How to value the proximity of different 2D distribution?

Dear MATLAB community,

I'm working in the field of laser physics and I want to know what methods can be used to evaluate the 2D Gaussian distribution?

I measured the intensity distribution of four light spots, as you can be seen from the figure below, 1 and 2 are relatively close to the Gaussian distribution, 3 is getting worse, and 4 is the worst. I have a 2D matrix of their intensity. Is there a function to evaluate their distribution closeness relative to a perfect 2D gaussian distribution (or matrix 1)? (such as f(matrix(1))=1, f(matrix(2))=0.9, f(matrix(3))=0.3, f(matrix(4))=0.1)

(Besides,here the relative intensity is more meaningful than the absolute intensity, so normalized matrices are used in the comparison. )

I have tested the Kullback–Leibler divergence function in the form D_KL(P||Q)=sum(P.*log(P./Q),'all','omitnan') (here P is the normalized 2D matrix 2,3,4 and Q is matrix 1), and the result are pretty pool

DKL =

1.0e+03 *

0 6.8731 5.6944 0.0839

I also tried the divergence function like div=var(A,1,'all'), and the result is not very satisfactory. A friend from the Department of Statistics told me to check functions like lillietest, chi2gof, and kstest . Well, those functions only accept one-dimensional data. This is not a serious problem, but I am really curious about how to get a simple description of such two-dimensional gaussian distribution,like those test function did to one-dimension data.

Sorry for my pool English and sincerely appreciate your answers :)

Here is several tested evaluation in my code:

load ls.mat

for i=1:4;

ene(i)=sum(sum(ls(:,:,i)));

labc=ls(:,:,i);

es=sort(labc(:));

enetop(i)=mean(es(end-100:end));

% enetop=median(es);

enefwhm(i)=sum(sum(ls(:,:,i)>=0.5*enetop(i)));

enes(i)=sum(sum(ls(:,:,i)>=0.1*enetop(i)));

eness(i)=sum(sum(ls(:,:,i)>=0.1*enetop(i)));

end

enerate=enefwhm./ene;

enerate1=enefwhm./enes;

enerate2=enefwhm./eness;

%DKL

lss=ls;

% lss(lss<=0.5*max(lss(:)))=0;

Q=lss(:,:,1)+1;

Q=Q/max(Q(:));

for k=1:4;

P=lss(:,:,k)+1;

P=P./max(P(:));

A=(P.*log(P./Q));

DKL(k)= sum(A,'all','omitnan');

end

%div

for j=1:4;

div(j)=var(ls(:,:,j),1,'all');

end

Answer 1

[Nipun]

Hi Zhe,

I understand that you want to evaluate the proximity of different 2D distributions. One way to do this is by calculating the Kullback-Leibler (KL) divergence or using metrics like the Bhattacharyya distance. Here’s an example using the Bhattacharyya distance:

% Example 2D distributions
dist1 = [mean1; cov1]; % Mean and covariance of distribution 1
dist2 = [mean2; cov2]; % Mean and covariance of distribution 2

% Define mean and covariance matrices
mean1 = [1; 2];
cov1 = [1 0.5; 0.5 2];
mean2 = [1.5; 2.5];
cov2 = [2 0.4; 0.4 1];

% Bhattacharyya distance calculation
meanDiff = mean1 - mean2;
covAvg = (cov1 + cov2) / 2;
distance = 0.125 * (meanDiff' / covAvg * meanDiff) + 0.5 * log(det(covAvg) / sqrt(det(cov1) * det(cov2)));

disp(distance);

This code calculates the Bhattacharyya distance between two 2D Gaussian distributions.

Hope this helps.

Regards,

Nipun