Graphics Programs Reference
In-Depth Information
are the mean values of the
x
and
y
data. The solution for the parameters is
y
x
i
x
x
i
y
i
x
i
y
i
−
−
nxy
a
=
x
i
b
=
x
i
(3.18)
−
n x
2
−
n x
2
These expressions aresusceptible to roundoff errors (the twoterms in each numerator
as well as in each denominator can be roughly equal). It is better to compute the
parametersfrom
y
i
(
x
i
−
x
)
x
i
(
x
i
−
b
=
a
=
y
−
xb
(3.19)
x
)
which areequivalenttoEqs. (3.18), but much less affectedbyrounding off.
Fitting Linear Forms
Consider the least-squares fit of the
linear form
m
f
(
x
)
=
a
1
f
1
(
x
)
+
a
2
f
2
(
x
)
+···+
a
m
f
m
(
x
)
=
a
j
f
j
(
x
)
(3.20)
j
=
1
whereeach
f
j
(
x
) is apredetermined function of
x
,calleda
basis function
.Substitution
into Eq. (3.13) yields
y
i
−
a
j
f
j
(
x
i
)
2
n
m
S
=
(a)
i
=
1
j
=
1
Thus Eqs. (3.14) are
2
n
y
i
−
a
j
f
j
(
x
i
)
f
k
(
x
i
)
m
∂
S
a
k
=−
=
0,
k
=
1
,
2
,...,
m
∂
i
=
1
j
=
1
Dropping the constant
(
−
2
)
and interchanging the order of summation, we get
n
f
j
(
x
i
)
f
k
(
x
i
)
a
j
=
m
n
f
k
(
x
i
)
y
i
,
k
=
1
,
2
,...,
m
j
=
1
i
=
1
i
=
1
Inmatrix notation these equations are
Aa
=
b
(3.21a)
where
n
n
A
kj
=
f
j
(
x
i
)
f
k
(
x
i
)
b
k
=
f
k
(
x
i
)
y
i
(3.21b)
i
=
1
i
=
1
Equations(3.21a), known as the
normal equations
of the least-squares fit, can be
solvedwith any of the methods discussedinChapter2.Note that the coefficient
matrix issymmetric, i.e.,
A
kj
=
A
jk
.
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