MATLAB: Non linear least square fitting

MATLAB

Hi,
I am really stuck trying to figure out how to fit a circle to some data points. I have watched plenty of videos on the topic but do not really understand. Could you please explain to me in the most simple way how to plot some random data points, for example x = [2; 3; -2; 1]; and y = [3; 2; 4; -1];, in matlab?
I have only come up with the following but do not know how to proceed: 
x = [2; 3; -2; 1];
y = [3; 2; 4; -1];
N = length(x);
% circle's equation x^2+y^2 = 2xc1+2yc2+c3
X = [ones(N,3),x];
Y=y;
phi = inv(X'*X)*X'*Y
plot(x,y,'*')
Thanks alot in advance!

Best Answer

Create noisy circular data
This section creates the noisy data we're fitting. 
% Create randome points along a circle
r = 22.52;                          % radius
cnt = [9.923, -2.42];               % (x,y) center
theta = rand(1,1000)*2*pi;          % Theta
xData = r*cos(theta)+cnt(1);        % x coordinates
yData = r*sin(theta)+cnt(2);        % y coordinates
% Add noise to coordinates
xData = xData + (rand(size(xData))*10-5); 
yData = yData + (rand(size(yData))*10-5);
Plot the input data
The only inputs are xData and yData. The rest of the variables above are unknowns.
% Plot the data with noise
clf()
plot(xData, yData, 'ko')
axis equal
grid on
Fit the xData,yData coordinates to a circle
This section uses nonlinear least squares fitting x = lsqnonlin(fun,x0). The first line defines the function to fit and is the equation for a circle. The second line are estimated starting points. See the link for more info on this function. The output circFit is a 1x3 vector defining the [x_center, y_center, radius] of the fitted circle. 
f = @(a) (xData-a(1)).^2 + (yData-a(2)).^2 - a(3).^2;
a0 = [mean(xData),mean(yData),max(xData)-mean(xData)]; 
circFit = lsqnonlin(f,a0);
Add the fitted circle to the noisy data
% use the circFit parameters to create the fitted circle
theta = linspace(0,2*pi,100);  % arbitrary spacing
xFit = circFit(3)*cos(theta) + circFit(1); 
yFit = circFit(3)*sin(theta) + circFit(2);
hold on
plot(xFit,yFit,'r-','LineWidth',3)
Compare the fitted values to the actual (noisy) values
Of course these values may differ when this code is repeated due to random noise.
This creates a table comparing the known and fitted results. 
T = array2table([[cnt,r];circFit],'VariableNames',{'xCnt','yCnt','radius'},...
    'RowNames',{'Real values','FitValues'})
T =
2×3 table
xCnt yCnt radius
______ ______ ______
Real values 9.923 -2.42 22.52
FitValues 9.8052 -2.548 22.926
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