showing the transformation converts problem

from Gaussian vector, (mean (1; 1) , cov ( 2 1 ; 1 1),

take 2 dimenttional smaple vector "randn(2, 1)" and uncorrelated Gaussian S ~ N (0, I)

and I habe to transformation X=UA^1/2S + µ cpnverts the uncorrelated Gaussian vector into the correlated Gaussian vector, where ∑ = UAU^T

please help me to find where to start from, I even dont know where to start

Answer 1

[Parag]

Hi, as per my understanding you are trying to do a transformation that converts an uncorrelated Gaussian vector S∼N(0,I) into a correlated Gaussian vector X∼N(μ,Σ). Let's break this down step by step.

We have a mean vector:

μ=[1 1]

and a covariance matrix:

Σ=[2 1

1 1]

The target is to generate samples from X using the transformation:

X= UA 1/2S +μ

where S∼N(0,I) , and Σ is decomposed as:

Σ=UAUT

where:

• U is an orthogonal matrix (eigenvectors of Σ ).
• A is a diagonal matrix (eigenvalues of Σ).
• A ½ is the square root of A (element-wise square root of eigenvalues).

To find U and A, compute the eigenvalues and eigenvectors of Σ.

Use randn(2,1) in MATLAB or to generate a standard normal vector S∼N(0,I).

Compute:

X=UA½S + μ

This converts the uncorrelated vector S into a correlated Gaussian sample.

Here’s a MATLAB implementation:

% Given mean and covariance
mu = [1; 1];
Sigma = [2 1; 1 1];

% Eigen decomposition of Sigma
[U, A] = eig(Sigma);

% Compute A^(1/2)
A_sqrt = sqrt(A);

% Generate an uncorrelated Gaussian vector
S = randn(2,1);

% Apply the transformation
X = U * A_sqrt * S + mu;

% Display the transformed sample
disp('Sample from correlated Gaussian distribution:');
disp(X);