Function Approximation

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# Function Approximation - PowerPoint PPT Presentation

Function Approximation. Function approximation (Chapters 13 &amp; 14) -- method of least squares -- minimize the residuals -- given data of points have noises -- the purpose is to find the trend represented by data. Function interpolation (Chapters 15 &amp; 16)

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Function Approximation
• Function approximation (Chapters 13 & 14)

-- method of least squares

-- minimize the residuals

-- given data of points have noises

-- the purpose is to find the trend represented by data.

• Function interpolation (Chapters 15 & 16)

-- approximating function match the given data exactly

-- given data of points are precise

-- the purpose is to find data between these points

### Chapter 13

Curve Fitting: Fitting a Straight Line

Least Square Regression
• Curve Fitting
• Statistics Review
• Linear Least Square Regression
• Linearization of Nonlinear Relationships
• MATLAB Functions

Wind Tunnel Experiment

Curve Fitting

Measure air resistance as a function of velocity

Regression and Interpolation

Curve fitting

(a) Least-squares regression

(b) Linear interpolation

(c) Curvilinear interpolation

Simple Statistics

Measurement of the coefficient of thermal expansion of structural steel [106 in/(inF)]

Mean, standard deviation, variance, etc.

Statistics Review
• Arithmetic mean
• Standard deviationabout the mean
• Coefficient of variation (c.v.)

1 6.485 0.007173 42.055

2 6.554 0.000246 42.955

3 6.775 0.042150 45.901

4 6.495 0.005579 42.185

5 6.325 0.059875 40.006

6 6.667 0.009468 44.449

7 6.552 0.000313 42.929

8 6.399 0.029137 40.947

9 6.543 0.000713 42.811

10 6.621 0.002632 43.838

11 6.478 0.008408 41.964

12 6.655 0.007277 44.289

13 6.555 0.000216 42.968

14 6.625 0.003059 43.891

15 6.435 0.018143 41.409

16 6.564 0.000032 43.086

17 6.396 0.030170 40.909

18 6.721 0.022893 45.172

19 6.662 0.008520 44.382

20 6.733 0.026669 45.333

21 6.624 0.002949 43.877

22 6.659 0.007975 44.342

23 6.542 0.000767 42.798

24 6.703 0.017770 44.930

25 6.403 0.027787 40.998

26 6.451 0.014088 41.615

27 6.445 0.015549 41.538

28 6.621 0.002632 43.838

29 6.499 0.004998 42.237

30 6.598 0.000801 43.534

31 6.627 0.003284 43.917

32 6.633 0.004008 43.997

33 6.592 0.000498 43.454

34 6.670 0.010061 44.489

35 6.667 0.009468 44.449

36 6.535 0.001204 42.706

 236.509 0.406514 1554.198

Coefficient of Thermal Expansion

Mean

Sum of the square of residuals

Standard deviation

Variance

Coefficient of variation

Histogram

Normal Distribution

• A histogram used to depict the distribution of data
• For large data set, the histogram often approaches the normal distribution (use data in Table 12.2)

Linear Regression

Fitting a straight line to observations

Small residual errors Large residual errors

Linear Regression
• Equation for straight line
• Difference between observation and line
• ei is the residual or error
Least Squares Approximation
• Minimizing Residuals (Errors)
• minimum average error (cancellation)
• minimum absolute error
• minimax error (minimizing the maximum error)
• least squares (linear, quadratic, ….)

Minimize Sum of Errors

Minimize Sum of Absolute Errors

Minimize the Maximum Error

Linear Least Squares
• Minimize total square-error
• Straight line approximation
• Not likely to pass all points if n > 2
Linear Least Squares
• Total square-error function: sum of the squares of the residuals
• Minimizing square-error Sr(a0 ,a1)

Solve for (a0 ,a1)

Linear Least Squares
• Minimize
• Normal equation y = a0 + a1x
Advantage of Least Squares
• Positive differences do not cancel negative differences
• Differentiation is straightforward
• weighted differences
• Small differences become smaller and large differences are magnified
Linear Least Squares
• Use sum( ) in MATLAB
Correlation Coefficient
• Sum of squares of the residuals with respect to the mean
• Sum of squares of the residuals with respect to the regression line
• Coefficient of determination
• Correlation coefficient
Correlation Coefficient
• Alternative formulation of correlation coefficient
• More convenient for computer implementation
Standard Error of the Estimate
• If the data spread about the line is normal
• “Standard deviation” for the regression line

Standard error of the estimate

No error if n = 2 (a0and a1)

Linear regression reduce the spread of data

Spread of data around the mean

Normal distributions

Spread of data around the best-fit line

Standard Deviation for Regression Line

Sy/x

Sy

Sy : Spread around the mean

Sy/x : Spread around the regression line

Example: Linear Regression

Standard deviation about the mean

Standard error of the estimate

Correlation coefficient

» x=1:7

x =

1 2 3 4 5 6 7

» y=[0.5 2.5 2.0 4.0 3.5 6.0 5.5]

y =

0.5000 2.5000 2.0000 4.0000 3.5000 6.0000 5.5000

» s=linear_LS(x,y)

a0 =

0.0714

a1 =

0.8393

x y (a0+a1*x) (y-a0-a1*x)

1.0000 0.5000 0.9107 -0.4107

2.0000 2.5000 1.7500 0.7500

3.0000 2.0000 2.5893 -0.5893

4.0000 4.0000 3.4286 0.5714

5.0000 3.5000 4.2679 -0.7679

6.0000 6.0000 5.1071 0.8929

7.0000 5.5000 5.9464 -0.4464

err =

2.9911

Syx =

0.7734

r =

0.9318

s =

0.0714 0.8393

Sum of squares of residuals Sr

Standard error of the estimate

Correlation coefficient

y =0.0714 + 0.8393 x

» x=0:1:7; y=[0.5 2.5 2 4 3.5 6.0 5.5];

Linear regression

y = 0.0714+0.8393x

Error :Sr = 2.9911

correlation coefficient : r = 0.9318

function [x,y] = example1

x = [ 1 2 3 4 5 6 7 8 9 10];

y = [2.9 0.5 -0.2 -3.8 -5.4 -4.3 -7.8 -13.8 -10.4 -13.9];

» [x,y]=example1;

» s=Linear_LS(x,y)

a0 =

4.5933

a1 =

-1.8570

x y (a0+a1*x) (y-a0-a1*x)

1.0000 2.9000 2.7364 0.1636

2.0000 0.5000 0.8794 -0.3794

3.0000 -0.2000 -0.9776 0.7776

4.0000 -3.8000 -2.8345 -0.9655

5.0000 -5.4000 -4.6915 -0.7085

6.0000 -4.3000 -6.5485 2.2485

7.0000 -7.8000 -8.4055 0.6055

8.0000 -13.8000 -10.2624 -3.5376

9.0000 -10.4000 -12.1194 1.7194

10.0000 -13.9000 -13.9764 0.0764

err =

23.1082

Syx =

1.6996

r =

0.9617

s =

4.5933 -1.8570

r = 0.9617

y = 4.5933  1.8570 x

Linear Least Square

y = 4.5933 1.8570 x

Error Sr = 23.1082

Correlation Coefficient r = 0.9617

» [x,y]=example2

x =

Columns 1 through 7

-2.5000 3.0000 1.7000 -4.9000 0.6000 -0.5000 4.0000

Columns 8 through 10

-2.2000 -4.3000 -0.2000

y =

Columns 1 through 7

-20.1000 -21.8000 -6.0000 -65.4000 0.2000 0.6000 -41.3000

Columns 8 through 10

-15.4000 -56.1000 0.5000

» s=Linear_LS(x,y)

a0 =

-20.5717

a1 =

3.6005

x y (a0+a1*x) (y-a0-a1*x)

-2.5000 -20.1000 -29.5730 9.4730

3.0000 -21.8000 -9.7702 -12.0298

1.7000 -6.0000 -14.4509 8.4509

-4.9000 -65.4000 -38.2142 -27.1858

0.6000 0.2000 -18.4114 18.6114

-0.5000 0.6000 -22.3720 22.9720

4.0000 -41.3000 -6.1697 -35.1303

-2.2000 -15.4000 -28.4929 13.0929

-4.3000 -56.1000 -36.0539 -20.0461

-0.2000 0.5000 -21.2918 21.7918

err =

4.2013e+003

Syx =

22.9165

r =

0.4434

s =

-20.5717 3.6005

Data in arbitrary order

Large errors !!

Correlation coefficient r = 0.4434

Linear Least Square: y =  20.5717 + 3.6005x

Linear regression

y =  20.5717 +3.6005x

Error Sr = 4201.3

Correlation r = 0.4434 !!

Linearization of Nonlinear Relationships
• Exponential equation
• Power equation

log : Base-10

Linearization of Nonlinear Relationships
• Saturation-growth-rate equation
• Rational function

Example 12.4: Power Equation

Transformed Data

log xi vs. log yi

y = 2 x 2

Power equation fit along with the data

x vs. y

>> x=[10 20 30 40 50 60 70 80];

>> y = [25 70 380 550 610 1220 830 1450];

>> [a, r2] = linregr(x,y)

a =

19.4702 -234.2857

r2 =

0.8805

y = 19.4702x  234.2857

12-12

>> x=[10 20 30 40 50 60 70 80];

>> y = [25 70 380 550 610 1220 830 1450];

>> linregr(log10(x),log10(y))

r2 =

0.9481

ans =

1.9842 -0.5620

log y = 1.9842 log x – 0.5620

y = (10–0.5620)x1.9842 =0.2742x1.9842

log x vs. log y

12-13

MATLAB Functions
• Least-square fit of nth-order polynomial

p = polyfit(x,y,n)

• Evaluate the value of polynomial using

y = polyval(p,x)

CVEN 302-501Homework No. 9
• Chapter 13
• Prob. 13.1 (20)& 13.2(20) (Hand Calculations)
• Prob. 13.5 (30) & 13.7(30) (Hand Calculation and MATLAB program)
• You may use spread sheets for your hand computation
• Due Oct/22, 2008 Wednesday at the beginning of the period