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// Copyright 2018 Developers of the Rand project.
// Copyright 2013 The Rust Project Developers.
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// option. This file may not be copied, modified, or distributed
// except according to those terms.
//! The Gamma and derived distributions.
// We use the variable names from the published reference, therefore this
// warning is not helpful.
#![allow(clippy::many_single_char_names)]
use self::ChiSquaredRepr::*;
use self::GammaRepr::*;
use crate::normal::StandardNormal;
use num_traits::Float;
use crate::{Distribution, Exp, Exp1, Open01};
use rand::Rng;
use core::fmt;
#[cfg(feature = "serde1")]
use serde::{Serialize, Deserialize};
/// The Gamma distribution `Gamma(shape, scale)` distribution.
///
/// The density function of this distribution is
///
/// ```text
/// f(x) = x^(k - 1) * exp(-x / θ) / (Γ(k) * θ^k)
/// ```
///
/// where `Γ` is the Gamma function, `k` is the shape and `θ` is the
/// scale and both `k` and `θ` are strictly positive.
///
/// The algorithm used is that described by Marsaglia & Tsang 2000[^1],
/// falling back to directly sampling from an Exponential for `shape
/// == 1`, and using the boosting technique described in that paper for
/// `shape < 1`.
///
/// # Example
///
/// ```
/// use rand_distr::{Distribution, Gamma};
///
/// let gamma = Gamma::new(2.0, 5.0).unwrap();
/// let v = gamma.sample(&mut rand::thread_rng());
/// println!("{} is from a Gamma(2, 5) distribution", v);
/// ```
///
/// [^1]: George Marsaglia and Wai Wan Tsang. 2000. "A Simple Method for
/// Generating Gamma Variables" *ACM Trans. Math. Softw.* 26, 3
/// (September 2000), 363-372.
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
pub struct Gamma<F>
where
F: Float,
StandardNormal: Distribution<F>,
Exp1: Distribution<F>,
Open01: Distribution<F>,
{
repr: GammaRepr<F>,
}
/// Error type returned from `Gamma::new`.
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub enum Error {
/// `shape <= 0` or `nan`.
ShapeTooSmall,
/// `scale <= 0` or `nan`.
ScaleTooSmall,
/// `1 / scale == 0`.
ScaleTooLarge,
}
impl fmt::Display for Error {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
f.write_str(match self {
Error::ShapeTooSmall => "shape is not positive in gamma distribution",
Error::ScaleTooSmall => "scale is not positive in gamma distribution",
Error::ScaleTooLarge => "scale is infinity in gamma distribution",
})
}
}
#[cfg(feature = "std")]
#[cfg_attr(doc_cfg, doc(cfg(feature = "std")))]
impl std::error::Error for Error {}
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
enum GammaRepr<F>
where
F: Float,
StandardNormal: Distribution<F>,
Exp1: Distribution<F>,
Open01: Distribution<F>,
{
Large(GammaLargeShape<F>),
One(Exp<F>),
Small(GammaSmallShape<F>),
}
// These two helpers could be made public, but saving the
// match-on-Gamma-enum branch from using them directly (e.g. if one
// knows that the shape is always > 1) doesn't appear to be much
// faster.
/// Gamma distribution where the shape parameter is less than 1.
///
/// Note, samples from this require a compulsory floating-point `pow`
/// call, which makes it significantly slower than sampling from a
/// gamma distribution where the shape parameter is greater than or
/// equal to 1.
///
/// See `Gamma` for sampling from a Gamma distribution with general
/// shape parameters.
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
struct GammaSmallShape<F>
where
F: Float,
StandardNormal: Distribution<F>,
Open01: Distribution<F>,
{
inv_shape: F,
large_shape: GammaLargeShape<F>,
}
/// Gamma distribution where the shape parameter is larger than 1.
///
/// See `Gamma` for sampling from a Gamma distribution with general
/// shape parameters.
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
struct GammaLargeShape<F>
where
F: Float,
StandardNormal: Distribution<F>,
Open01: Distribution<F>,
{
scale: F,
c: F,
d: F,
}
impl<F> Gamma<F>
where
F: Float,
StandardNormal: Distribution<F>,
Exp1: Distribution<F>,
Open01: Distribution<F>,
{
/// Construct an object representing the `Gamma(shape, scale)`
/// distribution.
#[inline]
pub fn new(shape: F, scale: F) -> Result<Gamma<F>, Error> {
if !(shape > F::zero()) {
return Err(Error::ShapeTooSmall);
}
if !(scale > F::zero()) {
return Err(Error::ScaleTooSmall);
}
let repr = if shape == F::one() {
One(Exp::new(F::one() / scale).map_err(|_| Error::ScaleTooLarge)?)
} else if shape < F::one() {
Small(GammaSmallShape::new_raw(shape, scale))
} else {
Large(GammaLargeShape::new_raw(shape, scale))
};
Ok(Gamma { repr })
}
}
impl<F> GammaSmallShape<F>
where
F: Float,
StandardNormal: Distribution<F>,
Open01: Distribution<F>,
{
fn new_raw(shape: F, scale: F) -> GammaSmallShape<F> {
GammaSmallShape {
inv_shape: F::one() / shape,
large_shape: GammaLargeShape::new_raw(shape + F::one(), scale),
}
}
}
impl<F> GammaLargeShape<F>
where
F: Float,
StandardNormal: Distribution<F>,
Open01: Distribution<F>,
{
fn new_raw(shape: F, scale: F) -> GammaLargeShape<F> {
let d = shape - F::from(1. / 3.).unwrap();
GammaLargeShape {
scale,
c: F::one() / (F::from(9.).unwrap() * d).sqrt(),
d,
}
}
}
impl<F> Distribution<F> for Gamma<F>
where
F: Float,
StandardNormal: Distribution<F>,
Exp1: Distribution<F>,
Open01: Distribution<F>,
{
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> F {
match self.repr {
Small(ref g) => g.sample(rng),
One(ref g) => g.sample(rng),
Large(ref g) => g.sample(rng),
}
}
}
impl<F> Distribution<F> for GammaSmallShape<F>
where
F: Float,
StandardNormal: Distribution<F>,
Open01: Distribution<F>,
{
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> F {
let u: F = rng.sample(Open01);
self.large_shape.sample(rng) * u.powf(self.inv_shape)
}
}
impl<F> Distribution<F> for GammaLargeShape<F>
where
F: Float,
StandardNormal: Distribution<F>,
Open01: Distribution<F>,
{
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> F {
// Marsaglia & Tsang method, 2000
loop {
let x: F = rng.sample(StandardNormal);
let v_cbrt = F::one() + self.c * x;
if v_cbrt <= F::zero() {
// a^3 <= 0 iff a <= 0
continue;
}
let v = v_cbrt * v_cbrt * v_cbrt;
let u: F = rng.sample(Open01);
let x_sqr = x * x;
if u < F::one() - F::from(0.0331).unwrap() * x_sqr * x_sqr
|| u.ln() < F::from(0.5).unwrap() * x_sqr + self.d * (F::one() - v + v.ln())
{
return self.d * v * self.scale;
}
}
}
}
/// The chi-squared distribution `χ²(k)`, where `k` is the degrees of
/// freedom.
///
/// For `k > 0` integral, this distribution is the sum of the squares
/// of `k` independent standard normal random variables. For other
/// `k`, this uses the equivalent characterisation
/// `χ²(k) = Gamma(k/2, 2)`.
///
/// # Example
///
/// ```
/// use rand_distr::{ChiSquared, Distribution};
///
/// let chi = ChiSquared::new(11.0).unwrap();
/// let v = chi.sample(&mut rand::thread_rng());
/// println!("{} is from a χ²(11) distribution", v)
/// ```
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
pub struct ChiSquared<F>
where
F: Float,
StandardNormal: Distribution<F>,
Exp1: Distribution<F>,
Open01: Distribution<F>,
{
repr: ChiSquaredRepr<F>,
}
/// Error type returned from `ChiSquared::new` and `StudentT::new`.
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
pub enum ChiSquaredError {
/// `0.5 * k <= 0` or `nan`.
DoFTooSmall,
}
impl fmt::Display for ChiSquaredError {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
f.write_str(match self {
ChiSquaredError::DoFTooSmall => {
"degrees-of-freedom k is not positive in chi-squared distribution"
}
})
}
}
#[cfg(feature = "std")]
#[cfg_attr(doc_cfg, doc(cfg(feature = "std")))]
impl std::error::Error for ChiSquaredError {}
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
enum ChiSquaredRepr<F>
where
F: Float,
StandardNormal: Distribution<F>,
Exp1: Distribution<F>,
Open01: Distribution<F>,
{
// k == 1, Gamma(alpha, ..) is particularly slow for alpha < 1,
// e.g. when alpha = 1/2 as it would be for this case, so special-
// casing and using the definition of N(0,1)^2 is faster.
DoFExactlyOne,
DoFAnythingElse(Gamma<F>),
}
impl<F> ChiSquared<F>
where
F: Float,
StandardNormal: Distribution<F>,
Exp1: Distribution<F>,
Open01: Distribution<F>,
{
/// Create a new chi-squared distribution with degrees-of-freedom
/// `k`.
pub fn new(k: F) -> Result<ChiSquared<F>, ChiSquaredError> {
let repr = if k == F::one() {
DoFExactlyOne
} else {
if !(F::from(0.5).unwrap() * k > F::zero()) {
return Err(ChiSquaredError::DoFTooSmall);
}
DoFAnythingElse(Gamma::new(F::from(0.5).unwrap() * k, F::from(2.0).unwrap()).unwrap())
};
Ok(ChiSquared { repr })
}
}
impl<F> Distribution<F> for ChiSquared<F>
where
F: Float,
StandardNormal: Distribution<F>,
Exp1: Distribution<F>,
Open01: Distribution<F>,
{
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> F {
match self.repr {
DoFExactlyOne => {
// k == 1 => N(0,1)^2
let norm: F = rng.sample(StandardNormal);
norm * norm
}
DoFAnythingElse(ref g) => g.sample(rng),
}
}
}
/// The Fisher F distribution `F(m, n)`.
///
/// This distribution is equivalent to the ratio of two normalised
/// chi-squared distributions, that is, `F(m,n) = (χ²(m)/m) /
/// (χ²(n)/n)`.
///
/// # Example
///
/// ```
/// use rand_distr::{FisherF, Distribution};
///
/// let f = FisherF::new(2.0, 32.0).unwrap();
/// let v = f.sample(&mut rand::thread_rng());
/// println!("{} is from an F(2, 32) distribution", v)
/// ```
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
pub struct FisherF<F>
where
F: Float,
StandardNormal: Distribution<F>,
Exp1: Distribution<F>,
Open01: Distribution<F>,
{
numer: ChiSquared<F>,
denom: ChiSquared<F>,
// denom_dof / numer_dof so that this can just be a straight
// multiplication, rather than a division.
dof_ratio: F,
}
/// Error type returned from `FisherF::new`.
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
pub enum FisherFError {
/// `m <= 0` or `nan`.
MTooSmall,
/// `n <= 0` or `nan`.
NTooSmall,
}
impl fmt::Display for FisherFError {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
f.write_str(match self {
FisherFError::MTooSmall => "m is not positive in Fisher F distribution",
FisherFError::NTooSmall => "n is not positive in Fisher F distribution",
})
}
}
#[cfg(feature = "std")]
#[cfg_attr(doc_cfg, doc(cfg(feature = "std")))]
impl std::error::Error for FisherFError {}
impl<F> FisherF<F>
where
F: Float,
StandardNormal: Distribution<F>,
Exp1: Distribution<F>,
Open01: Distribution<F>,
{
/// Create a new `FisherF` distribution, with the given parameter.
pub fn new(m: F, n: F) -> Result<FisherF<F>, FisherFError> {
let zero = F::zero();
if !(m > zero) {
return Err(FisherFError::MTooSmall);
}
if !(n > zero) {
return Err(FisherFError::NTooSmall);
}
Ok(FisherF {
numer: ChiSquared::new(m).unwrap(),
denom: ChiSquared::new(n).unwrap(),
dof_ratio: n / m,
})
}
}
impl<F> Distribution<F> for FisherF<F>
where
F: Float,
StandardNormal: Distribution<F>,
Exp1: Distribution<F>,
Open01: Distribution<F>,
{
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> F {
self.numer.sample(rng) / self.denom.sample(rng) * self.dof_ratio
}
}
/// The Student t distribution, `t(nu)`, where `nu` is the degrees of
/// freedom.
///
/// # Example
///
/// ```
/// use rand_distr::{StudentT, Distribution};
///
/// let t = StudentT::new(11.0).unwrap();
/// let v = t.sample(&mut rand::thread_rng());
/// println!("{} is from a t(11) distribution", v)
/// ```
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
pub struct StudentT<F>
where
F: Float,
StandardNormal: Distribution<F>,
Exp1: Distribution<F>,
Open01: Distribution<F>,
{
chi: ChiSquared<F>,
dof: F,
}
impl<F> StudentT<F>
where
F: Float,
StandardNormal: Distribution<F>,
Exp1: Distribution<F>,
Open01: Distribution<F>,
{
/// Create a new Student t distribution with `n` degrees of
/// freedom.
pub fn new(n: F) -> Result<StudentT<F>, ChiSquaredError> {
Ok(StudentT {
chi: ChiSquared::new(n)?,
dof: n,
})
}
}
impl<F> Distribution<F> for StudentT<F>
where
F: Float,
StandardNormal: Distribution<F>,
Exp1: Distribution<F>,
Open01: Distribution<F>,
{
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> F {
let norm: F = rng.sample(StandardNormal);
norm * (self.dof / self.chi.sample(rng)).sqrt()
}
}
/// The algorithm used for sampling the Beta distribution.
///
/// Reference:
///
/// R. C. H. Cheng (1978).
/// Generating beta variates with nonintegral shape parameters.
/// Communications of the ACM 21, 317-322.
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
enum BetaAlgorithm<N> {
BB(BB<N>),
BC(BC<N>),
}
/// Algorithm BB for `min(alpha, beta) > 1`.
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
struct BB<N> {
alpha: N,
beta: N,
gamma: N,
}
/// Algorithm BC for `min(alpha, beta) <= 1`.
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
struct BC<N> {
alpha: N,
beta: N,
delta: N,
kappa1: N,
kappa2: N,
}
/// The Beta distribution with shape parameters `alpha` and `beta`.
///
/// # Example
///
/// ```
/// use rand_distr::{Distribution, Beta};
///
/// let beta = Beta::new(2.0, 5.0).unwrap();
/// let v = beta.sample(&mut rand::thread_rng());
/// println!("{} is from a Beta(2, 5) distribution", v);
/// ```
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
pub struct Beta<F>
where
F: Float,
Open01: Distribution<F>,
{
a: F, b: F, switched_params: bool,
algorithm: BetaAlgorithm<F>,
}
/// Error type returned from `Beta::new`.
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
#[cfg_attr(feature = "serde1", derive(Serialize, Deserialize))]
pub enum BetaError {
/// `alpha <= 0` or `nan`.
AlphaTooSmall,
/// `beta <= 0` or `nan`.
BetaTooSmall,
}
impl fmt::Display for BetaError {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
f.write_str(match self {
BetaError::AlphaTooSmall => "alpha is not positive in beta distribution",
BetaError::BetaTooSmall => "beta is not positive in beta distribution",
})
}
}
#[cfg(feature = "std")]
#[cfg_attr(doc_cfg, doc(cfg(feature = "std")))]
impl std::error::Error for BetaError {}
impl<F> Beta<F>
where
F: Float,
Open01: Distribution<F>,
{
/// Construct an object representing the `Beta(alpha, beta)`
/// distribution.
pub fn new(alpha: F, beta: F) -> Result<Beta<F>, BetaError> {
if !(alpha > F::zero()) {
return Err(BetaError::AlphaTooSmall);
}
if !(beta > F::zero()) {
return Err(BetaError::BetaTooSmall);
}
// From now on, we use the notation from the reference,
// i.e. `alpha` and `beta` are renamed to `a0` and `b0`.
let (a0, b0) = (alpha, beta);
let (a, b, switched_params) = if a0 < b0 {
(a0, b0, false)
} else {
(b0, a0, true)
};
if a > F::one() {
// Algorithm BB
let alpha = a + b;
let beta = ((alpha - F::from(2.).unwrap())
/ (F::from(2.).unwrap()*a*b - alpha)).sqrt();
let gamma = a + F::one() / beta;
Ok(Beta {
a, b, switched_params,
algorithm: BetaAlgorithm::BB(BB {
alpha, beta, gamma,
})
})
} else {
// Algorithm BC
//
// Here `a` is the maximum instead of the minimum.
let (a, b, switched_params) = (b, a, !switched_params);
let alpha = a + b;
let beta = F::one() / b;
let delta = F::one() + a - b;
let kappa1 = delta
* (F::from(1. / 18. / 4.).unwrap() + F::from(3. / 18. / 4.).unwrap()*b)
/ (a*beta - F::from(14. / 18.).unwrap());
let kappa2 = F::from(0.25).unwrap()
+ (F::from(0.5).unwrap() + F::from(0.25).unwrap()/delta)*b;
Ok(Beta {
a, b, switched_params,
algorithm: BetaAlgorithm::BC(BC {
alpha, beta, delta, kappa1, kappa2,
})
})
}
}
}
impl<F> Distribution<F> for Beta<F>
where
F: Float,
Open01: Distribution<F>,
{
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> F {
let mut w;
match self.algorithm {
BetaAlgorithm::BB(algo) => {
loop {
// 1.
let u1 = rng.sample(Open01);
let u2 = rng.sample(Open01);
let v = algo.beta * (u1 / (F::one() - u1)).ln();
w = self.a * v.exp();
let z = u1*u1 * u2;
let r = algo.gamma * v - F::from(4.).unwrap().ln();
let s = self.a + r - w;
// 2.
if s + F::one() + F::from(5.).unwrap().ln()
>= F::from(5.).unwrap() * z {
break;
}
// 3.
let t = z.ln();
if s >= t {
break;
}
// 4.
if !(r + algo.alpha * (algo.alpha / (self.b + w)).ln() < t) {
break;
}
}
},
BetaAlgorithm::BC(algo) => {
loop {
let z;
// 1.
let u1 = rng.sample(Open01);
let u2 = rng.sample(Open01);
if u1 < F::from(0.5).unwrap() {
// 2.
let y = u1 * u2;
z = u1 * y;
if F::from(0.25).unwrap() * u2 + z - y >= algo.kappa1 {
continue;
}
} else {
// 3.
z = u1 * u1 * u2;
if z <= F::from(0.25).unwrap() {
let v = algo.beta * (u1 / (F::one() - u1)).ln();
w = self.a * v.exp();
break;
}
// 4.
if z >= algo.kappa2 {
continue;
}
}
// 5.
let v = algo.beta * (u1 / (F::one() - u1)).ln();
w = self.a * v.exp();
if !(algo.alpha * ((algo.alpha / (self.b + w)).ln() + v)
- F::from(4.).unwrap().ln() < z.ln()) {
break;
};
}
},
};
// 5. for BB, 6. for BC
if !self.switched_params {
if w == F::infinity() {
// Assuming `b` is finite, for large `w`:
return F::one();
}
w / (self.b + w)
} else {
self.b / (self.b + w)
}
}
}
#[cfg(test)]
mod test {
use super::*;
#[test]
fn test_chi_squared_one() {
let chi = ChiSquared::new(1.0).unwrap();
let mut rng = crate::test::rng(201);
for _ in 0..1000 {
chi.sample(&mut rng);
}
}
#[test]
fn test_chi_squared_small() {
let chi = ChiSquared::new(0.5).unwrap();
let mut rng = crate::test::rng(202);
for _ in 0..1000 {
chi.sample(&mut rng);
}
}
#[test]
fn test_chi_squared_large() {
let chi = ChiSquared::new(30.0).unwrap();
let mut rng = crate::test::rng(203);
for _ in 0..1000 {
chi.sample(&mut rng);
}
}
#[test]
#[should_panic]
fn test_chi_squared_invalid_dof() {
ChiSquared::new(-1.0).unwrap();
}
#[test]
fn test_f() {
let f = FisherF::new(2.0, 32.0).unwrap();
let mut rng = crate::test::rng(204);
for _ in 0..1000 {
f.sample(&mut rng);
}
}
#[test]
fn test_t() {
let t = StudentT::new(11.0).unwrap();
let mut rng = crate::test::rng(205);
for _ in 0..1000 {
t.sample(&mut rng);
}
}
#[test]
fn test_beta() {
let beta = Beta::new(1.0, 2.0).unwrap();
let mut rng = crate::test::rng(201);
for _ in 0..1000 {
beta.sample(&mut rng);
}
}
#[test]
#[should_panic]
fn test_beta_invalid_dof() {
Beta::new(0., 0.).unwrap();
}
#[test]
fn test_beta_small_param() {
let beta = Beta::<f64>::new(1e-3, 1e-3).unwrap();
let mut rng = crate::test::rng(206);
for i in 0..1000 {
assert!(!beta.sample(&mut rng).is_nan(), "failed at i={}", i);
}
}
}