Calculates the discrepancy of a design (B.M. Parker).
|Controls whether to print the discrepancy (
||Specifies the method to use to calculate the discrepancy (
||A variate of length two indicating which design points have swapped when updating the discrepancy criterion for the
||A matrix, or a pointer of variates, specifying the design points|
||Saves the discrepancy|
||Stores the distances, to allow fast updates with the
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.
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.
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 Lp discrepancy is a measure of non-uniformity of a design. More precisely, the Lp 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.)
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
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
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
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=1m |(xik – xjk)| )2
The L2 discrepancy is calculated according to the procedure of Hickernell (1988).
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.
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