I have a covariance matrix and mean values(OMEGA) for a multidimensional Gaussian distribution (3-dim) as follows (respectively),
COVAR = 1.0e-12 * 0.2498 -0.4832 0.2140 -0.4832 0.9543 -0.4456 0.2140 -0.4456 0.3245
OMEGA = 1.0e-06 * -0.0334 0.1460 -0.1079
The problem I am involved here is that of parameter estimation one. I am able to make the 2-dim contour plots (1,2-sigma) and marginalize them to get a 1-dim distribution for each parameters. The contour plot goes into the negative region too, which I want to get rid of as these parameters cannot take negative values. In a way, I am looking for a truncated Gaussian distribution (correct me if I am wrong!).
This is how, I remove the negative estimates from my data, and I replace them with zeros.
samples = mvnrnd(OMEGA,COVAR,100000); data = samples; data(data<0)=0;
Having done that, I can easily find the new mean and new covariance matrix pertaining to the modified data set.
OMEGA = mean(data); COVAR = cov(data);
I do not quite understand how to interpret these data. Is the new distribution still a multivariate Gaussian? It is definitely truncated.
I want my data or the contour plot to restrict to the positive quadrant only. The idea is to have a comparison with the Bayesian analysis where the prior is set up in such a way that the parameters do not take negative values (eg. we can think of a flat prior in the range [0 1], not containing negative values).
I should also mention how I make the contour plots. Since the initial data was Gaussian I have the privilege to compute the covariance matrix of (say) the first two parameters by removing the 3rd row and 3rd column as follows:
COV12(1,1) = COVAR(1,1); COV12(1,2) = COVAR(1,2); COV12(2,1) = COVAR(2,1); COV12(2,2) = COVAR(2,2); OM12(1) = OMEGA(1); OM12(2) = OMEGA(2);
Similar can be done for
ellipsedata(COV12,OM12,1000,[2.0 1.0]);%% 2-sigma and 1-sigma contour
I am not sure whether this is correct method to do for the new data, because of its validity only for a gaussian distribution.