nonlinear gaussian ssm

Moving from linear gaussian ssm, we relax the assumptions of linear dynamics (${\mathbf{F}}_t{\mathbf{z}}_t\to \mathbf{f}({\mathbf{z}}_t)$) and linear observation models (${\mathbf{H}}_t{\mathbf{z}}_t\to \mathbf{h}({\mathbf{z}}_t)$).

$$\begin{array}{rl}{\mathbf{z}}_{t+1}& =\mathbf{f}({\mathbf{z}}_t,{\mathbf{u}}_t)+{\mathbf{q}}_t,{\mathbf{q}}_t\sim \mathcal{N}(0,{\mathbf{Q}}_t)\\ {\mathbf{y}}_t& =\mathbf{h}({\mathbf{z}}_t,{\mathbf{u}}_t)+{\mathbf{r}}_t,{\mathbf{r}}_t\sim \mathcal{N}(0,{\mathbf{R}}_t)\end{array}$$

This model can be used to track an object performing nonlinear motion or in online learning using ssm for neural networks.