inside {multinomineq}

Check Whether Points are Inside a Convex Polytope

Description

Determines whether a point x is inside a convex poltyope by checking whether (1) all inequalities A*x <= b are satisfied or (2) the point x is in the convex hull of the vertices in V.

Usage

inside(x, A, b, V)

Arguments

x

a vector of length equal to the number of columns of A or V (i.e., a single point in D-dimensional space) or matrix of points/vertices (one per row).

A

a matrix with one row for each linear inequality constraint and one column for each of the free parameters. The parameter space is defined as all probabilities x that fulfill the order constraints A*x <= b.

b

a vector of the same length as the number of rows of A.

V

a matrix of vertices (one per row) that define the polytope of admissible parameters as the convex hull over these points (if provided, A and b are ignored). Similar as for A, columns of V omit the last value for each multinomial condition (e.g., a1,a2,a3,b1,b2 becomes a1,a2,b1). Note that this method is comparatively slow since it solves linear-programming problems to test whether a point is inside a polytope (Fukuda, 2004) or to run the Gibbs sampler.

See Also

Ab_to_V and V_to_Ab to change between A/b and V representation.

Examples

# linear order constraints:  x1<x2<x3<.5
A <- matrix(c(
  1, -1, 0,
  0, 1, -1,
  0, 0, 1
), ncol = 3, byrow = TRUE)
b <- c(0, 0, .50)

# vertices: admissible points (corners of polytope)
V <- matrix(c(
  0, 0, 0,
  0, 0, .5,
  0, .5, .5,
  .5, .5, .5
), ncol = 3, byrow = TRUE)

xin <- c(.1, .2, .45) # inside
inside(xin, A, b)
inside(xin, V = V)

xout <- c(.4, .1, .55) # outside
inside(xout, A, b)
inside(xout, V = V)