lecture 5 polynomial approximation
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數值方法 1
Polynomial InterpolationExampleLimitation
Polynomial Approximation (fitting)Line fittingQuadratic curve fittingPolynomial fitting
Lecture 5 Polynomial Lecture 5 Polynomial ApproximationApproximation
數值方法 2
PolynomialsPolynomials
P=poly([-4 0 3]); Polynomial determined byzeros in [-4,0,3]
x=linspace(-5,5);y=polyval(P,x);plot(x,y);
Plot polynomial P
數值方法 3
Sampling of paired dataSampling of paired data
P=poly([-4 0 3]); Produce a polynomialwith zeros [-4,0,3]
x=rand(1,6)*10-5;y=polyval(P,x);plot(x,y, 'ro');
Plot paired data
數值方法 4
Polynomial interpolationPolynomial interpolation
P=poly([-4 0 3]); n=6Produce a polynomialwith zeros [-4,0,3]
x=rand(1,n)*10-5;y=polyval(P,x);plot(x,y, 'ro');
Plot the interpolating polynomial
v=linspace(min(x),max(x));z=int_poly(v,x,y);plot(v,z,'k');
數值方法 5
Source codes
Demo_ipDemo_ip
數值方法 6
m=3, n=5m=3, n=5
>> demo_ipdata size n=5
數值方法 7
m=3, n=6m=3, n=6
>> demo_ipdata size n=6
數值方法 8
m=3, n=10m=3, n=10
>> demo_ipdata size n=10
數值方法 9
>> demo_ipdata size n=12
m=3, n=12m=3, n=12
數值方法 10
P=poly([-4 0 3]);n=6; Produce a polynomialwith zeros [-4,0,3]
x=rand(1,n)*10-5;y=polyval(P,x);nois=rand(1,n)*0.5-0.25;plot(x,y+nois, 'ro');
Plot the interpolating polynomial
v=linspace(min(x),max(x));z=int_poly(v,x,y+nois);plot(v,z,'k');
Sample with noiseSample with noise
數值方法 11
Demo_ip_noisyDemo_ip_noisy
source codes
數值方法 12
Noise dataNoise data
►Noise ratioNoise ratio Mean of abs(noise)/abs(signal) Mean of abs(noise)/abs(signal)
noise_ratio =
0.0074
數值方法 13
Numerical resultsNumerical results
Small noise causes intolerant interpolation
>> demo_ip_noisydata size n:12
noise_ratio =
0.0053
數值方法 14
Numerical resultsNumerical results
Small noise causes intolerant interpolation
>> demo_ip_noisydata size n:13
noise_ratio =
0.0031
數值方法 15
Interpolation Vs Interpolation Vs approximationapproximation
► An interpolating polynomial is expected to satisfy An interpolating polynomial is expected to satisfy all constraints posed by paired dataall constraints posed by paired data
► An interpolating polynomial is unable to retrieve an An interpolating polynomial is unable to retrieve an original target function when noisy paired data are original target function when noisy paired data are provided provided
► For noisy paired data, the goal of polynomial fitting For noisy paired data, the goal of polynomial fitting is revised to minimize the mean square is revised to minimize the mean square approximating errorapproximating error
數值方法 16
Polynomial approximationPolynomial approximation
►Given paired data, (xGiven paired data, (xii,y,yii), i=1,…,n,), i=1,…,n,
the approximating polynomial is the approximating polynomial is required to minimize the mean required to minimize the mean square error of approximating ysquare error of approximating yi i by by f(xf(xii))
2
1
))((1
E
n
iii yxf
n
數值方法 17
Polynomial fittingPolynomial fitting
fa1d_polyfit.m
數值方法 18
Line fittingLine fitting
►Minimizaing the mean square Minimizaing the mean square approximating errorapproximating error
2
1
)(1
b)E(a,
n
iii ybax
n
Target: baxy
數值方法 19
Line fittingLine fitting
Red:Approximating polynomial
>> fa1d_polyfitinput a function: x.^2+cos(x) :x+5keyin sample size:300polynomial degree:1
E =
8.3252e-004
a=1;b=5
數值方法 20
Line fittingLine fitting
Red:Approximating polynomial
a=3;b=1/2
>> fa1d_polyfitinput a function: x.^2+cos(x) :3*x+1/2keyin sample size:30polynomial degree:1
E =
0.0010
數值方法 21
Objective function IObjective function I
►Line fitting Line fitting
2
1fitting-Line1 )(
1b)(a,E)(E
n
iii ybax
n
Target: baxy
數值方法 22
►EE1 1 is a quadratic function of a and bis a quadratic function of a and b
►Setting derivatives of ESetting derivatives of E1 1 to zero to zero leads to leads to
a linear systema linear system 2
1fitting-Line1 )(
1b)(a,E)(E
n
iii ybax
n
數值方法 23
0)(2
0)(2
0
)(1
b)(a,E)(E
1
1
1
1
1
2
1fitting-Line1
n
iii
i
n
iii
n
iii
ybaxndb
dE
xybaxnda
dEd
dE
ybaxn
a linear system
數值方法 24
n
ii
n
ii
n
iii
n
ii
n
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n
iii
n
iiiii
n
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i
n
iii
ynbax
yxbxax
ybax
xybxax
ybaxndb
dE
xybaxnda
dE
11
111
2
1
1
2
1
1
1
1
0)(
0)(
0)(2
0)(2
數值方法 25
n
ii
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iii
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y
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111
2
數值方法 26
n
ii
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iii
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2
數值方法 27
Objective function IIObjective function II
►Quadratic polynomial fittingQuadratic polynomial fitting
2
1
2
gurveFittinQuadraticC2
)(1
c)b,(a,E)(E
n
iiii ycbxax
n
Target: cbxaxy 2
數值方法 28
►EE2 2 are quadraticare quadratic
►Setting derivatives of ESetting derivatives of E2 2 to zero to zero leads to leads to
a linear systema linear system
2
1
2
gurveFittinQuadraticC2
)(1
c)b,(a,E)(E
n
iiii ycbxax
n
數值方法 29
0)(2
0)(2
0)(2
0
)(1
c)b,(a,E)(E
1
23
1
22
2
1
22
2
2
1
2gurveFittinQuadraticC2
n
iiii
i
n
iiii
i
n
iiii
n
iiii
ycbxaxndc
dE
xycbxaxndb
dE
xycbxaxnda
dEd
dE
ycbxaxn
a linear system
數值方法 30
0)(
0)(
0)(
1
2
1
2
2
1
2
n
iiii
i
n
iiii
i
n
iiii
ycbxax
xycbxax
xycbxax
0)(
0)(
0)(
1
2
1
23
1
2234
n
iiii
n
iiiiii
n
iiiiii
ycbxax
xycxbxax
xycxbxaxˋ
數值方法 31
0)(
0)(
0)(
1
2
1
23
1
2234
n
iiii
n
iiiiii
n
iiiiii
ycbxax
xycxbxax
xycxbxaxˋ
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yncbxax
xycxbxax
xycxbxax
111
2
111
2
1
3
1
2
1
2
1
3
1
4
數值方法 32
n
ii
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iii
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4
數值方法 33
Quadratic polynomial Quadratic polynomial fittingfitting
►Minimization of an approximating Minimization of an approximating errorerror
2
1
2 )(1
c)b,E(a,
n
iiii ycbxax
n
Target: cbxaxy 2
數值方法 34
Quadratic poly fittingQuadratic poly fitting
input a function: x.^2+cos(x) :3*x.^2-2*x+1keyin sample size:20polynomial degree:2
E =
6.7774e-004a=3b=-2c=1
數值方法 35
Data driven polynomial Data driven polynomial approximationapproximation
►Minimization of Mean square error Minimization of Mean square error (mse)(mse)
►Data driven polynomial Data driven polynomial approximationapproximation f : a polynomial pf : a polynomial pmm
Polynomial degree m is less than data Polynomial degree m is less than data size nsize n
2
1
))((1
E
n
iiim yxp
n
數值方法 36
i allfor )( 0E
))((1
E 2
1
imi
n
iiim
xpy
yxpn
Special case: polynomial Special case: polynomial interpolationinterpolation
數值方法 37
POLYFIT: Fit polynomial to dataPOLYFIT: Fit polynomial to data
►polyfit(x,y,m)polyfit(x,y,m) x : input vectors or predictorsx : input vectors or predictors y : desired outputsy : desired outputs m : degree of interpolating polynomialm : degree of interpolating polynomial
►Use m to prevent from over-fittingUse m to prevent from over-fitting►Tolerance to noiseTolerance to noise
數值方法 38
POLYFIT: Fit polynomial to POLYFIT: Fit polynomial to datadata
P=poly([-4 0 3]); n=20;m=3; A polynomial determinedby zeros in [-4,0,3]
x=rand(1,20)*10-5;y=polyval(P,x);nois=rand(1,20)*0.5-0.25;plot(x,y+nois, 'ro');
Plot the interpolating polynomial
v=linspace(min(x),max(x));p=polyfit(x,y+nois,m);hold on;plot(v,polyval(p,v),'k');
數值方法 39
Non-polynomialNon-polynomial
► sinsin
fx=inline('sin(x)'); n=20;m=3
x=rand(1,n)*10-5;y=fx(x);nois=rand(1,n)*0.5-0.25;plot(x,y+nois, 'ro');hold on
數值方法 40
Under-fittingUnder-fitting
v=linspace(min(x),max(x));p=polyfit(x,y+nois,m);hold on;plot(v,polyval(p,v),'k');
m=3
Under-fitting due to approximating non-polynomial by low-degree polynomials
數值方法 41
Intolerant mseIntolerant mse
>> mean((polyval(p,x)-(y+nois)).^2)
ans =
0.0898
Under-fitting causes intolerant mean square error
數值方法 42
m=4;v=linspace(min(x),max(x));p=polyfit(x,y+nois,m);hold on;plot(v,polyval(p,v), 'b');
Under-fittingUnder-fitting
M=4
數值方法 43
>> fa1d_polyfitinput a function: x.^2+cos(x) :sin(x)keyin sample size:50polynomial degree:5
E =
0.0365
Fitting non-polynomialFitting non-polynomial
數值方法 44
>> fa1d_polyfitinput a function: x.^2+cos(x) :tanh(x+2)+sech(x)keyin sample size:30polynomial degree:5
E =
0.0097
Fitting non-polynomialFitting non-polynomial
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