678 



Abstract. -Recently, J. J. Pella 

 showed how the Kalman filter could be 

 applied to production modeling to esti- 

 mate the size and productivity of fish 

 stocks from a time series of catches and 

 relative abundance indices. We apply 

 these methods to the Deriso-Schnute 

 delay-difference equation. The Kalman 

 filter approach incorporates process 

 and measurement error naturally in 

 the model description. When the pro- 

 duction model is the delay-difference 

 equation, the error structure is particu- 

 larly attractive because process error 

 can be interpreted as simply the vari- 

 ance of recruitment, and measurement 

 error as the variance of the relative 

 abundance estimates. We derived prior 

 distributions of initial biomass in order 

 to begin the Kalman filter calculations. 

 Reanalysis of the data from the east- 

 ern tropical Pacific for yellowfin tuna, 

 Thunnus albacares, shows that model- 

 ing results can differ greatly depend- 

 ing on whether error is interpreted to 

 be process error or measurement error. 

 Simulation results show that nonlinear 

 least squares and Kalman filter esti- 

 mates agree well if data contain only 

 measurement error. In contrast, the 

 Kalman filter was clearly superior if 

 simulated data contained significant 

 amounts of process error. The presence 

 of process error positively biased biom- 

 ass estimates from both the nonlinear 

 least-squares and Kalman filter meth- 

 ods. The Kalman filter performed well 

 with Schnute's form of the delay-differ- 

 ence equation, even though this model 

 violates the assumption of independent 

 process error vectors. The Kalman fil- 

 ter also performed well when the vari- 

 ance ratio r was assumed known and 

 individual variances were estimated 

 from the data. However, it appeared dif- 

 ficult to estimate r as a parameter in 

 the maximum-likelihood estimation. 



Kalman filtering the delay-difference 

 equation: practical approaches 

 and simulations 



Daniel K. Kimura 

 James W. Balsiger 

 Daniel H. Ito 



Alaska Fisheries Science Center, National Marine Fisheries Service 



7600 Sand Point Way NE 



Seattle, WA981 15-0070 



e-mail address Dan.Kimura@noaa gov 



Manuscript accepted 27 June 1996. 

 Fishery Bulletin: 94: 678-691 ( 1996). 



Fishery production models are used 

 to model fishery population dynam- 

 ics when only catch (measured in 

 biomass units) and relative abun- 

 dance data are available. Originally, 

 only fishery catch-per-unit-of-effort 

 data (Schaefer, 1954; Pella and 

 Tomlinson, 1969) were used as rela- 

 tive abundance indices for fish 

 populations, but surveys often pro- 

 vide less biased abundance mea- 

 sures. Therefore, both time series 

 of fishery CPUE and survey abun- 

 dance indices are commonly used 

 estimates of relative abundance. 



Production models contrast with 

 age-structured models, such as 

 ADAPT (Gavaris, 1988) and Stock 

 Synthesis (Methot, 1989), that 

 model the population cohort in num- 

 bers at age and typically require 

 catch-at-age data. Age-structured 

 models allow for variation in re- 

 cruitment which has given this 

 class of model greater credence 

 among stock assessment scientists. 

 However, age data are technically 

 difficult to obtain, expensive, and 

 contain biases and variability that 

 can be difficult to interpret. In ad- 

 dition, age-structured models often 

 contain a large number of param- 

 eters, and sifting through possible 

 solutions when fitting multiple 

 sources of data can be subjective. 



Because of these data require- 

 ments and technical problems, pro- 

 duction models have remained of 



interest to stock assessment scien- 

 tists. Recent production model of- 

 ferings have included statistical re- 

 finements such as bootstrapping 

 (Prager, 1994), Bayesian analysis 

 (Hoenig et al., 1994), and the treat- 

 ment of statistical error (Polachek 

 et al., 1993). Two of the earliest pa- 

 pers to consider both process and 

 measurement error in population 

 dynamics modeling were those of 

 Ludwig and Walters ( 1981 ) followed 

 by Collie and Sissenwine (1983). 

 The Kalman filter approach, which 

 allows the consideration of both pro- 

 cess and measurement error, now 

 appears to be becoming standard for 

 assessments by either production or 

 age-structured models (Mendels- 

 sohn, 1988; Sullivan, 1992; Pella, 

 1993; Schnute, 1991, 1994). 



The methods described in this 

 paper meld two distinct pieces of 

 technology: the delay-difference 

 equation (Deriso, 1980; Schnute, 

 1985), and the Kalman filter (Kal- 

 man, 1960; Harvey, 1990; Pella, 

 1993). Collie and Walters (1991) 

 used the Kalman filter to predict 

 and update biomass estimates on 

 the basis of the delay-difference 

 equation but did not use the Kal- 

 man filter for parameter estimation. 

 The delay-difference equation has 

 deep roots in fishery modeling. In 

 fact, to a remarkable extent it can 

 encompass the fundamental para- 

 digms of age structure, exponential 



