# Sampling Random Numbers From The Truncated Multivariate Normal Distribution With Linear Constraints

### Description

This function generates random numbers from the truncated multivariate normal
distribution with mean equal to `mean`

and covariance matrix
`sigma`

and general linear constraints

*lower <= D x <= upper*

with either rejection sampling or Gibbs sampling.

### Usage

1 2 3 4 5 6 |

### Arguments

`n` |
Number of random points to be sampled. Must be an integer |

`mean` |
Mean vector (d x 1), default is |

`sigma` |
Covariance matrix (d x d), default is |

`lower` |
Vector of lower truncation points (r x 1),
default is |

`upper` |
Vector of upper truncation points (r x 1),
default is |

`D` |
Matrix for linear constraints (r x d), defaults to diagonal matrix (d x d), i.e. r = d. |

`algorithm` |
Method used, possible methods are the Fortan Gibbs sampler ("gibbs", default), the Gibbs sampler implementation in R ("gibbsR") and rejection sampling ("rejection") |

`...` |
additional parameters for Gibbs sampling, given to the internal method |

### Details

This method allows for *r > d* linear constraints, whereas `rtmvnorm`

requires a full-rank matrix D *(d x d)* and can only handle *r <= d* constraints at the moment.
The lower and upper bounds `lower`

and `upper`

are *(r x 1)*,
the matrix `D`

is *(r x d)* and x is *(d x 1)*.
The default case is *r = d* and *D = I_d*.

### Warning

This method will be merged with `rtmvnorm`

in one of the next releases.

### Author(s)

Stefan Wilhelm

### See Also

`rtmvnorm`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 | ```
## Not run:
################################################################################
#
# Example 5a: Number of linear constraints r > dimension d
#
################################################################################
# general linear restrictions a <= Dx <= b with x (d x 1); D (r x d); a,b (r x 1)
# Dimension d=2, r=3 linear constraints
#
# a1 <= x1 + x2 <= b2
# a2 <= x1 - x2 <= b2
# a3 <= 0.5x1 - x2 <= b3
#
# [ a1 ] <= [ 1 1 ] [ x1 ] <= [b1]
# [ a2 ] [ 1 -1 ] [ x2 ] [b2]
# [ a3 ] [ 0.5 -1 ] [b3]
D <- matrix(
c( 1, 1,
1, -1,
0.5, -1), 3, 2, byrow=TRUE)
a <- c(0, 0, 0)
b <- c(1, 1, 1)
# mark linear constraints as lines
plot(NA, xlim=c(-0.5, 1.5), ylim=c(-1,1))
for (i in 1:3) {
abline(a=a[i]/D[i, 2], b=-D[i,1]/D[i, 2], col="red")
abline(a=b[i]/D[i, 2], b=-D[i,1]/D[i, 2], col="red")
}
### Gibbs sampling for general linear constraints a <= Dx <= b
mean <- c(0, 0)
sigma <- matrix(c(1.0, 0.2,
0.2, 1.0), 2, 2)
x0 <- c(0.5, 0.2) # Gibbs sampler start value
X <- rtmvnorm2(n=1000, mean, sigma, lower=a, upper=b, D, start.value=x0)
# show random points within simplex
points(X, pch=20, col="black")
## End(Not run)
``` |

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