9.11 Problems

Use the information and data in the previous section to answer the following questions. Note that if any model is not converging, then you will need to increase the maxit parameter in the control argument/list that gets passed to MARSS(). For example, you might try control=list(maxit=2000).

1. Begin by fitting a reduced form of Equation (9.33) that includes only a time-varying level ($$\alpha_t$$) and observation error ($$v_t$$). That is,

\begin{align*} \text{log}(R_t) &= \alpha_t + \text{log}(S_t) + v_t \\ \text{log}(R_t/S_t) &= \alpha_t + v_t \end{align*}

This model assumes no density-dependent survival in that the number of recruits is an ascending function of spawners. Plot the ts of $$\alpha_t$$ and note the AICc for this model. Also plot appropriate model diagnostics. residuals() will return the innovations residuals for your fits.

1. Fit the full model specified by Equation (9.33). For this model, obtain the time series of $$\alpha_t$$, which is an estimate of the stock productivity in the absence of density-dependent effects. How do these estimates of productivity compare to those from the previous question? Plot the ts of $$\alpha_t$$ and note the AICc for this model. Also plot appropriate model diagnostics. ($$Hint$$: If you don’t want a parameter to vary with time, what does that say about its process variance?)

2. Fit the model specified by Equation (9.34) with the summer PDO index as the covariate (pdo_summer_t2). What is the mean level of productivity? Plot the ts of $$\delta_t$$ and note the AICc for this model. Also plot appropriate model diagnostics.

3. Fit the model specified by Equation (9.34) with the winter PDO index as the covariate (pdo_winter_t2). What is the mean level of productivity? Plot the ts of $$\delta_t$$ and note the AICc for this model. Also plot appropriate model diagnostics.

4. Based on AICc, which of the models above is the most parsimonious? Is it well behaved ($$i.e.$$, are the model assumptions met)? Plot the model forecasts for the best model. Is this a good forecast model?