.. _ProcWeylDesign:

Procedure: Generate a Weyl design
=================================

Description and Background
--------------------------

A Weyl design (also known as a Richtmyer design) is one of a number of
non-random space-filling designs suitable for defining a set of points
in the :ref:`simulator<DefSimulator>` input space for creating a
:ref:`training sample<DefTrainingSample>`.

The :math:`n` point Weyl design in :math:`p` dimensions is generated by a
generator set :math:`g=(g_1,\ldots,g_p)` of irrational numbers. See the
"Additional Comments" below for discussion of the choice of generators.

Inputs
------

-  Number of dimensions :math:`p`
-  Number of points desired :math:`n`
-  Set of irrational generators :math:`g_1,\ldots,g_p`

Outputs
-------

-  Weyl design :math:`D = \{x_1, x_2, \ldots, x_n\}`

Procedure
---------

For :math:`j=0,\ldots,n-1`, generate points as

.. math::
   x_{j+1}=\left(j\times g_1{\rm mod} 1, j\times g_2{\rm mod}
   1,\ldots,j\times g_d{\rm mod} 1\right).

Note that the operator "mod 1" here has the effect of returning the
fractional part of each number. For instance, if :math:`j=7` and :math:`g_1 =
\sqrt{2} = 1.414\ldots`, then

.. math::
   j\times g_1 = 9.89949\ldots

and so

.. math::
   j\times g_1 {\rm mod} 1 = 0.89949\ldots.

Additional Comments
-------------------

A potential problem with Weyl designs is the difficulty in finding
suitable generators. One suggestion is to let :math:`g_i` be the square
root of the :math:`i`-th prime, but this may not work well when :math:`p` is large.

References
----------

The following is a link to the repository for Matlab code for the Weyl
sequence in up to 100 dimensions:
`CPWeylSequence.m <https://virgo.aston.ac.uk/MUCM/WP3_1/code/CPWeylSequence.m>`_
(:ref:`disclaimer<MetaSoftwareDisclaimer>`).