MENDELSSOHN: ESTIMATING POPULATION SIZES 



I also suggest a new parameterization for the age- 

 specific exploitation rate q{a,t). If it is assumed that 

 q{a,t) = s{a)f{t), then the model is over determined. 

 I put a smoothness prior on f(t) in order to obtain 

 a tradeoff between the degree of smoothness mf{t) 

 as a function of time versus fidelity to the data. The 

 degree of differencing can be treated as a hyper- 

 parameter of the model to determine the optimal 

 amount of differencing given the data. 



An advantage of the approach of this paper is that 

 the calculations are straightforward and simple to 

 program, and explicit formulas are given for the op- 

 timal parameter values at each iteration of the EM 

 algorithm. Additional optimization software is not 

 required to perform the calculations. Moreover, 

 properties of the Kalman filter and the EM algo- 

 rithm are well known. There is a large literature giv- 

 ing variants of the filter to calculate sensitivity to 

 model misspecification, and recursive formulas for 

 the derivatives with respect to a given parameter 

 of the model also exist in the literature. 



It is also simple to include environmental variables 

 into the formulation, either as additional state vari- 

 ables or as fixed effects in the observation equation 

 or both (see Sallas and Harville [1988] on how to 

 estimate the fixed effect parameters within the con- 

 text of Kalman filtering). Thus the influence of the 

 environment can be modeled directly, rather than 

 resorting to the conventional practice of obtaining 

 population estimates first and correlating these 

 estimates with the environmental variables second. 



A disadvantage of my approach is that there is 

 no guarantee that any of the estimates of the under- 

 lying population sizes will be positive. The popula- 

 tion sizes are treated as normal random variables, 

 and it is quite possible for the additive corrections 

 in the filtered or smoothed estimates to make small 

 population sizes negative if the observation error is 

 large. P. Sullivan (fn. 3) has found that for a length- 

 based model the Kalman filtering approach works 

 best when there are pulses in the recruitment, that 

 is, when the population is not in equilibrium. The 

 likelihood surface is such that without recruitment 

 pulses it is difficult to estimate the parameters of 

 the growth-curve. Most fisheries are not in equilib- 

 rium, however. As the models in this paper do not 

 contain a growth-curve, it is unclear if a similar find- 

 ing will be valid. 



Some of my results suggest that the estimates are 

 sensitive to the form of the model chosen for the 

 population dynamics. This is not surprising, because, 

 unlike most missing data problems, the missing part 

 of the data is never observed directly, but only 



through the presumed form of the dynamics. For 

 example, when modeling catch (or catch per unit ef- 

 fort) against an environmental variable, catch data 

 often are not available for all periods. But there are 

 at least some periods when both variables are ob- 

 served, which can be used to estimate the relation- 

 ship between the two sets of variables. This rela- 

 tionship is used to produce the smoothed estimates 

 of the missing data. For estimating population sizes 

 from catch-at-age data, the a priori estimate of the 

 form of the observation equation replaces this em- 

 pirically derived estimate. 



In many of the references cited, multiplicative 

 errors are preferred in the observation equation 

 because variances appear to change with the size 

 of the population. My experience is that relaxing the 

 assumption of equal, uncorrelated errors appears to 

 at least partially take into account the observed 

 differences. If the model estimates are not satisfac- 

 tory, assuming additive, gaussian errors, then the 

 regular EM algorithm can be used to properly esti- 

 mate the smoothed estimates of the underlying 

 population. However, the EM algorithm requires the 

 complete data likelihood as well as the expectation 

 of the log-likelihood with respect to {y{l), . . .,y(T)). 

 In multiplicative models, assumptions about the 

 error structure can lead to very complicated multi- 

 variate distributions for the complete data due to 

 the Jacobian of the transformation. The conditional 

 expectation of the log-likelihood may have to be 

 evaluated by numerically integrating a nontrivial 

 multiple integral. Certainly, as a first pass, the 

 simpler techniques of this paper would appear to 

 have a lot to offer as an alternative. 



LITERATURE CITED 



Akaike, H. 



1979. Likelihood and the Bayes procedure. In J. M. Ber- 

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 Valencia. 

 Anderson, B. D. 0., and J. B. Moore. 



1979. Optimal filtering. Prentice Hall, Englewood Cliffs, 

 NJ, 357 p. 

 Ansley, C. F., and R. Kohn. 



1986. On the equivalence of two stochastic approaches to 



spline smoothing. In J. Gani and M. B. Priestly (editors), 



Time series and allied processes, p. 391-405. J. Appl. Prob. 



23A. 



Brillinger, D. R., J. Guckenheimer, p. Guttorp, and G. 



OSTER. 



1979. Empirical modeling of population time series data: The 

 case of age and density dependent vital rates. In Some 

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