# Low Discrepancy Sequences

### Description

A collection and description of functions to compute
Halton's and Sobol's low discrepancy sequences,
distributed in form of a uniform or normal distribution.

The functions are:

`runif.halton` | Uniform Halton sequence, |

`rnorm.halton` | Normal Halton sequence, |

`runif.sobol` | Uniform scrambled Sobol sequence, |

`rnorm.sobol` | Normal scrambled Sobol sequence, |

`runif.pseudo` | Uniform pseudo random numbers, |

`norma.pseudo` | Normal pseudo random numbers. |

### Usage

1 2 3 4 5 6 7 8 | ```
runif.halton(n, dimension, init)
rnorm.halton(n, dimension, init)
runif.sobol(n, dimension, init, scrambling, seed)
rnorm.sobol(n, dimension, init, scrambling, seed)
runif.pseudo(n, dimension, init)
rnorm.pseudo(n, dimension, init)
``` |

### Arguments

`dimension` |
an integer value, the dimension of the sequence. The maximum value for the Sobol generator is 1111. |

`init` |
a logical, if TRUE the sequence is initialized and restarts, otherwise not. By default TRUE. |

`n` |
an integer value, the number of random deviates. |

`scrambling` |
an integer value, if 1, 2 or 3 the sequence is scrambled otherwise not. If 1, Owen type type of scrambling is applied, if 2, Faure-Tezuka type of scrambling, is applied, and if 3, both Owen+Faure-Tezuka type of scrambling is applied. By default 0. |

`seed` |
an integer value, the random seed for initialization
of the scrambling process. By default 4711. On effective
if |

### Details

**Halton's Low Discrepancy Sequences:**

Calculates a matrix of uniform or normal deviated halton low
discrepancy numbers.

**Scrambled Sobol's Low Discrepancy Sequences:**

Calculates a matrix of uniform and normal deviated Sobol low
discrepancy numbers. Optional scrambling of the sequence can
be selected.

**Pseudo Random Number Sequence:**

Calculates a matrix of uniform or normal distributed pseudo
random numbers. This is a helpful function for comparing
investigations obtained from a low discrepancy series with
those from a pseudo random number.

### Value

All generators return a numeric matrix of size `n`

by `dimension`

.

### Note

The global variables `runif.halton.seed`

and
`runif.sobol.seed`

save the status to restart the
generators. Note, that only one instance of a generators
can be run at the same time.

The ACM Algorithm 659 implemented to generate scrambled
Sobol sequences is under the License of the ACM restricted
for academic and noncommerical usage. Please consult the
ACM License agreement included in the `doc`

directory.

### Author(s)

P. Bratley and B.L. Fox for the Fortran Sobol Algorithm 659,

S. Joe for the Fortran extension to 1111 dimensions,

Diethelm Wuertz for the Rmetrics **R**-port.

### References

Bratley P., Fox B.L. (1988);
*Algorithm 659: Implementing Sobol's Quasirandom
Sequence Generator*,
ACM Transactions on Mathematical Software 14, 88–100.

Joe S., Kuo F.Y. (1998);
*Remark on Algorithm 659: Implementing Sobol's Quaisrandom
Seqence Generator*.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | ```
## *.halton -
par(mfrow = c(2, 2), cex = 0.75)
runif.halton(n = 10, dimension = 5)
hist(runif.halton(n = 5000, dimension = 1), main = "Uniform Halton",
xlab = "x", col = "steelblue3", border = "white")
rnorm.halton(n = 10, dimension = 5)
hist(rnorm.halton(n = 5000, dimension = 1), main = "Normal Halton",
xlab = "x", col = "steelblue3", border = "white")
## *.sobol -
runif.sobol(n = 10, dimension = 5, scrambling = 3)
hist(runif.sobol(5000, 1, scrambling = 2), main = "Uniform Sobol",
xlab = "x", col = "steelblue3", border = "white")
rnorm.sobol(n = 10, dimension = 5, scrambling = 3)
hist(rnorm.sobol(5000, 1, scrambling = 2), main = "Normal Sobol",
xlab = "x", col = "steelblue3", border = "white")
## *.pseudo -
runif.pseudo(n = 10, dimension = 5)
rnorm.pseudo(n = 10, dimension = 5)
``` |