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
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
// 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 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, PartialEq)]
#[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),
        // 216-222. http://dx.doi.org/10.1145/42372.42381

        // 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();
    }

    #[test]
    fn binomial_distributions_can_be_compared() {
        assert_eq!(Binomial::new(1, 1.0), Binomial::new(1, 1.0));
    }
}