Restrepo and Powers: Application of robust regression to tuned stock assessment models 
153 
Table 3 
West Atlantic bluefin tuna, Thunnus thynnus, relative abundance indices (from ICCAT, 1995). The larval index is in relative 
biomass units, while all others are in relative numbers. The numbers below each index label are the ages or range of ages that 
each index is assumed to represent. TL = tended line, LL = longline, RR = rod and reel, GOM = Gulf of Mexico, NWA = Northwest 
Atlantic. 
Year 
Canada 
TL 
10 + 
Japan 
LLGOM 
10 + 
Japan 
LLNWA 
1-9 
Larval 
GOM 
8 + 
US 
LLGOM 
8 + 
US 
RR 
8 + 
US 
RR 
1-5 
1974 

1.4670 





1975 
— 
1.0200 
— 
— 
— 
— 
— 
1976 
— 
0.8960 
0.8134 
— 
— 
— 
— 
1977 
— 
0.6700 
1.7822 
1.7704 
— 
— 
— 
1978 
— 
0.9350 
1.4621 
4.2341 
— 
— 
— 
1979 
— 
0.9380 
0.5476 
— 
— 
— 
— 
1980 
— 
1.5130 
1.0327 
— 
— 
— 
1.2109 
1981 
2.3489 
0.5610 
1.4812 
0.9575 
— 
— 
0.1274 
1982 
2.1095 
— 
0.7121 
1.1008 
— 
— 
1.3417 
1983 
1.5621 
— 
0.5022 
0.8977 
— 
2.4703 
0.7816 
1984 
1.0718 
— 
0.8527 
0.4750 
— 
1.0949 
— 
1985 
0.5131 
— 
0.9967 
— 
— 
1.0483 
0.5366 
1986 
0.6157 
— 
0.5725 
0.1897 
— 
0.7324 
0.9995 
1987 
0.3991 
— 
1.1490 
0.3236 
1.7544 
0.6933 
1.2138 
1988 
0.6271 
— 
0.8773 
1.4146 
0.6842 
1.3195 
1.6059 
1989 
0.4561 
— 
0.7417 
0.5803 
1.0526 
0.6808 
1.3339 
1990 
0.2965 
— 
0.7754 
0.3446 
1.1404 
0.6204 
0.7331 
1991 
— 
— 
0.7523 
0.2652 
1.5614 
0.7694 
1.3277 
1992 
— 
— 
1.8813 
0.4464 
0.5263 
0.8727 
0.7968 
1993 
— 
— 
1.0675 
— 
0.2807 
0.6981 
0.9912 
requirements. Although the LS solutions for the lin- 
ear case (as described in the previous paragraph) can 
be accomplished with simple matrix manipulations, 
nonlinear LS solutions require iterative computa- 
tions. Stromberg (1993) presented a multistage al- 
gorithm for nonlinear regression that is similar to 
the one outlined above, succeeded by a direct mini- 
mization of the robust objective function by using the 
simplex search of Nelder and Mead (1965). Building 
upon Stromberg’s ideas, we reviewed algorithms for 
an LTS 1 solution to the bluefin tuna SPA. On the 
basis of these results and the work of Stromberg 
(1993), we adopted the algorithm below but acknowl- 
edge that there are many other possible fruitful op- 
tions to be explored, such as “simulated annealing” 
(Corana et al., 1987). Our algorithm uses the fact 
that the simplex search of Nelder and Mead (1965) 
requires p + 1 starting guesses, denoted by v vertices, 
for each of the p parameters being estimated. 
1 Find the LS estimate for the entire data set. The 
estimates (p) LS are used as starting guesses for 
step 2. 
2 Repeat s times: 
a) Set initial parameter guesses at random from 
within 10 times the (p) Lg estimates from step 1. 
b) Find the LTS estimates for the complete data 
set by using the starting values from step 2a. 
c) Restart step 2b until the objective function (ei- 
ther Eq. 2 or Eq. 3) does not change appreciably. 
d) Save the parameter estimates corresponding 
to the (p+l) LTS parameter sets with the lowest 
objective function value. 
3 Initialize the v vertices for the simplex search with 
the best (p+1) parameter sets from the s solutions 
from step 2 and find the LTS estimate for the en- 
tire data set. As in step 2, carry out restarts as 
needed. 
This algorithm is a direct robust minimization 
search that is initialized s times from a Monte Carlo 
grid centered around the LS solution. It is compu- 
tationally intensive, but this seems necessary given 
the multi-modal nature often encountered in the LTS 
or LMS objective function. For this study we used s 
= 500. For both the swordfish nonequilibrium pro- 
duction model and the bluefin tuna SPA analyses, 
step 2 involved 5 restarts on average and thus made 
the total number of minimizations greater than 
2,500. It should be noted that this search algorithm 
does not guarantee that a global minimum LTS so- 
lution is going to be found. Therefore, we favor mul- 
