Transcript Modelling stochastic fish stock dynamics using Markov Chain Monte

Modelling stochastic fish stock dynamics
using Markov Chain Monte Carlo
Reporter: Hsu Hsiang-Jung
Introduction
The precautionary approach has become a basis
concept in fish stock management.
The quantification of there uncertainties has
emphasised the need for developing stochastic
assessment approaches.
Estimates of parameters including process
variances and predicted stock number have been
obtained using likelihood Markov Chain Monte
Carlo.
Population dynamics models
Catch-at-age in numbers and effort data for
commercial fleets. (C f ,a , y and e f , y )
Catch-at-age in numbers without effort data
for the remaining part of total international
catches. (residual catches, Cres,a, y )
CPUE by age for research surveys. ( I s ,a , y )
Population dynamics models
N a1, y 1 | N a, y  N a, y exp( Z a, y ) exp( survival asurvival
)(1)
,y
Cres,a, y 
Fres,a, y
C f ,a , y 
e f , y q f ,a
Za, y
Z a, y
(1  e Za , y ) N a, y exp( resres, a, y)(2)
(1  e
Za, y
) N a , y exp( f  f ,a , y )  (3)
Population dynamics models
I s ,a, y  q s ,a N a, y exp( Z a, y
Ts
  s  s ,a , y )(4)
365
N min A, y  SSBy min A exp(  SSBy min A   recruit y min A ) y  min A (5)
N min A,1   start exp( initial )  (6)
N a ,1  N a 1,1 exp(  Z a 1,1 ) exp( initial a 1 ) a  min A (7)
Population dynamics models

2
res

2
process,res
 
2
process, f
 
2
process, s
2
f
2
s

2
sampling,res
(8)

2
sampling, f
(9)

2
sampling, s
(10)
Population dynamics models
“N” denotes the stock number.
“F” the fishing mortality .
“Z” the total mortality.
“  survival,  f ,  res and  s “ the standard deviations for the
survival and fishing processes.
Population dynamics models
“q” the catchability.
“e” the effort.
“T” the day of year when the survey takes place.
“s“ the standardised normal distribution.
Estimation methods
For complex models with strutural relationships between
variables and parameters, such as the stochastic survival
model considered, the so-called single component
Metropolis-Hastings or Gibbs sampling is an MCMC
method especially suitable for simulating the likelihood
function.
The difference between the MLE and this estimator lies in
the MLE being the maximum of the likelihood function
while the new estimator being the mean.
Simulation experiments
The catch observations were simulated in the following way:
1.The model used was the same as described by Equations (1)-(7).
2.The parameters , Θ, used were the values estimated applying data
described in the next section.
3.Fres,a,y and Ff,a,y were calculated, the latter using effort data and
catchabilities, qf,a.
4.NminA,1 was predicted by randomly darwing from the lognormal
distribution,(Equation(6))
5.Na,1 a=2,…,A were predicted by randomly drawing from the lognormal
distribution ,(Equation(7))
Simulation experiments
6. SSB1 was calculated.
7. For y = 2 recruitment NminA,ywas randomly drewn from
Equation(5).
8. For a = 2,…,A Na,y was randomly drawn from Equation(1a) and
SSBy calculated.
9. Steps 7and 8 were repeated as long as y < Y.
10. The catch observations, Cres,a,y, Cf,a,y and Is,a,y, were generated
from the lognormal distributions(Equations(2)-(4)).
Materials and software used
Catch-at and effort data for the Dutch and
English commercial beam trawl fleets.
Catch-at-age data for the combined fleet
without effort data.
Survey indices for the Dutch beam trawl
and the Sole Net Survey.
Materials and software used
The software package, WinBUGS1.4 was
used to simulate the posterior distributions
of the parameters.
Results
Results
Results
Results
2
2
 process
,

, Netherland
process, England and
2
 process
, res
Results
Results
Results of simulation experiments
 res , f and  s
Discussion
Our model of fish stock assessment which
includes stochastic survival and recruitment.
Errors associated with the catch-at-age by fleet
used in stock assessment consist of sampling errpr
and other errors denoted as process errors.
Discussion
The MCMC methodology, in particular the
single component Metropolis-Hastings and
graphical models.
It is easy to implement such complex
model in the WinBUGS program.