Dakota Reference Manual  Version 6.4
Large-Scale Engineering Optimization and Uncertainty Analysis
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Correlation among aleatory uncertain variables


Alias: none

Argument(s): REALLIST

Default: identity matrix (uncorrelated)


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 $n^2$ 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.


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).
# 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