The structure of the model

The probabilistic interpretation of the dynamics of the sources in the standard NSSM is

$$p(\mathbf{s}(t)\vert \mathbf{s}(t-1), \boldsymbol{\theta}) = N( \... ...s}(t); \; \mathbf{g}(\mathbf{s}(t-1)), \operatorname{diag}[\exp(\mathbf{v}_m)])$$ (5.49)

where $\mathbf{g}$ is the nonlinear dynamical mapping and $\operatorname{diag}[\exp(\mathbf{v}_m)]$ is the covariance matrix of the zero mean innovation process $\mathbf{i}(t) = \mathbf{s}(t)- \mathbf{g}(\mathbf{s}(t-1))$. The typical linear approach, applied to the nonlinear case, would use a different $\mathbf{g}$ and $\operatorname{diag}[\exp(\mathbf{v}_m)]$ for all the different states of the HMM.

The simplified model uses only one dynamical mapping $\mathbf{g}$ but has an own covariance matrix $\operatorname{diag}[\exp(\mathbf{v}_{M_j})]$ for each HMM state $j$. In addition to this, the innovation process is not assumed to be zero-mean but it has a mean depending on the HMM state. Mathematically this means that

$$p(\mathbf{s}(t)\vert \mathbf{s}(t-1), M_t = j, \boldsymbol{\theta... ...athbf{s}(t-1)) + \mathbf{m}_{M_j}, \operatorname{diag}[\exp(\mathbf{v}_{M_j})])$$ (5.50)

where $M_t$ is the HMM state, $\mathbf{m}_{M_j}$ and $\operatorname{diag}[\exp(\mathbf{v}_{M_j})]$ are, respectively, the mean and the covariance matrix of the innovation process for that state. The prior model of $s(1)$ remains unchanged.

Equation (5.50) summarises the differences between the switching NSSM and its components, as they were presented in Sections 5.1 and 5.2. The HMM ``output'' distribution is the one defined in Equation (5.50), not the data likelihood as in the ``stand-alone'' model. Similarly the model of the continuous hidden states in NSSM is slightly different from the one specified in Equation (5.25).