.. _ProcLatticeDesign: Procedure: Generate a lattice design ==================================== Description and Background -------------------------- A lattice design is one of a number of non-random space-filling designs suitable for defining a set of points in the :ref:`simulator` input space for creating a :ref:`training sample`. The :math:`n` point lattice in :math:`p` dimensions is generated by a positive integer generator set :math:`g=(g_1,\ldots,g_p)`. See the "Additional Comments" below for discussion of the choice of generators. Inputs ------ - Number of dimensions :math:`p` - Number of runs desired :math:`n` - Set of positive integer generators :math:`g_1,\ldots,g_p` Outputs ------- - Lattice design :math:`D = \{x_1, x_2, \ldots, x_n\}` Procedure --------- For :math:`j=0,\ldots,n-1`, generate lattice points as .. math:: x_{j+1}=\left(\frac{j}{n}g_1{\rm mod} 1,\ldots,\frac{j}{n}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, n=50` and :math:`g_1 = 13`, then .. math:: \frac{j}{n}g_1 = 1.82 and so .. math:: \frac{j}{n}g_1 {\rm mod} 1=0.82. Additional Comments ------------------- A potential problem with lattice designs is the difficulty in finding suitable generators for a lattice. A condition for generators is that :math:`g_1,\ldots,g_d` and :math:`n` should form a set of relatively prime numbers. However, this seems to be a necessary but not sufficient condition to obtain a lattice that fills the design space well. References ---------- Bates, R.A., Riccomagno, E., Schwabe, R., Wynn, H. (1998). The use of lattices in the design of high-dimensional experiments. IMS Lecture Notes, 34, 26-35. Sloan, I.H., Joe, S. (1994). Lattice methods for multiple integration. Clarendon Press, Oxford. Matlab code for generating lattice designs is available from `Ron Bates `_ (:ref:`disclaimer`).