The final stage in data preparation (after aggregating the data) is to expand the catch data to a distribution of weights using a length-weight relationship, then finally to scale the weights to match total landings.
Length-weight data is obtained from SVDBS (other data sources can be
substituted if necessary). A length-weight relationship is fit to the
data using fit_length_weight (as an alternative parameter
values from database svdbs.LENGTH_WEIGHT_COEFFICIENTS can be used, see
survdat::get_length_weight.
The relationship between the weight and length of fish i is
modelled as,
\[ W_i = \alpha L_i ^ \beta e^{z_i}\]
where \(\alpha\) and \(\beta\) are intercept and slope parameters (on log scale) and \(z_i \sim N(0,\sigma^2)\)
This model can be extended to :
\[ W_{ji} = \alpha L_{ji} ^ {\beta_j}
e^{z_i} \] for season j, j= 1, …, 4 (spring, summer, fall,
winter) or sex j, j = 1, …, 3 (0, 1, 2 categories) or extended further
to include season:sex combinations. These models are all nested and can
therefore be tested using standard statistical methods. See fit_length_weight
for fitting details
The fitted length-weight relationship is applied to the length
distributions and the aggregated landings are split accordingly using expand_landings_to_lengths.
All unclassified landings are then expanded using a similar method.
For each unique category (YEAR, QTR, NEGEAR, MARKET_CODE) we have both total landings (metric tons) and a sample of fish lengths for this category. The sampled fish lengths are converted to mean weights using the length-weight relationship. The resulting sample fish weights are mean weights (metric tons). The weight distribution of sampled fish is then scaled to the weight of total landings. This scaling assumes that the landed (commercial) fish have the same length distribution as the sampled fish.
For each unique category we define
\[expansion \; factor = \frac{total \; landings}{\sum sampled \; fish \; weigths} \] The expanded weights of each unique category are then scaled by this expansion factor to equal the total landings of the category.
\[expanded \; sample\; weights \; = expansion \; factor \;*\; sampled \; fish \; weights\]
The result is a distribution of weights based on lengths for each unique category.
The unclassified landings for a specific (YEAR, QTR, NEGEAR) are assumed to be a mixture of fish lengths similar to that in the observed MARKET_CODEs. Therefore the length distribution for the unclassified landings are assumed to have the same distribution as the length distributions of the landed fish in all of the known market codes combined. The sampled fish lengths (from the known market codes) are converted to weights using the length-weight relationship.
For each unique category we define
\[expansion \; factor = \frac{landings \; of \; unclassifieds}{\sum landings \; of \; known \; market \; codes } \]
leading to the expanded weights of the unclassified sample
\[expanded \; unclassified\; weights \; = expansion \; factor \;*\; fish \; weights \; of \; known \; market \; codes\]