UD 
NO ww 
-25 i 
-30F 
AIC Nx LOGu? 
a) 
ol 
a 
NO. OF PARAMETER (M + 1) 
Fig. A2.2. Behavior of AIC for Fig. A2.1. 
(From Yamanouchi, et al.4®) 
However, this procedure of guiding M from Eq. A2.6 does not give us the best 
values of the order, because Eq. A2.6 does not include the goodness of fit to the real 
structure of the data or the decrease in reliability with the increase in the number of 
parameters to be estimated. So, to show the penalty for this increase in the order, if we 
add the term 
2 X (no. of parameters) = 2(M + 2) (A2.7) 
to Eq. A2.6, that gives us the AIC, 
N 
AIC(M) = Nlog >’ (vi-Jm,i)” + 2(M + 2). (A2.8) 
t=1 
The number of parameters M + 2 comes from (M+1) 4;'s for do, a1: - - ay plus 1 
for o * that we must also estimate. The behavior of 2(M + 2) is shown in Fig. A2.2 by 
the mark 0 as a straight curve that increases linearly by M. Then AIC, expressed as the 
sum of N logo * and 2 (M + 2), behaves like the 4 in Fig. A2.2 and shows a minimum at 
a certain M, here M + 1 = 3 for the data shown in Fig. A2.1. Thus the order of the 
polynomial is determined as M = 2 by AIC criteria. 
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