Polynomial curve fitting

Hi,

I am trying to make polynomial curve fitting in sine wave. First I have created the wave and I took 10 samples, on which I add noise from a gaussian distribution.Now, I am trying to make curve fitting with a polynomial of 9th degree.I think that my results are wrong as the green curve is linear between the points.Does anybody know if am I correct?If there is a mistake I would like to inform me.

Thanks in advance

%%%%%%% code %%%%%%%%% t=0:0.001:1; k=sin(2*pi*t); plot(t,k); x=linspace(0,1,10);

for i=1:1:10 y(i)=sin(2*pi*x(i)); end;

r = randn(10,1);

for i=1:1:10 y_noise(i)=y(i)+r(i); end;

p = polyfit(x,y,9);

x2 = x; y2 = polyval(p,x2); plot(x,y,'o',x2,y2,t,k) grid on

disp(p)

RMSE=sqrt(mean((y-y2).^2+(x-x2).^2));

Answer 1

[Matt Fig]

You did it correctly. The curve is linear between the points because that is how MATLAB plots these things. If you zoom in high enough, there are lines connecting the points even on the curves that look smooth.

If you want your polynomial to look smooth, make the spacing smaller:

t = 0:0.001:1; 
k = sin(2*pi*t); 
x = linspace(0,1,10);
y = sin(2*pi*x);
r = randn(1,10);  % Some noise
y_noise = y + r;  % y with the noise added.
p = polyfit(x,y,9);
x2 = x;           % For error comparison.
y2 = polyval(p,x2);
x3  = linspace(0,1,100);    % For plotting.
y3 = polyval(p,x3);
plot(x,y,'ob',x3,y3,'r',t,k,'k') 
grid on
RMSE = sqrt(mean((y-y2).^2+(x-x2).^2));

.

.

.

EDIT

Notice that what you probably meant to do was something like this, which shows why higher order polynomial fitting is not always a good idea. As N gets larger, the fit gets better:

N = 10;
t = 0:0.001:1; 
k = sin(2*pi*t); 
x = linspace(0,1,10);
y = sin(2*pi*x);
r = randn(1,10)/N;  % Some noise, adjustable!!!
y_noise = y + r;  % y with the noise added.
p = polyfit(x,y_noise,9);
x2 = x;           % For error comparison.
y2 = polyval(p,x2);
x3  = linspace(0,1,100);    % For plotting.
y3 = polyval(p,x3);
plot(x,y_noise,'ob',t,k,'k',x3,y3,'r') 
legend({'Noisy Sin(2\pix)';'six(2\pix)';'Polyfit'})
grid on
RMSE = sqrt(mean((y-y2).^2+(x-x2).^2));

[john]

when I run this program:

t=0:0.001:1;

k=sin(2*pi*t);

plot(t,k);

x=linspace(0,1,10);

y = sin(2*pi*x);

r = randn(1,10); % Some noise

y_noise = y + r; % y with the noise added.

p = polyfit(x,y,2);

x3 = linspace(0,1,100); % For plotting.

y3 = polyval(p,x3);

plot(x,y,'ob',x3,y3,'r',t,k,'k')

grid on

RMSE=sqrt(mean((y-y3).^2+(x-x3).^2));

disp(p)

disp(RMSE)

I get this message:

??? Error using ==> minus

Matrix dimensions must agree.

Could you help me?

[Matt Fig]

Yes, if you want a better error estimate, look at more points:

N = 20;

t = 0:0.001:1;

k = sin(2*pi*t);

x = linspace(0,1,10);

y = sin(2*pi*x);

r = randn(1,10)/N; % Some noise

y_noise = y + r; % y with the noise added.

p = polyfit(x,y_noise,9);

x2 = x; % For error comparison.

y2 = polyval(p,x2);

x3 = t; % For plotting.

y3 = polyval(p,x3);

plot(x,y_noise,'ob',t,k,'k',x3,y3,'r')

legend({'Noisy Sin(2\pix)';'six(2\pix)';'Polyfit'})

grid on

RMSE = sqrt(mean((y-y2).^2+(x-x2).^2));

[Matt Fig]

Then you don't really need x2, but that is up to you...