Fit a semicircle in a distribution: MATLAB

Question

I have a set of data points which distribute around a semicircle. How can I fit a semicircle on that distribution in MATLAB? The data has two columns (X Y) and a number of rows (varying for different experiments).

Answer 1

One simple way would be to use the function fminunc which does unconstrained minimization.

Let's say you have data $x_i$ and $y_i$. Your hypothesis is that the fit on a circle, that is,

$$x_i^2 + y_i^2 = r^2 + \epsilon_i$$

where $r$ is to be determined. You can fit the radius by minimizing the sum of squared errors

$$S(r) = \sum_{i} e_i^2 = \sum_i (x_i^2 + y_i^2 - r^2)^2$$

and the optimal $r$ is the one which minimized this:

$$r^* = {\rm argmin}_r S(r)$$

First let's generate some data on a semicircle:

theta = pi * rand(100,1);
x = cos(theta) + 0.1 * randn(100,1);
y = sin(theta) + 0.1 * randn(100,1);

You can plot this and make sure it looks as you expect:

plot(x,y,'.'), grid on

Now you can find the radius that minimizes the error using fminunc. We expect the answer to be about 1, but we'll put 2 as our initial guess, so that the minimizer has to do some work:

r_opt = fminunc(@(r)sum((x.^2 + y.^2 - r^2).^2), 2);

And now you can plot this on top of your previous plot, to make sure it looks sensible:

hold on
xvals = linspace(-1,1,100);
plot(xvals, sqrt(r_opt^2-xvals.^2), 'r')