View source: R/discrepancyCriteria_cplus.R
discrepancyCriteria_cplus | R Documentation |
Compute discrepancy criteria. This function uses a C++ implementation of the function discrepancyCriteria
from package DiceDesign.
discrepancyCriteria_cplus(design,type='all')
design |
a matrix corresponding to the design of experiments.
The discrepancy criteria are computed for a design in the unit cube [0,1] | |||||||||||||||
type |
type of discrepancies (single value or vector) to be computed:
|
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.
A list containing the L2-discrepancies of the design
.
Laurent Gilquin
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.
The distance criterion provided by maximin_cplus
dimension <- 2
n <- 40
X <- matrix(runif(n*dimension),n,dimension)
discrepancyCriteria_cplus(X)
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