uncertain_correlation_matrix
Correlation among aleatory uncertain variables
Specification
Alias: None
Arguments: REALLIST
Default: identity matrix (uncorrelated)
Description
Aleatory uncertain variables may have correlations specified through
use of an 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.,
sampling), then the correlation matrix
specifies rank correlations [IC82]. When the method is a
reliability (i.e., local_reliability or global_reliability) or
stochastic expansion (i.e., polynomial_chaos or
stoch_collocation) method, then the correlation matrix specifies
correlation coefficients (normalized covariance) [HM00].
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 \(n^2\) entries where n is the
total number of aleatory uncertain variables. Ordering of the aleatory
uncertain variables is as shown in the input-specification-ordered
table in variables for normal, lognormal, …,
histogram_point.
When additional variable types are activated, they assume uniform distributions, and the ordering is as listed on variables.
Examples
Consider the following random variables, distributions and correlations:
\(X_1\) , normal, uncorrelated with others
\(X_2\) , normal, correlated with \(X_3\) , \(X_4\) and \(X_5\)
\(X_3\) , weibull , correlated with \(X_5\)
\(X_4\) , exponential, correlated with \(X_3\) , \(X_4\) and \(X_5\)
\(X_5\) , normal, correlated with \(X_5\) These correlations are captured by the following commands (order of the variables is respected).
uncertain_correlation_matrix
# ordering normal, exponential, weibull
# X_1 X_2 X_5 X_4 X_3
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 1.00
