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Fishery Bulletin 92(2). 1994 



Parameter estimation 



Parameter estimation for this model can be accom- 

 plished by a number of methods. The method pre- 

 sented here is a slight modification of one originated 

 by Pella (1967), later used by Pella and Tomlinson 

 (1969), and recently termed the "time-series method" 

 by Hilborn and Walters (1992). Although it is not 

 necessary to use equal time periods, the treatment 

 in the balance of this paper assumes, for simplicity, 

 that there are T equal time periods, indexed by x = 

 (1, 2, ..., T), and that a period is one year in duration. 

 The following symbols are used: 



/; 



F : 



population biomass at the start of year x 

 yield in biomass during year x 

 surplus production during year x, 

 fishing effort rate during year x, 

 fishing mortality rate during year x, 

 function of F', a = r - F T . 



Estimates of the first five of these quantities are rep- 

 resented by B T ,Y T ,P T ,f T , andF r . 



An important additional assumption is that, for 

 all x, F = qf ; in other words, that fishing mortality 

 rate is proportional to fishing effort rate and that 

 the catchability coefficient q is constant. (The as- 

 sumption of constant q is slightly relaxed later.) 



The data required for fitting are, for each time 

 period x, data on effort f z and the yield Y v where 

 x = II, 2, ..., T\ and T> 4. The parameters to be esti- 

 mated are r and K in Equation l,q, and B,, the bio- 

 mass at the beginning of the first year. The simplest 

 procedure accumulates residuals in yield. To perform 

 the estimation, the following algorithm is used: 



Al Obtain starting guesses for the four parameters. 



A2 Beginning with the current estimate of B v project 

 the population through time according to Equations 

 4a and 4b. For each year of the projection, com- 

 pute estimated yield from Equations 6a and 6b. 



A3 Compute the objective function to be minimized. 

 Assuming a multiplicative error structure in 

 yield, this is 



T 



y[io g (y r )-iog(y r )f. 



A4 Monitor the objective function for convergence. 

 If achieved, end. Otherwise, revise the param- 

 eter estimates (using a standard minimization 

 scheme) and continue at step A2. 



The simplex or "polytope" algorithm (Nelder and 

 Mead, 1965; Press et al., 1986) works well as the 

 minimization scheme in this application. Although 



not as rapid computationally as some other meth- 

 ods, the simplex algorithm is quite robust to start- 

 ing values and is easily manipulated (by restarts) to 

 avoid local minima (see Press et al., 1986, p. 292). 

 Rivard and Bledsoe (1978) used the Marquardt ( 1963) 

 algorithm successfully for estimation in a similar model. 

 The estimation method just described uses the re- 

 corded effort in each year to estimate yield. Alterna- 

 tively, one could use the recorded yield in each year 

 to estimate the fishing mortality rate (or equivalently, 

 the fishing effort rate). The solutions of Equations 

 6a and 6b for fishing mortality rate are 



when cc T * 0, or 



when a T =0. 



(8a) 



(8b) 



ln[l+)3B r ] 



To use this approach, one must revise the second and 

 third steps of the algorithm to become — 



A2' Beginning with the current estimate of B,, com- 

 pute the estimated fishing effort for each year 

 by solving Equation 8a or 8b and dividing by 

 q . Project the population to year-end with Equa- 

 tion 4. 



A3' Compute the objective function to be minimized. 

 Assuming a multiplicative error structure in 

 effort, this is 



X[log(/- r )-log(/ r ) 



r=l 



This is equivalent to minimizing the sums of 

 squared residuals in the logarithm of catch per 

 unit of effort, i.e. to minimizing 



£[iog(c T //;)-iog(c T //;) 



r=l 



A significant practical advantage of the second 

 approach is that it simplifies the analysis of data with 

 some missing data on effort. During parameter esti- 

 mation, effort is estimated for all years; for years of 

 missing effort, the contribution to the objective func- 

 tion is simply defined to be zero. In contrast, the 

 computations for the first approach are not possible 

 without data on effort for each year. 



Estimating effort from yield introduces two small 

 practical difficulties. The first difficulty is that Equa- 

 tion 8a is not an explicit solution for effort (because 

 a includes f), so it must be solved iteratively. This 

 is accomplished by putting a starting guess F T into 

 the right-hand side of the equation, solving, and sub- 

 stituting the result repeatedly until convergence is 

 achieved. A logical starting guess is F T = Y T / B r . 



