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:
\li \c halton or 
\li \c hammersley

Then three keywords are used to define the sequence and how it is sampled:
\li \c prime_base
\li \c sequence_start
\li \c 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 Halton1960,
\cite Halton1964, and 
\cite Kocis1997.
 
Faq::
See_Also::	method-dace, method-fsu_cvt, method-psuade_moat
