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\name{discrepancyCriteria}
\alias{discrepancyCriteria}
\title{Discrepancy measure}
\description{Compute discrepancy criteria. }
\usage{discrepancyCriteria(design,type='all')}
\arguments{
\item{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]\eqn{^d}.
If this condition is not satisfied the design is automatically rescaled.}
\item{type}{type of discrepancies (single value or vector) to be computed:
\tabular{ll}{
\code{'all'} \tab all type of discrepancies (default) \cr
\code{'C2'} \tab centered L2-discrepancy \cr
\code{'L2'} \tab L2-discrepancy \cr
\code{'L2star'} \tab L2star-discrepancy \cr
\code{'M2'} \tab modified L2-discrepancy \cr
\code{'S2'} \tab symmetric L2-discrepancy \cr
\code{'W2'} \tab wrap-around L2-discrepancy \cr
\code{'Mix2'} \tab mixture L2-discrepancy \cr
}
}
}
\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 \eqn{Vol(J)} the volume of a subset \eqn{J} of \eqn{[0; 1]^d} and \eqn{A(X; J)} the number of points of \eqn{X} falling in \eqn{J}, the \eqn{L2} discrepancy is:
\deqn{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}}{DL2 (X)^2 = \int_{[0,1]^d} [(A(X,J_{a,b})/n - Vol(J_{a,b})]^2 da db}
where \eqn{a = (a_{1}; ... ; a_{d})'}{a = (a1; ... ; ad)'}, \eqn{b = (b_{1};...; b_{d})'}{b = (b1;...; bd)'} and \eqn{J_{a,b} =
[a_{1}; b_{1}) \times ... \times [a_{d};b_{d})}{J_{a,b} =
[a1; b1) X ... X [ad;bd)}. The other L2-discrepancies are defined according to the same principle with different form from the subset \eqn{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 \code{design}.}
\references{
Fang K.T, Li R. and Sudjianto A. (2006) Design and Modeling for
Computer Experiments, \emph{Chapman & Hall}.
Fang K-T., Liu M-Q., Qin H. and Zhou Y-D. (2018) Theory and application of uniform
experimental designs. \emph{Springer}.
Franco J. (2008) Planification d'experiences numerique en phase exploratoire pour
la simulation des phenomenes complexes, \emph{PhD thesis, Ecole Nationale Superieure
des Mines de Saint Etienne}.
Hickernell F.J. (1998) A generalized discrepancy and quadrature error bound.
\emph{Mathematics of Computation}, \bold{67}, 299-322.
Pleming J.B. and Manteufel R.D. (2005) \emph{Replicated Latin Hypercube Sampling},
46th Structures, Structural Dynamics & Materials Conference, 16-21 April 2005, Austin
(Texas) -- AIAA 2005-1819.
}
\author{J. Franco, D. Dupuy & B. Iooss}
\seealso{distance criteria (\code{\link{coverage}}, \code{\link{meshRatio}},
\code{\link{mindist}} and \code{\link{phiP}})}
\examples{
dimension <- 2
n <- 40
X <- matrix(runif(n*dimension), n, dimension)
discrepancyCriteria(X)
}
\keyword{design}
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