Kimura etal.: Kalman filtering the delay-difference equation 



681 



parameters *F = (ln(5 1 ),ln(i? 1 ),ln(A)), because the 

 estimates by using Equation 7 may be suboptimal. 

 If the coefficient of variation is the same for all y t , 

 then all cv t can be set equal to one. 



We also briefly examined the possibility of estimat- 

 ing nonlinear least-squares parameters by modeling 

 both measurement and process error: 



SS 



£[(y ( -£4) 2 /<£ +($-*,)■ /ffj 



(8) 



Here, R x is the estimated mean recruitment as be- 

 fore, but R t t > 1, are allowed to vary. For Deriso's 

 form of the delay difference equation, differentiat- 

 ing Equation 8 with respect to R t and setting the re- 

 sult equal to zero yields 



R t = [(«! /r) + X(y t -XP t )]/[(!/ r)+ A 2 ] , 



(9) 



where r = <j 2 p I o 2 m , and P t is the delay-difference pro- 

 jection of biomass to year t prior to adding recruit- 

 ment. Estimating parameters by minimizing Equa- 

 tion 8 appeared unstable and was not pursued fur- 

 ther. This may be because mean recruitment R v and 

 the individual recruitments R t , both were estimated 

 from the data. With additional constraints, or if re- 

 cruitment indices were available, such an approach 

 might be useful. 



A have definite though unknown values, the abun- 

 dance index y t is observed only with error, which we 

 define as measurement error. The Kalman filter 

 method provides maximum-likelihood estimates of 

 B , R v and X, almost the same parameters as for the 

 nonlinear least-squares method but for the process 

 and measurement error model we have just de- 

 scribed. Note that we estimate S (see below) instead 

 of Bj because S cannot simply be defined to be equal 

 to B x in the equations which initialize the Kalman 

 filter method. For consistency, when comparing pa- 

 rameter estimates for nonlinear least-squares and 

 the Kalman filter methods, we shall compare only 

 estimates of B x (i.e. the one year projection of B ). 



Modeling and assumptions in addition to those of 

 nonlinear least squares are needed by the Kalman 

 filter. Because the Kalman filter is essentially a Baye- 

 sian procedure (Meinhold and Singpurwalla, 1983), 

 it requires a prior joint distribution for B and B_ v 

 In addition, in order to partition process and mea- 

 surement error, either the magnitude of process or 

 measurement error (or both) or their ratio must be 

 known or estimated. Nearly all of our simulations 

 will assume that process and measurement error 

 variances are known. Kalman filter estimation with- 

 out prior information concerning error variances 

 appears to be generally difficult. 



Maximum-likelihood estimation for 

 the Kalman filter 



We also applied the Kalman filter as described by 

 Pella (1993) to the delay-difference equation. Pella 

 (1993) credits Harvey (1990) for his own presenta- 

 tion, but we found Pella's presentation to be quite 

 adequate for applying the method. As described ear- 

 lier, the main reason for using the Kalman filter is 

 that it allows for process error in addition to mea- 

 surement error. The nonlinear least-squares esti- 

 mates of the previous section assume constant re- 

 cruitment and allow only for measurement error. 



The state transition equation of the Kalman filter 

 views the delay-difference equation (Eq. 2) as being 

 composed of both deterministic and stochastic com- 

 ponents. The determinisitic component of the pro- 

 cess assumes that the expected values of R t and R t x 

 are constant, say equal to R v If desired, we can also 

 assume virgin biomass when applying the Kalman 

 filter, i.e. we can estimate R 1 by using Equation 4. 

 This is analogous to the constant nonrandom recruit- 

 ment model fit by nonlinear least squares. The bio- 

 mass prediction given by Equation 2 is in error ow- 

 ing to variability in recruitment. This variability is 

 defined to be process error. Now, even though B t and 



Simulating datasets satisfying Kalman filter 

 assumptions 



In this section we describe simulation of relative 

 abundance indices that satisfy the assumptions of 

 the delay-difference equation and the Kalman filter. 

 As stated earlier, these simulations assume virgin 

 biomass (i.e. Equation 4 holds with B substituted 

 for Sj). This is the easiest way to avoid an initial 

 modeling bias in the simulation and model fits. Data 

 simulated in this manner will be fitted by using non- 

 linear least-squares and the delay-difference Kalman 

 filter methods without the assumption of virgin bio- 

 mass. To a large extent we followed the notation of 

 Pella ( 1993 ). Note that the "state space" of the sys- 

 tem is simply jargon for the unobservable true biom- 

 ass of the system. 



Initial conditions and assumptions The initial state 

 of the system is defined to be 



ffn = 



B 



We assume that a is unknown but has prior distri- 

 bution with mean 



