FISHERY BULLKTIN; VOL. 86. NO. 4 



of births and deaths at any time are available. In 

 their model, knowing the births is equivalent to 

 knowing the recruitment at each time period and 

 the deaths are observed directly without error. Also, 

 they appear to use the filtered estimates of the 

 population structure, while the smoothed estimates 

 are the minimum mean squared estimates given all 

 the data. P. Sullivan^ in his Ph.D. dissertation in- 

 dependently developed a length-based fishery model 

 using Kalman filtering and maximum likelihood 

 estimation. 



I reiterate that the models considered here assume 

 additive errors, while much of the existing literature 

 prefers multiplicative errors, particularly for the 

 observation equation. The basis of this preference 

 appears to be that models with multiplicative errors 

 have given a better "fit" to the data for other esti- 

 mation schemes. In these estimation schemes the 

 errors are assumed to have equal variances and are 

 assumed to be uncorrelated. The better fit found for 

 multiplicative errors in these algorithms may be due 

 to these assumptions on the error variances. In ef- 

 fect, the errors are being scaled by the observed data 

 which suggests that the assumption of equal vari- 

 ances is incorrect. In the present formulation I 

 assume additive errors, but I can allow for errors 

 with unequal variances, for errors that depend on 

 the size of either the observed catch n{a,t) or on the 

 unobserved population N{a,t), and for missing 

 observations, and I can obtain simple to compute 

 standard errors of the estimated underlying popu- 

 lation sizes. The algorithm is also simple to program. 

 If multiplicative errors are assumed, it is more dif- 

 ficult to calculate exact maximum likelihood esti- 

 mates. Approximate likelihood methods (such as the 

 extended Kalman filter) have known undesirable 

 properties. When the full richness of the assump- 

 tions allowed in additive error models is used, it is 

 an open question if multiplicative errors are to be 

 preferred. 



STATE SPACE MODELS 



The State-Space Model can be written in the form 

 (Jazwinski 1970; Anderson and Moore 1979; Ljung 

 and Soderstrom 1983): 



xit) = F{t)x{t - 1) + B{t)u{t -1) + Gw{t) (12) 



y{t) = H{t)xit) + v{t) (13) 



^P. Sullivan, Center of Quantitative Sciences, University of 

 Washington, Seattle, WA 98195, pers. commun. 1988. 



where x(t) = (xiit), . . .,x^,(0)' is the p-dimensional 

 unobserved state of the system; u{t) = (Wi(0. • • •- 

 UjXf))' is a p-dimensional vector of deterministic in- 

 puts; ^(0 = iy\{t). .  ■,y,f{t)y is the observed data 

 of the system; w{t) = {w^{t), . . .,w^,{t)y is a se- 

 quence of zero mean normal vectors with common 

 covariance matrix Q; v{t) = {v^{t), . . .,v,^{t)y is a 

 sequence of zero mean normal vectors with common 

 covariance matrix R\ and F, B, G, and H are ap- 

 propriately dimensioned matrices that may depend 

 on an unknown parameter vector B. Note that q, the 

 dimension of the observation vector, can be larger 

 than p, the dimension of the state vector. Thus 

 several different observation processes of the under- 

 lying dynamics are allowed. 



Using the same notation as in Equations (9) and 

 (10), the predicted, filtered, and smoothed estimates 

 of the state vector and the covariance matrices can 

 be calculated recursively as follows: for prediction 

 and filtering, 



x{t\t - 1) = F{t)x{t - l\t - 1) + Bu{t - 1) (14) 



P{t\t - 1) = F{t)P{t - l\t - l)Fity 



+ GQ{t)G^ (15) 



Kit) = P{t\t - l)H{ty 



X {H{t)P{t\t - l)H{ty + 7^(0) ' (16) 



xit\t) = xit\t - 1) + Kit) 



X iyit) - Hit)xit\t - 1)) (17) 



Pit\t) = Pit\t - 1) - Kit)Hit)Pit\t - 1) (18) 



where x(0|0) = ^i and P(0|0) = I, and for smoothing. 



Jit - 1) = Pit - l\t - l)HityiPit\t - l))-i (19) 



xit - 1\T) = xit - l\t - 1) + Jit - 1) 



X ixit\T) - xit\t - 1)) (20) 



Pit - 1\T) = Pit - l\t - 1) + Jit - 1) 



X iPit\T) - Pit\t - l))Jit- 1)^(21) 



The predicted state variable xit\t - 1) differs from 

 the quantity often used as the predicted value in the 

 fisheries literature in that it is based on the last 

 period's filtered estimate rather than on the last 



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