Lecture7-CurveFitting

7/28/2019 Lecture7-CurveFitting

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ENGG 407ENGG 407

Numerical Methods in EngineeringNumerical Methods in Engineering

Lecture 7Lecture 7

NMsfor Curve Fitting and

Yingxu Wang, Prof., PhD, PEng, FWIF, FICIC, SMIEEE, SMACM

Visiting Professor: MIT (2012), Stanford Univ. (2008), UC Berkeley (2008), Oxford Univ. (1995)

President, International Institute of Cognitive Informatics and Cognitive Computing (ICIC)

.Schulich School of Engineering, University of Calgary

Office: ICT542, Tel: 403 220 6141

[email protected]

http://www.enel.ucalgary.ca/People/wangyx/

ENGG407 - Numerical Methods in Engineering, Dr. Y. Wang, 25/02/13 L7-1


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1. Introduction1. Introduction

1. Introduction2. Linear curve fittin

- The least-square linear regression method3. Nonlinear curve fitting

-

- The logarithmic linear regression method

- The spline methods

4. Inter olation- Methods based on curve fitting

- The Lagrange polynomial methods

- Spline-based methods

5. MATLAB applications in curve fitt ing and interpolation6. Summary



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Curve Fitt ing and InterpolationCurve Fitt ing and Interpolation

urve ng

To find a numerical

(approximate) equation

that fits given data.

Interpolation

To find a numerical

approximation of an

internal oint based on

given data.

Extrapolation

To find a numerical approximation of an

external (extended) point based on given

data assumin that the trend is monotonic

orinvariant.


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NMs for Curve Fitting and InterpolationNMs for Curve Fitting and Interpolation

- Linearcurve fitting

- The least-square linear regression method

- Nonlinearcurve fitting

- Polynomial regression methods

- The spline methods Series of polynomial fits- The most important and practical method

-

Interpolation- Methods based on curve fitting- The Lagrange polynomial methods


1 0- Straight lines: ( )y x a x a

Typical formsof functions

11 1 0- Polynomials: ( ) ...

- Power functions: ( )

-

n nn n

p

px

y x a x a x a x a

y x kx



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TheThe MATLABMATLAB Curve Fitting InterfaceCurve Fitting Interface

140

100

Shape-preserving interpolant

y = 1.1*x - 22

y = 0.0079*x2 + 0.11*x + 0.7

- * 3 * 2 *

60

80. . . .

y = 3.5e-021*x4 - 1e-018*x3 + 0.0079*x2 + 0.11*x + 0.7

y = - 1.2e-029*x10 + 8.4e-027*x9 - 2.7e-024*x8 + 4.9e-022*x7 -

5.7e-020*x6 + 4.3e-018*x5 - 2.2e-016*x4 + 7.5e-015*x3 +

20

40

0.0079*x + 0.11*x + 0.7 data 1

spline

shape-preserving

linear

0

quadraticcubic

4th degree

10th degree

20 30 40 50 60 70 80 90 100 110 120-20


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Inductive Methodologies based on Curve Fitting andInductive Methodologies based on Curve Fitt ing and Extrapolation

The inductive methodology- To create a generic function based on limited observations of data.

The procedure of experimental case studies- Design a hypothesis (conceptual model) of a subject and related

- Collect data

- Inductively elicit the mathematical model

Usages

- Prove or validate a model (hypothesis)

- Calibrate (specify) parameters for a model



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2. Linear Curve Fitting2. Linear Curve Fitting

1. Introduction

2. Linear curve fitt in- The least-square linear regression method

3. Nonlinear curve fitting-









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LeastLeast--Square Regression for Linear FittingSquare Regression for Linear Fitting

Given a set of points ( , ) in a plain,i in x y

1 0by a linear function ?

- The residual: ( )

( )

i i i

n

wha

r

t is the best fit y f x a x a

y f x

n

1

- The total error: (t ii

E r

11

0)

- The absolute total error: ( )

i i

i

n n

y a x a

E r y a x a

2 2

1 01 1

1 1

The squares of total error: ( )

n n

s

i

i i ii

i

iE r y a x a

where the minimum or of total error ithe s

with a san o et ofptimization of the linear fit 2, , paires of data ( , ).i in x yn

r+/- may be

canceled inEt

Regression of

Eat is not unique

Es is proportional to eachri

and has no disadvantages asthose ofEtandEat;


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Mathematical Model of the Least Square Regression MethodMathematical Model of the Least Square Regression Method

2 21 0G iven [ ( )] ,

n n

i i iE r y a x a

1 010

the minimums or of the total errors are:

2 ( ) 0n

i ii

the

Ey a x a

a

min

1 011

2 ( ) 0

as an

ls n

i i ii

Ey a x a xa

optimiz

of the linear fit 2with a set of paires of data ( , ), .i in x yation n

1 01 1 1

2

0 1

0 1

n n n

i ii i i

ls n n n

x y

x xx xy

naa x a y

SL E

a x a x x

S a S

S a S a S

1 1 1

SPL et = M wh

i i i

lsSL E

1

0ere , ,yx

xyx xx

Sn SaP S M

SS Sa

01

Solving for , obtain:(1) ( )\

ls

P S a aMSL E P

f x x

xx y x xyS S S Sa

-1(2) P S 1 0

1 2

1( )

| |

xx x

xy

xx x

x x y

xx x

na a

nS S Sa nS S

M f x xS nS

M


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1 0G iven , 4,p T na a

E.g.1 The Least Square Linear Regression MethodE.g.1 The Least Square Linear Regression Method

4

[0 30 70 100]and

[0.94 1.05 1.17 1.28]

0 30 70 100 200

T

p

S x

1

4

1

0.94 1.05 1.17 1.28 4.44

i

y ii

S y

2 2 2 2 2

1

0 30 70 100 15800

xx ii

xy i

S x

S x y

4

0 0.94 30 1.05 70 1.17 100 1.28 241.40i 4 200 4.44

,200 15800 241.40

S olution 1:

i

yx

xyx xx

T T

Sn SS M

SS S

0

0 1, . , .

S olution 2:

xx

a aS

a

2 2

15800 4.44 241.40 2000.9428

4 15800 200

y xy x

xx x

S S S

nS S

1

1 0

2 2

. .0.0033

4 1580

0.00

0 200

T hus, the 33 0.942is: 8

xy x y

xx x

na

nS S

p a T a T

l east squ ar e r egression


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MATLAB: The Least Square Linear Regression MethodMATLAB: The Least Square Linear Regression Method

=, ,nx = length(x);Sx = sum (x);Sy = sum (y);Sxy = sum (x .* y);

Sxx = sum (x . 2);a1 = (nx*Sxy - Sx*Sy)/(nx*Sxx - Sx 2);

= * - * * - ^

% To print the calibrated function: f(x) = a1 x + a0;

fprintf ('%s %8.4f %s %8.4f \n', 'f(x) =', a1, 'x +', a0);f = x a1*x + a0 % To built the eneral functionplot(x, y,'*r-','markersize', 6); hold on;fplot(f, [0, 100]);xlabel('x'); ylabel('f(x)'); legend ('Sample data', 'Linear regression');

% General matrix solutionS = [nx, Sx; Sx, Sxx]; M = [Sy; Sxy];A = S \ M;



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Application ofApplication ofLeastSquareRegressionLeastSquareRegression

>> p = [0.94 0.96 1.0 1.05 1.07 1.09 1.14 1.17 1.21 1.24 1.28];

>> LeastSquareRegression (T, p)

T = 0.0034 T + 0.9336

a0 = 0.9336

a1 = 0.00341.25

1.3

1.1

1.15

1.2

re(a

tm)

1

1.05Pressu

0 10 20 30 40 50 60 70 80 90 1000.9

0.95

Sample dataLinear regression

emperature



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3. Nonlinear Curve Fitt ing3. Nonlinear Curve Fitt ing

1. Introduction

2. Linear curve fittin

- The least-square linear regression method 3. Nonlinear curve fitt ing-









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3.13.1 Polynomial Regression Methods

The uniform description of polynomials

11 1 0

An -order polynomial:

( ) ...m mm m

m

f x a x a x a x a

Types of polynomials- =

can fit a curve with , 1, points of sample data.n n m

, , .

- A least square linear regression is a special case of polynomial

regression where m = 1

- 2nd order: A quadratic parabola concave up/down determined byat east po nts o ata, m = , n > .

- 3rd order: m = 3, n > 4

A cubic polynomial

with one or more

bends determinedby at least 5 points of data.

Ver hi h order ol nomialmay result in wanders

in curve ends Not the higherthe better.


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Mathematical Model of the Polynomial Regression MethodMathematical Model of the Polynomial Regression Method

1

-m m

1 1 0with , , points of sample data,

...1

m m

n

n mn

1 2

1 1 01the square of total error is: [ ( .. )].

m m

i m i m i iiE y a x a ax ax

'

0, 0,1,2,...,

,

ls

E

E i n

In general LS formulation:S themodel matri

Based on , an SLE can be obtained as:lsE SP=MP theparameter vectorM themeasurementvector

.cf Ax=b

,1

- where is called ,, )(T

TTS S S ro -1T -1 T pseudo inversP=( M SS MS) S e

0 1- ... ,m



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Case 1: QuadraticCase 1: Quadratic Polynomial RegressionPolynomial Regression2

G iven a ( 2, 3) :m nQ uadratic P ol nomial a x a ax 2 2

2 1 01

[ ( )] , 3n

i i ii

E y a x a x a n

2 02 1 0

1

2 1

2 ( ) 0

2 0

i i i iio

n

y a x a x a xaE

a x a x a xE

11

2

22 2 (

i

i i

a

E

y a xa

2

1 01 ) 0

n

i ii a x a x

2

2

0 1 2

R ewri te , the following SL E is obtained:

x yx

ls

na S a S a S

E

2 3

2 3 4 2

0 1 2

0 1 2

2L E = x xyx x

x xx x y

a a aS a S a S a S

n n n n

w rehe xS x1 1 1

1

1

0 2

, , , and

thus [ , ]

ki y i i xy i ixi i i i

Ta a a

S y S x S x y

P = S \ MSP = M


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Case 2: HigherCase 2: Higher Order Polynomial RegressionsOrder Polynomial Regressions

4 3 2

Givenan 4 5 :m nmorder ol nomial a x a x xa x a a 4 3 2 2

4 3 2 1 01

[ ( )] , 5

n

i i i i ii

E y a x a x a x a x a n

0, 0,1,2,...,lsi

i nE

E

a

m ,sl

2 3 40 1 2 3 4 x yx x x

na S a S a S a S a S

2 3 4 5

2 3 4 5 6 2

0 1 2 3 4

0 1 2 3 4mSLE =

x xyx x x x

x x x x x x yS a S a S a S a S a S

3 4 5 6 7 3

4 5 6 7 8 4

0 1 2 3 4

0 1 3 42

x x x x x x y

x x x x x x yS a S a S a S a S a S

T

0 1 2 3 4



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MATLAB: The Polynomial Regression MethodMATLAB: The Polynomial Regression Method

, ,n = length (X);

for i = 1 : 2*m

= ^

endS(1,1) = n;

M(1,1) = sum(Y);, 2E.g.: m SP = M

for j = 2 : m+1

S(1, j) = xsum (j-1);

end

2

2 3

0 1 2

0 1 2

x yx

x xyx x

na S a S a S

S a S a S a S

for i = 2 : m+1

for j = 1 : m+1

S(i, j) = xsum (j+i-2);

2 3 4 20 1 2x x x x ya a a

enM(i,1) = sum (X .^ (i-1).* Y);

end



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Application ofApplication ofPolyRegressionPolyRegression

n =>> X = 0:0.4:6;

>> Y = [0 3 4.5 5.8 5.9 5.8 6.2 7.4 9.6 15.6 20.7 26.7 31.1 35.6 39.3 41.5];

>> [p] = PolyRegression (X, Y, m)p =

- . - a

12.8780 - a1

-10.1927 - a2. -

-0.2644 - a4

4 3 24 3 2 1 0

4 3 2

( )

0.2644 3.1185 10.1927 12.8780 0.2746

f x a x a x a x a x a

x x a x x



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3.2 The Logarithmic Linear Regression Method

A nonlinear problem may be transformed into a linear oneon the logarithmic space.

Treatment of axes

X : Y = linear : linear | linear : log | log : linear | log : log

Disadvantages

The accuracy / errorof results may be decreased / increased.

log



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Mathematical Model of the Logarithmic Linear Regression MethodMathematical Model of the Logarithmic Linear Regression Method

m

,its linear form is:

rebuilt:

ln( ) ln( ) ln( )

a x a

y m x b

0

1

( ) lnwhere anda

a my y

mx ,

its linear form is:

rebu

ln

il

( )

t:

ln( )

a x a

y mx b

1

( )where and

ln a my y

0a

0



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E.g.2 TheE.g.2 The LogarithmicLogarithmic Linear Regression MethodLinear Regression Method

1 Given the equation ( ) ,

1

RCr cexponential v t v e

1 0

rebuilt:

r c

r

RC

v a t a

ln( )

where and

r rav v

t t

1

1

1 1or C

RC a R

0

1 0

0 n or

I.e.: ( )

c c

rv t a t a

a v v e

1 1 aRC

0 c



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Application of theApplication of the LogarithmicLogarithmic Linear Regression MethodLinear Regression Method

% Lo arithmicLo arithmic linear re ressionlinear re ressiontexp = 2:2:30;

vexp = [9.7 8.1 6.6 5.1 4.4 3.7 2.8 2.4 2.0 1.6 1.4 1.1 0.85 0.69 0.6];

vexpLOG = log (vexp);

R = 5e6;

[a1, a0] = LinearRegration (texp, vexpLOG)Vc = exp(a0)

C = -1 / (R*a1)

t = 0 : 0.5 : 30;

Vr = Vc * exp(a1 * t);' ', , , ,

% Solutions

a1 = -0.1002

a0 = 2.4776Vc = 11.9131

C = 1.9968e-006 [uF]



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3.33.3 The PolynomialThe Polynomial SplineSpline FittingFitting Methods

- A long curve may be better fitted as multiple splines (pieces)

- Splines can be easily fitted by lower order polynomials, usually m= 3 or 2

- The wander roblem in curve ends can be avoided in iecewise fit tin

Types of polynomial splines- Linear spilines: m = 1, n > 2

- Quadratic spilines: m = 2, n > 3

- Cubic spilines: m = 3, n > 4



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Mathematical Model of PolynomialMathematical Model of Polynomial SplinesSplines

11 1 0( ) ...,

, 1m mm mk

n mf x a x a x a x a

1 2 11 1 21 2 11

[ , ,..Let [( ,..., ),( ,..., ),...,( ,..., )]., ]

k

k ki

n n k knxX X x x x x xX X X

1 i i

i

1 1 11 1

( ), // i.e.:n

f x x X x x x

2 2,... - are obtainedLow order polynomial fits

x x

:,

where knk k

otskn xots 1 1, 1,1 ,={ }i n i kx x x i x

: - = - -o skn to a n no a a po ns n



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The PolynomialThe Polynomial SplineSpline Fitting MethodsFitting Methods

function [ ] = PolySplineFitting (X1, Y1, X2, Y2)

p1 = polyfit (X1, Y1, 2);

fprintf ('fspline1(x) = %10.4f %s %10.4f %s %10.4f %s\n', p1(1), '*x^2 + (',

p1(2),')*x + (', p1(3), ')');

* ^ *

p2 = polyfit (X2, Y2, 2);fprintf ('fspline2(x) = %10.4f %s %10.4f %s %10.4f %s\n', p2(1), '*x^2 + (',

' * ' ' ', , ,

fspline2 = @(x2) p2(1)*x2^2 + p2(2)*x2 + p2(3);

grid on; hold on;

for xs1 = X1 1 : 0.01 : X2 1

ys1 = fspline1(xs1);

plot (xs1, ys1, 'k');

end

or xs = : . : eng


plot (xs2, ys2, 'g');

end

xlabel ('x - Segments');

ylabel ('y(x) - Splines');

legend ('Spline 1','Spline 2');


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% For iven x = 8 11 15 18 22 = 5 9 10 8 7

Application ofApplication ofPolySplineFittingPolySplineFitting

% Spline to 2 pieces

>> X1 = [8 11 15]

>> X2 = [15 18 22] 10

11

Data 1

Spline 1

>> Y1 = [5 9 10]

>> Y2 = [10 8 7]8

9

lines

Spline 2

6

7y(x)-

S

8 10 12 14 16 18 20 224

5

x - Se ments

>> PolySplineFitting (X1, Y1, X2, Y2)

fspline1(x) = -0.1548 *x^2 + ( 4.2738 )*x + ( -19.2857 )

* *. - . .



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4. Interpolation4. Interpolation

1. Introduction

2. Linear curve fittin- The least-square linear regression method










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4.1 Interpolation Methods Based on Curve Fitting4.1 Interpolation Methods Based on Curve Fitting

The result of a curve fitting, f(x) in [a, b], such asPolyRegression (X, Y, m), has already provided a solution

, , . . .

- This is known as the interpolation method based oncurve fitting.

- This also establishes the relationship between the

methods of curve fitt ing and interpolation.



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4.24.2 The Lagrange Polynomial Methods

The Lagrange polynomials are a form of polynomials forinterpolation for a given set of data.

- .

f(x) should be calculated forall x; so do forx int x afteran interpolation.



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Mathematical Model of the Lagrange MethodMathematical Model of the Lagrange Method

int 1Given ( ) at [ , ],nf x x x x

The representation of ( ) is:

( )n n n j

Lagrange polynomial f x

f x xx x

L xx

n11 1

where ( ) is called the .

( )

i j j i ji i

i

x x

Lagrange fuL nc ionx t

.g.:

2 1

1 2

( ) ( )

- 1st order LP: ( )

x x x x

f x y yx x x x

2 3 1 31 2

1 2 1 3 2 1 2 3

( )( ) ( )( )- 2nd order LP: ( )

( )( ) ( )( )

x x x x x x x xf x y y

x x x x x x x x

1 23

3 1 3 2

( )( )( )( )

x x x x yx x x x

...



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E.g.3 A 4E.g.3 A 4thth Order Lagrange Polynomial FittingOrder Lagrange Polynomial FittingGiven x = [1 2 4 5 7]

2 3 4 5 1 3 4 5

y = [52 5 -5 -40 10]

T he 4th order Lagrange polynomila is:

( )( )( )( ) ( )( )( )( )x x x x x x x x x x x x x x x x 1

1 2 1 3 1 4 1 5 2 1 2 3 2

( )( )( )( ) ( )( )(x x x x x x x x x x x x x 24 2 5

1 2 4 5 1 2 3 53 4

3 1 3 2 3 4 3 5 4 1 4 2 4 3 4 5

)( )

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

x x x

x x x x x x x x x x x x x x x xy y

x x x x x x x x x x x x x x x x

1 2 3 45

5 1 5 2 5 3 5 4

( )( )( )( )

( )( )( )( )

x x x x x x x xy

x x x x x x x x

52( 2)( 4)( 5)( 7) 5( 1)( 4)( 5)( 7)x x x x x x x x (1 2)(1 4)(1 5)(1 7) (2 1)(2 4)(2 5)(2 7)

5( 1)( 2)( 5)( 7) 40( 1)( 2)( 4)( 7)

(4 1)(4 2)(4 5)(4 7) (5 1)(5 2)(5 4)(5 7)

x x x x x x x x

10( 1)(

x x

2)( 4)( 5)

(7 1)(7 2)(7 4)(7 5)

0.72( 2)( 4)( 5)( 7) 0.17( 1)( 4)( 5)( 7)

x x

x x x x x x x x

int int

. .

0.06( 1)( 2)( 4)( 5)

L et 3, (

x x x x x x x x

x x x x

x f x

) 6.00


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MATLAB: The Lagrange Polynomial MethodMATLAB: The Lagrange Polynomial Method

function [Yint] = LagrangeINT(x, y, Xint)

for i = 1:nL(i) = 1;

or = :n

if j ~= i

L i = L i * Xint - x / x i - x ;end

end ( )n n n jx x

Yint = sum(y .* L);n

11 1 ( )i i j j i i jx x



34/56

Application of the Lagrange MethodApplication of the Lagrange Method

40

60

0

20

-40

-20

y

1 2 3 4 5 6 7-80

-60


x

4 34 3 Th S li B d M th d f I t l ti


35/56

4.34.3 The Spline-Based Methods for Interpolation

Inter olation b Pol nomial S lines

- Strategy: Polynomial spline fitting spline-based interpolation

Mathematical models

3 23 2 1 0

a) Polynomial spline fitting

, 1( )k

n mf x a x a x a x a

11 1 211 21

, 1,

2

1

1e , ..., , ...,

=

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

where { } - =,

k ki

knots i n i knots

n n k knx x x x x x

data pointsx x x N kn n kn

- ( -1)

k

k

1 1

23 2

1

2

sp nes:

( ),

( ),

i ii

k

f x x X

f x x X

x xx

13 2 1 0

...( ),

i i

k k

ii i

f x x

a x a x x

X

a a

int i

b) Interpolation in a s

(

pline

( ) i xx ff 3nt int 2int 3 2 1 0int int| )ix X a a x a ax x

MATLAB: The PolynomialMATLAB: The Polynomial SplineSpline MethodMethod


36/56

MATLAB: The PolynomialMATLAB: The Polynomial SplineSpline MethodMethod (1/2)(1/2)

function [Yi1, Yi2 ] = SplineFittingBasedInterpolation (X1, Y1, X2, Y2, Xi1, Xi2)

% Spline fitting



' * ' ' ', , ,

fspline1 = @(x1) p1(1)*x1^2 + p1(2)*x1 + p1(3);



p2(2),')*x + (', p2(3), ')');

fspline2 = @(x2) p2(1)*x2^2 + p2(2)*x2 + p2(3);

grid on; hold on;or xs = X : . : X


plot (xs1, ys1, 'k');

end

for xs2 = X2(1) : 0.01 : X2(length(X2))ys2 = fspline2(xs2);

plot (xs2, ys2, 'g');

en

xlabel ('x - Segments'); ylabel ('y(x) - Splines');

legend ('Spline 1','Spline 2'); ...

MATLAB: The PolynomialMATLAB: The Polynomial SplineSpline MethodMethod


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MATLAB: The PolynomialMATLAB: The Polynomial SplineSpline MethodMethod (2/2)(2/2)

=, , , , ,Xi1, Xi2)

% Inter olation

for xint1 = X1(1) : 0.5 : X2(1)

yint1 = fspline1(xint1);plot (X1, Y1, 'r*', xint1, yint1, 'b.')

end

for xint2 = X2(1) : 0.5 : X2(length(X2))y n = sp ne x n ;

plot (X2, Y2, 'r*', xint2, yint2, 'b.')

end

% Determine interpolation values

Yi1 = fspline1(Xi1);

= sp ne ;


Application ofApplication of SplineFittingBasedInterpolationSplineFittingBasedInterpolation


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% For iven x = 8 11 15 18 22 = 5 9 10 8 7

Application ofApplication ofSplineFittingBasedInterpolationSplineFittingBasedInterpolation

>> X1 = [8 11 15]

>> X2 = [15 18 22]

>> Y1 = [5 9 10] 10

11

Spline 1

Spline 2

>> Y2 = [10 8 7]

>> Xi1 = 10;Xi2 = 20 8

9

ines

6

7y(x)-

Spli

8 10 12 14 16 18 20 224

5

>> [Yi1, Yi2] = SplineFittingBasedInterpolation (X1, Y1, X2, Y2, Xi1, Xi2)

fs line1 x = -0.1548 *x^2 + 4.2738 *x + -19.2857

x - Segments

fspline2(x) = 0.0595 *x^2 + ( -2.6310 )*x + ( 36.0714 )

Yi1 = 7.9762Yi2 = 7.2619

4 44 4 D I t l ti dD I t l ti d C Fitti


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4.4 n4.4 n--D Interpolation andD Interpolation and Curve Fittingg

Matlab 2-D interpolation

Zint = interp2 (X, Y, Z, Xint, Yint)

Example[X, Y] = meshgrid(-3: .25 :3);

= pea s , ;

[Xint, Yint] = meshgrid(-3: .125 :3);

Zint = interp2 (X, Y, Z, Xint, Yint);

, , ,

hold,

mesh(Xint, Yint, Zint+15)

axis([-3 3 -3 3 -5 20])


Improved 3Improved 3 D Interpolation andD Interpolation and Curve Fittingg


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Improved 3Improved 3--D Interpolation andD Interpolation and Curve Fittingg

h0 = 0.25 h1 = 0.125 h2 = 0.01


Taylor Series Method for InterpolationTaylor Series Method for Interpolation


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Taylor Series Method for InterpolationTaylor Series Method for Interpolation

-200

0

200

400, - .>> F = 3*X.^2 + 4*X.*Y - 2*Y.^2

>> mesh (X, Y, F)

-10

-5

0

5

10

-10

-5

0

5

10

-400

>> Taylor1 = 20*(Y+3)-30

,x y x xy y

>> mes , , ay or

* ^ * ^10

-200

-100

0

100

200

200

400

- . - .

+ 4*(X-2).*(Y+3) + 20*(Y+3)-30

>> mesh (X, Y, Taylor2)

-10

-5

0

5

-10

-5

0

-5

0

5

10

-5

0

5

10

-400

-200

0

1 ! ( , )( , ) [ ( ) ( ) ]

nnk n kn f xyf x x a b

-10-10


0 ,k x a yn n x y

5 MATLAB A li ti i C Fitti d I t l ti5 MATLAB A li ti i C Fitti d I t l ti


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5. MATLAB Applications in Curve Fitting and Interpolation5. MATLAB Applications in Curve Fitting and Interpolation

1. Introduction








5. MATLAB applications in curve fitting and interpolation6. Summary


5 1 MATLAB B ilt5 1 MATLAB B ilt i F tii F ti


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5.1 MATLAB Built5.1 MATLAB Built--in Functionsin Functions

Curve fitting

polyfit (x, y, m)

interp1 (x, y, xi, method)

- Method: nearset linear default s line cubic chi- The last two may be used for extrapolation

interp2 (x, y, z, xi, yi, method)

, , , , , , ,

interpn (x1, x2, x3, , v, y1, y2, y3, , method)


Application ofApplication of polyfitpolyfit


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Application ofApplication ofpolyfitpolyfit

50

30

10

0 1 2 3 4 5 6

-10

0

Application ofApplication of interp1interp1


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Application ofApplication ofinterp1interp1




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Syntax

Zint = interp2 (X, Y, Z, Xint, Yint)

Example

[X, Y] = meshgrid(-3: .25 :3);

= pea s , ;

[Xint, Yint] = meshgrid(-3: .125 :3);

Zint = interp2 (X, Y, Z, Xint, Yint);

, , ,

hold,

mesh(Xint, Yint, Zint+15)

axis([-3 3 -3 3 -5 20])




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Syntax

VI = interp3 (X, Y, Z, V, XI, YI, ZI)

Example

[x,y,z,v] = flow(10);

[xi,yi,zi] = meshgrid

- -. . , . , .vi = interp3(x,y,z,v,xi,yi,zi);

slice(xi,yi,zi,vi,[6 9.5],2,[-2 .2]),


Application ofApplication of interpninterpn


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Application ofApplication ofinterpninterpn S ntax

VI = interpn (X1, X2, X3, ..., V, Y1, Y2, Y3, ...)

f = @(x,y,z,t) t.*exp(-x.^2 - y.^2 - z.^2);[x,y,z,t] = ndgrid(-1:0.2:1,-1:0.2:1,-1:0.2:1,0:2:10);

, , ,

[xi,yi,zi,ti] = ndgrid(-1:0.05:1,-1:0.08:1,

-1:0.05:1, ... 0:0.5:10);

= ' ', , , , , , , , ,nframes = size(ti, 4);

for j = 1:nframes

slice i :,:,:, , xi :,:,:, , zi :,:,:, , ...

vi(:,:,:,j),0,0,0);caxis([0 10]);

M(j) = getframe;

end


5 2 MATLAB Applications5 2 MATLAB Applications


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5.2 MATLAB Applications5.2 MATLAB Applications

Curve fitt ing

- Build a model of stopping distance of cars

- Build a model of bacteria growth over time

Interpolation

- Build a model of ocean water tem erature descent

Extrapolation

- Predicate ocean water temperature beyond sample data


E.g.4 Stopping Distances of CarsE.g.4 Stopping Distances of Cars


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E.g.4 Stopping Distances of CarsE.g.4 Stopping Distances of Cars

clear; clc

v=20:20:120;

d=[6 18 36 60 91 128];p = polyfi t (v, d, 2)

kModel=@ (x) p(1)*x.^2+p(2)*x+p(3);

vp = 20:120;

dp = kModel(vp);plot (vp, dp, v, d, '*')

xlabel('v (km/h)'), ylabel('d (m)')

>> p = 0.0079 0.1123 0.7000

% d(v) = 0.0079v

2

+ 0.1123v + 0.7000

E.g.5 Bacteria Growth RateE.g.5 Bacteria Growth Rate


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t0=0;

t=0:2:8;

y=log(Nt);

p=polyfit(t,y,1);

mu=p(1) 14

16

18

20

N0=exp(p(2)+mu*t0)

>> mu = 1.8837

>> =8

10

12

ln(Nt)

.

% f(t) = mu*t + ln(N0)

= 1.8837t + 3.65190 1 2 3 4 5 6 7 8

2

4

t

E.g.6 Ocean Water Temperature PredicationE.g.6 Ocean Water Temperature Predication


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>> D = [0 100 300 400 500 750 1000 1250 1500 1750];

= . . . . . .

>> WT_Int50 = interp1(D, WT, 50)

WT_Int50 = 21.8033

>> WT Int600 = inter 1 D WT 60020

22

_WT_Int600 = 13.4668

>> WT_Int1400 = interp1(D, WT, 1400)

WT_Int1400 = 4.7598 14

16

18

(oC)

>> WT_Extrap1800 =interp1(D, WT, 1800, 'pchip')

WT_Extrap1800 = 4.37258

10

12WT

_

interp1(D, WT, 2000, 'pchip')

WT_Extrap2000 = 4.32000 200 400 600 800 1000 1200 1400 1600 1800 2000

4

6

D (m)

E.g.7 Estimate the Absolute Zero by ExtrapolationE.g.7 Estimate the Absolute Zero by Extrapolation


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1 0G i v en , an dp a T a

1 0

[ 0 3 0 7 0 1 0 0 ][ 0 . 94 1 . 05 1 . 17 1 .28 ]

0 .0 0 3 3 0 .9 4 2 8

T

p

p a T a T

0

1

- - 0 . 9428L et 0 , - 2 8 5 .7 0 -2 7 3 .1 5 C

0 . 0 0 3 3

ap T

a

>> T = 0 10 20 30 40 50 60 70 80 90 100

>> P = [0.94 0.96 1.0 1.05 1.07 1.09 1.14 1.17 1.21 1.24 1.28];

>> p = polyfit (T, P, 1)

p = 0.0034 0.9336

= - . .

T0 = -274.5882 % Extrapolation

6 Summary6 Summary


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6. Summary6. Summary

1. Introduction








5. MATLAB applications in curve fitt ing and interpolation 6. Summary


Summary of Lecture 7Summary of Lecture 7


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Lecture 7. NMs for Curve Fitting and Interpolation

Overview Interpolation-- InductiveInductive modelingmodel ing o unctionso unctionsbased on sample databased on sample data

- Curve fitting- Interpolation

- Methods based on curve fitting

- The Lagrange polynomial methods- Spline-based methods

- Extrapolation

Linear curve fitting

- Least-squares estimation

MATLAB functions

- polyfit (x, y, m)

- inter 1 x xi method- e eas -square near

regression method

Nonlinear curve fitting

- interp2 (X, Y, Z, XI, YI)

- interp3 (X, Y, Z, V, XI, YI, ZI)

- interpn (X1, X2, X3, ..., V,-

method- Polynomial regression methods

- The spline method

, , , ...

Engineering applications- Case studies


Further ReadingFurther Reading


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gg

Numerical Methods for Engineers and Scientists- .


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