The CholeskyNormal.lng Model

Cholesky Factorization

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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

E[(L*XSN)'*(L*XSN)]

= 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;

Keywords:

Sampling | Cholesky Factorization | Multi-variate Normal |