logo
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
// 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 num_traits::{Float, FloatConst};
use crate::{Cauchy, Distribution, Standard};
use rand::Rng;
use core::fmt;

/// The Poisson distribution `Poisson(lambda)`.
///
/// This distribution has a density function:
/// `f(k) = lambda^k * exp(-lambda) / k!` for `k >= 0`.
///
/// # 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 = "serde1", 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)]
pub enum Error {
    /// `lambda <= 0` or `nan`.
    ShapeTooSmall,
}

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",
        })
    }
}

#[cfg(feature = "std")]
#[cfg_attr(doc_cfg, doc(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`.
    pub fn new(lambda: F) -> Result<Poisson<F>, Error> {
        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::zero();
            let mut p = F::one();
            while p > self.exp_lambda {
                p = p*rng.gen::<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.gen::<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, 0.5);
        test_poisson_avg_gen::<f64>(15.0, 0.5);
        test_poisson_avg_gen::<f32>(10.0, 0.5);
        test_poisson_avg_gen::<f32>(15.0, 0.5);
    }

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