MENDELSSOHN: ESTIMATING POPULATION SIZES 



^log|I| - ltr{^~'[P(0\T) + (NiO) - m) 



X (iV(0) - H)^]}, (6) 



a term due to the unobserved dynamics, 



^ 1 log I Q(0 

 2 (=1 



- ^ 1 «r{Q(0-' [(A'(<|r) - FNit - 1\T)) 

 2 ^=1 



X (A^(^|T) - FNit\TV 



+ Pit\T) + FP{t - 1\T)F' - Pit - 1\T) 



X F^ - FP(t, t - 1\T)]}, (7) 



and a term due to the observation process, 



J I log 1 Rit) I 



- - 1 tr{R{ty'[{nit) - H{t)Nit\T)) 

 2 (=1 



X {n{t) - Hit)Nit\T)y 

 + Hit)P{t\T)Hit)]} (8) 



where N{t\s) denotes 

 Nit\s) = E[Nit)\n{l),...Ms)l (9) 



P{t\s) denotes 

 P(t|s) = E[iNit) - Nit\s)) 



X (Nit) - N{t\s)y I nil), 

 ■■■Ms)], (10) 



and Pit, t - l|s) denotes 

 Pit,t - lis) = EliNit) - Nit\s)) 



X iNit - 1) - Nit - l\s)y 

 X ln(l),...,n(s)]. (11) 



As shown in Equations (6) through (8), the proper 

 estimates of the Nit) and the related covariance 

 matrices should be conditional expectations based 

 on all of the data rather than on only the data up 

 to time t. Assuming that all quantities are calculated 

 properly, estimates that include only the data up to 

 time t - 1 are termed "predicted" estimates, esti- 

 mates that include only the data up to time t are 

 termed "filtered" estimates, while estimates given 

 all the data are termed "smoothed" estimates. I 

 shall show below that the appropriate formulas for 

 predicted, filtered and smoothed estimates differ 

 significantly. Thus using Equation (5) as the likeli- 

 hood and treating the Nit) as parameters does not 

 produce proper estimates of the Nit). 



In the rest of this paper, I review state-space 

 models and methods for estimating both the param- 

 eters and the unobserved components of the model. 

 A very readable background for what follows is 

 chapter 3 in Shumway (1988). The estimation 

 scheme described does not require that the compo- 

 nents of R, Q, and 2 have equal variance and are 

 uncorrected. Explicit estimates of these matrices 

 are given. Then I show that a variety of age-based 

 models proposed in the literature can be formulated 

 as a state-space model, but that the formulations as 

 presented make the same error of treating the un- 

 observed components as parameters, and assume 

 zero covariance in the errors. Auxiliary information 

 as in Deriso et al. (1985) can be put into this for- 

 mat. And I show that the state-space formulation 

 can include multiple observations of the population, 

 but where some of the observations are missing. 

 This can arise when the population is observed from 

 fishing and from a variety of surveys, but some of 

 the surveys are not done every year. This is essen- 

 tially the problem discussed in Methot^. I give true 

 maximum likelihood estimates for this model, allow- 

 ing the different observation processes to have dif- 

 ferent error structures and estimate the relative 

 weight that should be given each. This is a sig- 

 nificant advance over the procedure in Methot 

 (fn. 2). 



A related paper is the analysis of Brillinger et al. 

 (1980) who use a modified Kalman filter and max- 

 imum likelihood estimation to estimate the average 

 birth and death rates and population structure of 

 Nicholson's blow-fly data when only total numbers 



^Methot, R. 1986. Synthetic estimates of historical abundance 

 and mortahty for northern anchovy, Engraulis mordax. Adm. 

 Rep. LJ-86-29. Southwest Fisheries Center La JoUa Laboratory, 

 National Marine Fisheries Service, NOAA, P.O. Box 271, La Jolla, 

 CA 92038. 



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