RIVARD and BLEDSOE: PARAMETER ESTIMATION FOR PELLATOMLINSON MODEL 



0.27. For the Pacific halibut data, p = 0.33. In 

 either case, p exhibits a relatively large coefficient 

 of variation when compared with the elements of 

 O. One could anticipate such results since p 

 reflects the "persistence" of fluctuations in popula- 

 tion size, and the estimation of p would therefore 

 require a longer catch history in order to achieve a 

 greater precision. But more importantly, the val- 

 ues of G and Var[OJ were not significantly altered 

 by the inclusion of the additional parameter. And 

 while the errors of any particular catch history 

 might indeed by correlated, the minimization 

 criterion (9) will provide satisfactory estimates of 

 B despite the fact that correlations do not enter 

 into its formulation. The limited results contained 

 herein suggest that serial correlation can be safely 

 ignored when the ratio (r - p)/r is near unity. 

 Under such a condition the estimation procedure 

 is robust with respect to the assumption of inde- 

 pendence of errors in actual data. 



CONCLUSION 



modifications of the model to incorporate such 

 hypothetical effects as migration or stock interac- 

 tions can be made easily. Of course, to the extent 

 that the esti mation procedure must rely strictly on 

 catch-effort data, it will be subject to the same 

 information uncertainties as any other method. 

 But within the basic estimation procedure, we can 

 combine the catch-effort data with prior informa- 

 tion and thereby reduce the uncertainties in our 

 estimates. The prior information can be any in- 

 formation on a state variable, s\xch.asB(t), or even 

 any prior knowledge of the coefficients as express- 

 ed by B ± Var(B). Suppose, for example, that we 

 have information from independent surveys on 

 stock density (acoustic surveys, indirect estima- 

 tion from knowledge of larval densities, or even 

 virtual population analysis from catch records). 

 Such surveys would then provide us with esti- 

 mates B^ each having a variance ViB, ), let us say, 

 at various times t. We can easily introduce such 

 information into the estimation procedure by 

 defining the new objective function 



The purpose of this paper has been to examine a 

 version of the Levenberg-Marquardt algorithm as 

 an alternative method for estimating the 

 coefficients of the generalized stock production 

 model. The parameter values obtained by this pro- 

 cedure are close to those obtained by previous 

 studies on yellowfin tuna and Pacific halibut. Ob- 

 viously, data requirements are such that a full 

 range of effort values (ranging over low and high 

 exploitation rates) are necessary to insure con- 

 vergence in the estimation procedure and to pro- 

 duce estimates with small variability. Our simu- 

 lations reveal that with the Levenberg-Marquardt 

 method both the estimates of coefficients and the 

 estimates of variances remain approximately un- 

 biased when white noise is considered. If present, 

 such bias is sufficiently small as to be obscured in 

 the variability associated with catch error. The 

 simulations also showed the range of variability in 

 parameter estimates that might be expected for 

 given levels of normally distributed error in catch 

 data. 



Because the parameters of interest appear 

 explicitly in the system equations, the estimation 

 procedure for the parameters also produces the 

 variance estimates directly. Moreover, the method 

 has a reliability and an efficiency of computation 

 somewhat greater than previous methods. And 

 since the estimation procedure relies on a numeri- 

 cally integrated system of differential equations. 



s(B) = s w.iY.-yy + s 



ill I j 



V -l^D _DX2 



{B.-B.y. (26) 



Introduction of the second term in the objective 

 function constrains the optimization and thereby 

 improves convergence. If the prior information 

 has extremely large variance, then this informa- 

 tion is of no value; the second term of Equation (26) 

 will tend to zero and the objective function then 

 reduces to Equation (9). In general, the alteration 

 permits the simultaneous employment of the two 

 state variables. Therefore, the final coefficients 

 are no longer based solely on catch and effort data; 

 their determination includes our knowledge of 

 previous stock densities. 



As observed here in a statistical setting, and 

 by Fletcher (1978a, b) in the exact analysis, 

 the Pella-Tomlinson system exhibits internal in- 

 stability in its parametric relationships. That 

 behavior arises from the variable nature of the 

 system's nonlinear! ty, which would not be particu- 

 larly detrimental if our problems were limited 

 strictly to the geometric syntheses of data by curve 

 fitting. But for the purposes of management and 

 preservation of stocks, the subject is elevated 

 partly at least to the status of parameter estima- 

 tion "where we look for procedures to obtain val- 

 ues of the parameters that not only fit the data 

 well, but also come on the average fairly close to 

 the true value" (Bard 1974). Although the Pella- 



533 



