BARTOOancl PARKER: STOCHASTIC AOE-FREUUKNCY ESTIMATION 



Tablk 1.— Input and estimated numbers-at-age for both the 

 deterministic (col. 3) and stochastic (col. 5) models, with the in- 

 put numbers-at-age in column 2. The differences between the 

 input numbers-at-age and the deterministic estimates are 

 given in column 4. 



greater than the maximum age, 10. Thirty-five 

 had lengths greater than L x and, consequently, 

 were not classifiable. 



BIAS RESOLUTION 



With estimated variance of length-at-age a sto- 

 chastic model can be built from the von Berta- 

 lanffy relationship: For any age the probability 

 of a specific length interval is the probability of 

 that interval taken over all length intervals con- 

 taining that age. Thus for all ages a probability 

 matrix ("P"-matrix) of dimension r by c can 

 be computed, where r = the number of rows, or 

 length intervals, and c — the number of columns, 

 or ages, then P (1,1) = (max. length, min. age). 

 If the number-at-age vector is "a" (ai = (min. 

 age)) and the number-at-length vector is L (L\ 

 = (max. length)), then 



P a = L. 



(3) 



And as long as r > c then the numbers-at-age vec- 

 tor can be uniquely solved via least-squares: 



a = (P'PY'P'L 



(4) 



Applying this stochastic method (Equation (4)) 

 to the previous example, the numbers-at-age 

 generated from the number-at-length vector are 

 given in column 5 of Table 1. Since the probabili- 

 ties of the P-matrix are the same as those used to 

 generate the number-at-length vector, it is not 

 surprising that the solution yields unbiased re- 

 sults. This computed example illustrates that the 

 stochastic method yields unbiased estimates of 

 age frequency. 



PACIFIC BONITO 



For the Pacific bonito, Sarda chiliensis, of the 

 eastern tropical Pacific, Campbell and Collins 

 (1975), using ages determined from otoliths, esti- 

 mated the von Bertalanffy growth parameters to 

 be L^ = 76 87 cm, t = -0.785 yr. and k = 0.6215. 

 Numbers-at-length for 1 cm intervals for ages I 

 through V are shown in Figure 1 with the corre- 

 sponding length-frequency plot in Figure 2. 

 These numbers represent the 1973 catch from 

 California waters and are a subset of the data 

 used to estimate the von Bertalanffy parameters. 

 This example serves to demonstrate bias and 

 illustrate application of the stochastic method. If 

 desired a variance-covariance matrix can be gen- 

 erated (Draper and Smith 1981) to estimate pre- 

 cision in the resulting age structure. 



80 r- 



70 - 



E 

 o 



h- 60 

 O 



z 



HI 



50 - 



40 



IV 



V 



AGE (years) 



Figure 1.— Numbers-at-age by length in 1 cm intervals for the 

 Pacific bonito. Sarda chiliensis. Data from 1973 California 

 landings (Campbell and Collins 1975). 



The length-frequency information and von 

 Bertalanffy parameters are used to generate 

 both deterministic and stochastic estimates of 

 numbers-at-age. The estimated length-at-age 



93 



