| // 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::ChiSquaredRepr::*; |
| use self::GammaRepr::*; |
| |
| use crate::normal::StandardNormal; |
| use num_traits::Float; |
| use crate::{Distribution, Exp, Exp1, Open01}; |
| use rand::Rng; |
| use core::fmt; |
| |
| /// 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<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")] |
| impl std::error::Error for Error {} |
| |
| #[derive(Clone, Copy, Debug)] |
| 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)] |
| 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)] |
| 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)] |
| 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)] |
| 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")] |
| impl std::error::Error for ChiSquaredError {} |
| |
| #[derive(Clone, Copy, Debug)] |
| 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)] |
| 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)] |
| 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")] |
| 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)] |
| 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 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<F> |
| where |
| F: Float, |
| StandardNormal: Distribution<F>, |
| Exp1: Distribution<F>, |
| Open01: Distribution<F>, |
| { |
| gamma_a: Gamma<F>, |
| gamma_b: Gamma<F>, |
| } |
| |
| /// 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 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")] |
| impl std::error::Error for BetaError {} |
| |
| impl<F> Beta<F> |
| where |
| F: Float, |
| StandardNormal: Distribution<F>, |
| Exp1: Distribution<F>, |
| Open01: Distribution<F>, |
| { |
| /// Construct an object representing the `Beta(alpha, beta)` |
| /// distribution. |
| pub fn new(alpha: F, beta: F) -> Result<Beta<F>, BetaError> { |
| Ok(Beta { |
| gamma_a: Gamma::new(alpha, F::one()).map_err(|_| BetaError::AlphaTooSmall)?, |
| gamma_b: Gamma::new(beta, F::one()).map_err(|_| BetaError::BetaTooSmall)?, |
| }) |
| } |
| } |
| |
| impl<F> Distribution<F> for Beta<F> |
| where |
| F: Float, |
| StandardNormal: Distribution<F>, |
| Exp1: Distribution<F>, |
| Open01: Distribution<F>, |
| { |
| fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> F { |
| let x = self.gamma_a.sample(rng); |
| let y = self.gamma_b.sample(rng); |
| x / (x + y) |
| } |
| } |
| |
| #[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(); |
| } |
| } |