Blurb::
Correlation among aleatory uncertain variables

Description::
Aleatory uncertain variables may have correlations specified through
use of an \c uncertain_correlation_matrix specification. This
specification is generalized in the sense that its specific meaning
depends on the nondeterministic method in use.

When the method is a
nondeterministic sampling method (i.e., \ref method-sampling), then the
correlation matrix specifies <i>rank correlations</i> 
\cite Iman1982.

When the method is 
a reliability (i.e., \c local_reliability or \c
global_reliability) or stochastic expansion (i.e., \c
polynomial_chaos or \c stoch_collocation) method, then the
correlation matrix specifies <i>correlation coefficients</i>
(normalized covariance) \cite Hal00.

In either of these cases, specifying the identity matrix results in
uncorrelated uncertain variables (the default). The matrix input
should be symmetric and have all \f$n^2\f$ entries where \e n is the
total number of aleatory uncertain variables.

Ordering of the aleatory uncertain variables is: 
<ol>
<li> normal </li>
<li> lognormal </li>
<li> uniform </li>
<li> loguniform </li>
<li> triangular </li>
<li> exponential </li>
<li> beta </li>
<li> gamma </li>
<li> gumbel </li>
<li> frechet </li>
<li> weibull </li>
<li> histogram bin </li>
<li> poisson </li>
<li> binomial </li>
<li> negative binomial </li>
<li> geometric </li>
<li> hypergeometric </li>
<li> histogram point </li>
</ol>

When additional variable types are activated, they assume uniform 
distributions, and the ordering is as listed on \ref variables.

Topics::	
Examples::
Consider the following random variables, distributions and correlations:
\li \f$X_1\f$, normal, uncorrelated with others
\li \f$X_2\f$, normal, correlated with \f$X_3\f$, \f$X_4\f$ and \f$X_5\f$
\li \f$X_3\f$, weibull , correlated with \f$X_5\f$
\li \f$X_4\f$, exponential, correlated with \f$X_3\f$, \f$X_4\f$ and \f$X_5\f$
\li \f$X_5\f$, normal, correlated with \f$X_5\f$
These correlations are captured by the following commands (order of the variables is respected).

\verbatim
uncertain_correlation_matrix
# ordering normal, exponential, weibull
# \f$X_1\f$ \f$X_2\f$ \f$X_5\f$ \f$X_4$\f \f$X_3\f$
1.00 0.00 0.00 0.00 0.00
0.00 1.00 0.50 0.24 0.78
0.00 0.50 1.00 0.00 0.20
0.00 0.24 0.00 1.00 0.49
0.00 0.78 0.20 0.49 1.0
\endverbatim 
Theory::
Faq::
See_Also::	
