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Using Cholesky Factorization to Generate (CholeskyNormal.lng)
Multi-variate Normal Random Variables with a
specified covariance matrix.
Using matrix notation, if XSN = a row vector of independent random variables with mean 0 and variance 1, and E[ ] is the expectation operator, and XSN' is the transpose of XSN, then its covariance matrix, E[ XSN'*XSN] = I, i.e., the identity matrix with 1's on the diagonal and 0's everywhere else.
Further, if we apply a linear transformation L, then
= L'*L*E[XSN'*XSN] = L'*L.
Thus, if L'*L = Q, then the L*XSN will have covariance matrix Q. Cholesky factorization is a way of finding a matrix L, so that L'*L = Q. So we can think of L as the matrix square root of Q.
If XSN is a vector of standard Normal random variables with mean 0 and variance 1, then L*XSN has a multivariate Normal distribution with covariance matrix L'*L;