FISHERY BULLETIN: VOL 76, NO 3 



e.g.. the coefficients of variation for the estimates 

 of maximum sustainable yield {rh ) were below G^c . 

 It is questionable, however, whether the data can 

 justify such precision. Variability of the exploited 

 population due to migration, to changes in fishery 

 regulations over time, and to expansion of the 

 fishing areas, as well as variability of q due to 

 learning by fishermen and to technological de- 

 velopments, are important factors underlying the 

 complexity of events influencing the serial catch- 

 effort information. In future research, alternative 

 forms of the model in which g is a variable 

 parameterized with respect to time will be 

 explored. Furthermore, in a randomly fluctuating 

 environment, equilibrium population levels (and 

 MSY, by extension) are not constant and the 

 equilibrium points are instead described by a 

 probabilistic cloud representing the equilibrium 

 probability distribution (May 1974). The knowl- 

 edge of this equilibrium probability distribution 

 would give us some idea of the probability of 

 achieving the desired management goal ( MSY, for 

 instance). 



W = 



(r-iy 



Y.Y^P 



Y,Y,P y,yp 



y.' : Kyy-'' 



YYp 

 1—1 f-< 



(r-l) 



Y.Y^p 



(r-2) 



Parameter p, constrained between and 1, is a 

 measure of the importance of lags and can be esti- 

 mated along with the parameters of the differen- 

 tial equations (1) and (3). Itcanbeseen that Equa- 

 tion (9) is a particular case of Equation (22), where 

 the off-diagonal elements of W are null. 



The Levenberg-Marquardt algorithm, as formu- 

 lated in Equation (11), is designed to minimize 

 directly a sum of squares of residuals as given by 

 Equation (9). In order to minimize Equation (22) 

 by using Equation (11), we must scale W by the 

 transformation 



DISCUSSION ON 

 ERROR STRUCTURE 



In the preceding examples, we found runs in the 

 time sequence plot of residuals. Those runs indi- 

 cate correlations among the residuals. Serial cor- 

 relation, as we usually find in applying production 

 models to catch data, indicates that the real sys- 

 tem is working differently than the presupposed 

 model and that some minor effects have been neg- 

 lected (such as age composition or environmental 

 factors). But as indicated by Draper and Smith 

 (1966), the effects of correlation can be ignored 

 when the ratio {r - p )/r tends to unity (p being the 

 number of estimated parameters). In certain situ- 

 ations, of course, this ratio is likely to become 

 small (tending to zero as r approaches p) and we 

 may want to consider weights (W,) which account 

 for both the inequality of variance and the correla- 

 tions. In our estimation procedure, the assumption 

 of uncorrelated error can be relaxed by writing 

 Equation (9) in the more general form (J. J. Pella, 

 pers. commun.) 



X = D~^ W D 



(23) 



SiO,p) = [Y-Y] W-i [Y-Yl' 



(22) 



where Y is the row vector of observed yields, Y is 

 the row vector of predicted yields, and W is the 

 symmetric, positive definite matrix 



532 



where D is a diagonal matrix having elements D, 

 1 r), and write W as 



y, (I 



W = D U A U'T D. 



(24) 



where U.\U^ is the eigenvalue and eigenvector 

 decomposition ofX. Note thatX is actually the 

 correlation matrix of errors. Therefore Equation 

 (22) becomes 



sie,p) 



= [Y-Y] D-i U A~i U'^ D~^ [Y 



Y]''. (25) 



Then Equation (25) has the same form as Equation 

 (9), where the weights iW,) are the square roots of 

 the eigenvalues of X and where the residuals are 

 given by [Y— Y J D ' U. Such a procedure requires, 

 however, diagnoalization of an r by r matrix. 

 Moreover, diagonalization must be repeated at 

 least p times for each iteration. This procedure 

 produces a 10-fold increase in computing time. 



Although an exhaustive study of all possible 

 stochastic effects on the model was not attempted, 

 some simulations were done to determine the 

 magnitude of error in parameter estimates due to 

 serial correlations of the e,. Results are given on 

 Tables 4 and 5. For the yellowfin tuna data, p = 



