ORDER 2 
Y-AXIS 
0.00 0.80 1.59 2.39 3.18 3.98 4.78 5.57 
X-AXIS 
Fig. A2.1. Fitting of polynomials to the observed data. 
(From Yamanouchi, et al.4®) 
All observed data include the statistical error. Accordingly, when x; are given, 
instead of Eq. A2.1, y; should be represented by a regressive polynomial, 
Vi = Ag + Q4X; + ax? +- ° -+ayx +e;. (A2.4) 
Here €; is a probability variable that follows the normal distribution with 0 mean and 
variance a2. Here, we can define that model fitting is the statistical selection of a model 
to fit best to data in statistical meaning. When we use Eq. A2.4 as the model, we can use 
the minimum AIC criteria or the MAIC method to determine the order M. When y; is 
approximated by a polynomial of order M as in Eq. A2.1, the coefficient ;’s has the 
values determined by the least squares method, 
N 
—L=min(1/N) >) Oi-yy,))- (A2.5) 
i=] 
Then let us investigate the behavior of this mean square of the residual error 
1< Yo 29) 
Wo ae =¢ 
1=1 
or N times the logarithm of Eq. A2.5 
Nlog ) Qi-Sy, = Nloga?. (A2.6) 
This value A2.6 is shown in Fig. A2.2 by the mark O and tells us that the larger the value 
of M, the smaller this value A2.6 is, or the larger the order M, the smaller the residual 
9) 
error G¢. 
262 
