volesti (version 1.0.0)

sample_points: Sample points from a convex Polytope (H-polytope, V-polytope or a zonotope) or use direct methods for uniform sampling from the unit or the canonical or an arbitrary \(d\)-dimensional simplex and the boundary or the interior of a \(d\)-dimensional hypersphere

Description

Sample N points with uniform or multidimensional spherical gaussian -centered in an internal point- target distribution. The \(d\)-dimensional unit simplex is the set of points \(\vec{x}\in \R^d\), s.t.: \(\sum_i x_i\leq 1\), \(x_i\geq 0\). The \(d\)-dimensional canonical simplex is the set of points \(\vec{x}\in \R^d\), s.t.: \(\sum_i x_i = 1\), \(x_i\geq 0\).

Usage

sample_points(P = NULL, N = NULL, distribution = NULL,
  WalkType = NULL, walk_step = NULL, exact = NULL, body = NULL,
  Parameters = NULL, InnerPoint = NULL)

Arguments

P

A convex polytope. It is an object from class (a) Hpolytope or (b) Vpolytope or (c) Zonotope.

N

The number of points that the function is going to sample from the convex polytope. The default value is \(100\).

distribution

Optional. A string that declares the target distribution: a) 'uniform' for the uniform distribution or b) 'gaussian' for the multidimensional spherical distribution. The default target distribution is uniform.

WalkType

Optional. A string that declares the random walk method: a) 'CDHR' for Coordinate Directions Hit-and-Run, b) 'RDHR' for Random Directions Hit-and-Run or c) 'BW' for Ball Walk. The default walk is 'CDHR'.

walk_step

Optional. The number of the steps for the random walk. The default value is \(\lfloor 10 + d/10\rfloor\), where \(d\) implies the dimension of the polytope.

exact

A boolean parameter. It should be used for the uniform sampling from the boundary or the interior of a hypersphere centered at the origin or from the unit or the canonical or an arbitrary simplex. The arbitrary simplex has to be given as a V-polytope. For the rest well known convex bodies the dimension has to be declared and the type of body as well as the radius of the hypersphere.

body

A string that declares the type of the body for the exact sampling: a) 'unit simplex' for the unit simplex, b) 'canonical simplex' for the canonical simplex, c) 'hypersphere' for the boundary of a hypersphere centered at the origin, d) 'ball' for the interior of a hypersphere centered at the origin.

Parameters

A list for the parameters of the methods:

• variance The variance of the multidimensional spherical gaussian. The default value is 1.

• dimension An integer that declares the dimension when exact sampling is enabled for a simplex or a hypersphere.

• radius The radius of the \(d\)-dimensional hypersphere. The default value is \(1\).

• BW_rad The radius for the ball walk.

InnerPoint

A \(d\)-dimensional numerical vector that defines a point in the interior of polytope P.

Value

A \(d\times N\) matrix that contains, column-wise, the sampled points from the convex polytope P.

References

R.Y. Rubinstein and B. Melamed, “Modern simulation and modeling” Wiley Series in Probability and Statistics, 1998.

A Smith, Noah and W Tromble, Roy, “Sampling Uniformly from the Unit Simplex,” Center for Language and Speech Processing Johns Hopkins University, 2004.

Art B. Owen, “Monte Carlo theory, methods and examples,” Art Owen, 2009.

Examples

# NOT RUN {
# uniform distribution from the 3d unit cube in V-representation using ball walk
P = GenCube(3, 'V')
points = sample_points(P, WalkType = "BW", walk_step = 5)

# gaussian distribution from the 2d unit simplex in H-representation with variance = 2
A = matrix(c(-1,0,0,-1,1,1), ncol=2, nrow=3, byrow=TRUE)
b = c(0,0,1)
P = Hpolytope$new(A,b)
points = sample_points(P, distribution = "gaussian", Parameters = list("variance" = 2))

# uniform points from the boundary of a 10-dimensional hypersphere
points = sample_points(exact = TRUE, body = "hypersphere", Parameters = list("dimension" = 10))

# 10000 uniform points from a 2-d arbitrary simplex
V = matrix(c(2,3,-1,7,0,0),ncol = 2, nrow = 3, byrow = TRUE)
P = Vpolytope$new(V)
points = sample_points(P, N = 10000, exact = TRUE)
# }