User`s guide

Parametric Fitting
3-7
To solve this equation for the unknown coefficients p
1
and p
2
, you write S as a
system of n simultaneous linear equations in two unknowns. If n is greater
than the number of unknowns, then the system of equations is overdetermined.
Because the least squares fitting process minimizes the summed square of the
residuals, the coefficients are determined by differentiating S with respect to
each parameter, and setting the result equal to zero.
The estimates of the true parameters are usually represented by b.
Substituting b
1
and b
2
for p
1
and p
2
, the previous equations become
where the summations run from i =1 to n. The normal equations are defined as
Solving for b
1
Sy
i
p
1
x
i
p
2
+()–()
2
i 1=
n
∑
=
p
1
∂
∂S
2 x
i
y
i
p
1
x
i
p
2
+()–()
i 1=
n
∑
– 0==
p
2
∂
∂S
2 y
i
p
1
x
i
p
2
+()–()
i 1=
n
∑
– 0==
x
i
y
i
b
1
x
i
b
2
+()–()
∑
0=
y
i
b
1
x
i
b
2
+()–()
∑
0=
b
1
x
i
2
∑
b
2
x
i
∑
+ x
i
y
i
∑
=
b
1
x
i
∑
nb
2
+ y
i
∑
=
b
1
nx
i
y
i
x
i
y
i
∑∑
–
∑
nx
i
2
x
i
∑


2
–
∑
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