Prista et al: Use of SARIMA models to assess data-poor fisheries 
175 
error (PE) (Mendelssohn, 1981; Hyndman and Koehler, 
2006). From these, RMSE was evaluated in the trans- 
formed scale to allow its comparison to a , and all others 
were computed in the more user-friendly original scale 
of the data. Additionally, we compared the forecasting 
performance of the SARIMA model against two simple 
naive forecasting models (naive model 1 or NM1, and 
naive model 2 or NM2) (Noakes et al., 1990; Stergiou 
et al., 1997). The latter represented ad hoc forecasting 
models likely to be used in data-poor fisheries with short 
time series of landings: with NM1, future landings were 
assumed to be equal to the landings registered in the 
previous year; and with NM2, future landings were 
assumed to be equal to the average monthly landings 
registered in the fitting period. We also evaluated the 
Kitanidis and Bras (1980) coefficient of persistence 
(P) that summarizes forecasting results by comparing 
them with those of a naive model where landings at 
time t + 1 are assumed equal to landings at time t. This 
coefficient takes values smaller than or equal to 1, with 
P=1 representing perfect model forecasts. 
Monitoring of fisheries 
SARIMA models predict the future on the assumption 
that the statistical properties of the process generating 
the data remain the same over time (Box et al., 2008). 
When framed within the perspective of statistical pro- 
cess control (e.g., Scandol, 2005; Box et al., 2008; Mesnil 
and Petitgas, 2009), this characteristic allows the pre- 
dictions of well-developed SARIMA models to be used 
as “guidelines” to monitor future observations. When 
a SARIMA model is found that appropriately fits the 
landings data, a significant departure of its forecasts 
from future observations can be seen as an indication 
that changes in the underlying fishery process have 
occurred (=out-of-control situation). In contrast, if such 
a significant departure does not take place, then there 
is no indication for such changes (= in-control situation). 
From a data-poor fisheries perspective, such a distinction 
means that if funding is limited and multiple fisheries 
require assessment, research and management efforts 
should be allocated to fisheries displaying out-of-control 
decreasing trends in production rather than to fisher- 
ies that remain stable or display in-control increasing 
trends (Scandol, 2003, 2005). 
The distinction between in-control and out-of-control 
landings requires a set of detection limits. To date, 
process-control detection limits for fisheries indicators 
have been derived mostly from historical reference da- 
ta (Scandol, 2003; Mesnil and Petitgas 2009; Petitgas, 
2009). However, most fisheries have only a few years 
of collected data and consequently historical limits 
are difficult to estimate. In such situations, model- 
based detection limits like the prediction intervals 
(Pis) of SARIMA models (Chatfield, 1993; Box et al., 
2008) provide easy-to-compute detection limits that 
explicitly take into account the correlation structure 
of the data. SARIMA Pis resemble confidence inter- 
vals for model forecasts and consist of upper and lower 
boundaries that encompass a 1-a probability region 
for future forecasts (Chatfield, 1993). Their main use 
is to convey the uncertainty around forecasts (De 
Gooijer and Hyndman, 2006). However, because pre- 
diction intervals encompass only future observations, 
as long as no structural changes take place in the 
underlying process (Chatfield, 1993), their boundar- 
ies can be used to monitor univariate data such as 
fisheries landings. 
To date, the prediction intervals (Pis) from SARIMA 
models have seldom been reported in fisheries literature 
and, when they have, with little detail and discussion 
(Table 1). To monitor the landings of the meagre fish- 
ery we used two types of Pis: single step Pis ( PI SS ^ ) 
and multistep Pis (PI ms /,). Single step Pis refer to a 
single monthly forecast (e.g., h = 3) and are useful for 
determining whether a specific monthly observation is 
an outlier at a given significance level a. Multistep Pis 
encompass a 1-a prediction region that is a simultane- 
ous PI for all observations registered up to a certain 
/?-step and are useful in detecting systematic depar- 
tures from historic al patterns. We calculated PI SS h as 
y/> ± t df,a/ 2 'J PMSE h where PMSE h is the expected mean 
squared prediction error at step h and df=N-DS-d-r 
(Chatfield, 1993; Harvey, 1989). In the calculation of 
multistep Pis, we used a conservative approach based 
on a first-order Bonfer roni ine quality, whereby PI ms h 
is given as y h ± t df a/2h J PMSE h and joint prediction in- 
tervals of, at least, 1-a around the point forecasts are 
obtained (Chan et al., 2004). 
Results 
Data modeling 
Large autocorrelations were recorded for lags 1, 2, 11, 
12, 23, and 24 with values 0.68, 0.32, 0.44, 0.46, 0.28 
and 0.31, respectively (Fig. 2). The sharp decrease in 
autocorrelation values after lag 2 (0.07 at lag 3) indi- 
cated no evidence of a long-term trend; consequently, 
there was no need to include a first-lag difference term 
in the SARIMA model structure (d= 0). In contrast, large 
autocorrelation values were registered at annual lags 
(and its multiples) which indicated the need to include 
a 12-month difference term in the models (S=12, D = 1) 
(Fig. 2). The ACF and PACF plots of the differenced 
series provided further support for these conclusions 
(Fig. 2). Accordingly, a SARIMA(p,0,q)x(P,l,Q) 12 was 
selected as the basic structure of the SARIMA candi- 
date set. 
Out of all models in the candidate set, a SARI- 
MA(0,0,5)x(l,l,0) 12 was selected as the best model 
for the meagre data (-2 In (L) = - 26.32, n- 48, r=7, 
AIC c =-9.52). This model had the following equation: 
(1+0.65, 10| R 12 ) vl i235= d+0.63, 19| £+0.56, 15) R 2 + 
0.51, 17| B 3 + oT93, 181 R 4 + 0^60, 21} B 5 )z t , 
with a noise variance estimate of &= 0.025 and 
