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Newton-Raphson search for fishing mortality

Source: R/model_int.R
calc_F.Rd

Performs a root finding routine to find the index of F that minimizes the difference between observed catch and the value predicted by the Baranov equation.

Usage

calc_F(
  Cobs,
  N,
  sel,
  wt,
  M,
  q_fs,
  delta = 1,
  na = dim(N)[1],
  nr = dim(N)[2],
  ns = dim(N)[3],
  nf = length(Cobs),
  Fmax = 2,
  nitF = 5L,
  trans = c("log", "logit")
)

Arguments

Cobs

Observed catch. Matrix [f, r]

N

Stock abundance at the beginning of the time step. Array [a, r, s]

sel

Selectivity. Array [a, f, s]

wt

Fishery weight at age. Array [a, f, s]

M

Instantaneous natural mortality. Units of per year [a, s]

q_fs

Relative catchability of stock s for fleet f. Defaults to 1 if missing. Matrix [f, s]

delta

Numeric, the duration of time in years corresponding to the observed catch, e.g., 0.25 is a quarterly time step.

na

Integer, number of age classes

nr

Integer, number of regions

ns

Integer, number of stocks

nf

Integer, number of fleets

Fmax

Numeric, the maximum Findex value

nitF

Integer, number of iterations for the Newton-Raphson routine

trans

Whether to perform the search in log or logit space

Value

A list containing:

  • F_afrs Fishing mortality array

  • F_ars Fishing mortality array (summed across fleets)

  • Z_ars Total mortality array

  • F_index Index of fishing mortality. Matrix [f, r]

  • CB_frs Catch (biomass) array

  • CN_afrs Catch (abundance) array

  • VB_afrs Vulnerable biomass at the beginning of the time step. Array

  • penalty Penalty term returned by posfun() when F_index exceeds Fmax

  • fn Difference between predicted and observed catch at the last iteration. Matrix [f, r]

  • gr Gradient of fn with respect to F_index in either log or logit space at the last iteration. Vector by [f, r]

Details

The predicted catch for fleet f in region r is $$ C^{\textrm{pred}}_{f,r} = \sum_s \sum_a v_{a,f,s} q_{f,s} F_{f,r} \dfrac{1 - \exp(-Z_{a,r,s})}{Z_{a,r,s}} N_{a,r,s} w_{a,f,s} $$

The Newton-Raphson routine minimizes \(f(x_{f,r}) = C_{f,r}^{\textrm{pred}} - C_{f,r}^{\textrm{obs}}\).

If trans = "log", \(F_{f,r} = \exp(x_{f,r})\).

If trans = "logit", \(F_{f,r} = F_{\textrm{max}}/(1 + \exp(x_{f,r}))\).

The gradient with respect to \(\vec{x}\) is $$ f'(x_{f,r}) = \sum_s \sum_a v_{a,f,s} q_{f,s} N_{a,r,s} w_{a,f,s} \left(\dfrac{\alpha\gamma}{\beta}\right)' $$

$$ \left(\dfrac{\alpha\gamma}{\beta}\right)' = \dfrac{(\alpha\gamma' + \alpha'\gamma)\beta - \alpha\gamma\beta'}{\beta^2} $$

where

\(\alpha_{f,r} = F_{f,r}\)
\(\beta_{a,r,s} = Z_{a,r,s} = M_{a,s} + \sum_f v_{a,f,s} q_{f,s} F_{f,r}\)
\(\gamma_{a,r,s} = 1 - \exp(-Z_{a,r,s})\)
\(\beta'_{a,f,r,s} = v_{a,f} q_{f,s} \alpha'_{f,r}\)
\(\gamma'_{a,f,r,s} = \exp(-Z_{a,r,s})\beta'_{a,f,r,s}\)

If trans = "log", \(\alpha'_{f,r} = \alpha_{f,r}\).

If trans = "logit", \(\alpha'_{f,r} = F_{\textrm{max}}\exp(-x_{f,r})/(1 + \exp(-x_{f,r}))^2\).

This function solves for \(\vec{x}\) by iterating until \(f(\vec{x})\) approaches zero, where the vector arrow indexes over fleet and region. In iteration \(i+1\): $$\vec{x}_{i+1} = \vec{x}_i - \dfrac{f(\vec{x}_i)}{f'(\vec{x}_i)}$$.

Author

Q. Huynh