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