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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 normal and derived distributions.
use crate::utils::ziggurat;
use num_traits::Float;
use crate::{ziggurat_tables, Distribution, Open01};
use rand::Rng;
use core::fmt;
/// Samples floating-point numbers according to the normal distribution
/// `N(0, 1)` (a.k.a. a standard normal, or Gaussian). This is equivalent to
/// `Normal::new(0.0, 1.0)` but faster.
///
/// See `Normal` for the general normal distribution.
///
/// Implemented via the ZIGNOR variant[^1] of the Ziggurat method.
///
/// [^1]: Jurgen A. Doornik (2005). [*An Improved Ziggurat Method to
/// Generate Normal Random Samples*](
/// Nuffield College, Oxford
///
/// # Example
/// ```
/// use rand::prelude::*;
/// use rand_distr::StandardNormal;
///
/// let val: f64 = thread_rng().sample(StandardNormal);
/// println!("{}", val);
/// ```
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(serde::Serialize, serde::Deserialize))]
pub struct StandardNormal;
impl Distribution<f32> for StandardNormal {
#[inline]
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> f32 {
// TODO: use optimal 32-bit implementation
let x: f64 = self.sample(rng);
x as f32
}
}
impl Distribution<f64> for StandardNormal {
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> f64 {
#[inline]
fn pdf(x: f64) -> f64 {
(-x * x / 2.0).exp()
}
#[inline]
fn zero_case<R: Rng + ?Sized>(rng: &mut R, u: f64) -> f64 {
// compute a random number in the tail by hand
// strange initial conditions, because the loop is not
// do-while, so the condition should be true on the first
// run, they get overwritten anyway (0 < 1, so these are
// good).
let mut x = 1.0f64;
let mut y = 0.0f64;
while -2.0 * y < x * x {
let x_: f64 = rng.sample(Open01);
let y_: f64 = rng.sample(Open01);
x = x_.ln() / ziggurat_tables::ZIG_NORM_R;
y = y_.ln();
}
if u < 0.0 {
x - ziggurat_tables::ZIG_NORM_R
} else {
ziggurat_tables::ZIG_NORM_R - x
}
}
ziggurat(
rng,
true, // this is symmetric
&ziggurat_tables::ZIG_NORM_X,
&ziggurat_tables::ZIG_NORM_F,
pdf,
zero_case,
)
}
}
/// The normal distribution `N(mean, std_dev**2)`.
///
/// This uses the ZIGNOR variant of the Ziggurat method, see [`StandardNormal`]
/// for more details.
///
/// Note that [`StandardNormal`] is an optimised implementation for mean 0, and
/// standard deviation 1.
///
/// # Example
///
/// ```
/// use rand_distr::{Normal, Distribution};
///
/// // mean 2, standard deviation 3
/// let normal = Normal::new(2.0, 3.0).unwrap();
/// let v = normal.sample(&mut rand::thread_rng());
/// println!("{} is from a N(2, 9) distribution", v)
/// ```
///
/// [`StandardNormal`]: crate::StandardNormal
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(serde::Serialize, serde::Deserialize))]
pub struct Normal<F>
where F: Float, StandardNormal: Distribution<F>
{
mean: F,
std_dev: F,
}
/// Error type returned from `Normal::new` and `LogNormal::new`.
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub enum Error {
/// The mean value is too small (log-normal samples must be positive)
MeanTooSmall,
/// The standard deviation or other dispersion parameter is not finite.
BadVariance,
}
impl fmt::Display for Error {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
f.write_str(match self {
Error::MeanTooSmall => "mean < 0 or NaN in log-normal distribution",
Error::BadVariance => "variation parameter is non-finite in (log)normal distribution",
})
}
}
#[cfg(feature = "std")]
#[cfg_attr(doc_cfg, doc(cfg(feature = "std")))]
impl std::error::Error for Error {}
impl<F> Normal<F>
where F: Float, StandardNormal: Distribution<F>
{
/// Construct, from mean and standard deviation
///
/// Parameters:
///
/// - mean (`μ`, unrestricted)
/// - standard deviation (`σ`, must be finite)
#[inline]
pub fn new(mean: F, std_dev: F) -> Result<Normal<F>, Error> {
if !std_dev.is_finite() {
return Err(Error::BadVariance);
}
Ok(Normal { mean, std_dev })
}
/// Construct, from mean and coefficient of variation
///
/// Parameters:
///
/// - mean (`μ`, unrestricted)
/// - coefficient of variation (`cv = abs(σ / μ)`)
#[inline]
pub fn from_mean_cv(mean: F, cv: F) -> Result<Normal<F>, Error> {
if !cv.is_finite() || cv < F::zero() {
return Err(Error::BadVariance);
}
let std_dev = cv * mean;
Ok(Normal { mean, std_dev })
}
/// Sample from a z-score
///
/// This may be useful for generating correlated samples `x1` and `x2`
/// from two different distributions, as follows.
/// ```
/// # use rand::prelude::*;
/// # use rand_distr::{Normal, StandardNormal};
/// let mut rng = thread_rng();
/// let z = StandardNormal.sample(&mut rng);
/// let x1 = Normal::new(0.0, 1.0).unwrap().from_zscore(z);
/// let x2 = Normal::new(2.0, -3.0).unwrap().from_zscore(z);
/// ```
#[inline]
pub fn from_zscore(&self, zscore: F) -> F {
self.mean + self.std_dev * zscore
}
/// Returns the mean (`μ`) of the distribution.
pub fn mean(&self) -> F {
self.mean
}
/// Returns the standard deviation (`σ`) of the distribution.
pub fn std_dev(&self) -> F {
self.std_dev
}
}
impl<F> Distribution<F> for Normal<F>
where F: Float, StandardNormal: Distribution<F>
{
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> F {
self.from_zscore(rng.sample(StandardNormal))
}
}
/// The log-normal distribution `ln N(mean, std_dev**2)`.
///
/// If `X` is log-normal distributed, then `ln(X)` is `N(mean, std_dev**2)`
/// distributed.
///
/// # Example
///
/// ```
/// use rand_distr::{LogNormal, Distribution};
///
/// // mean 2, standard deviation 3
/// let log_normal = LogNormal::new(2.0, 3.0).unwrap();
/// let v = log_normal.sample(&mut rand::thread_rng());
/// println!("{} is from an ln N(2, 9) distribution", v)
/// ```
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(serde::Serialize, serde::Deserialize))]
pub struct LogNormal<F>
where F: Float, StandardNormal: Distribution<F>
{
norm: Normal<F>,
}
impl<F> LogNormal<F>
where F: Float, StandardNormal: Distribution<F>
{
/// Construct, from (log-space) mean and standard deviation
///
/// Parameters are the "standard" log-space measures (these are the mean
/// and standard deviation of the logarithm of samples):
///
/// - `mu` (`μ`, unrestricted) is the mean of the underlying distribution
/// - `sigma` (`σ`, must be finite) is the standard deviation of the
/// underlying Normal distribution
#[inline]
pub fn new(mu: F, sigma: F) -> Result<LogNormal<F>, Error> {
let norm = Normal::new(mu, sigma)?;
Ok(LogNormal { norm })
}
/// Construct, from (linear-space) mean and coefficient of variation
///
/// Parameters are linear-space measures:
///
/// - mean (`μ > 0`) is the (real) mean of the distribution
/// - coefficient of variation (`cv = σ / μ`, requiring `cv ≥ 0`) is a
/// standardized measure of dispersion
///
/// As a special exception, `μ = 0, cv = 0` is allowed (samples are `-inf`).
#[inline]
pub fn from_mean_cv(mean: F, cv: F) -> Result<LogNormal<F>, Error> {
if cv == F::zero() {
let mu = mean.ln();
let norm = Normal::new(mu, F::zero()).unwrap();
return Ok(LogNormal { norm });
}
if !(mean > F::zero()) {
return Err(Error::MeanTooSmall);
}
if !(cv >= F::zero()) {
return Err(Error::BadVariance);
}
// Using X ~ lognormal(μ, σ), CV² = Var(X) / E(X)²
// E(X) = exp(μ + σ² / 2) = exp(μ) × exp(σ² / 2)
// Var(X) = exp(2μ + σ²)(exp(σ²) - 1) = E(X)² × (exp(σ²) - 1)
// but Var(X) = (CV × E(X))² so CV² = exp(σ²) - 1
// thus σ² = log(CV² + 1)
// and exp(μ) = E(X) / exp(σ² / 2) = E(X) / sqrt(CV² + 1)
let a = F::one() + cv * cv; // e
let mu = F::from(0.5).unwrap() * (mean * mean / a).ln();
let sigma = a.ln().sqrt();
let norm = Normal::new(mu, sigma)?;
Ok(LogNormal { norm })
}
/// Sample from a z-score
///
/// This may be useful for generating correlated samples `x1` and `x2`
/// from two different distributions, as follows.
/// ```
/// # use rand::prelude::*;
/// # use rand_distr::{LogNormal, StandardNormal};
/// let mut rng = thread_rng();
/// let z = StandardNormal.sample(&mut rng);
/// let x1 = LogNormal::from_mean_cv(3.0, 1.0).unwrap().from_zscore(z);
/// let x2 = LogNormal::from_mean_cv(2.0, 4.0).unwrap().from_zscore(z);
/// ```
#[inline]
pub fn from_zscore(&self, zscore: F) -> F {
self.norm.from_zscore(zscore).exp()
}
}
impl<F> Distribution<F> for LogNormal<F>
where F: Float, StandardNormal: Distribution<F>
{
#[inline]
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> F {
self.norm.sample(rng).exp()
}
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn test_normal() {
let norm = Normal::new(10.0, 10.0).unwrap();
let mut rng = crate::test::rng(210);
for _ in 0..1000 {
norm.sample(&mut rng);
}
}
#[test]
fn test_normal_cv() {
let norm = Normal::from_mean_cv(1024.0, 1.0 / 256.0).unwrap();
assert_eq!((norm.mean, norm.std_dev), (1024.0, 4.0));
}
#[test]
fn test_normal_invalid_sd() {
assert!(Normal::from_mean_cv(10.0, -1.0).is_err());
}
#[test]
fn test_log_normal() {
let lnorm = LogNormal::new(10.0, 10.0).unwrap();
let mut rng = crate::test::rng(211);
for _ in 0..1000 {
lnorm.sample(&mut rng);
}
}
#[test]
fn test_log_normal_cv() {
let lnorm = LogNormal::from_mean_cv(0.0, 0.0).unwrap();
assert_eq!((lnorm.norm.mean, lnorm.norm.std_dev), (-core::f64::INFINITY, 0.0));
let lnorm = LogNormal::from_mean_cv(1.0, 0.0).unwrap();
assert_eq!((lnorm.norm.mean, lnorm.norm.std_dev), (0.0, 0.0));
let e = core::f64::consts::E;
let lnorm = LogNormal::from_mean_cv(e.sqrt(), (e - 1.0).sqrt()).unwrap();
assert_almost_eq!(lnorm.norm.mean, 0.0, 2e-16);
assert_almost_eq!(lnorm.norm.std_dev, 1.0, 2e-16);
let lnorm = LogNormal::from_mean_cv(e.powf(1.5), (e - 1.0).sqrt()).unwrap();
assert_almost_eq!(lnorm.norm.mean, 1.0, 1e-15);
assert_eq!(lnorm.norm.std_dev, 1.0);
}
#[test]
fn test_log_normal_invalid_sd() {
assert!(LogNormal::from_mean_cv(-1.0, 1.0).is_err());
assert!(LogNormal::from_mean_cv(0.0, 1.0).is_err());
assert!(LogNormal::from_mean_cv(1.0, -1.0).is_err());
}
}