FISHKRY BULLETIN: VOL. m. NO. 4 



Shumway and Stoffer (1982a) gave a recursive for- 

 mula for calculating P(<,^ - 1|T) while performing 

 the backward smoothing recursion. 



If we assume that F is constrained to be of the 

 form F = TfiD where m is a constant and D is a 

 known matrix, then Shumway and Stoffer (1982b) 

 showed that 



Q = (C - rhBD' - mDB' + rh'^DAD') (27) 



m = 



tr{Q-^DAD') 



(28) 



Equations (27) and (28) can be solved by taking 

 an initial guess for m, then iteratively solving for 

 Q and m until the values converge. 



Finally, we can make explicit the effect of assum- 

 ing equal variances and no co variances for both w{t) 

 and v(t). For given estimates of Q and R, since both 

 are square, symmetric matrices, they can be fac- 

 tored as 



Q = UDU' 



R = LL' 



where U'ls an upper triangular matrix, Z) is a diag- 

 onal matrix, and L is the lower triangular square 

 root of R. To obtain an underlying population 

 dynamic that has an uncorrelated error vector w{t) 

 and uncorrelated observations with variances of 1, 

 we make the following transformations: 



G = GU 



y{t) = L-'yit) 



H{t) = L-'Hit) 



v{t) = L-h{t) 



and replace G, w, y, H, and v in Equations (14) 

 through (21) with the transformed values. Then v{t) 

 has covariance matrix 7, and 'w{t) has co variance 

 matrix D. The assumption that both the error in the 

 dynamics and in the observations are equal, further 

 constrains the values of D to be identical. This is 

 a very strong assumption. 



AN EXAMPLE 



As an example of these methods, I use the data 

 622 



for Pacific mackerel published in Parrish and 

 MacCall (1978). I emphasize that I am only using 

 these data for illustrative purposes and do not claim 

 to be making a careful, thorough reexamination of 

 the problem. Though m can be estimated using 

 Equation (28), I assume that the value of m is known 

 a priori. If I were to use a different value of m, it 

 would be difficult to judge to what extent the new 

 estimates differ solely due to the different mortal- 

 ity rate, rather than due to the estimation scheme. 

 I assume, as in the reference, that the mortality rate 

 m is equal to 0.5, so that the F matrix in my nota- 

 tion is a matrix with a value of 0.6065 in position 

 (i, i - 1), i = 2, . . . ,7, corresponding to the under- 

 lying dynamics for age groups 1 through 6. 



Recruitment in Pacific mackerel is highly variable. 

 I want to obtain estimates of recruitment that ac- 

 curately reflect this variability while still being con- 

 sistent with the observed data. Also, I do not want 

 to a priori assume a functional relationship between 

 recruitment and population size. To this end, I 

 assume that the recruitment time series, after tak- 

 ing differences of a given order, is a random vari- 

 able, i.e., 



V'^r(0 = w{t) 



(29) 



where w{t) is a. normal random variable with a mean 

 of zero and with an unknown variance o" and V''' 

 denotes A;th order finite differencing. Akaike (1979) 

 originally showed that this formulation is the dis- 

 crete equivalent of fitting a spline to the data (in 

 this case as a function of time), where the estimate 

 of the variance o" expresses the tradeoff between 

 the degree of smoothness in the fitted curve with 

 fidelity to the observed data. In this approach, k and 

 the variance of w{t) are treated as hyperparameters 

 of the model. A fitting criterion such as AIC is then 

 used to determine the best value of k given the data. 

 Following Kitagawa and Gersch (1984), I could use 

 smoothness priors to more generally decompose 

 recruitment as 



r{t) = Tit) + Sit) + 4(0 + wit) 



(30) 



where Tit) is a trend term (as in Equation (29)), Sit) 

 is a seasonal term, and |(0 is an irregular stationary 

 term. A decomposition such as Equation (30) would 

 be useful, for example, in modeling the monthly an- 

 choveta recruitment considered in Mendelssohn and 

 Mendo (1987). However, for convenience in this 

 paper, I restrict recruitment to be of the form in 

 Equation (29). 



