The stochastic noise in the observations is in this formulation
The log-likelihood function is now
To obtain the estimation equations, the log-likelihood is differentiated with respect to the parameters θ. The maximum is then defined at the point where these equations equal zero:
The parameters are catch rate parameter log(qNl) and the selection parameters. There are as many simultaneous estimation equations as there are parameters. We can solve the catch rate term directly, conditional on the selectivity estimate:
We can use this equation to substitute for log(qNl) in the other estimation equations. The estimation equations for the selection parameters can only be specified when the selection model has been decided upon. There are numerous models available. Here we use the log-normal (a two parameter model is used to illustrate the procedure). The selectivity for each length group is given by:
Differentiating with respect to each of the parameters we get:
We substitute these into the estimation equations:
These two estimation equations are generally not used since it is just as easy to maximise the log-likelihood function directly. It is possible to include the population (qN) in the parameters to be estimated directly, but this often leads to slow convergence because parameters are correlated. This may be solved by using different initial estimates.
It is possible to write a spreadsheet program that solved the two estimation equations. A simple way to do this would be to convert the solution of the two coupled equations into a single minimising problem. If the first equation is F(log k, ψ) = 0 and the second G(log k, ψ) = 0 then we can add the squared functions (avoiding negative values), so F2+G2 = min with respect to the parameter pair (log k, ψ). The minimum of the sum of squared values will be zero, with each of the terms equal to zero. However, in this case the approach has no advantage over maximising the likelihood directly.
The stochastic noise in the observations is now
The log-likelihood function now becomes.
The estimation equation becomes for the parameter θ in the selection model
and for the variance in the catch process σ2
with the solution
where n is the number of observations, i.e. number of length groups times number of mesh sizes. The θ parameters are log(q Nl) and the selection parameters φ.
The power transformation is a tool that approximately accounts for the error structure of the data (Elliott 1983). This approach is chosen in the regression framework since it allows a uniform way of dealing with a series of different error structures ranging from normal over Poisson to contagious error distributions. The cause of these error structures are both the stochastic nature of the experiment but also, and probably more importantly, the encounter process of fish with the gear.
The weighted least-squares fitting that is applied here is based on minimising
The power transformation of the observation is based on the standardised observation where c is the observation, ξ and σ are its mean and standard deviation respectively. Experience suggests that the standard deviation is a function of the mean value. This is modelled as σ=ξα with 0 ≤ α < 1 and therefore u ≈ c1-αlm - ξ1-αlm for c in the region of its mean value.
The least square estimators are derived from
Σ[c1-αlm - ξ1-αlm]2 = min
Introducing the selectivity model
ξlm = ΨlmNl
clm = ΨlmNl + noise we get
that can be solved
The selection Ψ is decomposed into normalised selection S and fishing power P as: Ψ = S P. With β = 1-α this gives the formula used in the framework.
If the data consist of more than one experiment, the population length distribution usually differs between experiments and a population distribution should be estimated for each experiment.
The standard variable u is, based on the central limit theorem, approximate normally distributed N(0,1) and the least square estimator ∑u2 = min is therefore approximately equal to the maximum likelihood estimator. This is the reason why the transformed least-squares and maximum likelihood estimators often are quite similar.
In the simple case of a Poisson distribution with α = 0.5, this simply reduces to the square root transformation for the least square fit. The stock estimator is then:
In the case of logarithmic distributed errors, the power transformation is not applicable the relevant transformation is the logarithmic. In this case, the stock estimator becomes
where M is the number of mesh sizes used in the experiment.