Curve Fitting
!
Process of finding simple analytical
function to approximate a set of data
!
Often from experimental measurements
!
Contains parameters that are adjusted to
agree with data
!
Usually two types of “fitting” functions
1.
Derived from fundamentals/physics
2.
Empirical equation that just matches
data well
1
Linear Regression
!
Straightline fit is simplest example
!
Mathematical expression:
!
“e” is error (or residual) between model and
observations
!
Error is discrepancy between the true value of “y”
and the approximate value: a
0
+a
1
x
01
ya
a
x
e
!"
"
ey
a
a
x
!#
#
2
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View Full DocumentCriteria for a “Best” Fit Options
1.
Line through the data that minimizes the sum of the residual
errors?
2.
Minimize the sum of the absolute values of the discrepancies?
3.
Minimize the maximum distance a
point falls from a line
$%
01
11
nn
ii
i
e
y
aa
x
!!
!#
#
&&
i
e
y
x
#
Has the effect of cancelling errors
Does not always yield a unique fit
Gives undue influence to an outlier
3
Better option  minimize sum of squares of residuals between
the measured “y” and calculated “y” with linear model
2
2
,,
m
o
d
0
1
1
n
ri
i
m
e
a
s
u
r
e
d
i
e
l
i
i
i
Se
yy
y
x
!
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 Spring '06
 Klaus
 Regression Analysis, Mathematical Expression, Nonlinear Relationships

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