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Fishery Bulletin 94(4). I 996 



a Q = E(a ) = 



B 



y B Oj 



and with covariance matrix 



P =S(a ) 



J\ To 



Estimators for y and y l are provided in Appendix 1 

 and only require prior estimates of p,co,a p and s . 

 To predict a v we assumed 



((l+p)s -ps s ) 



T n = 



«o = 



1 



' 1 -pcos ' 

 ,0 



,C - 



R^I-PCOSq) 







,and % = 



The state of the system at time £=1 is then a l = 

 T a +C + RqTIq, where n x is assumed to be a nor- 

 mal random variable with mean zero and variance 

 <r 2 p . For Deriso's form of the delay-difference equa- 

 tion, a>= 0, so that the value of n is irrelevant. How- 

 ever, for Schnute's form of the delay-difference equa- 

 tion we used n = because the data contained no 

 information concerning n . 



Projecting the state space Given the initial condi- 

 tions (above), and a catch sequence lc,|, the state 

 space variable a t can be projected indefinitely: 



a. = 



' B > ) 

 , B t-i 





(l+p)s ( -ps t s t _ x ' 

 1 



1 -pcos, ' 

 



A 



n t+i 

 n, 



R x (l- pios,) 

 



so that 



a, 



 T t a,+C t +R t r\ t . 



The process error vectors r/, are assumed to be inde- 

 pendent with 



E(r lt ) = 

 and with covariance matrix 



Q 



( < 



<> 

 o) 



Recall that T t requires a value for s t by first solving 

 Equation 5 for F r 



Addition of measurement error Given a catcha- 

 bility coefficient A, the observed variables can be de- 



scribed as y, = Za, +£,, where Z = (A 0),£,= e t , and 

 e t is assumed to be normally distributed with mean 

 zero and variance h = o 2 m . 



If expected recruitment is to be a function of stock 

 biomass in earlier years, the state space project vec- 

 tor C t must become a function of a t so that 



C t (a,) 



£(P, +1 )-pft)s,£(P,) 

 



If the delay between hatching and recruitment is to 

 be many years, then the dimension of a t must be in- 

 creased, so that the earlier biomass will be available 

 to calculate recruitment. 



So far we have described a statistical model which 

 includes process and measurement error and which 

 can be used to compute a realization of relative abun- 

 dance indices, say [y t \. In the implementation of the 

 Kalman filter given here, Deriso's form (co = 0) of the 

 delay-difference equation fully satisfies the assump- 

 tions of the model, but Schnute's (a) > 0) more realis- 

 tic form does not. This difference is due to the pro- 

 cess error vectors 



It 



which represent variability in recruitment and which 

 are assumed to be independent by the Kalman filter. 

 Because q t and q l+1 both contain n l+l , this assump- 

 tion is clearly violated for Schnute's form of the de- 

 lay-difference equation. This trade-off between real- 

 ism in the delay-difference equation and statistical 

 independence required of process error will be ex- 

 amined further with simulation. 



Kalman filter estimates 



If values of variance for process and measurement er- 

 ror are assumed, parameters T = |ln(B ),ln(/?, ),ln(A)] 

 can be estimated by the method of maximum likeli- 

 hood. In principle, the method is very similar to that 

 described for calculating nonlinear least-squares es- 

 timates. Given a value for T, initial values for a , P 

 can be specified as in the simulations; estimates of 

 biomass and its covariance matrix can then be fore- 

 cast \a lll _ l ,P lU _ l ! . The value of 4*, whose forecasts maxi- 

 mize the log-likelihood, is taken as the estimate. 



Because the Kalman filter incorporates process as 

 well as measurement error, estimation of biomass 

 occurs in two steps. 



1 Initial projections a t \t-i = T t -iat-i + C t -\ and 

 P /U _! = r,_ 1 P,_ 1 T,_ 1 '+r?,_ 1 Qr?,_ 1 ' are calculated on the 



