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# AFDISCREPANCY procedure

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

### Options

`PRINT` = string tokens Controls whether to print the discrepancy (`results`); default `resu` Specifies the method to use to calculate the discrepancy (`L2`, `maximin`, `entropy`); default `L2` 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 Saves the discrepancy 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 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.)

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=1m |(xikxjk)| )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.

Procedure: `AGSPACEFILLINGDESIGN`.

### Example

```CAPTION        'AFDISCREPANCY example',\