## 2D graphical tool for defect detection of Space-Filling Designs.

### Description

For a 2-dimensional design, the 2D radial scanning statistic (RSS) scans angularly the domain. In each direction, it compares the distribution of projected points to their theoretical distribution under the assumption that all design points are drawn from uniform distribution. For a d-dimensional design, all pairs of dimensions are scanned.The RSS detects the defects of low discrepancy sequences or orthogonal arrays, and can be used for selecting space-filling designs.

### Usage

rss2d(design, lower, upper, gof.test.type="greenwood",
gof.test.stat=NULL, transform=NULL, n.angle=360, graphics=1,
trace=TRUE, lines.lwd = 1, lines.lty = "dotted", ...)


### Arguments

 design a matrix or data.frame containing the d-dimensional design of experiments. The row no. i contains the values of the d input variables corresponding to simulation no. i lower the domain lower boundaries. upper the domain upper boundaries. gof.test.type an optional character indicating the kind of statistical test to be used to test the goodness-of-fit of the design projections to their theoretical distribution. Several tests are available, see unif.test.statistic. Default is "greenwood". gof.test.stat an optional number equal to the goodness-of-fit statistic at level 5%. Default is the modified test statistic for fully specified distribution (see details below). transform an optional character indicating what type of transformation should be applied before testing uniformity. Only one choice available "spacings", that lead to over-detection. Default - and recommended - is NULL. n.angle an optional number indicating the number of angles used. Default is 360 corresponding to a 0.5-degree discretization step. Note that the RSS curve is continuous. graphics an optional integer indicating whether a graph should be produced. If negative, no graph is produced. If superior to 2, the RSS curve only is plotted in the worst 2D coordinate subspace (corr. to the worst value of statistic). If 1 (default), the design is also added, with its projections onto the worst (oblique) axis. trace an optional boolean. Turn it to FALSE if you want no verbosity. lines.lwd optional number specifying the width of the straight lines involved in the graphical outputs (axis and projections) lines.lty optional character string specifying the type of the straight lines involved in the graphical outputs (axis and projections) ... optional graphical parameters of plot function, to draw the RSS curve.

### Value

a list with components:

 global.stat a matrix containing the values of the global statistic (equal to the maximum of statistic values over the RSS curve) for all pairs of dimensions. worst.case the worst pair of dimensions, that is the one that gives the worst value of global.stat. worst.dir the worst direction, that is the one that gives the worst value of the global statistic in the coordinate plane defined by worst.case. stat a vector of length n.angle containing the statitic values for each angle, in the coordinate plane defined by worst.case. angle a vector of length n.angle containing the corresponding angles used. curve a (2*n.angle)x2 matrix containing the discretized RSS curve. gof.test.stat the threshold at significance level 0.05 for the specified goodness-of-fit statistic. It is equal to the radius of the circle superimposed on the RSS figure.

O. Roustant

### References

Roustant O., Franco J., Carraro L., Jourdan A. (2010), A radial scanning statistic for selecting space-filling designs in computer experiments, MODA-9 proceedings.

D Agostino R.B., Stephens M.A. (1986), Goodness-of-fit techniques, CRC Press, New York.

unif.test.statistic, unif.test.quantile, rss3d

### Examples

## Detection of defects of Sobol designs

## requires randtoolbox package
library(randtoolbox)

## In 2D
type="l", col="red")

## In 8D
## All pairs of dimensions are tried to detect the worst defect
## (according to the specified goodness-of-fit statistic).
d <- 8
n <- 10*d