discrepancyCriteria_cplus {sensitivity}

Discrepancy measure

Description

Compute discrepancy criteria. This function uses a C++ implementation of the function discrepancyCriteria from package DiceDesign.

Usage

discrepancyCriteria_cplus(design,type='all')

Arguments

design

a matrix 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

Details

The discrepancy measures how far a given distribution of points deviates from a perfectly uniform one. Different discrepancies are available. 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_cplus 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)

Laurent Gilquin

References

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

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

The distance criterion provided by maximin_cplus

Examples

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