8.1 Overview

A multivariate autoregressive state-space (MARSS) model with covariate effects in both the process and observation components is written as: \[\begin{equation} \begin{gathered} \mathbf{x}_t = \mathbf{B}_t\mathbf{x}_{t-1} + \mathbf{u}_t + \mathbf{C}_t\mathbf{c}_t + \mathbf{w}_t, \text{ where } \mathbf{w}_t \sim \text{MVN}(0,\mathbf{Q}_t) \\ \mathbf{y}_t = \mathbf{Z}_t\mathbf{x}_t + \mathbf{a}_t + \mathbf{D}_t\mathbf{d}_t + \mathbf{v}_t, \text{ where } \mathbf{v}_t \sim \text{MVN}(0,\mathbf{R}_t) \end{gathered} \tag{8.1} \end{equation}\] where \(\mathbf{c}_t\) is the \(p \times 1\) vector of covariates (e.g., temperature, rainfall) which affect the states and \(\mathbf{d}_t\) is a \(q \times 1\) vector of covariates (potentially the same as \(\mathbf{c}_t\)), which affect the observations. \(\mathbf{C}_t\) is an \(m \times p\) matrix of coefficients relating the effects of \(\mathbf{c}_t\) to the \(m \times 1\) state vector \(\mathbf{x}_t\), and \(\mathbf{D}_t\) is an \(n \times q\) matrix of coefficients relating the effects of \(\mathbf{d}_t\) to the \(n \times 1\) observation vector \(\mathbf{y}_t\).

With the MARSS() function, one can fit this model by passing in model$c and/or model$d in the model argument as a \(p \times T\) or \(q \times T\) matrix, respectively. The form for \(\mathbf{C}_t\) and \(\mathbf{D}_t\) is similarly specified by passing in model$C and/or model$D. \(\mathbf{C}\) and \(\mathbf{D}\) are matrices and are specified as 2-dimensional matrices as you would other parameter matrices.