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# GRMULTINORMAL procedure

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

### Options

`NVALUES` = scalar Number of values to generate; default 1 The mean for the multivariate Normal distribution; default is a variate with values all equal to 0 The variance/covariance matrix for the multivariate Normal distribution; default is to use an identity matrix 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.

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.

Directive: `CALCULATE`.
Procedures: `GRCSR`, `GREJECTIONSAMPLE`, `GRLABEL`, `GRMNOMIAL`, `GRTHIN`, `GRTORSHIFT`, `SAMPLE`, `SVSAMPLE`.
Functions: `GRBETA`, `GRBINOMIAL`, `GRCHISQUARE`, `GRF`, `GRGAMMA`, `GRHYPERGEOMETRIC`, `GRLOGNORMAL`, `GRNORMAL`, `GRPOISSON`, `GRSAMPLE`, `GRSELECT`, `GRT`, `GRUNIFORM`.

Commands for: Calculations and manipulation.

### Example

```CAPTION 'GRMULTINORMAL example'; STYLE=meta
MATRIX  [ROWS=1000; COLUMN=5] random
VARIATE [NVALUES=5; VALUE=(1...5)] mu
SYMMETRICMATRIX [ROWS=5; VALUE=10,4,10,3,4,10,2,3,4,10,1,2,3,4,10] sigma,vcov
GRMULTINORMAL [NVALUES=1000; MEANS=mu; VCOVARIANCE=sigma; SEED=57653] random
CALC    vcov\$[1...5]   = VAR(random\$[*;1...5])
&       vcov\$[1;2...5] = COV(random\$[*;1]; random\$[*;2...5])
&       vcov\$[2;3...5] = COV(random\$[*;2]; random\$[*;3...5])
&       vcov\$[3;4...5] = COV(random\$[*;3]; random\$[*;4...5])
&       vcov\$[4;5]     = COV(random\$[*;4]; random\$[*;5])
PRINT   sigma,vcov; DECIMALS=2
```