A fundamental feature of probabilistic methods is the generation of random numbers with standard uniform distribution. The standard uniform distribution is a uniform distribution with a lower limit x_min = 0.0 and an upper limit x_max = 1.0. Methods for generating standard uniformly distributed random numbers are generally based on recursive calculations of the residues of modulus m from a linear transformation. Such a recursive relation is given by the equation:

where:

A set of random numbers with standard uniform distribution is obtained by normalizing the value calculated by (Equation 21–44) with the modulus m:

It is obvious from (Equation 21–44) that an identical set of random numbers will be obtained if the same start value for the seed s_i-1 is used. Therefore, the random numbers generated like that are also called “pseudo random” numbers. See Hammersley and Handscomb(308) for more details about the generation of random numbers with standard uniform distribution.

For probabilistic analyses, random numbers with arbitrary distributions such as the ones described in Statistical Distributions for Random Input Variables are needed. The most effective method to generate random number with any arbitrary distribution is the inverse transformation method. A set of random numbers for the random variable X having a cumulative distribution function F_x (x) can be generated by using a set of standard uniformly distributed random numbers according to (Equation 21–45) and transforming them with the equation:

Depending on the distribution type of the random variable X, the inverse cumulative distribution function can be calculated as described in Statistical Distributions for Random Input Variables.

Correlated random input variables must be dealt with by all probabilistic methods, if there are random input variables, the user has identified as being correlated with each other. In order to handle correlated random input variables it is necessary to transform the random variable values using the Nataf model. The Nataf model is explained in detail in Liu and Der Kiureghian(311)).

The direct Monte Carlo Simulation method is also called the crude Monte Carlo Simulation method. It is based on randomly sampling the values of the random input variables for each execution run. For the direct Monte Carlo Simulation method the random sampling has no memory, i.e., it may happen that one sampling point is relative closely located to one or more other ones. An illustration of a sample set with a sample size of 15 generated with direct Monte Carlo Simulation method for two random variables X₁ and X₂ both with a standard uniform distribution is shown in Figure 21.10: "Sample Set Generated with Direct Monte Carlo Simulation Method".

As indicated with the circle, there may be sample points that are located relatively close to each other.

For the Latin Hypercube Sampling technique the range of all random input variables is divided into n intervals with equal probability, where n is the number of sampling points. For each random variable each interval is “hit” only once with a sampling point. The process of generating sampling points with Latin Hypercube has a “memory” in the meaning that the sampling points cannot cluster together, because they are restricted within the respective interval. An illustration of a sample with a sample size of 15 generated with Latin Hypercube Sampling method for two random variables X₁ and X₂ both with a standard uniform distribution is shown in Figure 21.11: "Sample Set Generated with Latin Hypercube Sampling Method".

There are several ways to determine the location of a sampling point within a particular interval.

See Iman and Conover(309) for further details.

Location of Sampling Points Expressed in Probabilities

For central composite design the sampling points are located at five different levels for each random input variable. In order to make the specification of these levels independent from the distribution type of the individual random input variables, it is useful to define these levels in terms of probabilities. The five different levels of a central composite design shall be denoted with p_i, with i = 1, ... , 5.

A central composite design is composed of three different parts, namely:

A sample set based on a central composite design for three random variables X₁, X₂ and X₃ is shown in Figure 21.12: "Sample Set Based on a Central Composite Design".

For this example with three random input variables the matrix describing the location of the sampling points in terms of probabilities is shown in Table 21.1: "Probability Matrix for Samples of Central Composite Design".

Resolution of the Fractional Factorial Part

For problems with a large number of random input variables m, the number of sampling points is getting extensively large, if a full factorial design matrix would be used. This is due to the fact that the number of sampling points of the factorial part goes up according to 2^m in this case. Therefore, with increasing number of random variables it is common practice to use a fractional factorial design instead of a full factorial design. For a fractional factorial design, the number of the sampling points of the factorial part grows only with 2^m-f. Here f is the fraction of the factorial design so that f = 1 represents a half-factorial design, f = 2 represents a quarter-factorial design, etc. Consequently, choosing a larger fraction f will lead to a lower number of sampling points.

In a fractional factorial design the m random input variables are separated into two groups. The first group contains m - f random input variables and for them a full factorial design is used to determine their values at the sampling points. For the second group containing the remaining f random input variables defining equations are used to derive their values at the sampling points from the settings of the variables in the first group.

As mentioned above, we want to use the value of the random output parameters obtained at the individual sampling points for fitting a response surface. This response surface is an approximation function that is determined by a certain number of terms and coefficients associated with these terms. Hence, the fraction f of a fractional factorial design cannot become too large, because otherwise there would not be enough data points in order to safely and accurately determine the coefficients of the response surface. In most cases a quadratic polynomial with cross-terms will be used as a response surface model. Therefore, the maximum value for the fraction f must be chosen such that a resolution V design is obtained (here V stands for the Roman numeral 5). A design with a resolution V is a design where the regression coefficients are not confounded with each other. A resolution V design is given if the defining equation mentioned above includes at least 5 random variables as a total on both sides of the equation sign. Please see Montgomery(312) for details about fractional factorial designs and the use of defining equations.

For example with 5 random input variables X₁ to X₅ leads to a resolution V design if the fraction is f = 1. Consequently, a full factorial design is used to determine the probability levels of the random input variables X₁ to X₄. A defining equation is used to determine the probability levels at which the sampling points are located for the random input variable X₅. See Montgomery(312) for details about this example.

Location of Sampling Points Expressed in Random Variable Values

In order to obtain the values for the random input variables at each sampling point, the probabilities evaluated in the previous section must be transformed. To achieve this, the inverse transformation outlined under Common Features for all Probabilistic Methods can be used for non-correlated random variables. The procedure dealing with correlated random variables also mentioned under Common Features for all Probabilistic Methods can be used for correlated random variables.

Location of Sampling Points Expressed in Probabilities

For a Box-Behnken Matrix design, the sampling points are located at three different levels for each random input variable. In order to make the specification of these levels independent from the distribution type of the individual random input variables, it is useful to define these levels in terms of probabilities. The three different levels of a Box-Behnken Matrix design shall be denoted with p₁, with i = 1, ... , 3.

A Box-Behnken Matrix design is composed of two different parts, namely:

See Box and Cox(307) for further details. A sample set based on a central composite design for three random variables X₁, X₂ and X₃ is shown in Figure 21.13: "Sample Set Based on Box-Behnken Matrix Design".

For this example with three random input variables the matrix describing the location of the sampling points in terms of probabilities is shown in Table 21.2: "Probability Matrix for Samples of Box-Behnken Matrix Design".

Location of Sampling Points Expressed in Random Variable Values

In order to obtain the values for the random input variables at each sampling point, the same procedure is applied as mentioned above for the Central Composite Design.