412 



Fishery Bulletin 88(2). 1990 



ARIMA (j),d,q){P,D,Q)S, 



can be described by the following equation: 



{l-niBP){l-N^BP) 



a - B^iXl - B^>)X, = (l - UiBi)(l - U^BQ)et 



where X, = the value at time t, Bi' is a backward shift 

 operator that is used as Bt'Xj = X,_f,, rii, N^, Ui, and 

 Ui = arithmetic coefficients, e, = error term at time 

 t, p = order of autoregressive term (AR term), d = 

 degree of differencing involved to achieve stationar- 

 ity (I term), q = order of moving average term (MA 

 term), and 5 = seasonality (number of periods per 

 season); P. D, Q = seasonal terms (corresponding to 

 p,q,d respectively). 



Based on the examination of autocorrelation and par- 

 tial autocorrelation functions (not shown here) of the 

 original logarithmically transformed series, the follow- 

 ing model was fitted to the logarithms of the raw data, 



ARIMA (1,1,0)(1, 1,0)12 (Table 1), 



or, after substituting the autoregressive coefficients 

 and expanding the backward shift operator (Xf = 

 logarithm of raw data), 



Xt = 0.792X,_i-0.208X,_2 + 0.684X(_i2 



-0.862;^,_13-(-0.138X,_i4-h0.336X,_24 



-0.266X,_25 -0.07X,.2fi + e,. 



The unit of time (t) is one month. The model was 

 estimated using the approximate maximum-likelihood 

 algorithm of McLeod and Sales (1983). Parameters 

 were estimated using backcasting with length of 13. 



The examination of the cumulative periodogram of 

 the residuals (not shown here) (Box and Jenkins 1976) 

 indicated that residuals approximated random noise. 

 Actual catches for January 1985-December 1986, not 

 used in the development of the model, and forecasts 

 for those years are plotted in Figure 3. The coefficient 

 of determination (Table 1) was found to be r- = 0.94 

 for January 1985-December 1986 (for the untrans- 

 formed series). 



Mean absolute percentage error (MAPE) (Table 1) 

 for January 1985-December 1986 was 20.4%. Except 

 for March and October 1985 and February, October 

 and November 1986, when the absolute percentage 

 error (APE) was higher than 26% (Table 1), monthly 

 catches were predicted reasonably accurately within 

 an APE range of 2-26% (MAPE of the remaining 19 

 forecasts was 11%) (Table 1). 



Table 1 



Parameter estimates of the anchovy fishery model. MAPE 

 = mean absolute percentage error, APE = absolute percent- 

 age error, r ^ = coefficient of determination. 



APE = 



Absolute (actual catch at time t - forecast at time t ) 100 

 Actual catch at time t 



MAPE = mean of APE, and 



r -■ for 1985-86 

 Residual mean 

 MAPE 1985 

 MAPE 1986 

 MAPE 1985-86 

 APE > 26% 



March 1985 



October 1985 



February 1986 



October 1986 



November 1986 

 MAPE excluding APE > 26 



0.94 

 0.007 



13.3 



27.4 



20.4 



31.5 

 33.5 

 47.7 

 65.9 

 95.6 

 11 



Figure 3 



Comparison between actual monthly catches (D) of anchovy in Greek 

 waters during January 1985-December 1986 and the forecasts ( + ) 

 estimated from the model. 



The error in predicting the February-March and 

 October-November catches may be attributed to the 

 highly variable timing of the initiation of the inshore 

 (prespawning migration, which occurs sometime in 

 March) and the offshore migrations (in fall) both of 

 which are expected to be affected greatly by ocean- 



