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// Copyright 2018 Developers of the Rand project.
// Copyright 2016-2017 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 binomial distribution.
use crate::{Distribution, Uniform};
use rand::Rng;
use core::fmt;
use core::cmp::Ordering;
#[allow(unused_imports)]
use num_traits::Float;
/// The binomial distribution `Binomial(n, p)`.
///
/// This distribution has density function:
/// `f(k) = n!/(k! (n-k)!) p^k (1-p)^(n-k)` for `k >= 0`.
///
/// # Example
///
/// ```
/// use rand_distr::{Binomial, Distribution};
///
/// let bin = Binomial::new(20, 0.3).unwrap();
/// let v = bin.sample(&mut rand::thread_rng());
/// println!("{} is from a binomial distribution", v);
/// ```
#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = "serde1", derive(serde::Serialize, serde::Deserialize))]
pub struct Binomial {
/// Number of trials.
n: u64,
/// Probability of success.
p: f64,
}
/// Error type returned from `Binomial::new`.
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub enum Error {
/// `p < 0` or `nan`.
ProbabilityTooSmall,
/// `p > 1`.
ProbabilityTooLarge,
}
impl fmt::Display for Error {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
f.write_str(match self {
Error::ProbabilityTooSmall => "p < 0 or is NaN in binomial distribution",
Error::ProbabilityTooLarge => "p > 1 in binomial distribution",
})
}
}
#[cfg(feature = "std")]
#[cfg_attr(doc_cfg, doc(cfg(feature = "std")))]
impl std::error::Error for Error {}
impl Binomial {
/// Construct a new `Binomial` with the given shape parameters `n` (number
/// of trials) and `p` (probability of success).
pub fn new(n: u64, p: f64) -> Result<Binomial, Error> {
if !(p >= 0.0) {
return Err(Error::ProbabilityTooSmall);
}
if !(p <= 1.0) {
return Err(Error::ProbabilityTooLarge);
}
Ok(Binomial { n, p })
}
}
/// Convert a `f64` to an `i64`, panicking on overflow.
fn f64_to_i64(x: f64) -> i64 {
assert!(x < (core::i64::MAX as f64));
x as i64
}
impl Distribution<u64> for Binomial {
#[allow(clippy::many_single_char_names)] // Same names as in the reference.
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> u64 {
// Handle these values directly.
if self.p == 0.0 {
return 0;
} else if self.p == 1.0 {
return self.n;
}
// The binomial distribution is symmetrical with respect to p -> 1-p,
// k -> n-k switch p so that it is less than 0.5 - this allows for lower
// expected values we will just invert the result at the end
let p = if self.p <= 0.5 { self.p } else { 1.0 - self.p };
let result;
let q = 1. - p;
// For small n * min(p, 1 - p), the BINV algorithm based on the inverse
// transformation of the binomial distribution is efficient. Otherwise,
// the BTPE algorithm is used.
//
// Voratas Kachitvichyanukul and Bruce W. Schmeiser. 1988. Binomial
// random variate generation. Commun. ACM 31, 2 (February 1988),
// Threshold for preferring the BINV algorithm. The paper suggests 10,
// Ranlib uses 30, and GSL uses 14.
const BINV_THRESHOLD: f64 = 10.;
if (self.n as f64) * p < BINV_THRESHOLD && self.n <= (core::i32::MAX as u64) {
// Use the BINV algorithm.
let s = p / q;
let a = ((self.n + 1) as f64) * s;
let mut r = q.powi(self.n as i32);
let mut u: f64 = rng.gen();
let mut x = 0;
while u > r as f64 {
u -= r;
x += 1;
r *= a / (x as f64) - s;
}
result = x;
} else {
// Use the BTPE algorithm.
// Threshold for using the squeeze algorithm. This can be freely
// chosen based on performance. Ranlib and GSL use 20.
const SQUEEZE_THRESHOLD: i64 = 20;
// Step 0: Calculate constants as functions of `n` and `p`.
let n = self.n as f64;
let np = n * p;
let npq = np * q;
let f_m = np + p;
let m = f64_to_i64(f_m);
// radius of triangle region, since height=1 also area of region
let p1 = (2.195 * npq.sqrt() - 4.6 * q).floor() + 0.5;
// tip of triangle
let x_m = (m as f64) + 0.5;
// left edge of triangle
let x_l = x_m - p1;
// right edge of triangle
let x_r = x_m + p1;
let c = 0.134 + 20.5 / (15.3 + (m as f64));
// p1 + area of parallelogram region
let p2 = p1 * (1. + 2. * c);
fn lambda(a: f64) -> f64 {
a * (1. + 0.5 * a)
}
let lambda_l = lambda((f_m - x_l) / (f_m - x_l * p));
let lambda_r = lambda((x_r - f_m) / (x_r * q));
// p1 + area of left tail
let p3 = p2 + c / lambda_l;
// p1 + area of right tail
let p4 = p3 + c / lambda_r;
// return value
let mut y: i64;
let gen_u = Uniform::new(0., p4);
let gen_v = Uniform::new(0., 1.);
loop {
// Step 1: Generate `u` for selecting the region. If region 1 is
// selected, generate a triangularly distributed variate.
let u = gen_u.sample(rng);
let mut v = gen_v.sample(rng);
if !(u > p1) {
y = f64_to_i64(x_m - p1 * v + u);
break;
}
if !(u > p2) {
// Step 2: Region 2, parallelograms. Check if region 2 is
// used. If so, generate `y`.
let x = x_l + (u - p1) / c;
v = v * c + 1.0 - (x - x_m).abs() / p1;
if v > 1. {
continue;
} else {
y = f64_to_i64(x);
}
} else if !(u > p3) {
// Step 3: Region 3, left exponential tail.
y = f64_to_i64(x_l + v.ln() / lambda_l);
if y < 0 {
continue;
} else {
v *= (u - p2) * lambda_l;
}
} else {
// Step 4: Region 4, right exponential tail.
y = f64_to_i64(x_r - v.ln() / lambda_r);
if y > 0 && (y as u64) > self.n {
continue;
} else {
v *= (u - p3) * lambda_r;
}
}
// Step 5: Acceptance/rejection comparison.
// Step 5.0: Test for appropriate method of evaluating f(y).
let k = (y - m).abs();
if !(k > SQUEEZE_THRESHOLD && (k as f64) < 0.5 * npq - 1.) {
// Step 5.1: Evaluate f(y) via the recursive relationship. Start the
// search from the mode.
let s = p / q;
let a = s * (n + 1.);
let mut f = 1.0;
match m.cmp(&y) {
Ordering::Less => {
let mut i = m;
loop {
i += 1;
f *= a / (i as f64) - s;
if i == y {
break;
}
}
},
Ordering::Greater => {
let mut i = y;
loop {
i += 1;
f /= a / (i as f64) - s;
if i == m {
break;
}
}
},
Ordering::Equal => {},
}
if v > f {
continue;
} else {
break;
}
}
// Step 5.2: Squeezing. Check the value of ln(v) against upper and
// lower bound of ln(f(y)).
let k = k as f64;
let rho = (k / npq) * ((k * (k / 3. + 0.625) + 1. / 6.) / npq + 0.5);
let t = -0.5 * k * k / npq;
let alpha = v.ln();
if alpha < t - rho {
break;
}
if alpha > t + rho {
continue;
}
// Step 5.3: Final acceptance/rejection test.
let x1 = (y + 1) as f64;
let f1 = (m + 1) as f64;
let z = (f64_to_i64(n) + 1 - m) as f64;
let w = (f64_to_i64(n) - y + 1) as f64;
fn stirling(a: f64) -> f64 {
let a2 = a * a;
(13860. - (462. - (132. - (99. - 140. / a2) / a2) / a2) / a2) / a / 166320.
}
if alpha
> x_m * (f1 / x1).ln()
+ (n - (m as f64) + 0.5) * (z / w).ln()
+ ((y - m) as f64) * (w * p / (x1 * q)).ln()
// We use the signs from the GSL implementation, which are
// different than the ones in the reference. According to
// the GSL authors, the new signs were verified to be
// correct by one of the original designers of the
// algorithm.
+ stirling(f1)
+ stirling(z)
- stirling(x1)
- stirling(w)
{
continue;
}
break;
}
assert!(y >= 0);
result = y as u64;
}
// Invert the result for p < 0.5.
if p != self.p {
self.n - result
} else {
result
}
}
}
#[cfg(test)]
mod test {
use super::Binomial;
use crate::Distribution;
use rand::Rng;
fn test_binomial_mean_and_variance<R: Rng>(n: u64, p: f64, rng: &mut R) {
let binomial = Binomial::new(n, p).unwrap();
let expected_mean = n as f64 * p;
let expected_variance = n as f64 * p * (1.0 - p);
let mut results = [0.0; 1000];
for i in results.iter_mut() {
*i = binomial.sample(rng) as f64;
}
let mean = results.iter().sum::<f64>() / results.len() as f64;
assert!((mean as f64 - expected_mean).abs() < expected_mean / 50.0);
let variance =
results.iter().map(|x| (x - mean) * (x - mean)).sum::<f64>() / results.len() as f64;
assert!((variance - expected_variance).abs() < expected_variance / 10.0);
}
#[test]
fn test_binomial() {
let mut rng = crate::test::rng(351);
test_binomial_mean_and_variance(150, 0.1, &mut rng);
test_binomial_mean_and_variance(70, 0.6, &mut rng);
test_binomial_mean_and_variance(40, 0.5, &mut rng);
test_binomial_mean_and_variance(20, 0.7, &mut rng);
test_binomial_mean_and_variance(20, 0.5, &mut rng);
}
#[test]
fn test_binomial_end_points() {
let mut rng = crate::test::rng(352);
assert_eq!(rng.sample(Binomial::new(20, 0.0).unwrap()), 0);
assert_eq!(rng.sample(Binomial::new(20, 1.0).unwrap()), 20);
}
#[test]
#[should_panic]
fn test_binomial_invalid_lambda_neg() {
Binomial::new(20, -10.0).unwrap();
}
}