Nonlinear State-Space Models

In nonlinear state-space models, the observation vectors $\mathbf{x}(t)$, $t=1,2,\dots,T$, are assumed to have been generated from unobserved state (or source) vectors $\mathbf{s}(t)$. The model equations are

$$\mathbf{x}(t) = \mathbf{f}(\mathbf{s}(t )) + \mathbf{n}(t)$$ (1)

$$\mathbf{s}(t) = \mathbf{g}(\mathbf{s}(t-1)) + \mathbf{m}(t),$$ (2)

Both the mixing mapping $\mathbf{f}$ and the process mapping $\mathbf{g}$ are nonlinear. The noise model for both mixing and dynamical process is often assumed to be Gaussian

$$p(\mathbf{n}(t)) = \mathcal N\left[\mathbf{n}(t);\mathbf{0};\mathbf{\Sigma}_x\right]$$ (3)

$$p(\mathbf{m}(t)) = \mathcal N\left[\mathbf{m}(t);\mathbf{0};\mathbf{\Sigma}_s\right],$$ (4)

where $\mathbf{\Sigma}_x$ and $\mathbf{\Sigma}_s$ are the noise covariance matrices. In blind source separation, the mappings $\mathbf{f}$ and $\mathbf{g}$ are assumed to be unknown [1] but in this paper we concentrate on the case where they are known.