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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
// 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 Poisson distribution.
use Rng;
use distributions::{Distribution, Cauchy};
use distributions::utils::log_gamma;
/// The Poisson distribution `Poisson(lambda)`.
///
/// This distribution has a density function:
/// `f(k) = lambda^k * exp(-lambda) / k!` for `k >= 0`.
///
/// # Example
///
/// ```
/// use rand::distributions::{Poisson, Distribution};
///
/// let poi = Poisson::new(2.0);
/// let v = poi.sample(&mut rand::thread_rng());
/// println!("{} is from a Poisson(2) distribution", v);
/// ```
#[derive(Clone, Copy, Debug)]
pub struct Poisson {
lambda: f64,
// precalculated values
exp_lambda: f64,
log_lambda: f64,
sqrt_2lambda: f64,
magic_val: f64,
}
impl Poisson {
/// Construct a new `Poisson` with the given shape parameter
/// `lambda`. Panics if `lambda <= 0`.
pub fn new(lambda: f64) -> Poisson {
assert!(lambda > 0.0, "Poisson::new called with lambda <= 0");
let log_lambda = lambda.ln();
Poisson {
lambda,
exp_lambda: (-lambda).exp(),
log_lambda,
sqrt_2lambda: (2.0 * lambda).sqrt(),
magic_val: lambda * log_lambda - log_gamma(1.0 + lambda),
}
}
}
impl Distribution<u64> for Poisson {
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> u64 {
// using the algorithm from Numerical Recipes in C
// for low expected values use the Knuth method
if self.lambda < 12.0 {
let mut result = 0;
let mut p = 1.0;
while p > self.exp_lambda {
p *= rng.gen::<f64>();
result += 1;
}
result - 1
}
// high expected values - rejection method
else {
let mut int_result: u64;
// we use the Cauchy distribution as the comparison distribution
// f(x) ~ 1/(1+x^2)
let cauchy = Cauchy::new(0.0, 1.0);
loop {
let mut result;
let mut comp_dev;
loop {
// draw from the Cauchy distribution
comp_dev = rng.sample(cauchy);
// shift the peak of the comparison ditribution
result = self.sqrt_2lambda * comp_dev + self.lambda;
// repeat the drawing until we are in the range of possible values
if result >= 0.0 {
break;
}
}
// now the result is a random variable greater than 0 with Cauchy distribution
// the result should be an integer value
result = result.floor();
int_result = result as u64;
// this is the ratio of the Poisson distribution to the comparison distribution
// the magic value scales the distribution function to a range of approximately 0-1
// since it is not exact, we multiply the ratio by 0.9 to avoid ratios greater than 1
// this doesn't change the resulting distribution, only increases the rate of failed drawings
let check = 0.9 * (1.0 + comp_dev * comp_dev)
* (result * self.log_lambda - log_gamma(1.0 + result) - self.magic_val).exp();
// check with uniform random value - if below the threshold, we are within the target distribution
if rng.gen::<f64>() <= check {
break;
}
}
int_result
}
}
}
#[cfg(test)]
mod test {
use distributions::Distribution;
use super::Poisson;
#[test]
fn test_poisson_10() {
let poisson = Poisson::new(10.0);
let mut rng = ::test::rng(123);
let mut sum = 0;
for _ in 0..1000 {
sum += poisson.sample(&mut rng);
}
let avg = (sum as f64) / 1000.0;
println!("Poisson average: {}", avg);
assert!((avg - 10.0).abs() < 0.5); // not 100% certain, but probable enough
}
#[test]
fn test_poisson_15() {
// Take the 'high expected values' path
let poisson = Poisson::new(15.0);
let mut rng = ::test::rng(123);
let mut sum = 0;
for _ in 0..1000 {
sum += poisson.sample(&mut rng);
}
let avg = (sum as f64) / 1000.0;
println!("Poisson average: {}", avg);
assert!((avg - 15.0).abs() < 0.5); // not 100% certain, but probable enough
}
#[test]
#[should_panic]
fn test_poisson_invalid_lambda_zero() {
Poisson::new(0.0);
}
#[test]
#[should_panic]
fn test_poisson_invalid_lambda_neg() {
Poisson::new(-10.0);
}
}
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