q}}_t\sim \mathcal{N}(0,{\mathbf{Q}}_t)\\ {\mathbf{y}}_t& ={\mathbf{H}}_t{\mathbf{z}}_t+{\mathbf{u}}_t+{ϵ}_t,{ϵ}_t\sim \mathcal{N}(0,{\sigma }_t^2)\end{array}$$

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{z}_{t} = \begin{pmatrix} \text{global AR effect}_{t} \\ \text{global seasonal components}_{t} ... \\ \text{local AR effect site }1_{t} \\ \text{local seasonal components, site 1}_{t} ... \\ \text{local AR effect site }2_{t} \\ \text{local seasonal components, site 2}_{t} \\ \vdots \\ \text{local AR effect site }N_{t} \\ \text{local seasonal components, site N}_{t} \\ \end{pmatrix} $$with length $(1 + N_{\text{seasonal}}) + (1 + N_{\text{seasonal}}) \times N_{\text{sites}} = (1 + N_{\text{seasonal}})(1 + N_{\text{sites}})$. We can then use $\mathbf{H}_{t}$ to pick out the relevant components. For example, consider three sites ($N_\text{obs} = 3$) with three seasonal components (dummies here, but could be Fourier terms). Picking a timestep where the second seasonality component is active (transpose for readability):

\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}

- First row is all 1 as all three sites have the global autoregressive term - Second row is all 1 for the second component of global seasonality (first term is reference for identifiability) - R3: all 0 as third component of seasonality is not active on this timestep - R4: AR term for site 1 only - R5: seasonality component 2 for site 1 only - R6: seasonality component 3 is not active on this timestep - R7: AR term for site 2 only - R8: seasonality component 2 for site 2 only - R9: seasonality component 3 is not active on this timestep - R10: AR term for site 3 only - R11: seasonality component 3 for site 2 only - R12: seasonality component 3 is not active on this timestep Further modelling assumptions can be made: - $\mathbf{Q}_t \rightarrow \sigma_q$. Diagonal and static dynamics covariance, so no covariance in the autoregressive effects. - $\mathbf{F}_{t} = \mathbf{F} = \text{diag}(1, ..., 1, \rho_1, ..., \rho_N)$. Static diagonal dynamics matrix (vector), with perfect correlation for the global effects and measurable correlation for the local effects, and no cross-correlation. Site-specific covariates, such as holidays, can be implemented as $\mathbf{u}_t = \sum_{i} \mathbf{X}_t \odot \mathbf{\beta}$, where both the design matrix (of zeros and ones) and the covariates have shape $N_{\text{sites}} \times N_{\text{covariates}}$.