Prager: A nonequilibrium surplus-production model 



383 



and empirical Bayes methods for fitting production 

 models. Most of the extensions described by these 

 investigators could be combined with techniques pre- 

 sented here (e.g. tuning, bootstrapping), as required 

 for a particular analysis. 





1960 1965 1970 1975 1980 1985 1990 



120 I ' ' ' ' ' ' ' ' ' ] 



1960 1965 1970 1975 1980 1985 1990 



1960 1965 1970 1975 1980 1985 1990 

 Year 



Figure 2 



Goodness-of-fit of two production model analyses 

 loosely based on swordfish, Xiphias gladius, in the 

 North Atlantic Ocean. These analyses are illustra- 

 tive and are not intended as an assessment of sword- 

 fish. Model 2 differs from Model 1 in being tuned to 

 a hypothetical index of abundance. (A) Observed (o) 



and estimated ( ) fishing effort rate from Model 



1(B) Observed (o) and estimated ( ) effort rate 



from Model 2. (C) Observed (o) and estimated abun- 

 dance-index from Model 2. 



Autocorrelation 



Because catch and effort data are usually autocor- 

 related, the residuals from fitting — whether comput- 

 ed in yield or effort — may also be autocorrelated. A 

 matter of statistical concern is whether a method of 

 fitting that takes the autocorrelation into account 

 (such as one based on time-series analysis sensu Box 

 and Jenkins [1976]) might be more appropriate. Some 

 results relevant to this question were obtained by 

 Ludwig et al. ( 1988) in a study that used two differ- 

 ent objective functions to fit production models to 

 simulated data. The first was a total-least-squares 

 objective function, which did not take autocorrelation 

 into account; the second, an approximate-likelihood 

 objective function, which did. Ludwig et al. (1988) 

 found that the two methods produced very similar es- 

 timates; the authors concluded that the added complex- 

 ity of the approximate-likelihood method was probably 

 not warranted. In addition, the approximate-likelihood 

 method frequently failed to converge from poor start- 

 ing values. This does not mean that autocorrelation 

 should be ignored in all fisheries modeling; however, it 

 was not a major concern in the study cited. 



Process error 



The model presented here assumes that the produc- 

 tion of biomass is a deterministic function of the cur- 

 rent biomass; stochasticity occurs only in the obser- 

 vation of catch or effort or in the relation of fishing 

 effort to fishing mortality rate (if effort is being esti- 

 mated from catch). In reality, production is undoubt- 

 edly stochastic to some degree. In recognition of this, 

 fisheries models that explicitly incorporate process 

 error have been developed (e.g. Ludwig et al., 1988; 

 Sullivan, 1992). Because process errors are propa- 

 gated forward in time, it would seem that time- 

 series fisheries models (e.g. production models), 

 should include corrections for process errors, so that 

 the system can be modeled as correctly as possible. 

 Despite the undeniable logic of including process 

 error in fisheries models, there are also negative as- 

 pects, and the practical merit of including process 

 error in fisheries applications remains a topic for 

 research. The theory of models including process er- 

 ror was largely developed in process control (Kalman, 

 1960), a field in which large data sets are common. 

 Including both observation error and process error 

 in a model generally entails either estimating a large 

 number of nuisance parameters (the process errors), 

 making strong assumptions about the form or value 

 of the process error component, or both. In some 

 cases, the need to estimate additional parameters 

 can make it difficult or impossible to estimate pa- 

 rameters of interest, such as MSY, without additional 



