linear gaussian ssm

When the dynamics model and the observation model of a state space model are both Gaussian, we have a linear gaussian ssm,

The latent dynamics model is written as

$${\mathbf{z}}_{t+1}={\mathbf{F}}_t{\mathbf{z}}_t+{\mathbf{q}}_t,{\mathbf{q}}_t\sim \mathcal{N}(0,{\mathbf{Q}}_t)$$

with the observation model

$${\mathbf{y}}_t={\mathbf{H}}_t{\mathbf{z}}_t+{\mathbf{u}}_t+{\mathbf{r}}_t,{\mathbf{r}}_t\sim \mathcal{N}(0,{\mathbf{R}}_t)$$

variable	variable description	shape
${\mathbf{z}}_t$	state vector	(N_states, 1)
${\mathbf{y}}_t$	observation vector at time $t$	(N_obs, 1)
${\mathbf{F}}_t$	dynamics (transition) matrix	(N_states, N_states)
${\mathbf{Q}}_t$	covariance matrix of dynamics (system) noise	(N_states, N_states)
${\mathbf{H}}_t$	emission (observation) matrix	(N_obs, N_states)
${\mathbf{R}}_t$	covariance function for emission (observation) noise	(N_obs, N_obs)

${\mathbf{H}}_t$ extracts the relevant parts of the state vector ${\mathbf{z}}_t$. ${\mathbf{u}}_t$ are exogenous inputs.

Inference can be performed efficiently using kalman filtering and smoothing. These models have applications in object tracking and structural time series models.

See ssm resources.