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Function ApproximationFunction 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
Interpolation Interpolation and and
RegressionRegression
Chapter 13Chapter 13
Curve Fitting:Curve Fitting: Fitting a Straight LineFitting a Straight Line
Least Square RegressionLeast Square Regression Curve FittingCurve Fitting Statistics ReviewStatistics Review Linear Least Square RegressionLinear Least Square Regression Linearization of Nonlinear Linearization of Nonlinear
RelationshipsRelationships MATLAB FunctionsMATLAB Functions
Wind Tunnel ExperimentWind Tunnel Experiment
Measure air resistance as a function of velocity
Curve Fitting
(a) Least-squares regression
(b) Linear interpolation
(c) Curvilinear interpolation
Regression and InterpolationRegression and Interpolation
Curve fitting
Least-squares fit of a straight lineLeast-squares fit of a straight line
145083012206105503807025
8070605040302010v, m/s
F, N
Simple StatisticsSimple StatisticsMeasurement of the coefficient of thermal expansion of structural steel [106 in/(inF)]
535666766706592663366276
598649966216445645164036
703654266596624673366626
721639665646435662565556
655647866216543639965526
667632564956775655464856
......
......
......
......
......
......
Mean, standard deviation, variance, etc.
Statistics ReviewStatistics Review Arithmetic mean
Standard deviation about the mean
Variance (spread)
Coefficient of variation (c.v.)
n
yy i
2itt
y yyS 1n
Ss
;
1n
nyy
1n
yys
2
i2i
2i2
y
/
%.. 100y
svc y
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
2i
2i i y yyy i
Coefficient of Thermal ExpansionCoefficient of Thermal Expansion
%.%.
.%..
..
/).(./
..
.
..
64110056976
107770100
y
svc
0116147035
4065140
35
365092361981554
1n
nyys
107770136
4065140
1n
Ss
4065140yyS
5697636
509236
n
yy
y
22
i2i2
y
ty
2
it
i
Sum of the square of residuals
Standard deviation
Variance
Coefficient of variation
Mean
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)
HistogramHistogram
Normal Distribution
22
1( ) exp
22
x xp x
Regression and ResidualRegression and Residual
Linear RegressionLinear RegressionFitting a straight line to observations
Small residual errors Large residual errors
Equation for straight line
Difference between observation and line
ei is the residual or error
xaay 10
ii10i exaay
Linear RegressionLinear Regression
Least Squares ApproximationLeast Squares Approximation
Minimizing Residuals (Errors) minimum average error (cancellation) minimum absolute error minimax error (minimizing the maximum
error) least squares (linear, quadratic, ….)
)
n
1ii10i
n
1ii xaa(ye
n
1ii10i
n
1ii xaaye
Minimize the Maximum Error
Minimize Sum of Errors
Minimize Sum of Absolute Errors
Linear Least SquaresLinear Least Squares
Minimize total square-error Straight line approximation
Not likely to pass all points if n > 2
),( , , ),( , ),( , ),( 332211 nn yxyxyxyx
i10ii
10
xaaxfy
xaaxf
)(
)(
Linear Least SquaresLinear Least Squares
Total square-error function: sum of the squares of the residuals
Minimizing square-error Sr(a0 ,a1)
),( , , ),( , ),( , ),( 332211 nn yxyxyxyx
n
1i
2i10i
n
1i
2ir xaayeS )(
0a
S
0a
S
1
r
0
r
Solve for (a0 ,a1)
Linear Least SquaresLinear Least Squares Minimize
Normal equation y = a0 + a1x
n
1i
2i10i10r xaayaaS )(),(
n
1iii1
n
1i
2i0
n
1ii
n
1ii1
n
1ii0
n
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1
r
n
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r
yxaxax
yaxna
xxaay20a
S
xaay20a
S
n
xa
n
yxaya
xxn
yxyxna
i1
i10
2
i2i
iiii1
Advantage of Least SquaresAdvantage 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 SquaresLinear Least Squares Use sum( ) in MATLAB
SnS
SSnSa
SnS
SSSSa
S
S
a
a
SS
Sn
yS yxS
xS xS
let
2xxx
yxxy12
xxx
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xy
y
1
0
xxx
x
n
1iiyi
n
1iixy
n
1iix
n
1i
2ixx
,
,
,,
Correlation CoefficientCorrelation 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
n
ii
n
iit y
ny yyS
1
2
1
1 ;)(
2n
1ii10ir xaayS )(
t
rt
S
SSr
trt SSSr /)(2
Correlation CoefficientCorrelation Coefficient Alternative formulation of
correlation coefficient More convenient for
computer implementation
2i
2i
2i
2i
iiii
yynxxn
yxyxnr
)()(
))((
Standard Error of the EstimateStandard Error of the Estimate If the data spread about the line is normal “Standard deviation” for the regression line
2n
SS r
xy /
Standard error of the estimate
No error if n = 2 (a0 and a1)
Spread of data around the mean
Spread of data around the best-fit line
Linear regression reduce the spread of dataLinear regression reduce the spread of data
Normal distributions
Standard Deviation for Regression LineStandard Deviation for Regression Line
Sy/x
Sy
Sy : Spread around the mean
Sy/x : Spread around the regression line
2n
SS
1n
SS
rx/y
ty
Example: Linear RegressionExample: Linear Regression
5.119)5.5(7)0.6)(6()5.2)(5(
)0.4)(4()0.2)(3()5.2)(2()5.0)(1(yxS
1407654321xS
24/7n /Sy ; 0.245.50.65.30.40.25.25.0yS
428/7/nSx ; 287654321xS
iixy
22222222ixx
yiy
xix
99112S714322S5119S140S024S28S
199302908453849557
797206122603636066
589600051051725535
326503265001616044
3473004082069023
5625086220054522
1687057658501501
xaayyyyxxyx
rtxyxxyx
2i10i
2iii
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....
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)()(
Example: Linear RegressionExample: Linear Regression
8392857140a 0714285730a
5119
24
a
a
14028
287
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S
a
a
SS
Sn
10
1
0
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0
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.,.
.
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ty
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.)(
Standard deviation about the mean
Standard error of the estimate
773505
99112
2n
SS
99112xaayS
rxy
2i10ir
..
.)(
/
Correlation coefficient 932.0
7143.22
9911.27143.22
S
SSr
t
rt
Linear Least Square Regression Linear Least Square Regression
Modified MATLAB M-FileModified MATLAB M-File
» x=1:7x = 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 = 0.0714a1 = 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 -0.4464err = 2.9911Syx = 0.7734r = 0.9318s = 0.0714 0.8393 y =0.0714 + 0.8393 x
Sum of squares of residuals Sr
Standard error of the estimate
Correlation coefficient
» 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 = 4.5933a1 = -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 0.0764err = 23.1082Syx = 1.6996r = 0.9617s = 4.5933 -1.8570 y = 4.5933 1.8570 x
r = 0.9617
Linear Least Square
y = 4.5933 1.8570 x
Error Sr = 23.1082
Correlation Coefficient r = 0.9617
» [x,y]=example2x = 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 -0.2000y = 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 = -20.5717a1 = 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 21.7918err = 4.2013e+003Syx = 22.9165r = 0.4434s = -20.5717 3.6005
Correlation coefficient r = 0.4434
Linear Least Square: y = 20.5717 + 3.6005x
Data in arbitrary order
Large errors !!
Linear regression
y = 20.5717 +3.6005x
Error Sr = 4201.3Correlation r = 0.4434 !!
Linearization of Nonlinear RelationshipsLinearization of Nonlinear Relationships
Untransformed power equation
x vs. y
transformed datalog x vs. log y
Linearization of Linearization of Nonlinear RelationshipsNonlinear Relationships
Exponential equation
Power equation
iiii
11
x1
yx of instead yx use
xy
ey 1
,ln,
lnln
iiii
22
2
yx of instead yx use
x y
xy 2
,log,log
logloglog
log : Base-10
Linearization of Linearization of Nonlinear RelationshipsNonlinear Relationships
Saturation-growth-rate equation
Rational function
iii
i
4444
yx of instead y
1x use
xy
1
x
1y
,,
iiii
3
3
333
yx of instead y
1
x
1 use
x
11
y
1
x
xy
,,
Power equation fit along with the data
x vs. y
Transformed Data
log xi vs. log yi
Example 12.4: Power EquationExample 12.4: Power Equation
y = 2 x 2
12-12
>> 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 -234.2857r2 = 0.8805
y = 19.4702x 234.2857
12-13
>> x=[10 20 30 40 50 60 70 80];>> y = [25 70 380 550 610 1220 830 1450];>> linregr(log10(x),log10(y))r2 = 0.9481ans = 1.9842 -0.5620
log x vs. log y
log y = 1.9842 log x – 0.5620
y = (10–0.5620)x1.9842 = 0.2742 x1.9842
Least-square fit of nth-order polynomial p = polyfit(x,y,n)
Evaluate the value of polynomial using
y = polyval(p,x)
MATLAB FunctionsMATLAB Functions
n1n2n
21n
1 pxpxpxpxf )(
CVEN 302-501CVEN 302-501Homework No. 9Homework 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 You may use spread sheets for your hand
computationcomputation
Due Oct/22, 2008 Wednesday at the Due Oct/22, 2008 Wednesday at the beginning of the periodbeginning of the period