wrappedcauchy {cylcop}

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)

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.

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).

See Also

circular::dwrappedcauchy(), circular::rwrappedcauchy().

Examples

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)