Dakota Reference Manual  Version 6.4 Large-Scale Engineering Optimization and Uncertainty Analysis
uncertain_correlation_matrix

Correlation among aleatory uncertain variables

## Specification

Alias: none

Argument(s): 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 [51].

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)[41].

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 entries where n is the total number of aleatory uncertain variables.

Ordering of the aleatory uncertain variables is:

1. normal
2. lognormal
3. uniform
4. loguniform
5. triangular
6. exponential
7. beta
8. gamma
9. gumbel
10. frechet
11. weibull
12. histogram bin
13. poisson
14. binomial
15. negative binomial
16. geometric
17. hypergeometric
18. 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:

• , normal, uncorrelated with others
• , normal, correlated with , and
• , weibull , correlated with
• , exponential, correlated with , and
• , normal, correlated with These correlations are captured by the following commands (order of the variables is respected).
```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
```