Ellipse around the data in MATLAB

Question

I would like to reproduce the following figure in MATLAB:

There are two classes of points with X and Y coordinates. I'd like to surround each class with an ellipse with one parameter of standard deviation, which determine how far the ellipse will go along the axis.

The figure was created with another software and I don't exactly understand how it calculates the ellipse.

Here is the data I'm using for this figure. The 1st column is class, 2nd - X, 3rd - Y. I can use gscatter to draw the points itself.

A = [
    0   0.89287 1.54987
    0   0.69933 1.81970
    0   0.84022 1.28598
    0   0.79523 1.16012
    0   0.61266 1.12835
    0   0.39950 0.37942
    0   0.54807 1.66173
    0   0.50882 1.43175
    0   0.68840 1.58589
    0   0.59572 1.29311
    1   1.00787 1.09905
    1   1.23724 0.98834
    1   1.02175 0.67245
    1   0.88458 0.36003
    1   0.66582 1.22097
    1   1.24408 0.59735
    1   1.03421 0.88595
    1   1.66279 0.84183
];

gscatter(A(:,2),A(:,3),A(:,1))

FYI, here is the SO question on how to draw ellipse. So, we just need to know all the parameters to draw it.

Update:

I agree that the center can be calculated as the means of X and Y coordinates. Probably I have to use principal component analysis (PRINCOMP) for each class to determine the angle and shape. Still thinking...

Even if you don't know how the ellipse in the original figure is calculated, you could probably explain a little bit more what it is supposed to mean? It's not just surrounding the points as for the grey points for example, not all points are included in the ellipse. If you don't know anything, I would suggest the barycenter of the points as the center of the ellipse as a starting point. The second moment of area could give you the direction of the major axis. — groovingandi
@groovingandi: Basically it means, if the ellipse is drawn with standard deviation 3, with normal distribution it will include about 99% of data points. The ellipses on the figure were generated with STD=2. — yuk

Amro Amro · Accepted Answer · 2010-08-05T23:33:27

Consider the code:

%# generate data
num = 50;
X = [ mvnrnd([0.5 1.5], [0.025 0.03 ; 0.03 0.16], num) ; ...
      mvnrnd([1 1], [0.09 -0.01 ; -0.01 0.08], num)   ];
G = [1*ones(num,1) ; 2*ones(num,1)];

gscatter(X(:,1), X(:,2), G)
axis equal, hold on

for k=1:2
    %# indices of points in this group
    idx = ( G == k );

    %# substract mean
    Mu = mean( X(idx,:) );
    X0 = bsxfun(@minus, X(idx,:), Mu);

    %# eigen decomposition [sorted by eigen values]
    [V D] = eig( X0'*X0 ./ (sum(idx)-1) );     %#' cov(X0)
    [D order] = sort(diag(D), 'descend');
    D = diag(D);
    V = V(:, order);

    t = linspace(0,2*pi,100);
    e = [cos(t) ; sin(t)];        %# unit circle
    VV = V*sqrt(D);               %# scale eigenvectors
    e = bsxfun(@plus, VV*e, Mu'); %#' project circle back to orig space

    %# plot cov and major/minor axes
    plot(e(1,:), e(2,:), 'Color','k');
    %#quiver(Mu(1),Mu(2), VV(1,1),VV(2,1), 'Color','k')
    %#quiver(Mu(1),Mu(2), VV(1,2),VV(2,2), 'Color','k')
end

screenshot

EDIT

If you want the ellipse to represent a specific level of standard deviation, the correct way of doing is by scaling the covariance matrix:

STD = 2;                     %# 2 standard deviations
conf = 2*normcdf(STD)-1;     %# covers around 95% of population
scale = chi2inv(conf,2);     %# inverse chi-squared with dof=#dimensions

Cov = cov(X0) * scale;
[V D] = eig(Cov);

OP_DATA

Ellipse around the data in MATLAB

3 Answers

EDIT