11-appendix-e.rmd

# Simulation of Substock- and Year-Specific Maturity Schedules {#appendix-e}

\noindent
In the operating model of the simulation-estimation component of Chapter \@ref(ch4), maturity variability was modeled with more complexity than assumed by the most complex estimation model. This was deliberate and allowed assessment of estimation models that did not completely capture the complexity of the processes used to generate the data. Maturity was simulated to vary on average by substock (_i_._e_., some substocks would tend to mature at younger or earlier ages). Additionally, brood-year specific random maturity schedules were generated for each substock, though were simulated to be highly synchronous among substocks. I was unaware of a way to model such correlated dynamics using the Dirichlet random process, so I generated it using a hierarchical linear modeling approach. Note that some of the notation in this appendix uses symbols with different meanings in the main text and other appendices.

## Single substock, single year example

\noindent
First, define a vector of $n_a - 1$ coefficients:

\[
   \mathbf{X}=
  \left( {\begin{array}{ccc}
   \gamma_{0} & \gamma_1 & \gamma_2
  \end{array} } \right)
\]

\noindent
and a design matrix with rows and columns equal to $n_a - 1$:

\[
   \mathbf{X}=
  \left( {\begin{array}{ccc}
   1 & 0 & 0\\
   1 & 1 & 0\\
   1 & 0 & 1
  \end{array} } \right)
\]

\noindent
Then, combine them into a linear predictor on the logit scale:

$$\text{logit}(\boldsymbol{\psi}) = \mathbf{X}\boldsymbol{\gamma}$$

\noindent
The vector $\boldsymbol{\psi}$ contains the probability of maturity at each age, conditional on not having matured at any previous age for the first $n_a - 1$ possible ages-at-maturation. The elements of $\boldsymbol{\psi}$ can be converted to marginal probabilities of maturing at each age (elements of $\boldsymbol{p}$):

\[
  p_a = \left\{ \begin{array}{ll}
  \psi_a & \mbox{if $a = 1$} \\
  \psi_a(1 - \sum_{a=1}^{a-1} p_a) & \mbox{if 1 < $a$ < $n_a - 1$} \\
  1 - \sum_{a=1}^{a-1} p_a & \mbox{if $a = n_a$ } \\
  \end{array}
  \right. \\
\]

\noindent
These marginal probabilities can be then used to apportion recruitments occurring from brood year $y$ to the various calendar years of observation $t$.

## Extention to multiple substocks and years

\noindent
Alterations were made to the $\gamma_0$ parameter to be specific to each substock and year combination. Stock-level effects were randomly sampled:

$$\varepsilon_{j} \sim \text{N}(0, \sigma_{j}),$$

\noindent
then brood year- and substock-specific random effects were sampled:

$$\varepsilon_{y,1:n_j} \sim \text{MVN}(\varepsilon_{1:n_j}, \Sigma_{y}),$$

\noindent
to obtain the year- and substock-specific parameter:

$$\gamma_{0,y,j} = \gamma_0 + \varepsilon_{y,j}.$$

The following parameter values were used: $\gamma_0$ = -1.4, $\gamma_1$ = 1.4, $\gamma_2$ = 4, and $\sigma_{j}$ = 0.25. The covariance matrix $\Sigma_y$ was constructed with standard deviations for each substock equal to 0.2 and correlation equal to 0.9. These settings resulted in an average vector maturation probabilities of 0.2, 0.4, 0.37, and 0.03 for ages 4, 5, 6, and 7, respectively. An example is shown in Figure \@ref(fig:maturity-sim-fig).

\begin{figure}
  \centering
  \includegraphics{img/Ch4/maturity-sim-fig.jpg}
  \caption{Simulated probability of maturation-at-age for 13 simulated substocks over time. The horizontal dashed line represents maturity without substock or year random effects and the black solid represents the average across substocks. Note the highly correlated patterns in year-specific variability.}
  \label{fig:maturity-sim-fig}
\end{figure}

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