FISHERY BULLETIN: VOL. 76, NO. 3 



IwutiaJi 

 ^panajnzttui: • 



6. 



START 



Figure l. — information flow diagram 

 for the computer program written to es- 

 timate the coefficients of the 

 generalized production model. 



e. 



Levenberg-Marquardt 

 algorithm 



Runge-Kutta 

 algorithm 



Model: equations 

 (1) and (3) 



-i 



EWO 



[',] 



EUoiX 



O 



> Inionmcubion itouo 



> PlOQiam contAol ilow 

 Vautjo. -input 



Vzcla-ion aZgoiyCthm 



Calculation algo- 

 lithm 



an analytical form for the uncertainties in the 

 final values of the parameters (Bevington 1969). 

 By letting SiO) be the weighted residual sum of 

 squares for the final parameter estimates, how- 

 ever, the variance-covariance matrix of the esti- 

 mates (Bard 1974) can be approximated by 



imation in the neighborhood of O is appropriate. A 

 necessary and sufficient condition for the 

 F-distribution to be appropriate here is that dif- 

 ferences in true and estimated parameter values 

 are independent and approximately normally dis- 

 tributed with zero mean and equal variance. 



Ve = 



j^ J 



S(e)/(r-5). 



(12) 



Some idea of the joint variability of the parame- 

 ters can be obtained by evaluating the ellipsoidal 

 confidence region, based on the assumption that 

 the linearized form has validity around B ( Draper 

 and Smith 1966). The confidence region is then 

 given by 



[e-B] J'^ J[B-B]' 



F(5,r-5,l-a), 



(13) 



where F(5, r-5, l-«) is the standard tabulated 

 F-statistic. The ellipsoid is not a true confidence 

 region, of course, since the dependent variable, Y, 

 is a nonlinear function of B. The intervals ob- 

 tained are valid to the extent that a linear approx- 



DETERMINATION OF 

 STARTING VALUES 



In order to reduce the number of iterations re- 

 quired to minimize Equation (8), reasonably accu- 

 rate starting values should be employed. Starting 

 values can be calculated from a linearization and 

 simplification of the basic model. 



STEP 1. By using y,,y2, .. . , Y, and /■,, /a, . . . 

 fr, find an estimate of g from the Delury technique. 

 Note that this procedure generally underesti- 

 mates (? (see Ricker 1975). Correction forq will be 

 provided in step 4. 



STEP 2. Find estimates of B, + i from the equa- 

 tion 



B 



..1 = (^M.l^^..l,..2)/2' 



(14) 



526 



