gamma.rs

// Copyright 2018 Developers of the Rand project.
// Copyright 2013 The Rust Project Developers.
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// https://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or https://opensource.org/licenses/MIT>, at your
// option. This file may not be copied, modified, or distributed
// except according to those terms.

//! The Gamma and derived distributions.

use self::GammaRepr::*;
use self::ChiSquaredRepr::*;

use rand::Rng;
use crate::normal::StandardNormal;
use crate::{Distribution, Exp1, Exp, Open01};
use num_traits::Float;

/// 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.
///       DOI:[10.1145/358407.358414](https://doi.acm.org/10.1145/358407.358414)
#[derive(Clone, Copy, Debug)]
pub struct Gamma<N> {
    repr: GammaRepr<N>,
}

/// 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,
}

#[derive(Clone, Copy, Debug)]
enum GammaRepr<N> {
    Large(GammaLargeShape<N>),
    One(Exp<N>),
    Small(GammaSmallShape<N>)
}

// 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)]
struct GammaSmallShape<N> {
    inv_shape: N,
    large_shape: GammaLargeShape<N>
}

/// 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)]
struct GammaLargeShape<N> {
    scale: N,
    c: N,
    d: N
}

impl<N: Float> Gamma<N>
where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N>
{
    /// Construct an object representing the `Gamma(shape, scale)`
    /// distribution.
    #[inline]
    pub fn new(shape: N, scale: N) -> Result<Gamma<N>, Error> {
        if !(shape > N::zero()) {
            return Err(Error::ShapeTooSmall);
        }
        if !(scale > N::zero()) {
            return Err(Error::ScaleTooSmall);
        }

        let repr = if shape == N::one() {
            One(Exp::new(N::one() / scale).map_err(|_| Error::ScaleTooLarge)?)
        } else if shape < N::one() {
            Small(GammaSmallShape::new_raw(shape, scale))
        } else {
            Large(GammaLargeShape::new_raw(shape, scale))
        };
        Ok(Gamma { repr })
    }
}

impl<N: Float> GammaSmallShape<N>
where StandardNormal: Distribution<N>, Open01: Distribution<N>
{
    fn new_raw(shape: N, scale: N) -> GammaSmallShape<N> {
        GammaSmallShape {
            inv_shape: N::one() / shape,
            large_shape: GammaLargeShape::new_raw(shape + N::one(), scale)
        }
    }
}

impl<N: Float> GammaLargeShape<N>
where StandardNormal: Distribution<N>, Open01: Distribution<N>
{
    fn new_raw(shape: N, scale: N) -> GammaLargeShape<N> {
        let d = shape - N::from(1. / 3.).unwrap();
        GammaLargeShape {
            scale,
            c: N::one() / (N::from(9.).unwrap() * d).sqrt(),
            d
        }
    }
}

impl<N: Float> Distribution<N> for Gamma<N>
where StandardNormal: Distribution<N>, Exp1: Distribution<N>, Open01: Distribution<N>
{
    fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N {
        match self.repr {
            Small(ref g) => g.sample(rng),
            One(ref g) => g.sample(rng),
            Large(ref g) => g.sample(rng),
        }
    }
}
impl<N: Float> Distribution<N> for GammaSmallShape<N>
where StandardNormal: Distribution<N>, Open01: Distribution<N>
{
    fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N {
        let u: N = rng.sample(Open01);

        self.large_shape.sample(rng) * u.powf(self.inv_shape)
    }
}
impl<N: Float> Distribution<N> for GammaLargeShape<N>
where StandardNormal: Distribution<N>, Open01: Distribution<N>
{
    fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> N {
        // Marsaglia & Tsang method, 2000
        loop {
            let x: N = rng.sample(StandardNormal);
            let v_cbrt = N::one() + self.c * x;
            if v_cbrt <= N::zero() { // a^3 <= 0 iff a <= 0
                continue
            }

            let v = v_cbrt * v_cbrt * v_cbrt;
            let u: N = rng.sample(Open01);

            let x_sqr = x * x;
            if u < N::one() - N::from(0.0331).unwrap() * x_sqr * x_sqr ||
                u.ln() < N::from(0.5).unwrap() * x_sqr + self.d * (N::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)]
pub struct ChiSquared {
    repr: ChiSquaredRepr,
}

/// Error type returned from `ChiSquared::new` and `StudentT::new`.
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub enum ChiSquaredError {
    /// `0.5 * k <= 0` or `nan`.
    DoFTooSmall,
}

#[derive(Clone, Copy, Debug)]
enum ChiSquaredRepr {
    // 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<f64>),
}

impl ChiSquared {
    /// Create a new chi-squared distribution with degrees-of-freedom
    /// `k`.
    pub fn new(k: f64) -> Result<ChiSquared, ChiSquaredError> {
        let repr = if k == 1.0 {
            DoFExactlyOne
        } else {
            if !(0.5 * k > 0.0) {
                return Err(ChiSquaredError::DoFTooSmall);
            }
            DoFAnythingElse(Gamma::new(0.5 * k, 2.0).unwrap())
        };
        Ok(ChiSquared { repr })
    }
}
impl Distribution<f64> for ChiSquared {
    fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> f64 {
        match self.repr {
            DoFExactlyOne => {
                // k == 1 => N(0,1)^2
                let norm: f64 = 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)]
pub struct FisherF {
    numer: ChiSquared,
    denom: ChiSquared,
    // denom_dof / numer_dof so that this can just be a straight
    // multiplication, rather than a division.
    dof_ratio: f64,
}

/// Error type returned from `FisherF::new`.
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub enum FisherFError {
    /// `m <= 0` or `nan`.
    MTooSmall,
    /// `n <= 0` or `nan`.
    NTooSmall,
}

impl FisherF {
    /// Create a new `FisherF` distribution, with the given parameter.
    pub fn new(m: f64, n: f64) -> Result<FisherF, FisherFError> {
        if !(m > 0.0) {
            return Err(FisherFError::MTooSmall);
        }
        if !(n > 0.0) {
            return Err(FisherFError::NTooSmall);
        }

        Ok(FisherF {
            numer: ChiSquared::new(m).unwrap(),
            denom: ChiSquared::new(n).unwrap(),
            dof_ratio: n / m
        })
    }
}
impl Distribution<f64> for FisherF {
    fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> f64 {
        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)]
pub struct StudentT {
    chi: ChiSquared,
    dof: f64
}

impl StudentT {
    /// Create a new Student t distribution with `n` degrees of
    /// freedom.
    pub fn new(n: f64) -> Result<StudentT, ChiSquaredError> {
        Ok(StudentT {
            chi: ChiSquared::new(n)?,
            dof: n
        })
    }
}
impl Distribution<f64> for StudentT {
    fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> f64 {
        let norm: f64 = rng.sample(StandardNormal);
        norm * (self.dof / self.chi.sample(rng)).sqrt()
    }
}

/// 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)]
pub struct Beta {
    gamma_a: Gamma<f64>,
    gamma_b: Gamma<f64>,
}

/// Error type returned from `Beta::new`.
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub enum BetaError {
    /// `alpha <= 0` or `nan`.
    AlphaTooSmall,
    /// `beta <= 0` or `nan`.
    BetaTooSmall,
}

impl Beta {
    /// Construct an object representing the `Beta(alpha, beta)`
    /// distribution.
    pub fn new(alpha: f64, beta: f64) -> Result<Beta, BetaError> {
        Ok(Beta {
            gamma_a: Gamma::new(alpha, 1.)
                         .map_err(|_| BetaError::AlphaTooSmall)?,
            gamma_b: Gamma::new(beta, 1.)
                         .map_err(|_| BetaError::BetaTooSmall)?,
        })
    }
}

impl Distribution<f64> for Beta {
    fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> f64 {
        let x = self.gamma_a.sample(rng);
        let y = self.gamma_b.sample(rng);
        x / (x + y)
    }
}

#[cfg(test)]
mod test {
    use crate::Distribution;
    use super::{Beta, ChiSquared, StudentT, FisherF};

    #[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();
    }
}