# Non linear least square fitting

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Jenny Andersen on 9 Dec 2019
Commented: Jenny Andersen on 10 Dec 2019
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,'*')

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Jenny Andersen on 9 Dec 2019
What I meant was how to fit a circle to the data (non linear regression). With random I meant arbitrary points, and for example the points given above. I really just want to understand how to perform least square fitting in matlab, so the points does not really matter in this case. Without noise I guess.
Matt J on 9 Dec 2019
Jenny's answer converted to a comment:
I tried the following this time, but it does not seem to be working. Anyone knows why?
x = [2; 3; -2; 1];
y = [3; 2; 4; -1];
N = length(x);
plot (x,y,'*')
% circle's equation x^2+y^2 = 2xc1+2yc2+c3
x = c1+sqrt(-y^2+2ay+a^2+c
y = c1+sqrt(-y^2+2ax+a^2+c
X = [ones(N,3),x];
Y = y;
plot(X,Y)
Matt J on 9 Dec 2019
I have only come up with the following but do not know how to proceed ... phi = inv(X'*X)*X'*Y
You have tried to set up linear equations X*phi=Y and find its Ordinary Least Squares (OLS) solution. This is not a good approach here, because X and Y both have noise in them. But even if you were to do it this way, the better Matlab algorithm for OLS would be,
phi=X\Y

Adam Danz on 9 Dec 2019
Edited: Adam Danz on 9 Dec 2019
Create noisy circular data
This section creates the noisy data we're fitting.
% Create randome points along a circle
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
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.
'RowNames',{'Real values','FitValues'})
T =
2×3 table
______ ______ ______
Real values 9.923 -2.42 22.52
FitValues 9.8052 -2.548 22.926

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Jenny Andersen on 9 Dec 2019
Ok. Could you possibly show me the analytical alternatives to Adam's solution?
Jenny Andersen on 9 Dec 2019
Adam Danz on 9 Dec 2019
Thanks, @Matt J. I meant to mention alternative approaches to the special case when fitting a circle but by the time I finished typing out the answer, I had forgotten.
@Jenny Andersen, there are several option on the file exchange.
But Matt J's answer is even better since you won't need to use a function external to what Matlab already provides.

Matt J on 9 Dec 2019
Edited: Matt J on 9 Dec 2019
Ok. Could you possibly show me the analytical alternatives to Adam's solution?
See attached. Note that my method does not minimize the same fitting error metric as Adam's. If you prefer his, though (and maybe you should), it might be worth mentioning that I do observe somewhat better convergence when initialized with mine. Notice the smaller final "Norm of step" and "First-order optimality" when the iterations terminate.
f = @(a) (xData-a(1)).^2 + (yData-a(2)).^2 - a(3).^2;
opts=optimoptions('lsqnonlin','Display','iter');
a0 = [mean(xData),mean(yData),max(xData)-mean(xData)];
lsqnonlin(f,a0,[],[],opts);
Norm of First-order
Iteration Func-count f(x) step optimality
0 4 6.04793e+07 1.13e+07
1 8 1.69499e+07 3.881 6.82e+05
2 12 1.67336e+07 0.317084 4.6e+03
3 16 1.67336e+07 0.00219778 0.221
[c,R]=circfitAlgebraic([xData;yData]);
lsqnonlin(f,[c,R],[],[],opts);
Norm of First-order
Iteration Func-count f(x) step optimality
0 4 1.74094e+07 1.22e+06
1 8 1.67337e+07 0.555282 1.41e+04
2 12 1.67336e+07 0.00672023 2.07
3 16 1.67336e+07 9.87177e-07 0.00653

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Jenny Andersen on 10 Dec 2019
Thanks alot!