FISHERY BULLETIN: VOL. 86, NO. 4 



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Figure 6.— Spline estimates of f{t.). 



smoothness prior estimate has shifted some of the 

 variation in the observed data to variations in the 

 underlying population dynamics rather than varia- 

 tions in/(^). Further research needs to be done to 

 see which of these estimates is the most "valid". 



The EM algorithm in general can be sensitive to 

 the initial values given the parameters (see Wu 

 1983), and I have found that the fixed value of 1 can 

 also affect the estimates. Initial combinations of p<, 

 f{t), and s{a) that are totally inconsistent v^ith the 

 observed catch data can cause the algorithm to find 

 a local maximum. This can be avoided by ex- 

 perimenting with several, very different starting 

 values and determining if they converge to the same 

 estimates. 



If recruitment is thought to be a linear function 

 of the previous population size, then there is no 

 problem including this in the Kalman filter. If 

 recruitment is a nonlinear function of the previous 

 population, then the EKF can again be used to ap- 

 proximately determine the conditional expectations 

 needed for the EM algorithm. 



If information is available from a variety of 

 sources, say from fishing and from surveys, as in 

 Methot (fn. 2), then each of the vectors and matrices 

 can be partitioned to represent this situation. For 

 example, let y^it) be the observed catches from a 

 survey and yf{t) the observed catches from fishing. 

 Let y{ty = {y sit), y fit))', and partition the H matrix 

 similarly. Then the diagonal blocks of H will con- 



tain the observation dynamics for the survey and 

 for fishing, while the off-diagonal blocks will be zero. 

 Given these modifications, all the algorithms de- 

 scribed previously in this paper can be used to derive 

 estimates for this situation. 



Research or other surveys of the fishery usually 

 occur less frequently than does commercial fishing, 

 causing part of the vector y{t) to be missing at given 

 times periods. Shumway and Stoffer (1982a) and 

 Shumway (1988) gave a straightforward modifica- 

 tion of the Kalman filter for this case. 



DISCUSSION 



I have introduced a new method for estimating 

 population sizes from catch-at-age data that in- 

 cludes, if additive errors can be assumed, many of 

 the previous difference equation models. I show that 

 it is incorrect to treat the unobserved population 

 sizes as parameters to be estimated rather than as 

 missing data. I also show that the minimum mean 

 square estimates of the population sizes are the 

 smoothed estimates rather than the predicted 

 estimates suggested in many papers. The model 

 assumes neither equal variances in the errors in the 

 population dynamics nor in the observation errors 

 and does not require that the errors be uncorrected. 

 For Pacific mackerel, the smoothed estimates are 

 shown to be much more variable than the predicted 

 estimates. 



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