wrappedcauchy: Density, Distribution, Quantiles and Random Number Generation for the Wrapped Cauchy Distribution

Description

The distribution function (pwrappedcauchy()) and quantiles (qwrappedcauchy()) of the wrapped Cauchy distribution cannot be obtained analytically. They are therefore missing in the 'circular' package and are obtained here numerically. Random number generation (rwrappedcauchy()) and density (dwrappedcauchy()) don't need a numerical approximation and are provided here for consistency in parametrization with the other wrapped Cauchy functions.

Usage

rwrappedcauchy(n, location = 0, scale = 1)
dwrappedcauchy(theta, location = 0, scale = 1)
pwrappedcauchy(theta, location = 0, scale = 1, K = 100, check_prec = FALSE)
qwrappedcauchy(p, location = 0, scale = 1, K = 100, check_prec = FALSE)

Value

dwrappedcauchy() gives a vector of length length(theta) containing the density at theta.
pwrappedcauchy() gives a vector of length length(theta) containing the distribution function at the corresponding values of theta.
qwrappedcauchy() gives a vector of length length(p) containing the quantiles at the corresponding values of p.
rwrappedcauchy() generates a vector of length n containing the random samples, i.e. angles in $[-\pi, \pi)$.

Arguments

n: integer value, the number of random samples to be generated with rwrappedcauchy().
location: numeric value, the mean of the distribution.
scale: numeric value, the parameter tuning the spread of the density. It must be non-negative.
theta: numeric vector giving the angles where the density or distribution function is evaluated.
K: integer value, the number of "wraps" used in each direction to approximate the distribution.
check_prec: logical, whether to check if the precision of the numerical approximation with the current parameters is higher than 99%.
p: numeric vector giving the probabilities where the quantile function is evaluated.

Details

One could alternatively convert scale to rho via rho = exp(-scale) and use circular::rwrappedcauchy(theta, mu=location rho=rho) or circular::dwrappedcauchy(theta, mu=location rho=rho).

The wrapped Cauchy cdf, for which there is no analytical expression, is calculated by wrapping the Cauchy distribution $K$ times around the circle in each direction and summing the Cauchy cdfs at each point of the circle. Let $\Omega$ follow a Cauchy distribution and $\Theta$ a wrapped Cauchy distribution, where $\Theta$ can take values $\theta \in [-\pi,\pi)$. $Pr(\Theta \le \theta)$ is approximated as $$\sum^K_{k=-K}Pr(\Omega \le \theta+2\pi k)-Pr(\Omega \le -\pi+2\pi k).$$ The quantiles are calculated by numerical inversion.

Examples

Run this code

set.seed(123)

rwrappedcauchy(10, location = 0, scale =3)

dwrappedcauchy(c(0.1, pi), location = pi, scale =2)
circular::dwrappedcauchy(circular::circular(c(0.1,pi)), mu = circular::circular(pi), rho =exp(-2))

prob <- pwrappedcauchy(c(0.1, pi), location = pi, scale =2)
prob
qwrappedcauchy(prob, location = pi, scale =2)