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Fishery Bulletin 109(2) 
Table 2 
Candidate set of seasonal autoregressive integrated moving-average models. The “rule” column displays the mathematical 
expression used to determine the autoregressive components ( p ) and moving-average components ( q ) of the candidate models. 
“Max AR term” and “Max MA term” columns display the maximum autoregressive (AR) and moving-average (MA) lags included 
in the model equations, with respect to the original (x t ) and 12-month differenced log 10 -transformed mean-centered data 
(w t ='V 1 12 y t =V 1 12 (log 10 x f -4.022)), respectively. 
Model structure 
No. of models 
Rule 
Max AR term 
Max MA term 
(p,0,q)x(0,l,0) 12 
325 
q<25— p; p<24 
W t-24> X t-36 
Z t-12 
(p, 0 ,< 7 )x(l,l, 0 ) 12 
91 
q<13-p; p<12 
w t-24 > x t-36 
Z t-12 
(p, 0 ,< 7 )x( 0 ,l,l) 12 
91 
q<13-p; p<12 
w t-12 ’ x t-24 
Z t-24 
(p,0,q)x(l,l,l) 12 
1 
II 
© 
II 
o 
w t-12’ x t-24 
Z t-12 
structure so that only the nonstationarity of variance 
needs to be addressed before model fitting. The meagre 
time series (x t , t= 1, ... ,60) was seasonal and exhibited 
no trend (Fig. 1A), but annual variance-mean plots in- 
dicated an increase in variance with the series mean. 
To correct this, we evaluated Box-Cox transformations 
(Box and Cox, 1964) and found that a log 10 transforma- 
tion successfully stabilized the variance of the series. 
Accordingly, we log-transformed the data, subtracted 
its mean, and then used the mean-centered log-trans- 
formed data set ( y t , t= 1, ... ,60) as input to the SARIMA 
analyses (Fig. IB). 
Data modeling 
We fitted SARIMA models to the meagre data using a 
semi-automated approach based on a combination of the 
Box-Jenkins method with small-sample, bias-corrected 
Akaike information criteria (AIC c ) model selection (Roth- 
schild et ah, 1996; Brockwell and Davis, 2002). This 
approach involved three major steps: 1) selection of the 
candidate model set; 2) estimation of the model and 
determination of AIC c ; and 3) a diagnostic check. Details 
on the notation and model selection procedures used to 
fit SARIMA models to short time series are given in 
Appendices 1 and 2. 
Selection of the candidate model set was carried out 
by first analyzing sample estimates of the autocor- 
relation function (ACF) and partial autocorrelation 
function (PACF) in order to select the three major 
orders of the SARIMA models: d, D, and S. In the 
meagre case, we concluded that a configuration with 
d= 0, D = 1, and S = 12 should be adopted (see Results 
section). Consequently, a SARIMA(p,0,q)x(P,l,Q) 12 was 
selected as the basic model structure of the candidate 
set, with p, q, P, and Q left to vary. There is no a 
priori method to determine the maximum value that 
p, q, P , and Q can take, but the maximum orders of 
the models are obviously restricted by sample size. 
In our analysis, we conditioned p, q, P, and Q to the 
upper boundary max(p+q+SP+SQ) = 24 and p+q< 12 
(Table 2), which caused the maximum possible term 
of any SARIMA model to be x t _ 36 and the maximum 
possible number of parameters to be 13. We found 
this procedure to provide a good compromise between 
model complexity and the convergence of estimation 
algorithms. 
Model estimation was carried out by using maximum 
likelihood methods, after conditional sum of squares 
estimation of the starting values (Brockwell and Da- 
vis, 2002). Given the large number of models requiring 
estimation (Table 2), we developed a semi-automated 
software routine in R, vers. 2.5.1 (R Development Core 
Team, 2007) that estimated the models and output 
their AIC f values. This routine used several functions 
incorporated in the R packages “stats” (R Development 
Core Team, 2007), “tseries” (Trapletti and Hornik, 
2007), and “FinTS” (Graves, 2008). After estimation, 
the model with the minimum AIC c . was selected for 
further analysis. 
Diagnostic checks on the AIC c -selected model involved 
the following steps: 1) verification of the resemblance 
of residuals to white noise (ACF plots, Ljung-Box test, 
cumulative periodogram test); 2) tests on the normality 
of residuals (Jarque-Bera and Shapiro-Wilks tests); and 
3) confirmation of model stationarity, invertibility, and 
parameter redundancy (Shapiro et ah, 1968; Ljung and 
Box, 1978; Jarque and Bera, 1987; Box et al., 2008). All 
tests were carried out at a significance level of a=0.05. 
The variance explained by the model was determined 
as 1 - d 2 / c 2 (Stergiou, 1990a). 
Forecasts and model performance 
We evaluated 12 months of model forecasts, using the 
last month of the fitting data set as the forecast origin 
(i.e., April 2007). Forecasts were obtained in the mean- 
centered transformed scale ( y h , h = 1,...,12) and in the 
original scale of the data (x h , h- 1,...,12), after correcting 
for back-transformation bias (Pankratz, 1983). SARIMA 
model performance was assessed by comparing A-step 
forecasts [x h and y h ) with monthly landings observed 
between May 2007 and April 2008 ( x h and y h ). This was 
done by evaluating monthly forecast errors (e.g., e h - x h 
- x h ) and then considering a set of accuracy measures: 
1) annual root mean-square error (RMSE); 2) mean 
error (ME); 3) absolute percent error (APE /; ); 4) mean 
absolute percent error (MAPE); and 5) annual percent 
