1. Lecture 5 Polynomial Approximation
Polynomial Interpolation
Example
Limitation
Polynomial Approximation (fitting)
Line fitting
Quadratic curve fitting
Polynomial fitting
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2. Polynomials P=poly([-4 0 3]); x=linspace(-5,5); y=polyval(P,x);
Polynomial determined by
zeros in [-4,0,3]
P=poly([-4 0 3]);
x=linspace(-5,5);
y=polyval(P,x);
plot(x,y);
Plot polynomial P
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3. Sampling of paired data
Produce a polynomial
with zeros [-4,0,3]
P=poly([-4 0 3]);
x=rand(1,6)*10-5;
y=polyval(P,x);
plot(x,y, 'ro');
Plot paired data
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4. Polynomial interpolation
Produce a polynomial
with zeros [-4,0,3]
P=poly([-4 0 3]); n=6
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');
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5. Demo_ip
Source codes
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6. m=3, n=5
>> demo_ip
data size n=5
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7. m=3, n=6
>> demo_ip
data size n=6
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8. m=3, n=10
>> demo_ip
data size n=10
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9. m=3, n=12
>> demo_ip
data size n=12
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10. Sample with noise P=poly([-4 0 3]);n=6; x=rand(1,n)*10-5;
Produce a polynomial
with zeros [-4,0,3]
P=poly([-4 0 3]);n=6;
x=rand(1,n)*10-5;
y=polyval(P,x);
nois=rand(1,n)* ;
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');
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11. Demo_ip_noisy
source codes
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12. Noise data Noise ratio noise_ratio = 0.0074
Mean of abs(noise)/abs(signal)
noise_ratio =
0.0074
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13. Numerical results Small noise causes intolerant interpolation
>> demo_ip_noisy
data size n:12
noise_ratio =
0.0053
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14. Numerical results Small noise causes intolerant interpolation
>> demo_ip_noisy
data size n:13
noise_ratio =
0.0031
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15. Interpolation Vs approximation
An interpolating polynomial is expected to satisfy all constraints posed by paired data
An interpolating polynomial is unable to retrieve an original target function when noisy paired data are provided
For noisy paired data, the goal of polynomial fitting is revised to minimize the mean square approximating error
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16. Polynomial approximation
Given paired data, (xi,yi), i=1,…,n,
the approximating polynomial is required to minimize the mean square error of approximating yi by f(xi)
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17. Polynomial fitting
fa1d_polyfit.m
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18. Line fitting Target: Minimizaing the mean square approximating error
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19. Line fitting >> fa1d_polyfit input a function: x.^2+cos(x) :x+5
keyin sample size:300
polynomial degree:1
E =
8.3252e-004
Red:
Approximating polynomial
a=1;b=5
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20. Line fitting Red: Approximating polynomial >> fa1d_polyfit
input a function: x.^2+cos(x) :3*x+1/2
keyin sample size:30
polynomial degree:1
E =
0.0010
Red:
Approximating polynomial
a=3;b=1/2
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21. Objective function I
Line fitting
Target:
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22. E1 is a quadratic function of a and b
Setting derivatives of E1 to zero leads to
a linear system
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23. a linear system
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24. 數值方法

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27. Objective function II
Quadratic polynomial fitting
Target:
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28. Setting derivatives of E2 to zero leads to a linear system
E2 are quadratic
Setting derivatives of E2 to zero leads to
a linear system
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29. a linear system
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33. Quadratic polynomial fitting
Minimization of an approximating error
Target:
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34. Quadratic poly fitting
input a function: x.^2+cos(x) :3*x.^2-2*x+1
keyin sample size:20
polynomial degree:2
E =
6.7774e-004
a=3
b=-2
c=1
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35. Data driven polynomial approximation
Minimization of Mean square error (mse)
Data driven polynomial approximation
f : a polynomial pm
Polynomial degree m is less than data size n
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36. Special case: polynomial interpolation
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37. POLYFIT: Fit polynomial to data
polyfit(x,y,m)
x : input vectors or predictors
y : desired outputs
m : degree of interpolating polynomial
Use m to prevent from over-fitting
Tolerance to noise
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38. POLYFIT: Fit polynomial to data
A polynomial determined
by zeros in [-4,0,3]
P=poly([-4 0 3]); n=20;m=3;
x=rand(1,20)*10-5;
y=polyval(P,x);
nois=rand(1,20)* ;
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');
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39. Non-polynomial sin fx=inline('sin(x)'); n=20;m=3 x=rand(1,n)*10-5;
y=fx(x);
nois=rand(1,n)* ;
plot(x,y+nois, 'ro');hold on
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40. Under-fitting m=3 v=linspace(min(x),max(x));
p=polyfit(x,y+nois,m);hold on;
plot(v,polyval(p,v),'k');
Under-fitting due to approximating non-polynomial by low-degree polynomials
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41. Intolerant mse >> mean((polyval(p,x)-(y+nois)).^2) ans = 0.0898
Under-fitting causes intolerant mean square error
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42. Under-fitting M=4 m=4; v=linspace(min(x),max(x));
p=polyfit(x,y+nois,m);hold on;
plot(v,polyval(p,v), 'b');
M=4
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43. Fitting non-polynomial
>> fa1d_polyfit
input a function: x.^2+cos(x) :sin(x)
keyin sample size:50
polynomial degree:5
E =
0.0365
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44. Fitting non-polynomial
>> fa1d_polyfit
input a function: x.^2+cos(x) :tanh(x+2)+sech(x)
keyin sample size:30
polynomial degree:5
E =
0.0097
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