9.6 Analysis of salmon survival

Let’s see an example of a DLM used to analyze real data from the literature. Scheuerell and Williams (2005) used a DLM to examine the relationship between marine survival of Chinook salmon and an index of ocean upwelling strength along the west coast of the USA. Upwelling brings cool, nutrient-rich waters from the deep ocean to shallower coastal areas. Scheuerell & Williams hypothesized that stronger upwelling in April should create better growing conditions for phytoplankton, which would then translate into more zooplankton. In turn, juvenile salmon (“smolts”) entering the ocean in May and June should find better foraging opportunities. Thus, for smolts entering the ocean in year \(t\), \[\begin{equation} \tag{9.19} survival_t = \alpha_t + \beta_t f_t + v_t \text{ with } v_{t}\sim\,\text{N}(0,r), \end{equation}\] and \(f_t\) is the coastal upwelling index (cubic meters of seawater per second per 100 m of coastline) for the month of April in year \(t\).

Both the intercept and slope are time varying, so

\[\begin{align} \tag{9.20} \alpha_t &= \alpha_{t-1} + w_{\alpha,t} \text{ with } w_{\alpha,t} \sim \,\text{N}(0,q_{\alpha})\\ \beta_t &= \beta_{t-1} + w_{\beta,t} \text{ with } w_{\beta,t} \sim \,\text{N}(0,q_{\beta}). \end{align}\]

If we define \(\boldsymbol{\theta}_t = \begin{bmatrix}\alpha \\ \beta\end{bmatrix}_t\), \(\mathbf{G}_t = \mathbf{I}\), \(\mathbf{w}_t = \begin{bmatrix} w_\alpha \\ w_\beta\end{bmatrix}_t\), and \(\mathbf{Q} = \begin{bmatrix}q_\alpha& 0 \\ 0&q_\beta\end{bmatrix}\), we get Equation (9.3). If we define \(y_t = survival_t\) and \(\mathbf{F}_t = \begin{bmatrix}1 \\ f_t\end{bmatrix}\), we can write out the full DLM as a state-space model with the following form:

\[\begin{equation} \tag{9.21} y_t = \mathbf{F}_t^{\top}\boldsymbol{\theta}_t + v_t \text{ with } v_t\sim\,\text{N}(0,r)\\ \boldsymbol{\theta}_t = \mathbf{G}_t\boldsymbol{\theta}_{t-1} + \mathbf{w}_t \text{ with } \mathbf{w}_t \sim \,\text{MVN}(\mathbf{0},\mathbf{Q})\\ \boldsymbol{\theta}_0 = \boldsymbol{\pi}_0. \end{equation}\]

Equation (9.21) is equivalent to our standard MARSS model:

\[\begin{equation} \tag{9.22} \mathbf{y}_t = \mathbf{Z}_t\mathbf{x}_t + \mathbf{a} + \mathbf{v}_t \text{ with } \mathbf{v}_t \sim \,\text{MVN}(0,\mathbf{R}_t)\\ \mathbf{x}_t = \mathbf{B}_t\mathbf{x}_{t-1} + \mathbf{u} + \mathbf{w}_t \text{ with } \mathbf{w}_t \sim \,\text{MVN}(0,\mathbf{Q}_t)\\ \mathbf{x}_0 = \boldsymbol{\pi} \end{equation}\]

where \(\mathbf{x}_t = \boldsymbol{\theta}_t\), \(\mathbf{B}_t = \mathbf{G}_t\), \(\mathbf{y}_t = y_t\) (i.e., \(\mathbf{y}_t\) is 1 \(\times\) 1), \(\mathbf{Z}_t = \mathbf{F}_t^{\top}\), \(\mathbf{a} = \mathbf{u} = \mathbf{0}\), and \(\mathbf{R}_t = r\) (i.e., \(\mathbf{R}_t\) is 1 \(\times\) 1).