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
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
//! The geometric distribution `Geometric(p)`.

use crate::Distribution;
use core::fmt;
#[allow(unused_imports)]
use num_traits::Float;
use rand::Rng;

/// The [geometric distribution](https://en.wikipedia.org/wiki/Geometric_distribution) `Geometric(p)`.
///
/// This is the probability distribution of the number of failures
/// (bounded to `[0, u64::MAX]`) before the first success in a
/// series of [`Bernoulli`](crate::Bernoulli) trials, where the
/// probability of success on each trial is `p`.
///
/// This is the discrete analogue of the [exponential distribution](crate::Exp).
///
/// See [`StandardGeometric`](crate::StandardGeometric) for an optimised
/// implementation for `p = 0.5`.
///
/// # Density function
///
/// `f(k) = (1 - p)^k p` for `k >= 0`.
///
/// # Plot
///
/// The following plot illustrates the geometric distribution for various
/// values of `p`. Note how higher `p` values shift the distribution to
/// the left, and the mean of the distribution is `1/p`.
///
/// ![Geometric distribution](https://raw.githubusercontent.com/rust-random/charts/main/charts/geometric.svg)
///
/// # Example
/// ```
/// use rand_distr::{Geometric, Distribution};
///
/// let geo = Geometric::new(0.25).unwrap();
/// let v = geo.sample(&mut rand::thread_rng());
/// println!("{} is from a Geometric(0.25) distribution", v);
/// ```
#[derive(Copy, Clone, Debug, PartialEq)]
#[cfg_attr(feature = "serde1", derive(serde::Serialize, serde::Deserialize))]
pub struct Geometric {
    p: f64,
    pi: f64,
    k: u64,
}

/// Error type returned from [`Geometric::new`].
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub enum Error {
    /// `p < 0 || p > 1` or `nan`
    InvalidProbability,
}

impl fmt::Display for Error {
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
        f.write_str(match self {
            Error::InvalidProbability => {
                "p is NaN or outside the interval [0, 1] in geometric distribution"
            }
        })
    }
}

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

impl Geometric {
    /// Construct a new `Geometric` with the given shape parameter `p`
    /// (probability of success on each trial).
    pub fn new(p: f64) -> Result<Self, Error> {
        if !p.is_finite() || !(0.0..=1.0).contains(&p) {
            Err(Error::InvalidProbability)
        } else if p == 0.0 || p >= 2.0 / 3.0 {
            Ok(Geometric { p, pi: p, k: 0 })
        } else {
            let (pi, k) = {
                // choose smallest k such that pi = (1 - p)^(2^k) <= 0.5
                let mut k = 1;
                let mut pi = (1.0 - p).powi(2);
                while pi > 0.5 {
                    k += 1;
                    pi = pi * pi;
                }
                (pi, k)
            };

            Ok(Geometric { p, pi, k })
        }
    }
}

impl Distribution<u64> for Geometric {
    fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> u64 {
        if self.p >= 2.0 / 3.0 {
            // use the trivial algorithm:
            let mut failures = 0;
            loop {
                let u = rng.random::<f64>();
                if u <= self.p {
                    break;
                }
                failures += 1;
            }
            return failures;
        }

        if self.p == 0.0 {
            return u64::MAX;
        }

        let Geometric { p, pi, k } = *self;

        // Based on the algorithm presented in section 3 of
        // Karl Bringmann and Tobias Friedrich (July 2013) - Exact and Efficient
        // Generation of Geometric Random Variates and Random Graphs, published
        // in International Colloquium on Automata, Languages and Programming
        // (pp.267-278)
        // https://people.mpi-inf.mpg.de/~kbringma/paper/2013ICALP-1.pdf

        // Use the trivial algorithm to sample D from Geo(pi) = Geo(p) / 2^k:
        let d = {
            let mut failures = 0;
            while rng.random::<f64>() < pi {
                failures += 1;
            }
            failures
        };

        // Use rejection sampling for the remainder M from Geo(p) % 2^k:
        // choose M uniformly from [0, 2^k), but reject with probability (1 - p)^M
        // NOTE: The paper suggests using bitwise sampling here, which is
        // currently unsupported, but should improve performance by requiring
        // fewer iterations on average.                 ~ October 28, 2020
        let m = loop {
            let m = rng.random::<u64>() & ((1 << k) - 1);
            let p_reject = if m <= i32::MAX as u64 {
                (1.0 - p).powi(m as i32)
            } else {
                (1.0 - p).powf(m as f64)
            };

            let u = rng.random::<f64>();
            if u < p_reject {
                break m;
            }
        };

        (d << k) + m
    }
}

/// The standard geometric distribution `Geometric(0.5)`.
///
/// This is equivalent to `Geometric::new(0.5)`, but faster.
///
/// See [`Geometric`](crate::Geometric) for the general geometric distribution.
///
/// # Plot
///
/// The following plot illustrates the standard geometric distribution.
///
/// ![Standard Geometric distribution](https://raw.githubusercontent.com/rust-random/charts/main/charts/standard_geometric.svg)
///
/// # Example
/// ```
/// use rand::prelude::*;
/// use rand_distr::StandardGeometric;
///
/// let v = StandardGeometric.sample(&mut thread_rng());
/// println!("{} is from a Geometric(0.5) distribution", v);
/// ```
///
/// # Notes
/// Implemented via iterated
/// [`Rng::gen::<u64>().leading_zeros()`](Rng::gen::<u64>().leading_zeros()).
#[derive(Copy, Clone, Debug)]
#[cfg_attr(feature = "serde1", derive(serde::Serialize, serde::Deserialize))]
pub struct StandardGeometric;

impl Distribution<u64> for StandardGeometric {
    fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> u64 {
        let mut result = 0;
        loop {
            let x = rng.random::<u64>().leading_zeros() as u64;
            result += x;
            if x < 64 {
                break;
            }
        }
        result
    }
}

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

    #[test]
    fn test_geo_invalid_p() {
        assert!(Geometric::new(f64::NAN).is_err());
        assert!(Geometric::new(f64::INFINITY).is_err());
        assert!(Geometric::new(f64::NEG_INFINITY).is_err());

        assert!(Geometric::new(-0.5).is_err());
        assert!(Geometric::new(0.0).is_ok());
        assert!(Geometric::new(1.0).is_ok());
        assert!(Geometric::new(2.0).is_err());
    }

    fn test_geo_mean_and_variance<R: Rng>(p: f64, rng: &mut R) {
        let distr = Geometric::new(p).unwrap();

        let expected_mean = (1.0 - p) / p;
        let expected_variance = (1.0 - p) / (p * p);

        let mut results = [0.0; 10000];
        for i in results.iter_mut() {
            *i = distr.sample(rng) as f64;
        }

        let mean = results.iter().sum::<f64>() / results.len() as f64;
        assert!((mean - expected_mean).abs() < expected_mean / 40.0);

        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_geometric() {
        let mut rng = crate::test::rng(12345);

        test_geo_mean_and_variance(0.10, &mut rng);
        test_geo_mean_and_variance(0.25, &mut rng);
        test_geo_mean_and_variance(0.50, &mut rng);
        test_geo_mean_and_variance(0.75, &mut rng);
        test_geo_mean_and_variance(0.90, &mut rng);
    }

    #[test]
    fn test_standard_geometric() {
        let mut rng = crate::test::rng(654321);

        let distr = StandardGeometric;
        let expected_mean = 1.0;
        let expected_variance = 2.0;

        let mut results = [0.0; 1000];
        for i in results.iter_mut() {
            *i = distr.sample(&mut rng) as f64;
        }

        let mean = results.iter().sum::<f64>() / results.len() as f64;
        assert!((mean - 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 geometric_distributions_can_be_compared() {
        assert_eq!(Geometric::new(1.0), Geometric::new(1.0));
    }
}