discrepancyCriteria {DiceDesign}

Discrepancy measure

Description

Compute discrepancy criteria.

Usage

discrepancyCriteria(design,type='all')

Arguments

design

a matrix (or a data.frame) corresponding to the design of experiments. The discrepancy criteria are computed for a design in the unit cube [0,1]^d. If this condition is not satisfied the design is automatically rescaled.

type

type of discrepancies (single value or vector) to be computed:

'all'	all type of discrepancies (default)
'C2'	centered L2-discrepancy
'L2'	L2-discrepancy
'L2star'	L2star-discrepancy
'M2'	modified L2-discrepancy
'S2'	symmetric L2-discrepancy
'W2'	wrap-around L2-discrepancy
'Mix2'	mixture L2-discrepancy

Details

The discrepancy measures how far a given distribution of points deviates from a perfectly uniform one. Different L2 discrepancies are available in DiceDesign. For example, if we denote by Vol(J) the volume of a subset J of [0; 1]^d and A(X; J) the number of points of X falling in J, the L2 discrepancy is:

D_{L2} (X) = \left[ \int_{[0,1]^{2d}}{} \left( \frac{A(X,J_{a,b})}{n} - Vol (J_{a,b}) \right)^{2} da db \right]^{1/2}

where a = (a_{1}; ... ; a_{d})', b = (b_{1};...; b_{d})' and J_{a,b} = [a_{1}; b_{1}) \times ... \times [a_{d};b_{d}). The other L2-discrepancies are defined according to the same principle with different form from the subset J. Among all the possibilities, discrepancyCriteria implements only the L2 discrepancies because it can be expressed analytically even for high dimension.

Centered L2-discrepancy is computed using the analytical expression done by Hickernell (1998). The user will refer to Pleming and Manteufel (2005) to have more details about the wrap around discrepancy.

Value

A list containing the L2-discrepancies of the design.

Author(s)

J. Franco, D. Dupuy & B. Iooss

References

Fang K.T, Li R. and Sudjianto A. (2006) Design and Modeling for Computer Experiments, Chapman & Hall.

Fang K-T., Liu M-Q., Qin H. and Zhou Y-D. (2018) Theory and application of uniform experimental designs. Springer.

Franco J. (2008) Planification d'experiences numerique en phase exploratoire pour la simulation des phenomenes complexes, PhD thesis, Ecole Nationale Superieure des Mines de Saint Etienne.

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

Pleming J.B. and Manteufel R.D. (2005) Replicated Latin Hypercube Sampling, 46th Structures, Structural Dynamics & Materials Conference, 16-21 April 2005, Austin (Texas) – AIAA 2005-1819.

See Also

distance criteria (coverage, meshRatio, mindist and phiP)

Examples

dimension <- 2
n <- 40
X <- matrix(runif(n*dimension), n, dimension)
discrepancyCriteria(X)