The evolution in mathematical model fitting approaches has opened a variety of possi- bilities for new improved stock production models, and further assumptions regarding error structures.
Stock production model is a data modest fisheries model that can be useful if it can be made to work satisfactorily. Nonlinear parameter estimation using difference equations has become one of the elegant methods allowing growth, mortality and recruitment to be treated separately from biomass. Simple methods are useful as a basis. If additional information is available, more complex models can also be applied and compared
In addition, the stock production model is considered to be the most suitable type of stock assessment method for the present research, as regards its data requirements. Moreover, it should be the first step in quantitative stock assessment. Furthermore, uncertainty is present in every model and will not necessarily be reduced by making models more elaborate.
Underlying Theory and Methodology Chapter 2 The least squares method will be used for parameter estimation, since it has proved to be a useful, reasonably robust and potentially adequate simple method. Further analysis required for the validation of this method will also be conducted, using the bootstrapping method.
Chapter 3
The Conser Mixed Model
3.1
Introduction
Since the stock production model was first proposed (Graham, 1935, Schaefer, 1954), it has been widely applied to a variety of resources, including migrating fish (Goodyear and Prager, 2001, Prager, 2002), invertebrates (Polovina, 1989, Chen and Mont- gomery, 1999, D’Incao et al., 2002), temperate fish (Rose, 2004), and tropical fish species (Pella and Tomlinson, 1969). A range of different methods for parameter estimation have been proposed e.g. (Pella and Tomlinson, 1969, Fox, 1970, Schnute, 1977, Rivard and Bledsoe, 1978, Tsoa et al., 1985, Polacheck et al., 1993, Pella, 1993, Chen and Andrew, 1998, McAllister and Kirkwood, 1998, Meyer and Millar, 1999b, Prager, 2002, Schnute and Richards, 2002, Punt, 2003).
In practice, stock production models generally employ difference equations instead of differential equations. The formulation allows growth, mortality, and recruitment to be treated separately from biomass. Each parameter has a biological interpretation, separate from any other process. In principle, an independent parameter estimation is thus promoted (Conser, 1998) which can be conducted by using either additional data or previous studies. Despite being used in a different manner than the Bayesian framework, this is a way to incorporate prior information into the analysis. This
The Conser Mixed Model Chapter 3 approach avoids over-parameterisation which is commonly found when the number of independent parameters to be estimated is higher than the number of independent data sets (Shepherd, 1987).
Fitting time series difference equations with nonlinear minimisation is an elegant approach which avoids the unreliable assumption about stock equilibrium and pro- motes the inclusion of uncertainties, regarded as a essential features in the estimation processes (Hilborn and Peterman, 1996).
Production relationships were first written as difference equations by Walters and Hilborn (1976) and Hilborn (1979), when a multiple linear regression was proposed as the estimation method regarding catch-per-unit-effort and effort as independent variables.
A surplus production difference equation model is used as the starting point in this work. The production model was proposed by Shepherd (1987) and can be considered as one of the general class of models described by Schnute (1985). In order to estimate reliable parameters when fitting the model to available data, Shepherd (1987) treated catchability and natural mortality together with the intrinsic growth rate and pristine biomass as separate parameters. Furthermore, a re-parameterisation was presented in terms of current and pristine biomass, suggesting that reasonable ranges of two of those parameters ought to be selected to obtain a good fit.
In the Shepherd model (Shepherd, 1987), the natural mortality rate (M) can be fixed, as in more elaborate models such as virtual population analysis (Pope and Shepherd, 1985, Shepherd and Pope, 2002a). This procedure serves two different purposes. Firstly, it allows the results of stock production and age structured models to be more similar and so more easily comparable. Secondly, it clarifies the representation terms of biologically meaningful parameters and so increases the chance of obtaining convincing results, since prior estimates of natural mortality are often available from basic biological research.
The Conser Mixed Model Chapter 3 Conser (1998) pointed out the lack of a formal statistical modelling for parameter estimation in Shepherd (1987) and proposed a numerical framework considering both observation and process errors, hereafter called “Conser Mixed Model (CMM)”. Here the term “mixed” is used to describe a model which considers both observation and process error in the estimation procedure.
Using a composite objective function minimised by the Marquardt algorithm, Conser (1998) conducted a stock production model assessment on a sablefish stock (Anoplopo-
ma fimbria) caught off the USA Pacific coast.
However, the proposed statistical framework contains some inconsistencies in the assumptions and statistical configuration which will be explained in detail in the fol- lowing sections. In short, the CMM formulation of process error is not fully consistent with the biologically processes concept. Moreover, setting the same weight for process and observation errors, as in the CMM, does not guarantee a mixed model and fixing the stock resilience is not a justifiable assumption, since it has deep implications for the management actions. Further comments on these are made in the discussion of this chapter.
Therefore, the objective of this chapter is to reproduce the analysis of Conser (1998), analysing and pointing out the inconsistencies of the statistical model, and suggest- ing improvements. The following two sections describe the CMM and its parameter estimation method. In the section 3.4 results are analysed and structural and method- ological deficiencies are exhibited and discussed.