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8/7/2019 Lectures_on_Cuves_Fitting
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LECTURE - 7
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LEAST SQUARES
CURVE FITTING
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Motivation
Given a set of experimental data
x 1 2 3
y 5.1 5.9 6.3
The relationship between x
and y may not be clearwe want to find an
expression for f(x)1 2 3
8/7/2019 Lectures_on_Cuves_Fitting
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KEPPLER THIRD LAW OF
PLANETARY MOTION
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8/7/2019 Lectures_on_Cuves_Fitting
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Curve Fitting
Given a set of
tabulated data, find acurve or a function
that best representsthe data.
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EXPERIMENTAL ERROR
xi x1 x2 . xn
yi y1 y2 . yn
Given
The form of the function is assumed to
be known but the coefficients are
unknown.
kkk eyxf!
)(The difference is assumed to be the result of
experimental error
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ERROR
2
1
1
2
2
11
k
k
N
1kk
))(1
((f)EError-RMS
)(1)(EErrorAverage
)(max)(EErrorMaximum
N.k1for)(e)deviationscalled(alsoerrorcalledis
datarecordedtheand)f(xvaluetrueebetween th
differenceThedata.ofsetthe)}y,{(x
!
!
!
ee!
g
N
kk
N
kk
kk
kk
yxfN
yxfN
f
yxff
yxf
beLet
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LEAST- SQUARES LINE
INTRODUCTION
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LEAST- SQUARES LINE
The least square liney=f(x)=Ax+B is the line
that minimizes RMS-error
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Please write down the black boardsummary
FINDING LEAST-SQUARES LINE
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Determine the Unknowns
?),(minimizetoandobtainedoHo
)(),(
minimizetoba,indant toWe
0
2
baEba
bxafbaEN
k
kk !!
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Determine the Unknowns
?),(minimizetoandobtainwedoow
)(),(
minimizetoba,findwant toWe
0
2
baba
bxafban
i
ii
*
!* !
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Determine the Unknowns
0
),(
0),(
minimumfor theconditionNecessary
!x
x
!x
x
b
ba
aba
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Remember
!xx
!
!!
!!
n
kk
n
ki
n
kk
n
kk
xgaxga
axadx
d
11
11
)()(
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Example 1
!
!
!!x
x
!!x
x
!!!
!!
!
!
N
kkk
N
kk
N
kk
N
kk
N
kk
k
N
kkk
N
kkk
yxbxax
ybxaN
xybxabbaE
ybxaa
baE
11
2
1
11
1
1
EquationsNormal
20),(
20),(
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Example 1
!
!
!!
!!
!!!
N
k
k
N
k
k
N
k
k
N
k
k
N
k
k
N
k
k
N
k
kk
xbyN
a
xxN
yxyxN
b
11
2
11
2
111
1
givesEquationsNormaltheolving
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Example 1
k 1 2 3 sum
xk 1 2 3 6
yk 5.1 5.9 6.3 17.3
xk2 1 4 9 14
xk yk 5.1 11.8 18.9 35.8
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Example 1
60.04.5667
8.35146
3.1763
EquationsNormal
11
2
1
11
!!
!
!
!
!
!!!
!!
baSolving
ba
ba
yxbxax
ybxaN
N
k
kk
N
k
k
N
k
k
N
k
k
N
k
k
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Power Fit
AxY M!
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Example 2
data.fit theto
)cos()ln()(
formtheoffunctionafindtorequiredisIt
xecxbxaxf !
x 0.24 0.65 0.95 1.24 1.73 2.01 2.23 2.52
y 0.23 -0.23 -1.1 -0.45 0.27 0.1 -0.29 0.24
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Example 2
EquationsNormal
c
cbab
cba
acba
!x
*x
!
x
*x
!x*x
0),,(
0),,(
0),,(
minimumfor theconditionNecessary
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Example 2
equationsnormalthesolveandsumstheEvaluate
)()())((cos)()(ln
)(cos)()(cos)(cos)(cos)(ln
)(ln)()(ln)(cos)(ln)(ln
8
1
8
1
28
1
8
1
8
1
8
1
8
1
28
1
8
1
8
1
8
1
28
1
kkkk
k
k
x
k kk
x
k
x
k
x
k k
k
kk
x
k
k
k
kk
k
k
k
kk
x
k
kk
k
k
k
k
eyecexbexa
xyexcxbxxa
xyexcxxbxa
!!!!
!!!!
!!!!
!
!
!
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How do you judge performance?
best?select theyoudoow
data,fit thetofunctionsmoreorGiven two
sense.square
leastin thebesttheis)(smallerinresulting
functionTheone.eachfor)(computethen
functioneachforparameterstheDetermine
:
2
2
fE
fE
Answer
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Multiple Regression
Example:
Given the following data
It is required to determine afunction of two variables
f(x,t) = a + b x + c t
to explain the data that is bestin the least square sense.
t 0 1 2 3
x 0.1 0.4 0.2 0.2
f(x,t) 3 2 1 2
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SolutionofMultiple Regression
Construct , the sumof the square of theerror and derive the
necessary conditionsby equating thepartial derivativeswith respect to theunknown parameters
to zero then solvethe equations.
* t 0 1 2 3
x 0.1 0.4 0.2 0.2
f(x,t) 3 2 1 2
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SolutionofMultiple Regression
02),,(
02
),,(
02),,(conditionsNecessary
),,(
),(
4
1
4
1
4
1
4
1
2
!!x
*x
!!x
*x
!!x
*x
!*
!
!
!
!
!
i
i
iii
ii
iii
i
iii
i
iii
tfctbxac
cba
xfctbxab
cba
fctbxaa
cba
fctbxacba
ctbxatxf
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Nonlinear least squares problems
EXAMPLES
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data.fit thebestthatformtheoffuctionafind
bx
ae
ii
ii
i
bx
i
ibx
bx
i
ibx
i
ibx
ebayaeb
eyaea
yae
!
!
!
!!
x
*x
!!
x*x
!*
3
1
3
1
3
1
2
0
0
usingobtainedareEquationsNormal
Nonlinear Problem
Givenx 1 2 3
y 2.4 5. 9
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data.fit thebestthatformtheoffuctionafind
bx
ae
solve)easier to(useillWe
usingoInstead
)ln()ln(
bxln(a)ln(y)zDe ine
3
1
2
3
1
2
!
!
!*
!*
!!
!!
i
ii
ii
bx
ii
zbx
yae
yzandaet
i
E
E
Alternative Solution(LinearizationMethod)
Givenx 1 2 3
y 2.4 5. 9
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InconsistentSystem ofEquations
solutionNo
EquationsofsystemntinconsisteisThis
10
6
4
1.43
2221
equationsofsystemfollowingtheSolve
:Problem
2
1
!
x
x
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InconsistentSystem ofEquationsReasons
Inconsistent equations
may occur because of
errors in formulatingthe problem, errors in
collecting the data or
computational errors.
Solution if all lines intersect
at one point
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InconsistentSystem ofEquationsFormulationasa leastsquares problem
errorsquaresleasttheminimizetoandFind
10
6
4
1.43
22
21
asequationsthecan viewWe
21
3
2
1
2
1
xx
x
x
!
I
I
I
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Solution
12262.4936.6
)822416()62.3388()6.2484(0
)011.43(2.86)22(44)2(40
926.3628
)60248()6.2484()1882(0
)011.43(66)22(44)2(20
minimizetoandFind
)011.43(6)22(4)2(
21
21
2121212
21
21
2121211
21
221
221
221
!
!
!!x*x
!
!
!!x
*x
*
!*
xx
xx
xxxxxxx
xx
xx
xxxxxxx
xx
xxxxxx
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Solution
0.9799,2.0048
:
12262.4936.6
926.3628
:equationsNormal
21
21
21
!!
!
!
xx
Solution
xx
xx
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data.it thebestthatb)/(ax1ormtheouctionaind
!
!
!*
!*
!
!!
!
3
1
2
3
1
2
usewillWe
bax
1usingoInstead
1
bax1
ze ine
bax
1f(x)
i
ii
i
i
ii
zbax
y
yzLet
y
Examples(LinearizationMethod)
Givenx 1 2 3
y 0.23 .2 .14