Graphics Programs Reference
In-Depth Information
The terms
r
i
=
y
i
−
f
(
x
i
) in Eq. (3.13) arecalled
residuals
; theyrepresent the dis-
crepancybetween the datapoints and the fitting functionat
x
i
. The function
S
to be
minimizedisthus the sum of the squares of the residuals.Equations(3.14) are gener-
allynonlinear in
a
j
and may thus be difficult to solve. If the fitting functionis chosen
as a linear combination of specified functions
f
j
(
x
):
f
(
x
)
=
a
1
f
1
(
x
)
+
a
2
f
2
(
x
)
+···+
a
m
f
m
(
x
)
thenEqs. (3.14) arelinear. A typicalexample is apolynomialwhere
f
1
(
x
)
=
1,
f
2
(
x
)
=
x
,
x
2
, etc.
The spread of the dataabout the fitting curve isquantifiedbythe
standard devi-
ation
, definedas
f
3
(
x
)
=
S
n
σ
=
(3.15)
−
m
Note that if
n
m
, wehave
interpolation
, not curve fitting. In thatcase, both the
numerator and the denominatorinEq. (3.15) arezero, so that
=
σ
is meaningless, as it
shouldbe.
Fitting a Straight Line
Fitting a straight line
=
+
f
(
x
)
a
bx
(3.16)
to datais also known as
linear regression
. In thiscase the function to be minimizedis
n
bx
i
)
2
,
=
(
y
i
−
−
S
(
a
b
)
a
i
=
1
Equations(3.14) nowbecome
2
x
i
n
n
n
∂
S
a
=
1
−
2(
y
i
−
a
−
bx
i
)
=
−
y
i
+
na
+
b
=
0
∂
i
=
i
=
1
i
=
1
2
n
n
n
n
∂
S
x
i
b
=
1
−
2(
y
i
−
a
−
bx
i
)
x
i
=
−
x
i
y
i
+
a
x
i
+
b
=
0
∂
i
=
i
=
1
i
=
1
i
=
1
Dividing both equations by 2
n
and rearranging terms, we get
1
n
b
n
n
1
n
x
i
a
+
xb
=
y x
+
=
x
i
y
i
i
=
1
i
=
1
where
n
n
1
n
1
n
x
=
x
i
y
=
y
i
(3.17)
i
=
1
i
=
1
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