AFDISCREPANCY procedure

Calculates the discrepancy of a design (B.M. Parker).

Options

PRINT = string tokens	Controls whether to print the discrepancy (results); default resu
METHOD = string token	Specifies the method to use to calculate the discrepancy (L2, maximin, entropy); default L2
SWAP = variate	A variate of length two indicating which design points have swapped when updating the discrepancy criterion for the maximin or entropy criteria; default none

Parameters

DESIGN = matrices or pointers	A matrix, or a pointer of variates, specifying the design points
DISCREPANCY = scalars	Saves the discrepancy
DISTANCES = matrices	Stores the distances, to allow fast updates with the maximin or entropy criteria

Description

A space filling design is an experimental design for a number of runs, which each have a number of (usually) continuous factors. They are designed to ensure that the experiment is spread over the entire design space, so that large and potentially important regions are not ignored. AFDISCREPANCY can calculate a measure of the discrepancy of the design, that indicates how well it fills the space. This is used by the AGSPACEFILLINGDESIGN procedure to form a good design, that is, one with a low discrepancy.

The DESIGN parameter supplies either as a matrix with n rows and m columns, or a pointer with n variates each with m units, to specify a design with n points in a unit hypercube [0,1]^m.

The METHOD option specifies the criterion to use to measure the discrepancy of the design. The maximin criterion maximizes the minimum inter-point Euclidean distance. The entropy criterion minimizes -log |R|, where R is a measure of correlation between points in the design. The L_p discrepancy is a measure of non-uniformity of a design. More precisely, the L_p discrepancy measures the difference between the empirical cumulative distribution function of a design and the uniform cumulative distribution function. Here, we minimize the centred L₂ discrepancy. (See Fang et al. 2000.)

The DISTANCES option can supply a matrix to store a measure of the distance between the points in the designs for the maximin and entropy criteria. If a variate of two numbers is specified by the SWAP option, AFDISCREPANCY will update the distance criterion only for the design points that are changed, making a far faster procedure. This is used in the ESE algorithm adapted in AGSPACEFILLINGDESIGN.

By default the discrepancy is printed, but you can set option PRINT=* to suppress this. The discrepancy can be saved, in a scalar, using the DISCREPANCY option.

Options: PRINT, METHOD, SWAP.

Parameters: DESIGN, DISCREPANCY, DISTANCES.

Method

The maximin design maximizes the minimum Euclidean distance between points as described in Johnson et al. (1990). The entropy design maximizes |nR| where R is a Gaussian correlation matrix between design points. Thus, here we minimize a Gaussian correlation function. In a Bayesian context, minimizing the expected posterior entropy is equivalent to maximizing the prior entropy. See Koehler & Owen (1996). R here, for design points i and j, is defined as

exp( ∑_k=1^m |(x_ik – x_jk)| )²

The L₂ discrepancy is calculated according to the procedure of Hickernell (1988).

References

Fang, K.T., Lin, D.K., Winker, P. & Zhang, Y. (2000). Uniform design: theory and application. Technometrics, 42, 237-248.

Hickernell, F. (1998). A generalized discrepancy and quadrature error bound. Mathematics of Computation of the American Mathematical Society, 67, 299-322.

Johnson, M.E., Moore, L.M. & Ylvisaker, D. (1990). Minimax and maximin distance designs. Journal of Statistical Planning and Inference, 26, 131-148.

Koehler, J.R. & Owen, A.B. (1996). Computer experiments. Handbook of Statistics, 13, 261-308.

See also

Procedure: AGSPACEFILLINGDESIGN.

Example

CAPTION        'AFDISCREPANCY example',\
               !t('This generates a latin hypercube design in 3 dimensions',\
               'and 16 experimental runs, where discrepancy is minimized.');\
               STYLE=meta,plain
AGSPACEFILLINGDESIGN [PRINT=design; METHOD=latinhypercube; CENTRED=yes;\
               CRITERION=l2; NDIMENSIONS=3; NUNITS=16; SEED=101411] X
AFDISCREPANCY  [PRINT=results] DESIGN=X; DISCREPANCY=discrepancy