How to fit multivariate pdf and cdf from data

I have a set of simulated data from a Monte Carlo simulation which gives me a bivariate distribution. I can plot the results using histogram2, and I expect the results to be bivariate gaussian. How can I properly fit this 'empirical' data to get a normalized pdf and cdf which I can then integrate over to get some confidence intervals?

Answer 1

[Jeff Miller]

You don't need a bivariate histogram to fit the bivariate normal--just use the sample means and covariance matrix. Here's an example:

% Let's say your data are in an n,2 matrix called xy.
% Here is one randomly generated to use in the example.
muXY = [100, 200];
sigmaXY = [15^2, 5^2; 5^2, 20^2];
xy = mvnrnd(muXY,sigmaXY,10000);

% Here is your bivariate histogram:
figure; histogram2(xy(:,1),xy(:,2));

% Now estimate the parameters of the best-fitting Gaussian:
xybar = mean(xy);
xycovar=cov(xy);

% Plot the best-fitting bivariate pdf:
xsteps = min(xy(:,1)):1:max(xy(:,1));  % Adjust with step sizes appropriate for your
ysteps = min(xy(:,2)):1:max(xy(:,2));  % x and y values.
[X,Y] = meshgrid(xsteps,ysteps);
F = mvnpdf([X(:) Y(:)],xybar,xycovar);  % Note that xybar and xycovar are used here.
F = reshape(F,length(ysteps),length(xsteps));
figure; surf(xsteps,ysteps,F);
caxis([min(F(:))-.5*range(F(:)),max(F(:))]);
xlabel('x'); ylabel('y'); zlabel('Probability Density');

[supernoob]

Thanks, this is really helpful. Once I have the pdf I need to integrate in polar coordinates over a chosen area. Once I have that done, this problem is solved!

[Tim Fulcher]

Many thanks for this Jeff. Very useful.

Answer 2

[dpb]

They're in the BinCounts property of the object or you can just use the old histcounts2.

ADDENDUM

Ah, ok. I've not tried in Matlab, seems a definite lack of no prepared function indeed...

The Answer_108846 implies one can do it with MLE using the supplied functions for pdf/cdf.

Attach your data and I'll try to see if I can give it a go later on...btw, you'll probably get much better fit using the raw data than histogram bin counts.

[supernoob]

Thanks. I realized this shortly after posting the question and deleted that part, sorry for the confusion. I still can't figure out how to fit these values to get a bivariate gaussian pdf. gmdistribution is not the correct choice.