MENDELSSOHN: F.STIMATIN(; I'OPILATION SIZES 



period's predicted estimate. (For example, Deriso 

 et al. (1985) suggested using backward VPA or 

 cohort analysis to obtain predicted values.) The 

 filtered estimate x{t |0 is a weighted average of the 

 predicted value of x{t\t - 1) and the observed er- 

 ror in estimating y{t), where the weighting term 

 K{t) (the Kalman gain matrix) is regression-like. 

 Similarly, the covariance of the estimate, as meas- 

 ured by P{t \t - 1), increases due to the prediction, 

 but decreases by the amount K{t)H{t)P{t\t - 1) 

 after the observation has been made. The smoothed 

 estimate x{t\T), which is the correct estimate of the 

 underlying population (it satisfies the conditional ex- 

 pectation), is found by a backward recursion on the 

 filtered estimates, where the filtered estimate is ad- 

 justed by a regression on the error between the 

 smoothed and predicted estimates of the following 

 period. Thus the smoothed estimates correct the 

 predicted estimates both by the error in predicting 

 the observed data as well as by the error in pre- 

 dicting the underlying population when using the 

 filtered estimates of the underlying population. The 

 square roots of the diagonal terms of the various 

 P matrices produced by the Kalman recursions are 

 the standard errors of the predicted, filtered, and 

 smoothed estimates of the population. 



Equations (12) and (13) are the basic form of the 

 state-space model. It is a simple extension to the 

 model to allow any of the matrices F, B, G, or H 

 to be nonlinear functions of the past values of the 

 yit) (see, for example, Shiryayev (1984), section 

 VI. 7), to allow the error vectors w{t) and v{t) to de- 

 pend on past values of the y{t) (Shiryayev 1984), or 

 to allow the v{t) to depend on the underlying state 

 vector x{t) (Zehnwirth 1988). 



In a typical fisheries problem, the matrix H{t) 

 represents fishing. If some age-specific measure of 

 effort E{a,t) is known, then H(t) is a diagonal 

 matrix with E{a,t) on the diagonals. Or it may be 

 assumed that the exploitation rate is of the form 

 s{a)E{t), where E{t) is known and the s(a) values 

 are to be estimated. Then for given values of s(a), 

 the matrix H(t) has s{a)E{t) on its diagonals. The 

 matrix F{t) is formed in a similar manner to repre- 

 sent the population dynamics. 



In some parameterizations, it is assumed that a 

 known vector is subtracted from the state vector 

 either before or after the effect of F on the popu- 

 lation. For example, the known vector might be the 

 catch from the previous time period. The extension 

 to the Kalman filter in this case is straightforward, 

 an example of which can be found in Jazwinski 

 (1970). Essentially, all predicted estimates of the 



state are corrected by the constant amount. The 

 covariance and gain calculations are unaffected by 

 the known vector. 



The Kalman filter. Equations (14) through (21), 

 assumes that the matrices F, B, G, H, R, Q, and I, 

 and the vector jj. are known. For fisheries problems, 

 the matrices F and H usually depend on a set of 

 parameters to be estimated (e.g., F = ml), and R, 

 Q, and ^ are to be estimated. Let be a vector con- 

 taining the parameters that F and H depend on, and 

 let = (0,i?,Q,/i) be the total parameters of the 

 model. Shumway and Stoffer (1982a) showed that 

 conditional on 0, the complete data likelihood is 

 given by Equation (4). They apply a result of Demp- 

 ster et al. (1977), which shows that maximum like- 

 lihood estimates of the parameters can be obtained 

 by finding the conditional expectation (the E-step) 

 of the complete data likelihood with respect to the 

 missing "data" (in this case the missing data are the 

 sufficient statistics of the normal distribution) and 

 alternately estimating the expected value of the 

 missing data, and then maximing the likelihood (the 

 M-step) using the completed data. Shumway and 

 Stoffer (1982a) showed that the expected conditional 

 log-likelihood is given by Equations (6) through (8). 

 All of the terms in this likelihood, for a given value 

 of 0, can be found by the Kalman filter. Moreover, 

 given these values, the maximization problem is a 

 deterministic one. 



If we assume that the matrix F is independent of 

 time and unrestricted, then Shumway and Stoffer 

 (1982a) showed that the maximization step is accom- 

 plished by setting 



F = 5,(l)5,_i(0)-i 



Q = (5,(0) - S,{l)Sf_\{0)St(iy)/T 



T 



R = T-'^ [{y{t) - H{t)x{t\T) 



t=\ 



X {y{t) - H{t)x{t\T)y 

 + H{t)P{t\T)H{ty] 

 M = x{0\T). 



T 



where 5,(j) = I iPit,t - j\T) 



+ xit\T)xit -JlTY). 



(22) 

 (23) 



(24) 

 (25) 



(26) 

 621 



