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



679 



survival, and von Bertalanffy growth. The Kalman 

 filter (Kalman, 1960), on the other hand, originated 

 in engineering where it has been widely used in con- 

 trol theory and quality control. The basic idea of 

 Kalman recursive filtering (Meinhold and Sing- 

 purwalla, 1983) is that the current state of the sys- 

 tem (i.e. the current fish biomass) can be estimated 

 from the system's past (biomass) estimates in two 

 steps: a forecast step and an update step. The fore- 

 cast estimate is made prior to the current observa- 

 tion of relative abundance; the updated estimate is 

 made following the current observation of relative 

 abundance. The updated estimate of biomass is the 

 modeler's best estimate of the "true" biomass and is, 

 roughly speaking, a weighted average between the 

 forecast and observed abundance values. The up- 

 dated biomass estimate is used to forecast the next 

 biomass estimate, which then is updated with the 

 next observed value. The absolute and relative mag- 

 nitude of process and measurement error, which are 

 assumed normal and whose values are generally as- 

 sumed by the filter, largely determine the result of 

 the update step. If there is little process error as- 

 sumed by the filter, updating will not substantially 

 change the forecast biomass estimate. 



For fishery modeling, the most salient feature of 

 the Kalman filter is its allowance for both process 

 and measurement error. In the implementation of 

 the delay-difference equation with the Kalman fil- 

 ter, process error can be interpreted as the variance 

 of recruitment, 1 and measurement error as the vari- 

 ance of the estimate of relative abundance. Thus the 

 Kalman filter method allows for variation in recruit- 

 ment, a property that heretofore seemed to be exclu- 

 sive to age-structured models. 



The implementation of the Kalman filter presented 

 here is due to Pella (1993), who applied the method 

 to a simple recruitment model ascribed to Schnute 

 (1991) and to the generalized production model of 

 Pella and Tomlinson ( 1969). For comparative purposes, 

 we also fitted the delay-difference equation with ordi- 

 nary nonlinear least squares, which in this case as- 

 sumes that data contain only measurement error. 



The delay-difference population 

 dynamics model 



The delay-difference equation is a simple biomass- 

 based model that contains the core dynamics of age- 



structured models (Deriso, 1980; Schnute, 1985). 

 Thus under many conditions the population dynam- 

 ics described by the delay-difference equation should 

 agree with that of age-structured models (e.g. vir- 

 tual population analysis). The applications of the 

 delay-difference equation we propose require the fol- 

 lowing parameter values and information: 



1 The instantaneous natural mortality rate M. 



2 Brody 2 growth parameters p and co. 



3 Annual catches in biomass c t , t=l,...n. 



4 Annual survey biomass estimates y t , either rela- 

 tive (catchability unknown) or absolute (catcha- 

 bility known) abundance indices, with a few miss- 

 ing values allowed. 



The natural mortality rate, M, is always difficult 

 to know but estimates or educated guesses are usu- 

 ally available. The Brody growth parameters can be 

 estimated with nonlinear least squares on relatively 

 small aged samples that provide weight-at-age data 

 (Schnute, 1985): 



W' y 



i+j-) 



,=(o k _ 1 +((o k -co k _ 1 )a-p l ] )i 



(1-p) for7>0. 



(1) 



Here, W i+; is the observed weight of a k+j yr-old fish, 

 o^j, co k , and p are parameters to be estimated, and k 

 is the age at recruitment. Our application of the de- 

 lay-difference equation requires co = co k _ 1 /co k so that 

 co = co h _ l l thf.. The difference between Deriso's and 

 Schnute's forms of growth (and hence their delay- 

 difference equations) is that Deriso's original equa- 

 tion requires unrealistically that co k _ 1 = so that 

 co = 0. A simple substitution in Schnute's more gen- 

 eral formulations of growth and delay-difference 

 equation results in Deriso's forms of growth and de- 

 lay-difference equation. 



Catches and fishing effort are usually among the 

 first statistics collected from a fishery. Survey esti- 

 mates of stock biomass sometimes come later through 

 significant cost and effort by fisheries management 

 agencies (Gunderson, 1993). If insufficient numbers 

 of survey estimates are available, fishery CPUE can 

 provide the biomass indices. However, fishery CPUE 

 data may bias model estimates because catchability 

 often changes over time. Survey data provide either 

 relative (catchability unknown) or absolute (catcha- 

 bility known) biomass abundance indices. Absolute 

 abundance estimates for a few years will typically 



1 A reviewer noted that process error can also arise from other 

 factors, such as variability in growth and survival, and other 

 sources that make the deterministic delay-difference popula- 

 tion dynamics incorrect. 



2 A reviewer pointed out that "Ford" may be a better name for 

 these growth parameters (Ricker, 1975). 



