17 t Data Processing - Istiartoistiarto.staff.ugm.ac.id/docs/dpt/DPT9 Regression and Interpolation... · Data Processing Techniques Curve Fitting: Regression and Interpolation 3-t-17

UniversitasGadjahMadaDepartmentofCivilandEnvironmentalEngineeringMasterofEngineeringinNaturalDisasterManagement

DataProcessingTechniquesCurveFitting:RegressionandInterpolation

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1

CurveFitting• Reference• Chapra,S.C.,CanaleR.P.,1990,NumericalMethodsforEngineers,2ndEd.,McGraw-HillBookCo.,NewYork.• Chapter11and12,pp.319-398.

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CurveFitting• Alineorcurvethatrepresentsanumberofdatapoints• Therearetwomethodstofindsuchlineorcurve• Regression• Interpolation

• Engineeringapplications• Trendanalysis• Hypothesistesting

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RegressionvsInterpolation• Regression• Thedatashowsignificanterrorsornoise• Tofindasinglecurvethatrepresentgeneraltrendofthedata• Regressionline(curve)doesnotneedtopasseverydatapoint

• Interpolation• Thedataareaccurate• Tofindacurveorcurvesthatencompass(es)everydatapoint• Toestimatevaluesbetweendatapoints

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RegressionandInterpolation• Extrapolation• Similartointerpolationbutappliedtooutsiderangeofdatapoints• Notrecommended

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CurveFittingtoMeasuredData• Trendanalysis• Useofdatatrend(measurements,experiments)toestimatevalues

• Ifthedataareaccurate,useinterpolationtechnique• Ifthedatashownoise,useregressiontechnique

• Hypothesistesting• Comparisonbetweentheoreticalvalueswithcomputedones

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Recall:StatisticalParameters• Arithmeticmean

• Standarddeviation

• Variance

• Coefficientofvariation

represen

tdatadistrib

ution

!!sy

2 =Stn−1

!!y = 1

nyi∑

!!sy =

Stn−1 !!

St = yi −y( )2

∑

!!c.v.=

syy100%

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ProbabilityDistribution

X

freq

NormalDistributiononeofdatadistributionsthatisfrequentlyencounteredinengineering

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REGRESSIONSimpleLinearRegression

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Regression:Least-squareMethod• Tofindasinglecurveorfunction(approximate)thatrepresentsthegeneraltrendofthedata• Thedatashowsignificanterror• Thecurvedoesnotneedtopasseverydatapoint

• Methods• Linearregression(simplelinearregression)• Linearizedexpressions• Polynomialregression• Multiplelinearregression• Non-linearregression

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Regression:Least-squareMethod• How• Spreadsheet(MSExcel)• Computerprogram

• MatLab• Freeware

• Octave• Scilab• Freemat

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SimpleLinearRegression• Tofindastraightlinethatrepresentsthegeneraltrendofdatapoints:(x0,y0),(x1,y1),…,(xn,yn)

• MSExcel• =INTERCEPT(y,x)• =SLOPE(y,x)

yreg =a0 + a1xa0 : intercepta1 : slope

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SimpleLinearRegression• Errororresidual• Discrepanciesbetweenactualvalueofy (y data)andapproximatevalueofy (yreg)accordingtolinearexpressiona0 +a1x

• Minimizethesumofsquaredresidues

!!e= y− a0 +a1x( )

!!min Sr!

"#$=min ei

2!"

#$=min yi −a0 −a1xi( )

2

∑!"'#$(

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SimpleLinearRegression• Howtofinda0 anda1?• DifferentiatetheequationofSr twice;firstlyw.r.ta0 andlastlyw.r.ta1• Seteachofthetwoequationstozero• Solvetheequationsfora0 anda1

!!

∂Sr∂a0

=−2 y−a0 −a1xi( )∑ =0

∂Sr∂a1

=−2 y−a0 −a1xi( )xi∑ =0

!!

a1 =n xiyi∑ − xi∑ yi∑n xi

2∑ − xi∑( )2

a0 = y −a1 x

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Example#1i xi yi =f(xi)0 1 0.51 2 2.52 3 23 4 44 5 3.55 6 66 7 5.5

01234567

0 1 2 3 4 5 6 7

y=f(x

)

X

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Example#1i xi yi xi yi xi2 yreg (yi−yreg)2 (yi−ymean)2

0 1 0.5 0.5 1 0.910714 0.168686 8.576531

1 2 2.5 5 4 1.75 0.5625 0.862245

2 3 2.0 6 9 2.589286 0.347258 2.040816

3 4 4.0 16 16 3.428571 0.326531 0.326531

4 5 3.5 17.5 25 4.267857 0.589605 0.005102

5 6 6.0 36 36 5.107143 0.797194 6.612245

6 7 5.5 38.5 49 5.946429 0.199298 4.290816

∑= 28 24.0 119.5 140 ∑= 2.991071 22.71429

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Example#1

a1 =n xiyi∑ − xi∑ yi∑

n xi2∑ − xi∑( )

2=

7 119.5( )−28 24( )7 140( )− 28( )2

= 0.839286

!!

y = 247=3.4

x = 287= 4

a0 =3.4−0.839286 4( ) =0.071429

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Example#1

01234567

0 1 2 3 4 5 6 7 8

Y

X

data

regression

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Error• Error• Standarderrormagnitude

• Noticeitssimilaritywithstandarddeviation

!!sy x =

Srn−2

!!sy =

Stn−1 !!

St = yi −y( )2

∑

!!Sr = yi −a0 −a1xi( )

2

∑

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Error• Diffrencebetweenthetwo“errors”signifiesanimprovementofthepredictionorareductionoferror

r2 =St − Sr

St

=1−Sr

St

r =n xiyi∑ − xi∑( ) yi∑( )

n xi2∑ − xi∑( )

2n yi

2∑ − yi∑( )2

coefficientofdetermination

correlationcoefficient

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Error

!!

Sr = yi −a0 −a1xi( )2

∑ =2.991071

St = yi −y( )2

∑ =22.71429

!!

r2 =1−SrSt=1− 2.991971

22.71429=0.868318

r =0.931836

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Example#2i xi yi =f(xi)0 1 5.51 2 62 3 3.53 4 44 5 25 6 2.56 7 0.5

01234567

0 1 2 3 4 5 6 7

y=f(x

)

X

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REGRESSIONRegressionofLinearizedExpression

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LinearRegression• Linearizednon-linearequations• Logarithmiceq.à lineareq.• Exponentialeq.à lineareq.• nthorderpolynomialeq.(n >1)à lineareq.• etc.

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LinearRegression

x

y ln y

1

ln a

!y =aebx

!!lny = lna+bx

x

b

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LinearRegression

1

x

y log y

logx

b!y =axb

!!logy = loga+blogx

!!loga

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LinearRegression

1/y

1!y =a x

b+ x

1/x

y

x

!!

1y=b+ xax

=1a+ba1x

!!1 a!b a

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REGRESSIONPolynomialRegression

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PolynomialRegression• Someengineeringdata,althoughexhibitingamarkedpattern,ispoorlyrepresentedbyastraightline• Method1:Coordinatetransformation(linearizednon-lineareq.)• Method2:Polynomialregression

• Themth-degreepolynomial

• Thesumofthesquaresoftheresiduals!!y =a0 +a1x+a2x

2 +...+amxm

!!Sr = ei

2

i=1

n

∑ = yi −a0 −a1xi +a2xi2 +...+amxi

m( )2

i=1

n

∑

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• Theleast-squaremethodextendedtofitthedatatoanmth-degreepolynomial

• Theseequationscanbesetequaltozeroandrearrangedtodevelopasetofnormalequations

!!

∂Sr∂a0

=−2 yi −a0 −a1xi +a2xi2 +...+amxi

m( )i=1

n

∑

∂Sr∂a1

=−2 xi yi −a0 −a1xi +a2xi2 +...+amxi

m( )i=1

n

∑

∂Sr∂a2

=−2 xi2 yi −a0 −a1xi +a2xi

2 +...+amxim( )

i=1

n

∑

.

.

.∂Sr∂am

=−2 xim yi −a0 −a1xi +a2xi

2 +...+amxim( )

i=1

n

∑

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!!

a0n+a1 xii=1

n

∑ +a2 xi2

i=1

n

∑ +...+am xim

i=1

n

∑ = yii=1

n

∑

a0 xii=1

n

∑ +a1 xi2

i=1

n

∑ +a2 xi3

i=1

n

∑ +...+am xim+1

i=1

n

∑ = xiyii=1

n

∑

a0 xi2

i=1

n

∑ +a1 xi3

i=1

n

∑ +a2 xi4

i=1

n

∑ +...+am xim+2

i=1

n

∑ = xi2yi

i=1

n

∑

.

.

.

a0 xim

i=1

n

∑ +a1 xim+1

i=1

n

∑ +a2 xim+2

i=1

n

∑ +...+am xi2m

i=1

n

∑ = ximyi

i=1

n

∑

§ Therearem+1linearequationshavingm+1unknowns,i.e.a0,a1,a2,…,am

§ Theselinearequationscanbesimultaneouslysolvedbyusingmethodssuchas• Gausselimination• Gauss-Jordan• Jacobiiteration• Matrixinversion

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Example• Fitasecond-orderpolynomialtothedatainthetableontheright

• Answer

xi yi0 2.1

1 7.7

2 13.6

3 27.2

4 40.9

5 61.1

!!y =a0 +a1x+a2x2

!!

y =2.47857+2.35929x+1.86071x2

r2 =1−SrSt=1− 3.74657

2513.39=0.99851

r =0.99925

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REGRESSIONMultipleLinearRegression

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MultipleLinearRegression• Supposethedependentvariabely isalinearfunctionoftwoindependentvariablesx1 andx2

• Thebestvaluesofthecoefficientsaredeterminedbysettingupthesumofthesquaresoftheresiduals

!!y =a0 +a1x1+a2x2

!!Sr = yi −a0 −a1x1i −a2x2i( )

2

i=1

n

∑

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MultipleLinearRegression

!!

∂Sr∂a0

=−2 yi −a0 −a1x1i −a2x2i( )i=1

n

∑

∂Sr∂a1

=−2 x1i yi −a0 −a1x1i −a2x2i( )i=1

n

∑

∂Sr∂a2

=−2 x2i yi −a0 −a1x1i −a2x2i( )i=1

n

∑

§ Differentiatingthisequationw.r.teachoftheunknowncoefficients

!!

a0n+a1 x1ii=1

n

∑ +a2 x2ii=1

n

∑ = yii=1

n

∑

a0 x1ii=1

n

∑ +a1 x1i2

i=1

n

∑ +a2 x1i x2ii=1

n

∑ = x1iyii=1

n

∑

a0 x2ii=1

n

∑ +a1 x1i x2ii=1

n

∑ +a2 x2i2

i=1

n

∑ = x2iyii=1

n

∑

§ Equatingthedifferentialstozeroandexpressingtheresultedequationasasetofsimultaneouslineareqsyield

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MultipleLinearRegression

!!

n x1ii=1

n

∑ x2ii=1

n

∑

x1ii=1

n

∑ x1i2

i=1

n

∑ x1i x2ii=1

n

∑

x2ii=1

n

∑ x1i x2ii=1

n

∑ x22

i=1

n

∑

"

#

$$$$$$$$

%

&

''''''''

a0a1a2

(

)**

+**

,

-**

.**

=

yii=1

n

∑

x1i yii=1

n

∑

x2i yii=1

n

∑

(

)

****

+

****

,

-

****

.

****

§ Writteninmatrixform 3-Oct-17

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Example• Findthebestlinearequationthatfitstothedatainthetableontheright

• Answer

x1 x2 y

0 0 5

2 1 10

2.5 2 9

1 3 0

4 6 3

7 2 27!!

y =5+4x1 −3x2R2 =1

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MultipleLinearRegression• Multiplelinearregressioncanbeusefulinthederivationofpowerequationsofthegeneralform

• Suchequationsareextremelyusefulwhenfittingexperimentaldata• Inordertousethemultiplelinearregression,theequationistransformedbytakingitslogarithmtoyield

!!y =a0x1a1x2

a2 ...xmam

!!logy = loga0 +a1 logx1+a2 logx2 +...+am logxm

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REGRESSIONGeneralLinearLeastSquares

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GeneralLinearLeastSquares• Thethreetypesofregressionthathavebeenpresented,i.e.simplelinear,polynomial,andmultiplelinearcanbeexpressedinageneralleast-squaresmodel

• wherez0,z1,…,zm arem+1differentfunctions• m+1isthenumberofindependentvariables• n+1isthenumberofdatapoints

• Theaboveexpressioncanbewritteninamatrixform

!!y =a0z0 +a1z1+a2z2 +...+amzm

!Y{ }= Z!" #

$ A{ }

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GeneralLinearLeastSquares!Y{ }= Z!" #

$ A{ }

!!

Z!" #$=

a01 a11 . . . am1

a02 a12 . . . am2

. . .

. . .

. . .a0n a1n amn

!

"

%%%%%%%%

#

$

&&&&&&&&

§ {Y}containstheobservedvaluesofthedependentvariables

§ [Z]iaamatrixoftheobservedvaluesoftheindependentvariables

§ {A}containstheunkowncoefficients

!Z!" #$TZ!" #$ A{ }= Z!" #

$TY{ }

!!Sr = yi − ajzji

j=1

m

∑#

$%%

&

'((

2

i=1

n

∑

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GeneralLinearLeastSquares

!Z!" #$TZ!" #$ A{ }= Z!" #

$TY{ }

§ Solutionstrategy• LUdecomposition• Cholesky’smethod• Matrixinverseapproach

!!A{ }= Z!" #

$TZ!" #$

!"%

#$&

−1

Z!" #$TY{ }

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INTERPOLATIONNewtonMethodLagrangeMethod

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Interpolation

linear quadratic cubic

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Interpolation• Situation• Needtoestimateintermediatevaluesbetweenprecisedatapoints.• Themostcommonmethodusedforthispurposeispolynomialinterpolation

• Generalformulaforannth-orderpolynomialis

• Thereisonlyonepolynomialofordern orlessthatpassesthroughalln+1datapoints.

!!f x( ) =a0 +a1x+a2x2 +...+anxn

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Interpolation• Solutionfornthorderpolynomialrequiresn+1datapoints• Availablemethodstofindnthorderpolynomialthatinterpolatesn + 1 datapointsare:• NewtonMethod• LagrangeMethod

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LinearInterpolation:NewtonMethod

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )001

0101

01

01

0

01

xxxxxfxf

xfxf

xxxfxf

xxxfxf

---

+=

--

=--

f(x)

f(x1)

f1(x)

f(x0)

x0 x1x

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QuadraticInterpolation:NewtonMethod

( ) ( ) ( )( )

( ) ( ) ( )!2

212021102010

12021022

20110

1020102

210

xbxxbxbbxxbxbb

xxbxxbxxbxbxbxbb

xxxxbxxbbxf

aaa

+--++-=

--++-+=

--+-+=

"" #"" $%""" #""" $%

( ) 22102 xaxaaxf ++=

ïî

ïí

ì

=--=+-=

22

120211

1020100

ba

xbxbba

xxbxbba

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QuadraticInterpolation:NewtonMethod

( ) ( ) ( ) ( )[ ] [ ] [ ]

12

0112012

12

01

01

12

12

2,

,,xxxxfxxf

xxxfxx

xxxfxf

xxxfxf

b-

--==

---

---

=

( )00 xfb =

( ) ( ) [ ]0101

011 , xxf

xxxfxf

b =--

=

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PolynomialInterpolation:NewtonMethod

( ) ( ) ( )( ) ( )110010 ...... ----++-+= nnn xxxxxxbxxbbxf

( )[ ][ ]

[ ]011

0122

011

00

,,...,,

.

.

.

,,

,

xxxxfb

xxxfb

xxfb

xfb

nnn -=

===

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[ ] ( ) ( )

[ ] [ ] [ ]

[ ] [ ] [ ]0

02111011

,...,,,...,,,,...,,

,,,,

,

xxxnxfxxxf

xxxxf

xx

xxfxxfxxxf

xx

xfxfxxf

n

nnnnnn

ki

kjjikji

ji

jiji

--

=

--

=

--

=

----

( ) ( ) ( ) [ ] ( )( ) [ ]( )( ) ( ) [ ]01110

012100100

,...,,...

...,,,

xxxfxxxxxx

xxxfxxxxxxfxxxfxf

nnn

n

-----++--+-+=

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i xi f(xi)Computational Steps

1st 2nd 3rd

0 x0 f(x0) f[x1,x0] f[x2,x1,x0] f[x3,x2,x1,x0]

1 x1 f(x1) f[x2,x1] f[x3,x2,x1]

2 x2 f(x2) f[x3,x2]

3 x3 f(x3)

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PolynomialInterpolation:LagrangeMethod

( ) ( ) ( )

( ) Õ

å

¹=

=

--

=

=

n

ijj ji

ji

n

iiin

xx

xxxL

xfxLxf

0

0

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Example

i xi f(xi)

0 1 1.5

1 4 3.1

2 5 6

3 6 2.101234567

0 1 2 3 4 5 6 7

f(x)

X

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SPLINEINTERPOLATIONLinearSplineQuadraticSplineCubicSpline

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SplineInterpolation• Forn+1datapointsà nth-orderinterpolatingpolynomials• Thereisacasewhereafunctionisgenerallysmoothbutundergoesanabruptchangesomewherealongtheregionofinterest• Higher-orderpolynomials,n >>,tendtoswingthroughwildoscillationsinthevicinityofanabruptchange

• Lower-orderpolynomial,n <<,mightbetterrepresentthedatapattern• Lower-orderpolynomials:splineinterpolation

• Linearsplines(n =1)• Quadraticsplines(n =2)• Cubicsplines(n =3)

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PolynomialvsSplineInterpolations

n =1n » n =1n »

§ nth-orderpolynomial

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LinearSplines• 1st-orderspline:straightline• Ordereddatapoints:x0,x1,x2,…,xn

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) nnnnn xxxxxmxfxf

xxxxxmxfxf

xxxxxmxfxf

££-+=

££-+=££-+=

---- 1111

21111

10000

.

.

.

( ) ( )ji

jii xx

xfxfm

--

=+

+

1

1

slope:

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LinearSplines• Linearsplines• Theyarethereforeidenticaltolinearinterpolation• Thedrawbackoflinearsplinesisthattheyarenotsmooth• Atdatapointswheretwosplinesmeet(calledaknot),theslopechangesabruptly

• Thefirstderivativeofthefunctionisdiscontinuousatknots• Theabovedeficiencyisovercomebyusinghigher-orderpolynomialsplinesthatensuresmoothnessattheknotsbyequatingderivativesatthesepoints

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QuadraticSplines• Quadraticsplines• Inordertoensurethatthemthderivativesarecontinuousattheknots,asplineofatleastm+1ordermustbeused

• 3rdorderpolynomialsorcubicsplinesthatensurecontinuousfirstandsecondderivativesaremostfrequentlyusedinpractice.• Thediscontinuousthirdandfourthderivativescannotusuallybedetectedvisually,thustheycanbeignored

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60

QuadraticSplines• Objective:toderivea2nd-orderpolynomialforeachintervalbetweendatapoints• Thosepolynomialshavetoshowcontinuousfirstderivativeatdatapoints

• Thegeneralformulaofa2nd-orderpolynomial

• Forn+1datapoints(i =0,1,2,…,n)therearen intervals,sothatthereare3n unknownconstants(ai,bi,ci;i =1,2,…,n) toevaluate• Requires3n equations

( ) iii cxbxaxf ++= 2

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QuadraticSplines• The3n equations

1. Thesplinecurvesintersecttheknots,thusthesplinesati-1andiintervalsmeetatdatapoint[xi-1,f(xi-1)]

2. Thefirstsplinecurvepassesthroughthefirstdatapoint(i =1)andthelastsplinecurvepassesthroughtheendpoint(i =n)

i =2,3,…,n2(n- 1)eqs.

( )( )11

21

1111211

---

------

=++

=++

iiiiii

iiiiii

xfcxbxa

xfcxbxa

2 eqs.( )( )nnnnnn xfcxbxa

xfcxbxa

=++

=++2

0101201

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62

QuadraticSplines• The3n equations

3. Thegradients(thefirstderivatives)ofthesplinecurveattheinteriorknotsareequal

4. Assumethatthesecondderivativeiszeroatthefirstdatapoint

i =2,3,…,n(n- 1)eqs.!!

!f x( ) =2ax+b ⇒ 2ai−1xi−1+bi−1 =2aixi−1+bi

1eq.0=ia

asaconsequence,thefirsttwodatapoints(i =0andi =1)areconnectedwithastraightline

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63

QuadraticSplines• The3n equations2(n – 1)+2+(n – 1)+1=3n

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64

CubicSplines• Objective:toderivea3rd-orderpolynomialforeachintervalbetweendatapoints• Thosepolynomialshavetoshowcontinuousfirstandsecondderivativesatdatapoints

• Thegeneralformulaofa3rd-orderpolynomial

• Forn+1datapoints(i =0,1,2,…,n)therearen intervals,sothatthereare4n unknownconstants(ai,bi,ci,di;i =1,2,…,n) toevaluate• Requires4n equations

!!f x( ) =aix3+bix2 +ci x+di

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65

CubicSplines• The4n equations

1. Thesplinecurvesintersecttheknots,thusthesplinesati-1andiintervalsmeetatdatapoint[xi-1,f(xi-1)]à (2n – 2)eqs.

2. Thefirstsplinecurvepassesthroughthefirstdatapoint(i =1)andthelastsplinecurvepassesthroughtheendpoint(i =n)à 2eqs.

3. Thegradients(thefirstderivatives)ofthesplinecurveattheinteriorknotsareequalà (n – 1)eqs.

4. Thesecondderivativesofthesplinecurveattheinteriorknotsareequalà (n – 1)eqs.

5. Thesecondderivativesattheendknotsarezeroà 2eqs.

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CubicSplines• The4n equations• Thefifthconditionbringstothefollowingconsequence

• Thesplinecurvesatthefirstandlastintervalsarestraightlines• thefirsttwodatapointsareconnectedbyastraightline• thelasttwodatapointsareconnectedbyastraightline

• Thereisanalternativecondition• Thesecondderivativesattheendknotsareknown

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CubicSplines• The4n equations2(n – 2)+2+(n – 1)+(n – 1)+2=4n

• Itispossibletodomathematicalmanipulationssothatthecubicsplinethatrequires(n – 1)equationstoevaluateà refertoChapraandCanale(1990),pp.395-396.

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CubicSplines( ) ( )

( ) ( ) ( )( ) ( )

( )( )

( )( ) ( )

( )( )

( )( ) ( )111

1

11

1

1

31

1

13

1

1

6

6

66

---

-

--

-

-

--

-

-

-

-úû

ùêë

é -¢¢-

-+

-úû

ùêë

é -¢¢-

-+

--¢¢

+--¢¢

=

iiii

ii

i

iiii

ii

i

iii

ii

ii

ii

xxxxxf

xxxf

xxxxxf

xxxf

xxxxxf

xxxxxf

xf

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )[ ] ( ) ( ) ( )[ ]iiii

iiii

iiiiiiiii

xfxfxx

xfxfxx

xfxxxfxxxfxx

--

+--

=¢¢-+¢¢-+¢¢-

--

++

++-+--

11

11

111111

66

2

!!

n!intervals!!f x0( ) =0!!f xn( ) =0

"

#$$

%$$

⇒ n−1( ) !equation

2unknownsateachinterval:

( ) ( )ii xfxf ¢¢¢¢ - dan 1

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3-Oct-17

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