ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 26 Regression Analysis-Chapter 17.

ECIV 301

Programming & Graphics

Numerical Methods for Engineers

Lecture 26

Regression Analysis-Chapter 17

Curve Fitting

Often we are faced with the problem…

x y0.924 -0.003880.928 -0.00743

0.93283 0.005690.93875 0.00188

0.94 0.01278

-0.01

-0.005

0

0.005

0.01

0.015

0.92 0.925 0.93 0.935 0.94 0.945

what value of y corresponds to x=0.935?

Curve FittingQuestion 2: Is it possible to find a simple and convenient formula that represents data approximately ?

-0.01

-0.005

0

0.005

0.01

0.015

0.92 0.925 0.93 0.935 0.94 0.945

e.g. Best Fit ?

Approximation

Experimental Measurements

Strain

Str

ess

Experimental Measurements

Strain

Str

ess

BEST FIT CRITERIA

Strain

y S

tres

s

xaaxl 10)(

ii

iii

xaay

xlye

10

)(

Error at each Point

Best Fit => Minimize Error

n

iii

n

ielimeasuredi

n

ii

xaay

yye

1

210

1

2mod,,

1

2

Best Strategy

Best Fit => Minimize Error

n

iii

n

ii xaaye

1

210

1

2

Objective:

What are the values of ao and a1

that minimize ?

n

iie

1

2

Least Square Approximation

101

210

1

2 ,aaSxaaye r

n

iii

n

ii

In our case

Since xi and yi are known from given data

02,

110

0

10

n

iii

r xaaya

aaS

02,

110

1

10

n

iiii

r xxaaya

aaS

Least Square Approximation

n

iii

n

ii

n

ii xyxaxa

11

21

10

n

ii

n

ii yxana

1110

2 Eqtns 2 Unknowns

Least Square Approximation

xaya 10

2

11

2

1111

n

ii

n

ii

n

ii

n

ii

n

iii

xxn

yxyxna

n

xx

n

ii

1

n

yy

n

ii

1

Example

y = 0.8393x + 0.0714

0

1

2

3

4

5

6

7

0 2 4 6 8

Exper 1

Linear (Exper 1)

Example

y = 0.8329x + 0.0714

-1

0

1

2

3

4

5

6

7

0 2 4 6 8

Exper. 2

Linear (Exper. 2)

Quantification of Error

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8

Exper 1

Average

42.37

241

n

yy

n

ii

Quantification of Error

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8

Exper 1

Average

n

iit yyS

1

2

Quantification of Error

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8

Exper 1

Average

n

iit yyS

1

2

1n

Ss ty

Quantification of Error

n

iit yyS

1

2

1n

Ss ty

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8

Exper 1

Average

Standard Deviation Shows Spread Around mean Value

y = 0.8393x + 0.0714

0

1

2

3

4

5

6

7

0 2 4 6 8

Quantification of Error

n

iii

n

iir xaayeS

1

210

1

2

Quantification of Error

n

iii

n

iir xaayeS

1

210

1

2

2/ n

Ss r

xy

“Standard Deviation” for Linear Regressiony = 0.8393x + 0.0714

0

1

2

3

4

5

6

7

0 2 4 6 8

Quantification of Error

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8

Exper 1

Average

n

iit yyS

1

2

y = 0.8393x + 0.0714

0

1

2

3

4

5

6

7

0 2 4 6 8

n

iiir xaayS

1

210Better Representation

Less Spread

Quantification of Error

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8

Exper 1

Average

n

iit yyS

1

2

y = 0.8393x + 0.0714

0

1

2

3

4

5

6

7

0 2 4 6 8

n

iiir xaayS

1

210

t

rt

S

SSr

2

Coefficient of Determination

t

rt

S

SSrr

2

Correlation Coefficient

Linearized Regressionxbeay 1

1

xba

eay xb

11

1

ln

lnln 1

BxA

1

1ln

bB

aA

The Exponential Equation

Linearized Regression2

2bxay

xba

xay b

2210

21010

log

loglog 2

BxA

2

210log

bB

aA

The Power Equation

Linearized Regression

xb

xay

33

33

3 111

axa

b

y

BxA

3

3

3

1

a

bB

aA

The Saturation-Growth-Rate Equation

Polynomial Regression

exaxaay 2210

A Parabola is Preferable

Polynomial Regression

210r

n

1i

222i10i

n

1i

2i

a,a,aS

xaxaay

e

Minimize

Polynomial Regression

0xaxaay2a

a,a,aS n

1i

2i2i10i

0

210r

0xxaxaay2a

a,a,aS n

1ii

2i2i10i

1

210r

0xxaxaay2a

a,a,aS n

1i

2i

2i2i10i

2

210r

Polynomial Regression

i22i1i0 yaxaxa)n(

ii23i1

2i0i yxaxaxax

i2i2

4i1

3i0

2i yxaxaxax

3 Eqtns 3 Unknowns

Polynomial Regression

i2i

ii

i

2

1

0

4i

3i

2i

3i

2ii

2ii

yx

yx

y

a

a

a

xxx

xxx

xxn

Use any of the Methods we Learned

Polynomial Regression

n

1i

222i10i210r xaxaaya,a,aS

With a0, a1, a2 known the Total Error

3n

Ss r

xy Standard Error

t

rt2

S

SSr

Coefficient of

Determination

Polynomial Regression

n

1i

2mmi10im10r xaxaaya,a,aS

For Polynomial of Order m

1mn

Ss r

xy Standard Error

t

rt2

S

SSr

Coefficient of

Determination