The mean, variance, covariance, and correlation coefficients of a multiple subscripted parameter are computed (using the *MOPER command). Refer to Kreyszig([162]) for the basis of the following formulas. All operations are performed on columns to conform to the database structure. The covariance is assumed to be a measure of the association between columns.

The following notation is used:

where:

[x] = starting matrix

i = row index of first array parameter matrix

j = column index of first array parameter matrix

m = number of rows in first array parameter matrix

n = number of columns in first array parameter matrix

subscripts s, t = selected column indices

[S] = covariance matrix n x n

[c] = correlation matrix n x n

= variance

The mean of a column is:

(16–18)

The covariance of the columns s and t is:

(16–19)

The variance, , of column s is the diagonal term S_ss of the covariance matrix [S]. The equivalent common definition of variance is:

(16–20)

The correlation coefficient is a measure of the independence or dependence of one column to the next. The correlation and mean operations are based on Hoel([163]) (and initiated when CORR is inserted in the Oper field of the *MOPER command).

Correlation coefficient:

(16–21)

value S of the terms of the coefficient matrix range from -1.0 to 1.0 where:

A vector can be filled with a random sample of real numbers based on a uniform distribution with given lower and upper bounds (using RAND in the Func field on the *VFILL command) (see Figure 16.2: Uniform Density):

(16–22)

where:

Figure 16.2:  Uniform Density

The numbers are generated using the URN algorithm of Swain and Swain([161]). The initial seed numbers are hard coded into the routine.

A vector may be filled with a random sample of real numbers based on a Gaussian distribution with a known mean and standard deviation (using GDIS in the Func field on the *VFILL command).

First, random numbers P(x), with a uniform distribution from 0.0 to 1.0, are generated using a random number generator. These numbers are used as probabilities to enter a cumulative standard normal probability distribution table (Abramowitz and Stegun([160])), which can be represented by Figure 16.3: Cumulative Probability Function or the Gaussian distribution function:

Figure 16.3:  Cumulative Probability Function

(16–23)

where:

The table maps values of P(x) into values of x, which are standard Gaussian distributed random numbers from -5.0 to 5.0, and satisfy the Gaussian density function (Figure 16.4: Gaussian Density):

Figure 16.4:  Gaussian Density

(16–24)

where:

The x values are transformed into the final Gaussian distributed set of random numbers, with the given mean and standard deviation, by the transformation equation:

(16–25)

A vector may be filled with a random sample of real numbers based on a triangular distribution with a known lower bound, peak value location, and upper bound (using TRIA in the Func field on the *VFILL command).

First, random numbers P(x) are generated as in the Gaussian example. These P(x) values (probabilities) are substituted into the triangular cumulative probability distribution function:

(16–26)

where:

which is solved for values of x. These x values are random numbers with a triangular distribution, and satisfy the triangular density function (Figure 16.5: Triangular Density):

Figure 16.5:  Triangular Density

(16–27)

A vector may be filled with a random sample of real numbers based on a beta distribution with known lower and upper bounds and α and β parameters (using BETA in the Func field on the *VFILL command).

First, random numbers P(x) are generated as in the Gaussian example. These random values are used as probabilities to enter a cumulative beta probability distribution table, generated by the program. This table can be represented by a curve similar to (Figure 16.3: Cumulative Probability Function), or the beta cumulative probability distribution function:

(16–28)

The table maps values of P(x) into x values which are random numbers from 0.0 to 1.0. The values of x have a beta distribution with given α and β values, and satisfy the beta density function (Figure 16.6: Beta Density):

Figure 16.6:  Beta Density

(16–29)

where:

a = lower bound (input as CON1 on *VFILL command)

b = upper bound (input as CON2 on *VFILL command)

α = alpha parameter (input as CON3 on *VFILL command)

β = beta parameter (input as CON4 on *VFILL command)

B (α, β) = beta function

f(t) = beta density function

The x values are transformed into the final beta distributed set of random numbers, with given lower and upper bounds, by the transformation equation:

(16–30)

A vector may be filled with a random sample of real numbers based on a gamma distribution with a known lower bound for α and β parameters (using GAMM in the Func field on the *VFILL command).

First, random numbers P(x) are generated as in the Gaussian example. These random values are used as probabilities to enter a cumulative gamma probability distribution table, generated by the program. This table can be represented by a curve similar to Figure 16.7: Gamma Density, or the gamma cumulative probability distribution function:

(16–31)

where:

The table maps values of P(x) into values of x, which are random numbers having a gamma distribution with given α and β values, and satisfy the gamma distribution density function (Figure 16.7: Gamma Density):

Figure 16.7:  Gamma Density

(16–32)

where:

α = alpha parameter of gamma function (input as CON2 on *VFILL command)

β = beta parameter of gamma density function (input as CON3 on *VFILL command)

a = lower bound (input as CON1 on *VFILL command)

The x values are relocated relative to the given lower bound by the transformation equation:

(16–33)