The generative model

The Bayesian model follows the standard NSSM defined in Equation (4.13):

$$\begin{split}\mathbf{s}(t+1) &= \mathbf{g}(\mathbf{s}(t)) + \... ...{x}(t)&= \mathbf{f}(\mathbf{s}(t)) + \mathbf{n}(t). \end{split}$$ (5.18)

The basic structure of the model can be seen in Figure 5.1.

Figure 5.1: The nonlinear state-space model.

The nonlinear functions $\mathbf{f}$ and $\mathbf{g}$ of the NSSM are modelled by MLP networks. The networks have one hidden layer and can thus be written as in Equation (4.15):

$$\mathbf{f}(\mathbf{s}(t)) = \mathbf{B}\tanh [ \mathbf{A}\mathbf{s}(t)+ \mathbf{a} ] + \mathbf{b}$$ (5.19)

$$\mathbf{g}(\mathbf{s}(t)) = \mathbf{s}(t)+ \mathbf{D}\tanh [ \mathbf{C}\mathbf{s}(t)+ \mathbf{c} ] + \mathbf{d}$$ (5.20)

where $\mathbf{A}, \mathbf{B}, \mathbf{C}$ and $\mathbf{D}$ are the weight matrices of the networks and $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and $\mathbf{d}$ are the bias vectors.

Because the data are assumed to be generated by a continuous dynamical system, it is reasonable to assume that the values of the hidden states $\mathbf{s}(t)$ do not change very much from one time index to the next one. Therefore the MLP network representing the function $\mathbf{g}$ is used to model only the change.