Kimura et al.: Kalman filtering the delay-difference equation 



683 



basis of the value of 4 / , initial conditions, and the 

 data (Cj, ... , c ( _j)and (y v ... y,^). Recall again that 

 T^j requires a value for s t _ r that can be estimated 

 by first solving the catch equation (Eq. 5) for F t _ v 

 As described earlier, if expected recruitment is to 

 be a function of earlier biomass estimates, then 

 C,_j must be a function of a t _ v 



2 If an observation ofy t exists, the initial projections 



a, =a 



and P 



, + P^Z'f-Hy, 



tU _ x are updated on 



the basis of 



Za ,i,-i> and P t =P tU _ 1 - 



= ZP M _^Z' +h . If an ob- 



^o =P I< 



7o 



7i 

 7o 



rA, 



P I ' o is dependent only on p, co, and s . 



P^Z'f^ZPtU-u where f t 



servation of y t does not exist, further projections 

 can be made by using Equation 1, with a tU _ x and can be estimated from 



P t \ t _i used in place of a, and P t . Therefore, Kalman 

 filter estimation can accommodate missing rela- 

 tive abundance indices provided they are few in 

 number. 



where A 



The Kalman recursions are then carried forward just 

 as would have been done if no reparametrization had 

 been performed. However, the covariance matrices 

 \P t \ t -i and PJ are now automatically scaled versions 

 lP*-i=P«-i'< and P; =P t lo 2 J, and f;=f,lo 2 m . 

 The vectors a tui and a t are unaffected by the scal- 

 ing which cancels in their equations. The parameter 

 estimates are dependent only on the value of r, not 

 the variance components o 2 and o 2 m . However, o\ 



3 The log-likelihood is calculated to be 



where n is the number of y, observed (also a 2 p = ro 2 m ). 

 The log-likelihood to be maximized (see Appendix 2) 

 then becomes 



ln(L(y,T)) : 



| ln(2/r). 



1 _n_ .. n 2 



where v t = y t -Za tU _ v and/) = ZP tU _ X Z' + h. 



As with the nonlinear least-squares method, the 

 Kalman filter estimate of 4* was also calculated in 

 two stages. First, we searched a grid of possible ini- 

 tial Aq values, say from 0.5 to 1.5 at 0.1 intervals. 

 For each fixed A value, we then estimated the con- 

 ditional maximum-likelihood estimate *¥* =[ln(Bo>, 

 \n(R 1 )] by a quasi-Newton method. We then picked 

 the A* and T* having the largest conditional likeli- 

 hood, and used that as an initial estimate for find- 

 ing the unconditional maximum likelihood estimate 

 NP = [ln(B ),ln(i? 1 ),ln(A)], again by using a quasi- 

 Newton method. The asymptotic covariance matrix 

 of the maximum-likelihood estimates can be esti- 

 mated from the inverse Hessian of minus the log- 

 likelihood. 



Generally, our Kalman filter estimates were made 

 with the assumption that process and measurement 

 error variances were known. However, under the 

 Kalman filter, if r = o 2 p I a 2 m is known, then all model 

 parameters, including o 2 m and hence a 2 p , can be es- 

 timated from the data. Following Pella (1993), we 

 reparameterized h = 1 and 



Q 



r 0\ 

 r 



ln(L c (y,»P)) = — [ln(2;r) + l]- 

 l£ln(/;*)-£ln<<7£). 



(10) 



/=i 



It can be noted that 



The question arises whether r can be estimated as 

 just another parameter by using the method of maxi- 

 mum likelihood? Although estimation of r appears 

 possible in theory, our experience suggests that this 

 will be impractical for many data sets. The reason is 

 that the likelihood function appears to be insensi- 

 tive to r (Fig. 1). 



Data analysis and simulations 



We reanalyzed the data set for yellowfin tuna, 

 Thunnus albacares (Table 1), taken from Pella and 

 Tomlinson ( 1969 ) by using methods described in this 

 paper. Ages read from otoliths of yellowfin tuna (Wild, 

 1986) indicate that length at age is nearly linear. This 

 implies exponential growth in weight at age, but we 

 shall assume linear growth in weight at age (i.e. p = 1.0, 

 co = 0.0). Because fish rapidly disappear from the popula- 

 tion at age 4 yr, we assume M = 0.60. Therefore, we as- 

 sume constant growth with a large natural mortal- 

 ity rate. Analysis by Pella (1993) indicates a biom- 

 ass range of from 600 to 1,400 million pounds and 

 an annual sustainable yield of 193 million pounds. 

 Our data analyses (i.e. fits to the data in Table 1) 

 assume 



Dl a 2 , =46,o 2 p =32,775 as reported by Pella 

 (1993). These values suggest that most of the error 



