GRMULTINORMAL procedure

Generates multivariate Normal pseudo-random numbers (P.W. Goedhart & K.L. Moore).


Options

NVALUES = scalar
Number of values to generate; default 1

MEANS = variate
The mean for the multivariate Normal distribution; default is a variate with values all equal to 0

VCOVARIANCE = symmetric matrix
The variance/covariance matrix for the multivariate Normal distribution; default is to use an identity matrix

SEED = scalar
Seed to generate the random numbers; default 0 continues an existing sequence or initializes the sequence automatically if no random numbers have been generated in this job


Parameters

NUMBERS = pointers or matrices
Saves the random numbers as either a pointer to a set of variates or a matrix


Description

GRMULTINORMAL generates pseudo-random numbers from a multivariate Normal distribution Np(μ, Σ). The mean μ is specified by the option MEANS as a variate of length p; the variance-covariance matrix Σ is specified by the option VCOVARIANCE as a symmetric matrix with p rows and columns; and the option NVALUES specifies the number of values n to be generated. Note that VCOVARIANCE must be positive semi-definite.

   The numbers can be saved using the NUMBERS parameter, in either a pointer to a set of variates, or a matrix. If the NUMBERS structure or structures are already declared, their dimensions must be compatible with the settings of the NVALUES, MEANS and VCOVARIANCE options. The dimensions are also used, if necessary, to set defaults for the options. By default, MEANS is taken to be a variate of zero values, and VCOVARIANCE is taken to be the identity matrix. If the setting of NUMBERS is not already declared, if will be defined as a pointer to a set of variates with dimensions deduced from the option settings.


Options: NVALUES, MEANS, VCOVARIANCE.

Parameter: NUMBERS.


Method

Pseudo-random numbers from a multivariate Normal distribution are generated by forming a matrix Y of columns of univariate Normal random numbers, using the Box-Muller method (Box & Muller 1958), followed by a linear transformation

X = A Y + μ,

where A is calculated by a CHOLESKI decomposition, AA′ = Σ. (See, for example, Johnson 1987 pages 52-55, Tong 1990 pages 181-186).


Action with RESTRICT

Variates that have been restricted will receive output from GRMULTINORMAL only in those units that are not excluded by the restriction. Values in the excluded units remain unchanged. Note that the NVALUES option must equal the full size of the variates. Restrictions on the MEANS variate are ignored.


References

Box, G.E.P. & Muller, M.E. (1958). A note on generation of normal deviates. Annals of Mathematical Statistics, 28, 610-611.

Johnson, M.E. (1987). Multivariate Statistical Simulation. John Wiley & Sons, New York.

Tong, Y.L. (1990). The Multivariate Normal Distribution. Springer-Verlag, New York.