172 
Fishery Bulletin 1 14(2) 
Catch (t) 
Limit (m) 
m 
<$> 25 
I * 36 
B 47 
V 59 
O 70 
O 80 
A 9? 
+ 104 
Figure 4 
Linear relationships between uncorrected predicted biomasses, obtained from models with different 
depths to set the limits between small and large tunas, and the real catch for the 21 samples collected 
by a commercial Spanish purse seiner in the central and eastern Atlantic Ocean between 2009 and 
2011. The coefficients of correlation and determination of these relationships were used to select the 
optimum depth limit between small and large tunas for the particular case of this study. 
specific case of these 21 fishing sets in the Atlantic 
Ocean, the best values for coefficients of correlation (r) 
and determination (r 2 ) (r=0.85, r 2 - 0.73; Table 1, Fig. 
4) provided by the optimization process led us to recon- 
sider and modify the original selection of vertical depth 
limits. Because of a known potential overlap between 
species and fish sizes in the vertical depth zones and 
the higher number of small tuna than large tuna in- 
dividuals occurring at DFADs, we opted to keep only 
the TS and weight values of small tunas for the whole 
depth range (i.e. , corresponding to a depth limit of 115 
m (Table 1 and Fig. 4). This is the more ecologically 
coherent choice because the presence of small tunas at 
DFADs usually exceeds 95% of the catch (Fonteneau et 
al., 2013), which was also supported by the proportion 
of small tunas found in our 21 fishing sets. 
Defining an error function In order to reduce uncer- 
tainty and improve the accuracy of biomass estima- 
tion with the new method, the uncorrected predicted 
biomass was compared and calibrated to the 21 real 
catches. Error (in t) of the new method was modeled 
with different regression models (polynomials of order 
2 and 3, generalized linear models [GLM], and gener- 
alized additive models [GAM]; Hastie and Tibshirani, 
[1990]; Venables and Dichmont, [2004]; Wood, [2006]) 
(Fig. 5) as a function of the uncorrected predicted bio- 
mass, which was corrected as follows: 
B c = B u - f(B u ) + £, (4) 
where B c = the corrected predicted biomass; 
B u = the uncorrected predicted biomass of the new 
method; and 
f(B u ) - the error modeled following different re- 
gression methods as a function of uncor- 
rected predicted biomass (/IB U ]=-0.318 
B u 2 +0.9951 B u -17.598 for the polynomial 
of order 2; /TBJ=-0.0346 B„ 3 +0.7223 B u 2 - 
6.7657 B u — 5.5385 for the polynomial of 
order 3; /IB J=-4.7605 B w -3.0981 for the 
GLM; and /IB J=absolute error of B u ~s(Bu) 
as a generic formula for the GAM. Note 
that no explicit expression for the estimat- 
ed smooth terms (s) is available for GAMs 
(see Hastie and Tibshirani, 1990). For the 
GAM case, values used to correct B u were 
extracted by using the predict. gam func- 
tion of the mgcv package (vers. 1.7.29); 
and £ is the assumed error (0 in this case). 
All the regression models were fitted by 
using the mgcv package, vers. 1.7.29, for R 
software, vers. 3. 2.1 (R Core Team, 2015). 
The prediction capacity of each model was measured 
by computing both r and r 2 between the corrected pre- 
dicted biomass and the 21 real catches, and by using 
