evalContour: Evaluating Convex Polytopes

In modQR: Multiple-Output Directional Quantile Regression

Evaluating Convex Polytopes

Description

Given the system of inequalities AAMat%*%ZVec <= BBVec describing a convex polytope/contour with an interior point IPVec in the Euclidean space of dimension two to six, this function identifies all nonredundant constraints and computes some characteristics of the resulting convex polytope such as its vertices, facets, volume and surface area.

Usage

evalContour(AAMat, BBVec = NULL, IPVec = NULL)

Arguments

AAMat	the constraints matrix from the system of inequalities defining the convex polytope. It should be a numeric matrix with two to six columns.
BBVec	the right-hand side from the system of inequalities defining the convex polytope. It should be a numeric column vector of the same length as the first column of AAMat.
IPVec	an interior point of the investigated convex polytope. This argument can be omitted or set equal to a numeric column vector of the same length as the first row of AAMat. If IPVec is NULL or if given IPVec does not lie well inside the convex contour, then a well-positioned interior point is searched for internally, which may slow down the computation and make it less reliable.

Details

This function is included to be used for evaluating (regression) quantile contours or their cuts.

In fact, the function analyzes not the polytope itself, but its regularized intersection with the zero-centered hypercube of the edge length 2 000 that is employed as an artificial bounding box to avoid the problems with unbounded contours. The regularization consists of rounding the vertices (i.e., all of their coordinates) of such an intersection to the seventh decimal digit and of considering only the polytope determined by all the distinct rounded vertices for the final analysis.

Value

evalContour returns a list with the following components describing the resulting convex polytope:

Status	0 - OK. 2 - the contour seems virtually empty. 3 - the search for a well-positioned interior point IPVec failed. 4 - the number of input parameters is too low. 5 - AAMat is not a numeric matrix with two to six columns. 6 - BBVec is not a numeric column vector of the right length. 7 - IPVec is not a numeric column vector of the right length.
TVVMat	the matrix with clearly distinct contour vertices (in rows).
TKKMat	the matrix with clearly distinct elementary facets (in rows). Each row contains the indices of the rows of TVVMat where the facet vertices are stored. Each facet has the same number of vertices equal to the number of columns of AAMat. See also help(convhulln) for the meaning of TKKMat.
NumF	the number of clearly distinct contour facets.
NumV	the number of clearly distinct contour vertices.
Vol	the volume of the contour (the area in 2D).
Area	the surface area of the contour (the circumference in 2D and the surface in 3D).

Examples

##- a simple example using a tilted zero-centered square 
AAMat <- rbind(c(-1,-1), c(1,-1), c(1,1), c(-1,1))
BBVec <- c(1, 1, 1, 1)
IPVec <- c(0, 0)
CST <- evalContour(AAMat, BBVec, IPVec)
print(CST)

##- computing and evaluating the 0.15-quantile contour of 199 
##random points uniformly distributed in the unit square 
##centered at zero
Tau  <- 0.15
YMat <- matrix(runif(2*199, -0.5, 0.5), 199, 2)
C <- compContourM1u(Tau, YMat)
CST <- evalContour(-C$CharST$HypMat[,1:2], -C$CharST$HypMat[,3])
print(CST)

##See also the examples ExampleA to ExampleE for some 
##more elaborate ways of computing, evaluating and 
##plotting the (regression) quantile contours.

modQR documentation built on May 11, 2022, 5:18 p.m.