Restrepo and Powers: Application of robust regression to tuned stock assessment models 
155 
tually all of the 500 solutions converged to the same 
value. The outlier detection criteria identified five of 
the original 27 data points (19%), all of which oc- 
curred before 1981 (Fig. 1). Predicted index values 
with both the LTS and trimmed LS solutions were 
higher than the LS predictions prior to the late 1980’s 
and lower than LS predictions in recent years (Fig. 1). 
Predicted relative biomass values with either the LTS 
or trimmed LS solutions are higher than the initial LS 
solution (ICCAT, 1995), particularly in the 1990’s (Fig. 
2) and suggest less of a decline in the population. Abso- 
lute biomass predictions with the LTS method were 
generally higher than those from the initial LS solu- 
tion, whereas trimmed LS solutions were lower (Fig. 
2). The trimmed LS solution results in biomass levels 
Q Q - Ll J I II I i — LJ — LJ — l_l — I — I — L .] 1 I I L_l L_J I L_J LJ L_i_l L_J I 
70 75 80 85 90 95 00 05 
0 Ll 1 i i i i i i i i i i i 
70 75 80 85 90 95 00 05 
Year 
Figure 2 
Predicted biomass relative to biomass at maximum sus- 
tainable yield (B/B MSY , top panel) and absolute biomass 
(bottom panel) resulting from LS, LTS, and trimmed LS 
solutions. The left side of the graphs show the production 
model estimates. The right sides of the graphs are projec- 
tions made with the fishing mortality rate at maximum 
sustainable yield (F MSY ) and with the fishing mortality rate 
in 1993 (F 93). Ascending limbs were projected by using 
F msy . descending limbs by using F 93. 
that are lower than those in the other two methods; 
however, the decline over the time series is less. Bio- 
mass projections were made under two strategies: 1) a 
recovery strategy in which future fishing mortality rate 
was fixed at the value that would produce maximum 
sustainable yield and 2) a status quo strategy in which 
the fishing mortality would be fixed at the 1993 level. 
The LTS and trimmed LS projections indicate that both 
recovery and decline is not as rapid as that predicted 
from the initial LS solution (Fig. 2). 
The robust regression techniques applied here tend 
to provide a better fit to the index data points in re- 
cent years at the expense of the data points in the 
earlier years of the series. Indeed, several of the 
points identified through the outlier detection pro- 
cess were those data points for which there was much 
debate regarding variability and bias (ICCAT, 1995). 
However, some of the data points identified here were 
not identified by ICCAT (1995); therefore, we reem- 
phasize the point that the selection of outliers should 
be based on objective criteria. 
Bluefin tuna SPA 
As mentioned before, a high-breakdown robust re- 
gression objective function can possess multiple 
minima. Figure 3 illustrates this point with the LTSj 
objective function plotted around ± 50% of the final 
estimate for one of the parameters, while all other 
parameter values were fixed at their solution. The fig- 
ure highlights the need for an exhaustive search ow- 
ing to the multimodal nature of the response surface. 
Figure 4 shows the observed indices of relative 
abundance in the first column, the scaled residuals 
o 
0 05 -I 1 1 
0.25 0.5 0 75 
Parameter estimate (1CT 4 ) 
Figure 3 
Trimmed squares objective function (Eq. 2) plotted around 
the solution for one of the parameters estimated in the 
bluefin tuna sequential population assessment (catch- 
ability for the U.S. rod and reel large fish index, Table 3). 
The plot shows that multiple local minima can occur in 
robust regression problems. 
