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cauchy.rs
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cauchy.rs
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use crate::distribution::{Continuous, Univariate};
use crate::statistics::*;
use crate::{Result, StatsError};
use rand::distributions::Distribution;
use rand::Rng;
use std::f64;
/// Implements the [Cauchy](https://en.wikipedia.org/wiki/Cauchy_distribution)
/// distribution, also known as the Lorentz distribution.
///
/// # Examples
///
/// ```
/// use statrs::distribution::{Cauchy, Continuous};
/// use statrs::statistics::Mode;
///
/// let n = Cauchy::new(0.0, 1.0).unwrap();
/// assert_eq!(n.mode(), 0.0);
/// assert_eq!(n.pdf(1.0), 0.1591549430918953357689);
/// ```
#[derive(Debug, Copy, Clone, PartialEq)]
pub struct Cauchy {
location: f64,
scale: f64,
}
impl Cauchy {
/// Constructs a new cauchy distribution with the given
/// location and scale.
///
/// # Errors
///
/// Returns an error if location or scale are `NaN` or `scale <= 0.0`
///
/// # Examples
///
/// ```
/// use statrs::distribution::Cauchy;
///
/// let mut result = Cauchy::new(0.0, 1.0);
/// assert!(result.is_ok());
///
/// result = Cauchy::new(0.0, -1.0);
/// assert!(result.is_err());
/// ```
pub fn new(location: f64, scale: f64) -> Result<Cauchy> {
if location.is_nan() || scale.is_nan() || scale <= 0.0 {
Err(StatsError::BadParams)
} else {
Ok(Cauchy { location, scale })
}
}
/// Returns the location of the cauchy distribution
///
/// # Examples
///
/// ```
/// use statrs::distribution::Cauchy;
///
/// let n = Cauchy::new(0.0, 1.0).unwrap();
/// assert_eq!(n.location(), 0.0);
/// ```
pub fn location(&self) -> f64 {
self.location
}
/// Returns the scale of the cauchy distribution
///
/// # Examples
///
/// ```
/// use statrs::distribution::Cauchy;
///
/// let n = Cauchy::new(0.0, 1.0).unwrap();
/// assert_eq!(n.scale(), 1.0);
/// ```
pub fn scale(&self) -> f64 {
self.scale
}
}
impl Distribution<f64> for Cauchy {
fn sample<R: Rng + ?Sized>(&self, r: &mut R) -> f64 {
self.location + self.scale * (f64::consts::PI * (r.gen::<f64>() - 0.5)).tan()
}
}
impl Univariate<f64, f64> for Cauchy {
/// Calculates the cumulative distribution function for the
/// cauchy distribution at `x`
///
/// # Formula
///
/// ```ignore
/// (1 / π) * arctan((x - x_0) / γ) + 0.5
/// ```
///
/// where `x_0` is the location and `γ` is the scale
fn cdf(&self, x: f64) -> f64 {
(1.0 / f64::consts::PI) * ((x - self.location) / self.scale).atan() + 0.5
}
}
impl Min<f64> for Cauchy {
/// Returns the minimum value in the domain of the cauchy
/// distribution representable by a double precision float
///
/// # Formula
///
/// ```ignore
/// NEG_INF
/// ```
fn min(&self) -> f64 {
f64::NEG_INFINITY
}
}
impl Max<f64> for Cauchy {
/// Returns the maximum value in the domain of the cauchy
/// distribution representable by a double precision float
///
/// # Formula
///
/// ```ignore
/// INF
/// ```
fn max(&self) -> f64 {
f64::INFINITY
}
}
impl Entropy<f64> for Cauchy {
/// Returns the entropy of the cauchy distribution
///
/// # Formula
///
/// ```ignore
/// ln(γ) + ln(4π)
/// ```
///
/// where `γ` is the scale
fn entropy(&self) -> f64 {
(4.0 * f64::consts::PI * self.scale).ln()
}
}
impl Median<f64> for Cauchy {
/// Returns the median of the cauchy distribution
///
/// # Formula
///
/// ```ignore
/// x_0
/// ```
///
/// where `x_0` is the location
fn median(&self) -> f64 {
self.location
}
}
impl Mode<f64> for Cauchy {
/// Returns the mode of the cauchy distribution
///
/// # Formula
///
/// ```ignore
/// x_0
/// ```
///
/// where `x_0` is the location
fn mode(&self) -> f64 {
self.location
}
}
impl Continuous<f64, f64> for Cauchy {
/// Calculates the probability density function for the cauchy
/// distribution at `x`
///
/// # Formula
///
/// ```ignore
/// 1 / (πγ * (1 + ((x - x_0) / γ)^2))
/// ```
///
/// where `x_0` is the location and `γ` is the scale
fn pdf(&self, x: f64) -> f64 {
1.0 / (f64::consts::PI
* self.scale
* (1.0 + ((x - self.location) / self.scale) * ((x - self.location) / self.scale)))
}
/// Calculates the log probability density function for the cauchy
/// distribution at `x`
///
/// # Formula
///
/// ```ignore
/// ln(1 / (πγ * (1 + ((x - x_0) / γ)^2)))
/// ```
///
/// where `x_0` is the location and `γ` is the scale
fn ln_pdf(&self, x: f64) -> f64 {
-(f64::consts::PI
* self.scale
* (1.0 + ((x - self.location) / self.scale) * ((x - self.location) / self.scale)))
.ln()
}
}
#[cfg_attr(rustfmt, rustfmt_skip)]
#[cfg(test)]
mod test {
use std::f64;
use crate::statistics::*;
use crate::distribution::{Univariate, Continuous, Cauchy};
use crate::distribution::internal::*;
fn try_create(location: f64, scale: f64) -> Cauchy {
let n = Cauchy::new(location, scale);
assert!(n.is_ok());
n.unwrap()
}
fn create_case(location: f64, scale: f64) {
let n = try_create(location, scale);
assert_eq!(location, n.location());
assert_eq!(scale, n.scale());
}
fn bad_create_case(location: f64, scale: f64) {
let n = Cauchy::new(location, scale);
assert!(n.is_err());
}
fn test_case<F>(location: f64, scale: f64, expected: f64, eval: F)
where F: Fn(Cauchy) -> f64
{
let n = try_create(location, scale);
let x = eval(n);
assert_eq!(expected, x);
}
fn test_almost<F>(location: f64, scale: f64, expected: f64, acc: f64, eval: F)
where F: Fn(Cauchy) -> f64
{
let n = try_create(location, scale);
let x = eval(n);
assert_almost_eq!(expected, x, acc);
}
#[test]
fn test_create() {
create_case(0.0, 0.1);
create_case(0.0, 1.0);
create_case(0.0, 10.0);
create_case(10.0, 11.0);
create_case(-5.0, 100.0);
create_case(0.0, f64::INFINITY);
}
#[test]
fn test_bad_create() {
bad_create_case(f64::NAN, 1.0);
bad_create_case(1.0, f64::NAN);
bad_create_case(f64::NAN, f64::NAN);
bad_create_case(1.0, 0.0);
}
#[test]
fn test_entropy() {
test_case(0.0, 2.0, 3.224171427529236102395, |x| x.entropy());
test_case(0.1, 4.0, 3.917318608089181411812, |x| x.entropy());
test_case(1.0, 10.0, 4.833609339963336476996, |x| x.entropy());
test_case(10.0, 11.0, 4.92891951976766133704, |x| x.entropy());
}
#[test]
fn test_mode() {
test_case(0.0, 2.0, 0.0, |x| x.mode());
test_case(0.1, 4.0, 0.1, |x| x.mode());
test_case(1.0, 10.0, 1.0, |x| x.mode());
test_case(10.0, 11.0, 10.0, |x| x.mode());
test_case(0.0, f64::INFINITY, 0.0, |x| x.mode());
}
#[test]
fn test_median() {
test_case(0.0, 2.0, 0.0, |x| x.median());
test_case(0.1, 4.0, 0.1, |x| x.median());
test_case(1.0, 10.0, 1.0, |x| x.median());
test_case(10.0, 11.0, 10.0, |x| x.median());
test_case(0.0, f64::INFINITY, 0.0, |x| x.median());
}
#[test]
fn test_min_max() {
test_case(0.0, 1.0, f64::NEG_INFINITY, |x| x.min());
test_case(0.0, 1.0, f64::INFINITY, |x| x.max());
}
#[test]
fn test_pdf() {
test_case(0.0, 0.1, 0.001272730452554141029739, |x| x.pdf(-5.0));
test_case(0.0, 0.1, 0.03151583031522679916216, |x| x.pdf(-1.0));
test_almost(0.0, 0.1, 3.183098861837906715378, 1e-14, |x| x.pdf(0.0));
test_case(0.0, 0.1, 0.03151583031522679916216, |x| x.pdf(1.0));
test_case(0.0, 0.1, 0.001272730452554141029739, |x| x.pdf(5.0));
test_almost(0.0, 1.0, 0.01224268793014579505914, 1e-17, |x| x.pdf(-5.0));
test_case(0.0, 1.0, 0.1591549430918953357689, |x| x.pdf(-1.0));
test_case(0.0, 1.0, 0.3183098861837906715378, |x| x.pdf(0.0));
test_case(0.0, 1.0, 0.1591549430918953357689, |x| x.pdf(1.0));
test_almost(0.0, 1.0, 0.01224268793014579505914, 1e-17, |x| x.pdf(5.0));
test_case(0.0, 10.0, 0.02546479089470325372302, |x| x.pdf(-5.0));
test_case(0.0, 10.0, 0.03151583031522679916216, |x| x.pdf(-1.0));
test_case(0.0, 10.0, 0.03183098861837906715378, |x| x.pdf(0.0));
test_case(0.0, 10.0, 0.03151583031522679916216, |x| x.pdf(1.0));
test_case(0.0, 10.0, 0.02546479089470325372302, |x| x.pdf(5.0));
test_case(-5.0, 100.0, 0.003183098861837906715378, |x| x.pdf(-5.0));
test_almost(-5.0, 100.0, 0.003178014039374906864395, 1e-17, |x| x.pdf(-1.0));
test_case(-5.0, 100.0, 0.003175160959439308444267, |x| x.pdf(0.0));
test_case(-5.0, 100.0, 0.003171680810918599756255, |x| x.pdf(1.0));
test_almost(-5.0, 100.0, 0.003151583031522679916216, 1e-17, |x| x.pdf(5.0));
test_case(0.0, f64::INFINITY, 0.0, |x| x.pdf(-5.0));
test_case(0.0, f64::INFINITY, 0.0, |x| x.pdf(-1.0));
test_case(0.0, f64::INFINITY, 0.0, |x| x.pdf(0.0));
test_case(0.0, f64::INFINITY, 0.0, |x| x.pdf(1.0));
test_case(0.0, f64::INFINITY, 0.0, |x| x.pdf(5.0));
test_case(f64::INFINITY, 1.0, 0.0, |x| x.pdf(-5.0));
test_case(f64::INFINITY, 1.0, 0.0, |x| x.pdf(-1.0));
test_case(f64::INFINITY, 1.0, 0.0, |x| x.pdf(0.0));
test_case(f64::INFINITY, 1.0, 0.0, |x| x.pdf(1.0));
test_case(f64::INFINITY, 1.0, 0.0, |x| x.pdf(5.0));
}
#[test]
fn test_ln_pdf() {
test_case(0.0, 0.1, -6.666590723732973542744, |x| x.ln_pdf(-5.0));
test_almost(0.0, 0.1, -3.457265309696613941009, 1e-14, |x| x.ln_pdf(-1.0));
test_case(0.0, 0.1, 1.157855207144645509875, |x| x.ln_pdf(0.0));
test_almost(0.0, 0.1, -3.457265309696613941009, 1e-14, |x| x.ln_pdf(1.0));
test_case(0.0, 0.1, -6.666590723732973542744, |x| x.ln_pdf(5.0));
test_case(0.0, 1.0, -4.402826423870882219615, |x| x.ln_pdf(-5.0));
test_almost(0.0, 1.0, -1.837877066409345483561, 1e-15, |x| x.ln_pdf(-1.0));
test_case(0.0, 1.0, -1.144729885849400174143, |x| x.ln_pdf(0.0));
test_almost(0.0, 1.0, -1.837877066409345483561, 1e-15, |x| x.ln_pdf(1.0));
test_case(0.0, 1.0, -4.402826423870882219615, |x| x.ln_pdf(5.0));
test_case(0.0, 10.0, -3.670458530157655613928, |x| x.ln_pdf(-5.0));
test_almost(0.0, 10.0, -3.457265309696613941009, 1e-14, |x| x.ln_pdf(-1.0));
test_case(0.0, 10.0, -3.447314978843445858161, |x| x.ln_pdf(0.0));
test_almost(0.0, 10.0, -3.457265309696613941009, 1e-14, |x| x.ln_pdf(1.0));
test_case(0.0, 10.0, -3.670458530157655613928, |x| x.ln_pdf(5.0));
test_case(-5.0, 100.0, -5.749900071837491542179, |x| x.ln_pdf(-5.0));
test_case(-5.0, 100.0, -5.751498793201188569872, |x| x.ln_pdf(-1.0));
test_case(-5.0, 100.0, -5.75239695203607874116, |x| x.ln_pdf(0.0));
test_case(-5.0, 100.0, -5.75349360734762171285, |x| x.ln_pdf(1.0));
test_case(-5.0, 100.0, -5.759850402690659625027, |x| x.ln_pdf(5.0));
test_case(0.0, f64::INFINITY, f64::NEG_INFINITY, |x| x.ln_pdf(-5.0));
test_case(0.0, f64::INFINITY, f64::NEG_INFINITY, |x| x.ln_pdf(-1.0));
test_case(0.0, f64::INFINITY, f64::NEG_INFINITY, |x| x.ln_pdf(0.0));
test_case(0.0, f64::INFINITY, f64::NEG_INFINITY, |x| x.ln_pdf(1.0));
test_case(0.0, f64::INFINITY, f64::NEG_INFINITY, |x| x.ln_pdf(5.0));
test_case(f64::INFINITY, 1.0, f64::NEG_INFINITY, |x| x.ln_pdf(-5.0));
test_case(f64::INFINITY, 1.0, f64::NEG_INFINITY, |x| x.ln_pdf(-1.0));
test_case(f64::INFINITY, 1.0, f64::NEG_INFINITY, |x| x.ln_pdf(0.0));
test_case(f64::INFINITY, 1.0, f64::NEG_INFINITY, |x| x.ln_pdf(1.0));
test_case(f64::INFINITY, 1.0, f64::NEG_INFINITY, |x| x.ln_pdf(5.0));
}
#[test]
fn test_cdf() {
test_almost(0.0, 0.1, 0.006365349100972796679298, 1e-16, |x| x.cdf(-5.0));
test_almost(0.0, 0.1, 0.03172551743055356951498, 1e-16, |x| x.cdf(-1.0));
test_case(0.0, 0.1, 0.5, |x| x.cdf(0.0));
test_case(0.0, 0.1, 0.968274482569446430485, |x| x.cdf(1.0));
test_case(0.0, 0.1, 0.9936346508990272033207, |x| x.cdf(5.0));
test_almost(0.0, 1.0, 0.06283295818900118381375, 1e-16, |x| x.cdf(-5.0));
test_case(0.0, 1.0, 0.25, |x| x.cdf(-1.0));
test_case(0.0, 1.0, 0.5, |x| x.cdf(0.0));
test_case(0.0, 1.0, 0.75, |x| x.cdf(1.0));
test_case(0.0, 1.0, 0.9371670418109988161863, |x| x.cdf(5.0));
test_case(0.0, 10.0, 0.3524163823495667258246, |x| x.cdf(-5.0));
test_case(0.0, 10.0, 0.468274482569446430485, |x| x.cdf(-1.0));
test_case(0.0, 10.0, 0.5, |x| x.cdf(0.0));
test_case(0.0, 10.0, 0.531725517430553569515, |x| x.cdf(1.0));
test_case(0.0, 10.0, 0.6475836176504332741754, |x| x.cdf(5.0));
test_case(-5.0, 100.0, 0.5, |x| x.cdf(-5.0));
test_case(-5.0, 100.0, 0.5127256113479918307809, |x| x.cdf(-1.0));
test_case(-5.0, 100.0, 0.5159022512561763751816, |x| x.cdf(0.0));
test_case(-5.0, 100.0, 0.5190757242358362337495, |x| x.cdf(1.0));
test_case(-5.0, 100.0, 0.531725517430553569515, |x| x.cdf(5.0));
test_case(0.0, f64::INFINITY, 0.5, |x| x.cdf(-5.0));
test_case(0.0, f64::INFINITY, 0.5, |x| x.cdf(-1.0));
test_case(0.0, f64::INFINITY, 0.5, |x| x.cdf(0.0));
test_case(0.0, f64::INFINITY, 0.5, |x| x.cdf(1.0));
test_case(0.0, f64::INFINITY, 0.5, |x| x.cdf(5.0));
test_case(f64::INFINITY, 1.0, 0.0, |x| x.cdf(-5.0));
test_case(f64::INFINITY, 1.0, 0.0, |x| x.cdf(-1.0));
test_case(f64::INFINITY, 1.0, 0.0, |x| x.cdf(0.0));
test_case(f64::INFINITY, 1.0, 0.0, |x| x.cdf(1.0));
test_case(f64::INFINITY, 1.0, 0.0, |x| x.cdf(5.0));
}
#[test]
fn test_continuous() {
test::check_continuous_distribution(&try_create(-1.2, 3.4), -1500.0, 1500.0);
test::check_continuous_distribution(&try_create(-4.5, 6.7), -5000.0, 5000.0);
}
}