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.