uppose the random vector
is a collection of iid (independent identically distributed) standard normal
variables. The joint distribution of
is given by the function
The
linear combination
with any nondegenerate matrix
and vector
is called "the multivariate normal variable
".
According to the results of the previous section, the joint distribution of
is given by the
expression
The matrix
is a positivedefinite symmetric matrix and the covariance matrix of
.
In practice, one usually observes an approximation of
from historical data and faces the problem of
reconstruction.
There is more then one way to represent
as a product of some matrix and its transpose. A particular representation
with a lowdiagonal matrix
is called "Choleski decomposition" of
.
The matrix
may be reconstructed from
through an efficient recursive procedure. See
[Numerical]
. An open source implementation of Cholesky
decomposition is available in numpy.linalg module of Python programming
language.
