Part 2. Short Examples

The section shows a series of short examples for different types of models specified in MARSS structure. See the ATSA book for a wide variety of detailed examples and explanations.

The MARSS model has the form: \[\begin{equation*} \begin{gathered} \mathbf{x}_t = \mathbf{B}_t\mathbf{x}_{t-1} + \mathbf{u}_t + \mathbf{C}_t\mathbf{c}_t + \mathbf{G}_t\mathbf{w}_t, \text{ where } \mathbf{w}_t \sim \,\text{MVN}(0,\mathbf{Q}_t)\\ \mathbf{y}_t = \mathbf{Z}_t\mathbf{x}_t + \mathbf{a}_t + \mathbf{D}_t\mathbf{d}_t + \mathbf{H}_t\mathbf{v}_t, \text{ where } \mathbf{v}_t \sim \,\text{MVN}(0,\mathbf{R}_t)\\ \mathbf{x}_1 \sim \,\text{MVN}(\boldsymbol{\pi},\boldsymbol{\Lambda}) \text{ or } \mathbf{x}_0 \sim \,\text{MVN}(\boldsymbol{\pi},\boldsymbol{\Lambda}) \end{gathered} \tag{5.2} \end{equation*}\]

The \(\mathbf{x}\) on the left are the hidden states. The \(\mathbf{y}\) on the left are the observed data. Missing values are allowed in \(\mathbf{y}\).