tdaunif: Uniform manifold samplers for topological data analysis

Description

Generate uniform random samples from embedded manifolds, optionally with noise.

Details

This package assembles functions that generate samples of points uniformly from the surfaces of embedded manifolds. An embedding is a one-to-one continuous map f:M\to X from a manifold M to a Euclidean coordinate space X, and each function relies on a parameterization of M given by a continuous bijective function p:S\to f(M) that may identify some points of s (boundary or interior) to produce the topology of M. (This means that the inverse of p may not be continuous.)

Sampling points P uniformly from S and mapping the sample to f(M) may produce a non-uniform sample p(P) due to differences in the local sampling rate per unit interior (length, area, volume, etc.), quantified as the Jacobian (higher-order derivative) of p. tdaunif uses two techniques to correct for this:

• The more numerical (brute-force) technique is to compute the Jacobian on the parameter space and oversample locally at a rate proportional to the Jacobian. This oversampling is done via rejection sampling as illustrated by Diaconis, Holmes, and Shahshahani (2013).

• The more analytic technique is to invert the Jacobian symbolically in order to define an interior-preserving parameterization q:S\to f(M), as illustrated for 2-manifolds by Arvo (2001). Sampling P uniformly on S then produces a uniform sample q(P) on f(M). The interior-preserving map also enables stratified sampling on the manifold via stratification of the parameter space.

Multivariate Gaussian noise in the coordinate space can be added to any sample.

Author(s)

Jason Cory Brunson

Brandon Demkowicz

Sanmati Choudhary

References

J Arvo (2001) Stratified Sampling of 2-Manifolds. SIGRAPH 2001 (State of the Art in Monte Carlo Ray Tracing for Realistic Image Synthesis), Course Notes, Vol. 29. https://www.cs.princeton.edu/courses/archive/fall04/cos526/papers/course29sig01.pdf

P Diaconis, S Holmes, and M Shahshahani (2013) Sampling from a Manifold. Advances in Modern Statistical Theory and Applications: A Festschrift in honor of Morris L. Eaton, 102–125. doi:10.1214/12-IMSCOLL1006

See Also

Useful links:

• https://tdaverse.github.io/tdaunif/

• Report bugs at https://github.com/tdaverse/tdaunif/issues

Sample (with noise) from annuli

Description

These functions generate uniform samples from annuli with in 2-dimensional space with major radius 1, optionally with noise.

Usage

sample_annulus(n, r = 0.5, bins = 1L, sd = 0)

sample_disk(n, bins = 1L, sd = 0)

Arguments

Details

The sample is generated by an area-preserving parameterization of the annulus. This parameterization was derived through the method for sampling 2-manifolds as described by Arvo (2001).

References

J Arvo (2001) Stratified Sampling of 2-Manifolds. SIGRAPH 2001 (State of the Art in Monte Carlo Ray Tracing for Realistic Image Synthesis), Course Notes, Vol. 29. https://www.cs.princeton.edu/courses/archive/fall04/cos526/papers/course29sig01.pdf

Examples

set.seed(99812L)

# Uniformly sampled unit disk in 2-space
x <- sample_disk(1800, sd = 0)
plot(x, asp = 1, pch = 19, cex = .5)

# Uniformly sampled unit disk in 2-space with Gaussian noise
x <- sample_disk(1800, sd = .1)
plot(x, asp = 1, pch = 19, cex = .5)

# Uniformly sampled annulus
x <- sample_annulus(100, r = .4, sd = 0)
plot(x, asp = 1, pch = 19, cex = .5)

Sample (with noise) from archimedean spirals and swiss rolls

Description

These functions generate uniform samples from archimedean spirals in 2-dimensional space or from swiss rolls in 3-dimensional space, optionally with noise.

Usage

sample_arch_spiral(n, ar = 1, arms = 1L, min_wrap = 0, max_wrap = 1, sd = 0)

sample_swiss_roll(
  n,
  ar = 1,
  arms = 1L,
  min_wrap = 0,
  max_wrap = 1,
  width = 2 * pi,
  sd = 0
)

Arguments

Details

The archimedean spiral starts at the origin and wraps around itself such that radial distances between all the spiral branches are equal. The specific parameterization was taken from Koeller (2002). The uniform sample is generated through a rejection sampling process as described by Diaconis, Holmes, and Shahshahani (2013).

The swiss roll sampler is patterned after one in drtoolbox and extended from the archimedean spiral sampler.

References

Koeller, J. (2002). Spirals. Retrieved July 18, 2019, from http://www.mathematische-basteleien.de/spiral.htm

P Diaconis, S Holmes, and M Shahshahani (2013) Sampling from a Manifold. Advances in Modern Statistical Theory and Applications: A Festschrift in honor of Morris L. Eaton, 102–125. doi:10.1214/12-IMSCOLL1006

Examples

set.seed(77151L)

#Uniformly sampled archimedean spiral in 2-space, with 1 wrap
x <- sample_arch_spiral(360, min_wrap = 0, max_wrap = 1)
plot(x, asp = 1, pch = 19, cex = .5)

#Uniformly sampled archimedean spiral in 2-space, with 1 wrap 
#and aspect ratio of 2:1
x <- sample_arch_spiral(360, ar = 2, min_wrap = 0, max_wrap = 1)
plot(x, asp = 1, pch = 19, cex = .5)

#Uniformly sampled archimedean spiral in 2-space, with 1 wrap 
#and aspect ratio of 1:2
x <- sample_arch_spiral(360, ar = 0.5, min_wrap = 0, max_wrap = 1)
plot(x, asp = 1, pch = 19, cex = .5)

#Uniformly sampled archimedean spiral in 2-space, with 5 wraps
x <- sample_arch_spiral(360, min_wrap = 0, max_wrap = 5)
plot(x, asp = 1, pch = 19, cex = .5)

#Uniformly sampled archimedean spiral in 2-space, with 5 wraps and starting from
#2 wraps
x <- sample_arch_spiral(360, min_wrap = 2, max_wrap = 5)
plot(x, asp = 1, pch = 19, cex = .5)

#Uniformly sampled archimedean spiral, from 1 to 2 wraps, with 3 arms
x <- sample_arch_spiral(360, arms = 3, min_wrap = 1, max_wrap = 2)
plot(x, asp = 1, pch = 19, cex = .5)

#Uniformly sampled archimedean spiral in 2-space, with 1 wrap and noise with a
#standard deviation of 0.1
x <- sample_arch_spiral(360, min_wrap = 0, max_wrap = 1, sd = 0.1)
plot(x, asp = 1, pch = 19, cex = .5)

#Uniformly sampled swiss roll in 3-space, from 0 to 1 wraps and width 2*pi
x <- sample_swiss_roll(720, width = 2*pi)
pairs(x, asp = 1, pch = 19, cex = .5)
pca <- prcomp(x)
plot(x %*% pca$rotation, asp = 1, pch = 19, cex = .5)

#Uniformly sampled swiss roll in 3-space, from 0 to 1 wraps and width 2*pi
x <- sample_swiss_roll(720, ar = 2, width = 2*pi)
pairs(x, asp = 1, pch = 19, cex = .5)
pca <- prcomp(x)
plot(x %*% pca$rotation, asp = 1, pch = 19, cex = .5)

Sample (with noise) from circles

Description

These functions generate uniform and stratified samples from configurations of circles of radius 1 in 2- or 3-dimensional space, optionally with noise.

Usage

sample_circle(n, bins = 1L, sd = 0)

sample_circles_interlocked(n, bins = 1L, sd = 0)

Arguments

Details

The function sample_circle() uses the usual sinusoidal parameterization from the unit interval to the unit circle.

The function sample_circles_interlocked() effectively samples from a pair of circles and rotates them in 3-dimensional space so that they are interlocked (perpendicular to each other).

Both functions are length-preserving and admit stratification. If bins = 2, the stratification for the latter function will simply be between the two interlocked circles.

Examples

set.seed(14312L)

# circle in 2-space
x <- sample_circle(120, sd = .1)
plot(x, asp = 1, pch = 19, cex = .5)

# interlocked circles in 3-space
x <- sample_circles_interlocked(120, sd = .1)
pairs(x, asp = 1, pch = 19, cex = .5)

Sample (with noise) from ellipses

Description

These functions generate uniform samples from configurations of ellipses of major or minor radius 1 in 2- or 3-dimensional space, optionally with noise.

Usage

sample_ellipse(n, ar = 1, sd = 0)

sample_cylinder_elliptical(n, ar = 1, width = 1, sd = 0)

Arguments

Details

The function sample_ellipse() uses the usual sinusoidal parameterization from the unit interval to an ellipse with radii 1 and 1/ar. The uniform sample is generated through a rejection sampling process as described by Diaconis, Holmes, and Shahshahani (2013).

Examples

set.seed(97205L)

# ellipses in 2-space
x <- sample_ellipse(120, ar = 6)
plot(x, asp = 1, pch = 19, cex = .5)
x <- sample_ellipse(120, ar = 1/6)
plot(x, asp = 1, pch = 19, cex = .5)

# ellipses in 2-space
x <- sample_ellipse(120, ar = 6, sd = .1/6)
plot(x, asp = 1, pch = 19, cex = .5)
x <- sample_ellipse(120, ar = 1/6, sd = .1)
plot(x, asp = 1, pch = 19, cex = .5)

# cylinders in 3-space
x <- sample_cylinder_elliptical(120, ar = 1)
pairs(x, asp = 1, pch = 19, cex = .5)
x <- sample_cylinder_elliptical(120, ar = 3, width = 2*pi)
pairs(x, asp = 1, pch = 19, cex = .5)

Sample (with noise) from Klein bottles

Description

These functions generate uniform samples from Klein bottles in 4-dimensional space, optionally with noise.

Usage

sample_klein_tube(n, ar = 2, sd = 0)

sample_klein_flat(n, ar = 1, bump = 0.1, sd = 0)

Arguments

Details

The function sample_klein_tube() uses the Möbius tube parameterization obtained from the now-defunct Encyclopédie des Formes Mathématiques Remarquables and currently documented on Wikipedia.

The function sample_klein_flat() uses a flat parameterization based on that of the torus, as presented on Wikipedia.

Both uniform samples are generated through a rejection sampling process as described by Diaconis, Holmes, and Shahshahani (2013).

References

P Diaconis, S Holmes, and M Shahshahani (2013) Sampling from a Manifold. Advances in Modern Statistical Theory and Applications: A Festschrift in honor of Morris L. Eaton, 102–125. doi:10.1214/12-IMSCOLL1006

Examples

set.seed(834L)

# Klein bottle tube embedding in 4-space
x <- sample_klein_tube(120, sd = .05)
pairs(x, asp = 1, pch = 19, cex = .5)

# Klein bottle flat torus-based embedding in 4-space
x <- sample_klein_flat(120, sd = .05)
pairs(x, asp = 1, pch = 19, cex = .5)

Sample (with noise) from lemniscates (figure eights)

Description

These functions generate uniform samples from lemniscates (figure eights) in 2-dimensional space, optionally with noise.

Usage

sample_lemniscate_gerono(n, sd = 0)

Arguments

Details

These functions use a simple parameterization from the unit circle to the lemniscate of Gerono, as presented on Wikipedia. The uniform sample is generated through a rejection sampling process as described by Diaconis, Holmes, and Shahshahani (2013).

References

P Diaconis, S Holmes, and M Shahshahani (2013) Sampling from a Manifold. Advances in Modern Statistical Theory and Applications: A Festschrift in honor of Morris L. Eaton, 102–125. doi:10.1214/12-IMSCOLL1006

Examples

set.seed(12051L)

# Uniformly sampled figure eight in 2-space
x <- sample_lemniscate_gerono(720)
plot(x, asp = 1, pch = 19, cex = .5)

# Naively sampled figure eight, for comparison
theta <- runif(n = 720, min = 0, max = 2*pi)
x_naive <- cbind(x = cos(theta), y <- cos(theta) * sin(theta))
plot(x_naive, asp = 1, pch = 19, cex = .5)

# Uniformly sampled figure eight in 2-space with Gaussian noise
x <- sample_lemniscate_gerono(720, 0.1)
plot(x, asp = 1, pch = 19, cex = .5)

Sample (with noise) from Möbius strips

Description

These functions generate uniform samples from Möbius strips in 3-dimensional space, optionally with noise.

Usage

sample_mobius_rotoid(n, ar = 2, sd = 0)

Arguments

Details

The function sample_mobius_rotoid() uses the rotoid Möbius surface parameterization obtained from the now-defunct Encyclopédie des Formes Mathématiques Remarquables. Uniform samples are generated through a rejection sampling process as described by Diaconis, Holmes, and Shahshahani (2013). The Jacobian determinant was symbolically computed and simplified using Mathematica v13.3.1.0.

References

P Diaconis, S Holmes, and M Shahshahani (2013) Sampling from a Manifold. Advances in Modern Statistical Theory and Applications: A Festschrift in honor of Morris L. Eaton, 102–125. doi:10.1214/12-IMSCOLL1006

Examples

set.seed(5898L)

# Möbius strip embedding in 3-space
x <- sample_mobius_rotoid(180, ar = 1.5, sd = 0.02)
pairs(x, asp = 1, pch = 19, cex = .5)

# color-code points by coordinates
x_ran <- apply(x, 2L, range)
x_col <- rgb((x[, 1L] - x_ran[1L, 1L]) / (x_ran[2L, 1L] - x_ran[1L, 1L]),
             (x[, 2L] - x_ran[1L, 2L]) / (x_ran[2L, 2L] - x_ran[1L, 2L]),
             (x[, 3L] - x_ran[1L, 3L]) / (x_ran[2L, 3L] - x_ran[1L, 3L]))
pairs(x, asp = 1, pch = 19, cex = .75, col = x_col)

# shape-code by octant
x_pch <- apply(x > 0, 1L, \(l) sum(2L ^ seq(0L, 2L) * l))
pairs(x, asp = 1, cex = 1, pch = x_pch)

Add noise to a sample

Description

This function adds Gaussian noise to coordinate data contained in a matrix. It is called by samplers to introduce noise when sd is passed a positive value.

Usage

add_noise(x, sd = 0)

Arguments

Sample (with noise) from real projective planes

Description

These functions generate uniform samples from real projective planes in 4-dimensional space, optionally with noise.

Usage

sample_projective_plane(n, sd = 0)

Arguments

Details

The real projective plane only embeds into a Euclidean space of dimension at least 4. This embedding is adapted from Wikipedia. The uniform sample is generated through a rejection sampling process as described by Diaconis, Holmes, and Shahshahani (2013).

References

P Diaconis, S Holmes, and M Shahshahani (2013) Sampling from a Manifold. Advances in Modern Statistical Theory and Applications: A Festschrift in honor of Morris L. Eaton, 102–125. doi:10.1214/12-IMSCOLL1006

Examples

set.seed(22764L)

# real projective plane embedding in 4-space
x <- sample_projective_plane(120)
pairs(x, asp = 1, pch = 19, cex = .5)

Custom uniform rejection samplers

Description

These functions create rejection samplers, and uniform manifold samplers based on them, using user-provided parameterization and Jacobian functions.

Usage

make_rejection_sampler(
  parameterization,
  jacobian,
  min_params,
  max_params,
  max_jacobian
)

Arguments

Details

The rejection sampling technique of Diaconis, Holmes, and Shahshahani (2013) uses a parameterized embedding from a parameter space to a coordinate space and relies on a formula for its jacobian determinant. The parameterization must be a function that takes vector arguments of equal length and returns a coordinate matrix of the same number of rows. The jacobian must be a function that takes the same arguments and returns a scalar value. The parameters must range from their respective minima min_params to their respective maxima max_params. max_jacobian must be provided, though it may be larger than the theoretical maximum of the jacobian determinant.

References

P Diaconis, S Holmes, and M Shahshahani (2013) Sampling from a Manifold. Advances in Modern Statistical Theory and Applications: A Festschrift in honor of Morris L. Eaton, 102–125. doi:10.1214/12-IMSCOLL1006

Examples

set.seed(47569L)

# parameterization and Jacobian for Klein bottle tube embedding
klein_parameterization <- function(theta, phi) {
  cbind(
    w = (1 + .5 * cos(theta)) * cos(phi),
    x = (1 + .5 * cos(theta)) * sin(phi),
    y = .5 * sin(theta) * cos(phi/2),
    z = .5 * sin(theta) * sin(phi/2)
  )
}
klein_jacobian <- function(theta, phi) {
  unname(.5 * sqrt((1 + .5 * cos(theta)) ^ 2 + (.5 * .5 * sin(theta)) ^ 2))
}
# custom sampler based on these functions
klein_sampler <- make_rejection_sampler(
  klein_parameterization,
  klein_jacobian,
  max_params = c(theta = 2*pi, phi = 2*pi),
  max_jacobian = klein_jacobian(cbind(theta = 0))
)
# compare custom sampler to `sample_klein_tube()`
pairs(klein_sampler(n = 360), asp = 1, pch = 19, cex = .5)
pairs(sample_klein_tube(n = 360, ar = 2), asp = 1, pch = 19, cex = .5)

Sample (with noise) from a sphere

Description

These functions generate uniform samples from a sphere of radius 1 and dimension 2 in 3-space, or in arbitrary dimension in 1-higher-dimensional space, optionally with noise.

Usage

sample_2hemisphere(n, bins = 1L, sd = 0)

sample_2sphere(n, bins = 1L, sd = 0)

sample_sphere(n, dim = 1, sd = 0)

Arguments

Details

The function sample_sphere() is adapted from sphereUnif() in the TDA package. It uses stats::rnorm() to sample from a multivariate Gaussian and normalizes the resulting coordinates.

The function sample_2hemisphere() uses an area-preserving parameterization of the upper hemisphere, and sample_2sphere() uses two of these samples, one reflected over the horizontal plane, to produce a sample from a sphere. The parameterization was derived through the method for sampling 2-manifolds as described by Arvo (2001).

References

J Arvo (2001) Stratified Sampling of 2-Manifolds. SIGRAPH 2001 (State of the Art in Monte Carlo Ray Tracing for Realistic Image Synthesis), Course Notes, Vol. 29. https://www.cs.princeton.edu/courses/archive/fall04/cos526/papers/course29sig01.pdf

Examples

set.seed(50253L)

# 1-sphere in 2-space
x <- sample_sphere(120, dim = 1, sd = .1)
pairs(x, asp = 1, pch = 19, cex = .5)

# 2-sphere in 3-space
x <- sample_sphere(120, dim = 2, sd = .1)
pairs(x, asp = 1, pch = 19, cex = .5)

# 3-sphere in 4-space
x <- sample_sphere(120, dim = 3, sd = .1)
pairs(x, asp = 1, pch = 19, cex = .5)

# 4-sphere in 5-space
x <- sample_sphere(120, dim = 4, sd = .1)
pairs(x, asp = 1, pch = 19, cex = .5)

Stratified sample of any unit dimensional space

Description

These functions generate stratified samples of any dimension including the unit line segment in 1-dimensional space, the unit square in 2-space, the unit cube in 3-space.

Usage

sample_strat_segment(n, bins)

sample_strat_square(n, bins)

sample_strat_cube(n, bins)

sample_stratify(n, bins, dim)

Arguments

Details

(Details.)

Examples

set.seed(28522L)

#Stratified sample in 1-dimension with 10 intervals
values <- sample_strat_segment(13, 10)
x <- cbind(values, rep(0, 13))
plot(x, asp = 1, pch = 19, cex = .5, xlab = 'x', ylab = '')
segments(x0 = seq(0, 1, .1), y0 = -1, y1 = 1)

#Stratified sample of a unit square with 100 cells
x <- sample_strat_square(110, 10)
plot(x, asp = 1, pch = 19, cex = .5, xlab = 'x', ylab = 'y')
segments(x0 = seq(0, 1, .1), y0 = 0, y1 = 1)
segments(y0 = seq(0, 1, .1), x0 = 0, x1 = 1)

#Stratified sample of a unit cube with 27 cells
x <- sample_strat_cube(27, 3)
#Bird's eye view of the cube
plot(x[, c(1, 2)], asp = 1, pch = 19, cex = .5, xlab = 'x', ylab = 'y')
segments(x0 = seq(0, 1, 1/3), y0 = 0, y1 = 1)
segments(y0 = seq(0, 1, 1/3), x0 = 0, x1 = 1)
#Side view of the cube
plot(x[,c(2,3)], asp = 1, pch = 19, cex = .5, xlab = 'y', ylab = 'z')
segments(x0 = seq(0,1,1/3),y0 = 0, y1 = 1)
segments(y0 = seq(0,1,1/3),x0 = 0, x1 = 1)

#All of the same illustrations, but only using sample_stratify()

#Stratified sample in 1-dimension with 10 intervals
values <- sample_stratify(13,10,1)
x <- cbind(values,rep(0,13))
plot(x, asp = 1, pch = 19, cex = .5, xlab = 'x', ylab = '')
segments(x0 = seq(0,1,.1),y0 = -1, y1 = 1)

#Stratified sample of a unit square with 100 cells
x <- sample_stratify(110,10, 2)
plot(x, asp = 1, pch = 19, cex = .5, xlab = 'x', ylab = 'y')
segments(x0 = seq(0,1,.1),y0 = 0, y1 = 1)
segments(y0 = seq(0,1,.1),x0 = 0, x1 = 1)

#Stratified sample of a unit cube with 27 cells
x <- sample_stratify(27,3, 3)
#Bird's eye view of the cube
plot(x[,c(1,2)], asp = 1, pch = 19, cex = .5, xlab = 'x', ylab = 'y')
segments(x0 = seq(0,1,1/3),y0 = 0, y1 = 1)
segments(y0 = seq(0,1,1/3),x0 = 0, x1 = 1)
#Side view of the cube
plot(x[,c(2,3)], asp = 1, pch = 19, cex = .5, xlab = 'y', ylab = 'z')
segments(x0 = seq(0,1,1/3),y0 = 0, y1 = 1)
segments(y0 = seq(0,1,1/3),x0 = 0, x1 = 1)

#Stratified sample of a unit 4-cube with 81 cells
x <- sample_stratify(81, 3, 4)
#One view of the cube
plot(x[,c(1,2)], asp = 1, pch = 19, cex = .5, xlab = 'x', ylab = 'y')
segments(x0 = seq(0,1,1/3),y0 = 0, y1 = 1)
segments(y0 = seq(0,1,1/3),x0 = 0, x1 = 1)
#Another view of the cube
plot(x[,c(2,3)], asp = 1, pch = 19, cex = .5, xlab = 'y', ylab = 'z')
segments(x0 = seq(0,1,1/3),y0 = 0, y1 = 1)
segments(y0 = seq(0,1,1/3),x0 = 0, x1 = 1)

Sample (with noise) from tori

Description

These functions generate uniform samples from configurations of tori of primary radius 1 in 3-dimensional space, optionally with noise.

Usage

sample_torus_tube(n, ar = 2, sd = 0)

sample_tori_interlocked(n, ar = 2, sd = 0)

sample_torus_flat(n, ar = 1, sd = 0)

Arguments

Details

The function sample_torus_tube() uses the tubular parameterization into 3-dimensional space documented at MathWorld.

The function sample_torus_flat() uses a flat parameterization (having zero Gaussian curvature) into 4-dimensional space, as presented on Wikipedia.

The function sample_tori_interlocked() samples from two tubular tori interlocked in the same way as sample_circles_interlocked().

All uniform samples are generated through a rejection sampling process as described by Diaconis, Holmes, and Shahshahani (2013).

References

P Diaconis, S Holmes, and M Shahshahani (2013) Sampling from a Manifold. Advances in Modern Statistical Theory and Applications: A Festschrift in honor of Morris L. Eaton, 102–125. doi:10.1214/12-IMSCOLL1006

Examples

set.seed(33183L)

# torus tube embedding in 3-space
x <- sample_torus_tube(120, sd = .05)
pairs(x, asp = 1, pch = 19, cex = .5)

# torus flat embedding in 4-space
x <- sample_torus_flat(120, sd = .05)
pairs(x, asp = 1, pch = 19, cex = .5)

# interlocked tubular tori in 3-space
x <- sample_tori_interlocked(360, ar = 6, sd = .01)
pairs(x, asp = 1, pch = 19, cex = .5)

Sample (with noise) from trefoil knot

Description

These functions generate uniform samples from trefoil knot in 3-dimensional space, optionally with noise.

Usage

sample_trefoil(n, sd = 0)

Arguments

Details

The trefoil knot is the simplest nontrivial knot and contains three unique crossings in three dimensional space. This uniform sample is generated by rejection sampling. This process allows for simulation of random samples from the trefoil knot distribution, by using random samples from a more convenient distribution. It applies rejection/acceptance criterion such that the samples that are accepted are to be distributed as if they were from the target distribution. The uniform sample is generated through a rejection sampling process as described by Diaconis, Holmes, and Shahshahani (2013).

References

P Diaconis, S Holmes, and M Shahshahani (2013) Sampling from a Manifold. Advances in Modern Statistical Theory and Applications: A Festschrift in honor of Morris L. Eaton, 102–125. doi:10.1214/12-IMSCOLL1006

Examples

set.seed(73398L)

# Uniformly sampled trefoil knot in 3-space
x <- sample_trefoil(180)
pairs(x, asp = 1, pch = 19, cex = .5, col = "#00000077")

# Uniformly sampled trefoil knot in 3-space, with Gaussian noise
x <- sample_trefoil(180, sd = .1)
pairs(x, asp = 1, pch = 19, cex = .5, col = "#00000077")

Sample (with noise) from planar triangles

Description

This function generates uniform and stratified samples from configurations of planar triangles in 2-dimensional space, optionally with noise.

Usage

sample_triangle_planar(n, triangle, bins = 1, sd = 0)

Arguments

Details

The sample is generated by an area-preserving parameterization of the planar triangle. This parameterization was derived through the method for sampling 2-manifolds as described by Arvo (2001).

References

J Arvo (2001) Stratified Sampling of 2-Manifolds. SIGRAPH 2001 (State of the Art in Monte Carlo Ray Tracing for Realistic Image Synthesis), Course Notes, Vol. 29. https://www.cs.princeton.edu/courses/archive/fall04/cos526/papers/course29sig01.pdf

Examples

set.seed(23004L)

#Uniformly sampled equilateral planar triangle in 2-space
equilateral_triangle <- cbind(c(0,0), c(0.5,sqrt(3)/2), c(1,0))
x <-  sample_triangle_planar(10000, equilateral_triangle)
plot(x, asp = 1, pch = 19, cex = .25)

#Stratified sample of equilateral planar triangle in 2-space with 100 bins
equilateral_triangle <- cbind(c(0,0), c(0.5,sqrt(3)/2), c(1,0))
x <-  sample_triangle_planar(10000, equilateral_triangle, bins = 100)
plot(x, asp = 1, pch = 19, cex = .25)

#Uniformly sampled equilateral planar triangle in 2-space with Gaussian noise
equilateral_triangle <- cbind(c(0,0), c(0.5,sqrt(3)/2), c(1,0))
x <-  sample_triangle_planar(10000, equilateral_triangle, sd = .1)
plot(x, asp = 1, pch = 19, cex = .25)