In many practical hydrosystems engineering problems, random variables often are statistically and physically dependent. Furthermore, distribution types for the random variables involved can be a mixture of different distributions, of which the corresponding joint PDF or CDF is difficult to establish. As a practical alternative, to replicate such systems properly, the Monte Carlo simulation should be able to preserve the correlation relationships among the stochastic variables and their marginal distributions.
In a multivariate setting, the joint PDF represents the complete information describing the probabilistic structures of the random variables involved. When the joint PDF or CDF is known, the marginal distribution and conditional distributions can be derived, from which the generation of multivariate random variates can be made straightforwardly in the framework of Rosenblatt (1952). However, in most practical engineering problems involving multivariate random variables, the derivation of the joint CDF generally is difficult, and the availability of such information is rare. The level of difficulty, in both theory and practice, increases with the number of random variables and perhaps even more so by the type of corresponding distributions. Therefore, more often than not, one has to be content with preserving incomplete information represented by the marginal distribution of each individual random variable and the correlation structure. In doing so, the difficulty of requiring a complete joint PDF in the multivariate Monte Carlo simulation is circumvented.
To generate correlated random variables with a mixture of marginal distributions, a methodology adopting a bivariate distribution model was first suggested by Li and Hammond (1975). The practicality of the approach was advanced by Der Kiureghian and Liu (1985), who, based on the Nataf bivariate distribution model (Nataf, 1962), developed a set of semiempirical formulas so that the necessary calculations to preserve the original correlation structure in the normal transformed space are reduced (see Table 4.5). Chang et al. (1994) used this set of formulas, which transforms the correlation coefficient of a pair of nonnormal random variables to its equivalent correlation coefficient in the bivariate standard normal space, for multivariate simulation. Other practical alternatives, such as the polynomial normal transformation (Vale and Maurelli, 1983; Chen and Tung, 2003), can serve the same purpose. Through a proper normal transformation, the multivariate Monte Carlo simulation can be performed in a correlated standard normal space in which efficient algorithms, such as those described in Sec. 6.5.2, can be applied.
The Monte Carlo simulation that preserves marginal PDFs and correlation structure of the involved random variables consists of following two basic steps:
Step 1. Transformation to a standard normal space. Through proper normal transformation, the operational domain is transformed to a standard normal space in which the transformed random variables are treated as if they were multivariate standard normal with the correlation matrix Rz. As a result, multivariate normal random variates can be generated by the techniques described in Sec. 6.5.2.
Step 2. Inverse transformation. Once the standardized multivariate normal random variates are generated, then one can do the inverse transformation
Xk = ^[Ф( Zk)] for k = 1,2,…, K (6.37)
to compute the values of multivariate random variates in the original space.
