## Composition data

### Catch at age

The catch (in numbers) $C$ at age for fleet $f$ is $C_{y,a,f} = \dfrac{v_{y,a,f} F_{y,f}}{Z_{y,a}} N_{y,a} (1 - \exp[-Z_{y,a}]).$

If the model is conditioned on catch, then $F_{y,f}$ can be estimated as parameters or solved iteratively to match the observed catch. If the model is conditioned on effort, then $F_{y,f} = q_f E_{y,f},$ where $E_{y,f}$ is the observed effort and $q_f$ is the scaling coefficient.

The proportion of the catch-at-age is $p_{y,a,f} = \dfrac{C_{y,a,f}}{\sum_a C_{y,a,f}}.$

### Catch at length

The catch at length is calculated assuming a normally distributed length-at-age $P(\ell,a)$, where $C_{y,\ell,f} = \sum_a C_{y,a,f} P(\ell|a)$ and

$P(\ell|a) = \begin{cases} \phi(L'_{\ell+1}) & \ell = 1\\ \phi(L'_{\ell+1}) - \phi(L'_\ell) & \ell = 2, \ldots, L - 1,\\ 1 -\phi(L'_\ell) & \ell = L \end{cases}$ with $L'_{\ell}$ as the length at the lower boundary of length bin $\ell$ and $\phi(L'_{\ell})$ as the cumulative distribution function of a normal variable with mean $\tilde{L}_{y,a}$ (the expected mean length at age $a$) and standard deviation $\tilde{L}_{y,a} \times CV^L$ ($CV^L$ is the coefficient of variation in mean length at age).

The proportion of the catch-at-length is $p_{y,\ell,f} = \dfrac{C_{y,\ell,f}}{\sum_{\ell}C_{y,\ell,f}}.$