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``````// Copyright 2018 Developers of the Rand project.
// Copyright 2016-2017 The Rust Project Developers.
//
// option. This file may not be copied, modified, or distributed
// except according to those terms.

//! The Poisson distribution `Poisson(λ)`.

use crate::{Cauchy, Distribution, Standard};
use core::fmt;
use num_traits::{Float, FloatConst};
use rand::Rng;

/// The [Poisson distribution](https://en.wikipedia.org/wiki/Poisson_distribution) `Poisson(λ)`.
///
/// The Poisson distribution is a discrete probability distribution with
/// rate parameter `λ` (`lambda`). It models the number of events occurring in a fixed
/// interval of time or space.
///
/// This distribution has density function:
/// `f(k) = λ^k * exp(-λ) / k!` for `k >= 0`.
///
/// # Known issues
///
/// See documentation of [`Poisson::new`].
///
/// # Plot
///
/// The following plot shows the Poisson distribution with various values of `λ`.
/// Note how the expected number of events increases with `λ`.
///
/// ![Poisson distribution](https://raw.githubusercontent.com/rust-random/charts/main/charts/poisson.svg)
///
/// # Example
///
/// ```
/// use rand_distr::{Poisson, Distribution};
///
/// let poi = Poisson::new(2.0).unwrap();
/// let v = poi.sample(&mut rand::thread_rng());
/// println!("{} is from a Poisson(2) distribution", v);
/// ```
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
pub struct Poisson<F>
where
F: Float + FloatConst,
Standard: Distribution<F>,
{
lambda: F,
// precalculated values
exp_lambda: F,
log_lambda: F,
sqrt_2lambda: F,
magic_val: F,
}

/// Error type returned from [`Poisson::new`].
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
// Marked non_exhaustive to allow a new error code in the solution to #1312.
#[non_exhaustive]
pub enum Error {
/// `lambda <= 0`
ShapeTooSmall,
/// `lambda = ∞` or `lambda = nan`
NonFinite,
}

impl fmt::Display for Error {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
f.write_str(match self {
Error::ShapeTooSmall => "lambda is not positive in Poisson distribution",
Error::NonFinite => "lambda is infinite or nan in Poisson distribution",
})
}
}

#[cfg(feature = "std")]
impl std::error::Error for Error {}

impl<F> Poisson<F>
where
F: Float + FloatConst,
Standard: Distribution<F>,
{
/// Construct a new `Poisson` with the given shape parameter
/// `lambda`.
///
/// # Known issues
///
/// Although this method should return an [`Error`] on invalid parameters,
/// some (extreme) values of `lambda` are known to return a [`Poisson`]
/// object which hangs when [sampled](Distribution::sample).
/// Large (less extreme) values of `lambda` may result in successful
/// sampling but with reduced precision.
/// See [#1312](https://github.com/rust-random/rand/issues/1312).
pub fn new(lambda: F) -> Result<Poisson<F>, Error> {
if !lambda.is_finite() {
return Err(Error::NonFinite);
}
if !(lambda > F::zero()) {
return Err(Error::ShapeTooSmall);
}
let log_lambda = lambda.ln();
Ok(Poisson {
lambda,
exp_lambda: (-lambda).exp(),
log_lambda,
sqrt_2lambda: (F::from(2.0).unwrap() * lambda).sqrt(),
magic_val: lambda * log_lambda - crate::utils::log_gamma(F::one() + lambda),
})
}
}

impl<F> Distribution<F> for Poisson<F>
where
F: Float + FloatConst,
Standard: Distribution<F>,
{
#[inline]
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> F {
// using the algorithm from Numerical Recipes in C

// for low expected values use the Knuth method
if self.lambda < F::from(12.0).unwrap() {
let mut result = F::one();
let mut p = rng.random::<F>();
while p > self.exp_lambda {
p = p * rng.random::<F>();
result = result + F::one();
}
result - F::one()
}
// high expected values - rejection method
else {
// we use the Cauchy distribution as the comparison distribution
// f(x) ~ 1/(1+x^2)
let cauchy = Cauchy::new(F::zero(), F::one()).unwrap();
let mut result;

loop {
let mut comp_dev;

loop {
// draw from the Cauchy distribution
comp_dev = rng.sample(cauchy);
// shift the peak of the comparison distribution
result = self.sqrt_2lambda * comp_dev + self.lambda;
// repeat the drawing until we are in the range of possible values
if result >= F::zero() {
break;
}
}
// now the result is a random variable greater than 0 with Cauchy distribution
// the result should be an integer value
result = result.floor();

// 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 = F::from(0.9).unwrap()
* (F::one() + comp_dev * comp_dev)
* (result * self.log_lambda
- crate::utils::log_gamma(F::one() + result)
- self.magic_val)
.exp();

// check with uniform random value - if below the threshold, we are within the target distribution
if rng.random::<F>() <= check {
break;
}
}
result
}
}
}

#[cfg(test)]
mod test {
use super::*;

fn test_poisson_avg_gen<F: Float + FloatConst>(lambda: F, tol: F)
where
Standard: Distribution<F>,
{
let poisson = Poisson::new(lambda).unwrap();
let mut rng = crate::test::rng(123);
let mut sum = F::zero();
for _ in 0..1000 {
sum = sum + poisson.sample(&mut rng);
}
let avg = sum / F::from(1000.0).unwrap();
assert!((avg - lambda).abs() < tol);
}

#[test]
fn test_poisson_avg() {
test_poisson_avg_gen::<f64>(10.0, 0.1);
test_poisson_avg_gen::<f64>(15.0, 0.1);

test_poisson_avg_gen::<f32>(10.0, 0.1);
test_poisson_avg_gen::<f32>(15.0, 0.1);

// Small lambda will use Knuth's method with exp_lambda == 1.0
test_poisson_avg_gen::<f32>(0.00000000000000005, 0.1);
test_poisson_avg_gen::<f64>(0.00000000000000005, 0.1);
}

#[test]
#[should_panic]
fn test_poisson_invalid_lambda_zero() {
Poisson::new(0.0).unwrap();
}

#[test]
#[should_panic]
fn test_poisson_invalid_lambda_infinity() {
Poisson::new(f64::INFINITY).unwrap();
}

#[test]
#[should_panic]
fn test_poisson_invalid_lambda_neg() {
Poisson::new(-10.0).unwrap();
}

#[test]
fn poisson_distributions_can_be_compared() {
assert_eq!(Poisson::new(1.0), Poisson::new(1.0));
}
}
``````