1
Week 5
Week 5
Curve Fitting & Interpolation
Week 5
Objectives
• Use leastsquare regression to fit
a
straight line to measured data
• Use polynomial regression to fit
polynomials to the measured data
Week 5
Curve Fitting
• There are two approaches for curve fitting:
leastsquares regression
&
interpolation
•
Leastsquares regression
: When the data exhibits a
significant degree of error, then the strategy is to derive
a single curve that represents the general trend of the
data
•
Interpolation
: when the data is known to be very
precise, the basic approach is to fit a curve or a series of
curves that pass directly through each of the points.
Estimation of values between wellknown discrete
points is called interpolation
Week 5
Curve fitting
Interpolation
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Week 5
Leastsquares regression
• Leastsquares regression uses basic concepts of
statistics: mean, standard deviation, residual
sum of the squares, and confidence intervals
• Arithmetic mean: sum of the individual data points
divided by the number of points
• Standard deviation: the square root of ratio of total
sum of the squares of the residuals and (n1)
n
y
y
i
∑
=
()
1
2
−
−
=
∑
n
y
y
s
i
y
Week 5
Leastsquares fit of a straight line
• A straight line is fitted to a set of observed
points (
x
1
,
y
1
), (
x
2
,
y
2
) … , (
x
n
,
y
n
)
y = a
0
+ a
1
x
• Let e be the error or residual between the true
value of y and approximate value
∑∑
−
−
=
⎪
⎪
⎭
⎪
⎪
⎬
⎫
−
−
=
−
−
=
−
−
=
i
i
i
n
n
n
x
a
a
y
e
or
x
a
a
y
e
x
a
a
y
e
x
a
a
y
e
1
0
1
0
2
1
0
2
2
1
1
0
1
1
L
L
L
L
L
L
Week 5
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 Three '09
 WeihuaLi
 Regression Analysis, 20 30 40, a1 ∑ xi

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