ON THE EFFECTS OF CIRCULAR BOLT PATTERNS ON THE ...

of the dependent variable predicted from the best fit equation is x i , for any particular set of values,
'
li
'
2i
'
3i
'
X , X , X ,..., X ni , while it is measured (or directly determined) that the value isx. i Deviation of the
predicted value from the measured value is given by:
'
i
'
'
' '
C C x C x C x
xi x xi
... n ni
(5.7)
0
1 1i
The sum of the squares,S for m number of data is given by:
S
m
i1
' 2
xx i
i
2
2i
' (5.8)
'
The unknown coefficients C 0 , C1,
C2,...,
Cn
are determined by minimizing the quality S with respect
to each coefficient; in other words, by setting it equal to zero, as shown below.
S
C
'
0
S
C
1
S
C
This will result in 1
2
S
...
C
n
0
95
(5.9)
n linear simultaneous equations from which the coefficients C 0 , C1,
C2,...,
Cn
can be determined. To determine C , the anti-logarithm of 0
0 C must be found.
A “goodness of fit” of the prediction equation is a comparison of S, the sum of the squares, and the
deviations for the constant term C0 above. The constant term model is:
C0
'
S (5.10)
and the sum of the squares of this model can be written as
S
0
m
i1
in which
' ' 2
xix regression” and the ratio
also be written:
2
R 1
0
(5.11)
'
x 0 is the mean. The difference between S 0 and S is called as the “sum of squares due to
0
S
S 0 2
is called as the “coefficient of multiple determination,” R which can
S
0
S
(5.12)
S
'