APPENDIX A2 
POLYNOMIAL MODEL FITTING TO OBSERVED DATA*® 
Given pairs of data {X;, ¥;},i=1- - -N, we often want to express Y; as a 
polynomial of X;. 
If we fit yy.;, a polynomial of x; of order M, as 
Pei = Ag+ aX; + nx ++ > - + aye (A2.1) 
p= 1, 
we can estimate the coefficients dp, a,- : - d,, by the least squares method. Namely, 
computing the mean square of the difference 
N 
1 A 
Ou= W > 0% —Jmiy, (A2.2) 
i=1 
and find the ao, a) - - dm that makeo, minimum. Then from 
80 9 ay+(> xi)a1 +: +(S au = > yi 
0ao 
(A2.3) 
it «0 (Sixi)ao+ (Sx? )ai + : + (Sal ay = ic 
80 (Sx a9 + (Satay +: + (S32 \ay = ye. | 
day 
The do: - - ay are calculated as the solution of the simultaneous Eq. A2.3. 
Here, the most serious problem is determining the order. We can make yy; as close 
as we wish to y; by making the order M very large. However, if M is too large, yi 
follows, even to the random error of the observations. Of course, if M is too small, yy; 
sometimes neglects the variations in the data that really exist, as shown in Fig. A2.1. 
Order 1 is too small, but order 5 seems too large. 
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