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Fishery Bulletin 94(4). 1996 



Figure 1 



The likelihood profile I minus ln(L c )) for estimating Kalman 

 filter parameters as a function of the variance ratio 

 ( r = a 2 p /a 2 m ) and lambda (A), showing that the likelihood 

 profile appeared insensitive to the value of variance ratio. 

 The assumed parameters were ft, = 250, A= l,p= 1, 0)=0, 

 Af = 0.60, s = exp(-0.60), r=1.0, variances assumed to be 

 <T 2 = 100 a 2 = 100 , with n = 100 yr of simulated data. 



is process error (i.e. variance of recruitment) as op- 

 posed to measurement error (i.e. error in measuring 

 the relative abundance index). 



D2 ct„ 2 , = 5,000 o 2 p = 1,000. These values suggest 

 that most of the error is measurement error as op- 

 posed to process error. 



Under the heading "Simulating datasets satisfy- 

 ing Kalman filter assumptions," we describe a de- 

 tailed model for simulating data that satisfy the 

 Kalman filter assumptions. In this paper all simula- 

 tions are roughly related to the yellowfin tuna dataset 

 (i.e. 34 yr of simulated data with similar model pa- 

 rameter values). To do this we specified values for 

 natural mortality, Af , Brody growth parameters p and 

 co, expected recruitment R x , catchability A, and vari- 

 ances a 2 and a 2 n . Assuming virgin biomass (Eq. 4), 

 values for ( Af, p, co, o p ,R x ) provide us with the mean 

 virgin biomass and its covariance matrix over neigh- 

 boring years (a ,P ). The simulation is initialized by 

 generating biomass estimates, i.e. bivariate normal 

 random deviates having the prior expectation a and 

 covariance matrix P . These initializations assumed 



s = exp (-Af ). Because trends in the catch data can SI of n = 1,000 o l p = 1. Measurement error only, 



affect modeling results, we also simulated the catch 

 data by assuming that catches were lognormal (see 

 Kimura, 1989), ln(c,) ~ JV(p, o 2 ), with E(c, ) = c , and 

 var( c, ) = cv 2 c 2 , with c = 141.8 (average from Table 

 1), and cv = 0.2. 



Three simulations were replicated 100 times so 

 that we could examine the bias, variance, and root 

 mean square error (i.e. the square root of mean 

 square error) of Kalman filter and nonlinear least- 

 squares parameter estimates when the true param- 

 eter values were known. For these simulations we 

 assumed that the stock was initially a virgin biom- 

 ass (R } = 250, X = 1, p = 1, co= 0, M = 0.60), and the 

 catch data were generated as lognormal random de- 

 viates LN(c= 141.8, cv = 0.2). We assumed that data 

 sets had only measurement error, only process er- 

 ror, or a mixture of both. 



52 o 2 m = 1 o 2 , = 1000. Process error only. 



53 a 2 = 500 o 2 = 500 . Measurement and process 



