# lemniscates: Sample (with noise) from lemniscates (figure eights) In tdaunif: Uniform Manifold Samplers for Topological Data Analysis

## Description

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

## Usage

 `1` ```sample_lemniscate_gerono(n, sd = 0) ```

## Arguments

 `n` Number of observations. `sd` Standard deviation of (independent multivariate) Gaussian noise.

## 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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```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) ```

tdaunif documentation built on Oct. 26, 2020, 5:08 p.m.