Blurb::
Design of Computer Experiments - Quasi-Monte Carlo sampling
Description::
Quasi-Monte Carlo methods
produce low discrepancy sequences, especially if one is interested
in the uniformity of projections of the point sets onto
lower dimensional faces of the hypercube (usually 1-D: how well do
the marginal distributions approximate a uniform?)

This method generates sets of uniform random variables on the
interval [0,1]. If the user specifies lower and upper bounds for a
variable, the [0,1] samples are mapped to the [lower, upper] interval.

The user must first choose the sequence type:

- ``halton`` or
- ``hammersley``

Then three keywords are used to define the sequence and how it is sampled:

- ``prime_base``
- ``sequence_start``
- ``sequence_leap``

Each of these has defaults, so specification is optional.
Topics::
package_fsudace, design_and_analysis_of_computer_experiments
Examples::

Theory::
The quasi-Monte Carlo sequences of Halton and Hammersley are deterministic
sequences determined by a set of prime bases.
Generally, we recommend that the user leave the default
setting for the bases, which are the lowest primes.
Thus, if one wants to generate a sample set for 3 random variables,
the default bases used are 2, 3, and 5 in the Halton sequence.
To give an example of how these sequences look, the Halton sequence
in base 2 starts with points 0.5, 0.25, 0.75, 0.125, 0.625, etc.
The first few points in a Halton base 3 sequence are
0.33333, 0.66667, 0.11111, 0.44444, 0.77777, etc. Notice that the Halton
sequence tends to alternate back and forth, generating a point closer to zero
then a point closer to one. An individual sequence is based on a radix
inverse function defined on a prime base. The prime base determines
how quickly the [0,1] interval is filled in. Generally, the lowest
primes are recommended.

The Hammersley sequence is the same as the Halton sequence, except the values
for the first random variable are equal to 1/N, where N is the number of
samples. Thus, if one wants to generate a sample set of 100 samples for 3
random variables, the first random variable has values 1/100, 2/100, 3/100,
etc. and the second and third variables are generated according to a Halton
sequence with bases 2 and 3, respectively.

For more information about
these sequences, see :cite:p:`Halton1960`,
:cite:p:`Halton1964`, and
:cite:p:`Kocis1997`.
Faq::

See_Also::
