Correlation among aleatory uncertain variables


  • Alias: None

  • Arguments: 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 [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.


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
  # 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