Kimura etal.: Kalman filtering the delay-difference equation 



685 



In order to understand the distribution of bias in the 

 estimated biomass trends, we made histograms of 

 the residuals Z t =B t - B, (true) for years t=l, 17, and 

 34, for both the Kalman filter and nonlinear least- 

 squares estimates. 



Finally, two specialized simulations were repli- 

 cated 100 times: 



SP1 



co>0: 



i?! = 250, X = \p = 0.75, co = 0.75, 

 M = 0.3, o 2 m = \ol = 1000 



SP2 



r fixed: 



i? x = 250, A = 1, p = 1, co = 0, 



M 



0.6, o" m =500, a; =500. 



With these simulations the stock was assumed to be ini- 

 tially a virgin biomass. The purpose of simulation SP1 

 was to examine how severely the correlation in process 

 error ( r\ t ) induced by Schnute's form of the delay-differ- 

 ence equation ( co > ) degrades the parameter estimates 

 from the Kalman filter. The purpose of SP2 was to ex- 

 amine whether estimating parameters by assuming only 

 r = o" p I a m = 1.0 was known (i.e. by maximizing Equa- 

 tion 10), would degrade the Kalman filter estimates. 



Results 



Our results from reanalyzing the data set in Table 1 

 with variance assumptions described as Dl and D2 



are shown in Figure 2. The variance assumption Dl, 

 that of Pella (1993), is close to assuming only pro- 

 cess error. The result is estimated biomass trends 

 that are scaled but that exactly trace the relative 

 abundance indices (Fig. 2A). Our model fit is very 

 similar to that reported by Pella ( 1993) and indicates 

 that, despite great differences in the underlying state 

 transition models (delay-difference versus general- 

 ized production), the Kalman filter method, with 

 large process error, can provide quite similar esti- 

 mates of biomass trends. The nonlinear least-squares 

 fit assumes only measurement error and differs con- 

 siderably from the Kalman filter fit. In the analysis 

 with variance assumed to be D2, the Kalman filter 

 assumes that error is predominantly measurement 

 error but also contains a realistic component of pro- 

 cess error. The result is Kalman filter estimates of 

 relative abundance that are very similar to nonlin- 

 ear least-squares estimates (Fig. 2B). 



Results from replicated simulations based on as- 

 sumptions S1-S3 are shown in Table 2. Under the 

 variance assumption SI, we have only measurement 

 error. Under this assumption, the Kalman filter and 

 nonlinear least-squares methods performed similarly. 

 Root mean square error ( RMSE ) estimates, compared 

 with sample standard deviations of parameter esti- 

 mates, show that both methods have little bias but 

 that the inverse Hessian of minus the log-likelihood 

 appears to give estimates of the uncertainty of pa- 

 rameter estimates that are biased low (i.e. are more 

 similar to standard deviation than RMSE ). Residual 

 plots (Fig. 3) indicate biomass traces have symmetri- 

 cal error. 



Kalman filter 

 Least squares 



30 34 33 42 



58 62 66 70 



46 50 54 58 62 66 



Year 



Figure 2 



Fits to yellowfin tuna, Thunnus albacares, assuming p = 1, <u = 0, M = 0.60, s„ = exp(-0.60l, and data from the 

 eastern Pacific Ocean (Tablet). (A) Kalman filter fit assuming cr =46 a] =32,775 as in Pella ( 1993). (B) Kalman 

 filter by assuming predominantly measurement error a 2 m = 5,000 o\ = 1,000. Nonlinear least-squares estimates 

 are the same in both (A) and (B); biomass is in millions of pounds. 



