Compute discrepancy criteria. This function uses a C++ implementation of the function discrepancyCriteria from package DiceDesign.
discrepancyCriteria_cplus(design,type='all')A list containing the L2-discrepancies of the 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 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 |
Laurent Gilquin
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.
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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