Graphics Programs Reference
In-Depth Information
T
(
◦
C)
0
21
.
1
37
.
8
54
.
4
71
.
1
87
.
8
100
µ
k
(10
−
3
m
2
/
.
.
.
.
.
.
.
s)
1
79
1
13
0
696
0
519
0
338
0
321
0
296
10
◦
, 30
◦
, 60
◦
and 90
◦
C.
20.
The table showshow the relative density
Interpolate
µ
k
at
T
=
ρ
of air varies with altitude
h
.Deter-
mine the relative density of air at 10
.
5 km.
.
.
.
.
.
.
h
(km)
0
1
525
3
050
4
575
6
10
7
625
9
150
ρ
1
0
.
8617
0
.
7385
0
.
6292
0
.
5328
0
.
4481
0
.
3741
3.4
Least-Squares Fit
Overview
If the dataareobtained from experiments, they typically contain a significant amount
of randomnoise duetomeasurementerrors. The task of curve fitting istofind a
smooth curvethat fits the datapoints “on the average.” Thiscurve should have a
simple form (e.g. a low-orderpolynomial), so astonot reproduce the noise.
Let
f
(
x
)
=
f
(
x
;
a
1
,
a
2
,...,
a
m
)
be the function that istobe fitted to the
n
datapoints (
x
i
,
n
. The
notation implies that wehave a function of
x
thatcontains the parameters
a
j
,
j
y
i
),
i
=
1
,
2
,...,
=
,
,...,
<
.
The form of
f
(
x
) is determinedbeforehand, usually
from the theory associatedwith the experimentfromwhich the dataisobtained. The
onlymeansofadjusting the fit is the parameters.For example, if the datarepresent the
displacements
y
i
of an overdampedmass-spring systemattime
t
i
, the theory suggests
the choice
f
(
t
)
1
2
m, where
m
n
a
1
te
−
a
2
t
. Thuscurve fitting consists of two steps: choosing the form
of
f
(
x
), followedbycomputation of the parametersthat produce the best fittothe
data.
This brings us to the question: what is meant by “best”fit? If the noise isconfined
to the
y
-coordinate, the most commonlyusedmeasure is the
least-squares fit
, which
minimizes the function
=
n
f
(
x
i
)]
2
S
(
a
1
,
a
2
,
...,
=
[
y
i
−
a
m
)
(3.13)
i
=
1
with respect to each
a
j
. Therefore, the optimal values of the parameters are givenby
the solution of the equations
∂
S
a
k
=
0,
k
=
1
,
2
,...,
m
(3.14)
∂
Search WWH ::
Custom Search