Multivariate Guassian Distribution
I want to learn Multivariate Gussian distribution so I written the following code.*I am implementing following formula Iam expecting this type of plot Where I am doing mistake? clear all clc % Taking two guassian random variables x=randn(1000,1); y=randn(1000,1); X=[x y]; X=X'; d=size(X,1); % find means of x,y mx=mean(x); my=mean(y); mumat=[mx my]'; mumat=repmat(mumat,1,size(X,2)); Dif_mat=X-mumat; % The above step (Dif_mat) is (X-mu) in the formula cov_mat=cov(X'); % covariance matrix det_cv=det(cov_mat); % det of cov matrix inv_cov=inv(cov_mat); % inverse of cov matrix % scale term before exp in forumala scale=((2*pi)^(d/2))*sqrt((abs(det_cv))); scale=inv(scale); % Mahabolis distance in formula MB=Dif_mat'*cov_mat*Dif_mat; % find the final probability p=scale*exp((-1/2)*MB); surf(x,y,p)
Neeta Dsouza answered .
2025-11-20
You're really trying to do two things here. The first is, you have some random data and you want to fit it to a multivariate normal distribution. Your approach to this part works, although it can be streamlined:
n = 1000; d=2; X = randn(n,2);
Get mean and covariance:
mumat=mean(X); cov_mat=cov(X);
The second part is plotting the resulting distribution. Here you need a regular grid for your variables, not the random values you generated above:
x = -3:.2:3; y = -3:.2:3; [X,Y] = meshgrid(x,y); X = X-mumat(1); Y = Y-mumat(2);
Combine X and Y in a way that each row represents one 2D variable.
Z = [X(:) Y(:)];
Now calculate the probabilities.
scale=((2*pi)^(d/2))*sqrt(abs(det(cov_mat))); p = zeros(length(Z),1); for ii=1:length(Z) p(ii) = exp(-Z(ii,:)/cov_mat*Z(ii,:)'/2)/scale; end
Reshape and plot.
p = reshape(p,length(x),length(y));
surf(x,y,p)
xlabel('x'), ylabel('y')
For example
Lets Compute and plot the pdf of a bivariate normal distribution with parameters mu = [0 0] and Sigma = [0.25 0.3; 0.3 1].
Define the parameters mu and Sigma.
mu = [0 0]; Sigma = [0.25 0.3; 0.3 1];%initialisation
Now Creating a grid of evenly spaced points in two-dimensional space
x1 = -3:0.2:3; x2 = -3:0.2:3; [X1,X2] = meshgrid(x1,x2); X = [X1(:) X2(:)];
Now we will evaluate the pdf of the normal distribution at the grid points.
y = mvnpdf(X,mu,Sigma); y = reshape(y,length(x2),length(x1));
Now plotting
surf(x1,x2,y)
caxis([min(y(:))-0.5*range(y(:)),max(y(:))])
axis([-3 3 -3 3 0 0.4])
xlabel('x1')
ylabel('x2')
zlabel('Probability Density')
