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 L_{2} 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 L_{2} 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