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</pre><pre class="rust"><code><span class="comment">// Copyright 2018 Developers of the Rand project.
// Copyright 2016-2017 The Rust Project Developers.
//
// Licensed under the Apache License, Version 2.0 &lt;LICENSE-APACHE or
// https://www.apache.org/licenses/LICENSE-2.0&gt; or the MIT license
// &lt;LICENSE-MIT or https://opensource.org/licenses/MIT&gt;, at your
// option. This file may not be copied, modified, or distributed
// except according to those terms.
</span><span class="doccomment">//! The binomial distribution.
</span><span class="kw">use crate</span>::{Distribution, Uniform};
<span class="kw">use </span>rand::Rng;
<span class="kw">use </span>core::fmt;
<span class="kw">use </span>core::cmp::Ordering;
<span class="attribute">#[allow(unused_imports)]
</span><span class="kw">use </span>num_traits::Float;
<span class="doccomment">/// 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 &gt;= 0`.
///
/// # Example
///
/// ```
/// use rand_distr::{Binomial, Distribution};
///
/// let bin = Binomial::new(20, 0.3).unwrap();
/// let v = bin.sample(&amp;mut rand::thread_rng());
/// println!(&quot;{} is from a binomial distribution&quot;, v);
/// ```
</span><span class="attribute">#[derive(Clone, Copy, Debug)]
#[cfg_attr(feature = <span class="string">&quot;serde1&quot;</span>, derive(serde::Serialize, serde::Deserialize))]
</span><span class="kw">pub struct </span>Binomial {
<span class="doccomment">/// Number of trials.
</span>n: u64,
<span class="doccomment">/// Probability of success.
</span>p: f64,
}
<span class="doccomment">/// Error type returned from `Binomial::new`.
</span><span class="attribute">#[derive(Clone, Copy, Debug, PartialEq, Eq)]
</span><span class="kw">pub enum </span>Error {
<span class="doccomment">/// `p &lt; 0` or `nan`.
</span>ProbabilityTooSmall,
<span class="doccomment">/// `p &gt; 1`.
</span>ProbabilityTooLarge,
}
<span class="kw">impl </span>fmt::Display <span class="kw">for </span>Error {
<span class="kw">fn </span>fmt(<span class="kw-2">&amp;</span><span class="self">self</span>, f: <span class="kw-2">&amp;mut </span>fmt::Formatter&lt;<span class="lifetime">&#39;_</span>&gt;) -&gt; fmt::Result {
f.write_str(<span class="kw">match </span><span class="self">self </span>{
Error::ProbabilityTooSmall =&gt; <span class="string">&quot;p &lt; 0 or is NaN in binomial distribution&quot;</span>,
Error::ProbabilityTooLarge =&gt; <span class="string">&quot;p &gt; 1 in binomial distribution&quot;</span>,
})
}
}
<span class="attribute">#[cfg(feature = <span class="string">&quot;std&quot;</span>)]
#[cfg_attr(doc_cfg, doc(cfg(feature = <span class="string">&quot;std&quot;</span>)))]
</span><span class="kw">impl </span>std::error::Error <span class="kw">for </span>Error {}
<span class="kw">impl </span>Binomial {
<span class="doccomment">/// Construct a new `Binomial` with the given shape parameters `n` (number
/// of trials) and `p` (probability of success).
</span><span class="kw">pub fn </span>new(n: u64, p: f64) -&gt; <span class="prelude-ty">Result</span>&lt;Binomial, Error&gt; {
<span class="kw">if </span>!(p &gt;= <span class="number">0.0</span>) {
<span class="kw">return </span><span class="prelude-val">Err</span>(Error::ProbabilityTooSmall);
}
<span class="kw">if </span>!(p &lt;= <span class="number">1.0</span>) {
<span class="kw">return </span><span class="prelude-val">Err</span>(Error::ProbabilityTooLarge);
}
<span class="prelude-val">Ok</span>(Binomial { n, p })
}
}
<span class="doccomment">/// Convert a `f64` to an `i64`, panicking on overflow.
</span><span class="kw">fn </span>f64_to_i64(x: f64) -&gt; i64 {
<span class="macro">assert!</span>(x &lt; (core::i64::MAX <span class="kw">as </span>f64));
x <span class="kw">as </span>i64
}
<span class="kw">impl </span>Distribution&lt;u64&gt; <span class="kw">for </span>Binomial {
<span class="attribute">#[allow(clippy::many_single_char_names)] </span><span class="comment">// Same names as in the reference.
</span><span class="kw">fn </span>sample&lt;R: Rng + <span class="question-mark">?</span>Sized&gt;(<span class="kw-2">&amp;</span><span class="self">self</span>, rng: <span class="kw-2">&amp;mut </span>R) -&gt; u64 {
<span class="comment">// Handle these values directly.
</span><span class="kw">if </span><span class="self">self</span>.p == <span class="number">0.0 </span>{
<span class="kw">return </span><span class="number">0</span>;
} <span class="kw">else if </span><span class="self">self</span>.p == <span class="number">1.0 </span>{
<span class="kw">return </span><span class="self">self</span>.n;
}
<span class="comment">// The binomial distribution is symmetrical with respect to p -&gt; 1-p,
// k -&gt; 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
</span><span class="kw">let </span>p = <span class="kw">if </span><span class="self">self</span>.p &lt;= <span class="number">0.5 </span>{ <span class="self">self</span>.p } <span class="kw">else </span>{ <span class="number">1.0 </span>- <span class="self">self</span>.p };
<span class="kw">let </span>result;
<span class="kw">let </span>q = <span class="number">1. </span>- p;
<span class="comment">// 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.
</span><span class="kw">const </span>BINV_THRESHOLD: f64 = <span class="number">10.</span>;
<span class="kw">if </span>(<span class="self">self</span>.n <span class="kw">as </span>f64) * p &lt; BINV_THRESHOLD &amp;&amp; <span class="self">self</span>.n &lt;= (core::i32::MAX <span class="kw">as </span>u64) {
<span class="comment">// Use the BINV algorithm.
</span><span class="kw">let </span>s = p / q;
<span class="kw">let </span>a = ((<span class="self">self</span>.n + <span class="number">1</span>) <span class="kw">as </span>f64) * s;
<span class="kw">let </span><span class="kw-2">mut </span>r = q.powi(<span class="self">self</span>.n <span class="kw">as </span>i32);
<span class="kw">let </span><span class="kw-2">mut </span>u: f64 = rng.gen();
<span class="kw">let </span><span class="kw-2">mut </span>x = <span class="number">0</span>;
<span class="kw">while </span>u &gt; r <span class="kw">as </span>f64 {
u -= r;
x += <span class="number">1</span>;
r <span class="kw-2">*</span>= a / (x <span class="kw">as </span>f64) - s;
}
result = x;
} <span class="kw">else </span>{
<span class="comment">// Use the BTPE algorithm.
// Threshold for using the squeeze algorithm. This can be freely
// chosen based on performance. Ranlib and GSL use 20.
</span><span class="kw">const </span>SQUEEZE_THRESHOLD: i64 = <span class="number">20</span>;
<span class="comment">// Step 0: Calculate constants as functions of `n` and `p`.
</span><span class="kw">let </span>n = <span class="self">self</span>.n <span class="kw">as </span>f64;
<span class="kw">let </span>np = n * p;
<span class="kw">let </span>npq = np * q;
<span class="kw">let </span>f_m = np + p;
<span class="kw">let </span>m = f64_to_i64(f_m);
<span class="comment">// radius of triangle region, since height=1 also area of region
</span><span class="kw">let </span>p1 = (<span class="number">2.195 </span>* npq.sqrt() - <span class="number">4.6 </span>* q).floor() + <span class="number">0.5</span>;
<span class="comment">// tip of triangle
</span><span class="kw">let </span>x_m = (m <span class="kw">as </span>f64) + <span class="number">0.5</span>;
<span class="comment">// left edge of triangle
</span><span class="kw">let </span>x_l = x_m - p1;
<span class="comment">// right edge of triangle
</span><span class="kw">let </span>x_r = x_m + p1;
<span class="kw">let </span>c = <span class="number">0.134 </span>+ <span class="number">20.5 </span>/ (<span class="number">15.3 </span>+ (m <span class="kw">as </span>f64));
<span class="comment">// p1 + area of parallelogram region
</span><span class="kw">let </span>p2 = p1 * (<span class="number">1. </span>+ <span class="number">2. </span>* c);
<span class="kw">fn </span>lambda(a: f64) -&gt; f64 {
a * (<span class="number">1. </span>+ <span class="number">0.5 </span>* a)
}
<span class="kw">let </span>lambda_l = lambda((f_m - x_l) / (f_m - x_l * p));
<span class="kw">let </span>lambda_r = lambda((x_r - f_m) / (x_r * q));
<span class="comment">// p1 + area of left tail
</span><span class="kw">let </span>p3 = p2 + c / lambda_l;
<span class="comment">// p1 + area of right tail
</span><span class="kw">let </span>p4 = p3 + c / lambda_r;
<span class="comment">// return value
</span><span class="kw">let </span><span class="kw-2">mut </span>y: i64;
<span class="kw">let </span>gen_u = Uniform::new(<span class="number">0.</span>, p4);
<span class="kw">let </span>gen_v = Uniform::new(<span class="number">0.</span>, <span class="number">1.</span>);
<span class="kw">loop </span>{
<span class="comment">// Step 1: Generate `u` for selecting the region. If region 1 is
// selected, generate a triangularly distributed variate.
</span><span class="kw">let </span>u = gen_u.sample(rng);
<span class="kw">let </span><span class="kw-2">mut </span>v = gen_v.sample(rng);
<span class="kw">if </span>!(u &gt; p1) {
y = f64_to_i64(x_m - p1 * v + u);
<span class="kw">break</span>;
}
<span class="kw">if </span>!(u &gt; p2) {
<span class="comment">// Step 2: Region 2, parallelograms. Check if region 2 is
// used. If so, generate `y`.
</span><span class="kw">let </span>x = x_l + (u - p1) / c;
v = v * c + <span class="number">1.0 </span>- (x - x_m).abs() / p1;
<span class="kw">if </span>v &gt; <span class="number">1. </span>{
<span class="kw">continue</span>;
} <span class="kw">else </span>{
y = f64_to_i64(x);
}
} <span class="kw">else if </span>!(u &gt; p3) {
<span class="comment">// Step 3: Region 3, left exponential tail.
</span>y = f64_to_i64(x_l + v.ln() / lambda_l);
<span class="kw">if </span>y &lt; <span class="number">0 </span>{
<span class="kw">continue</span>;
} <span class="kw">else </span>{
v <span class="kw-2">*</span>= (u - p2) * lambda_l;
}
} <span class="kw">else </span>{
<span class="comment">// Step 4: Region 4, right exponential tail.
</span>y = f64_to_i64(x_r - v.ln() / lambda_r);
<span class="kw">if </span>y &gt; <span class="number">0 </span>&amp;&amp; (y <span class="kw">as </span>u64) &gt; <span class="self">self</span>.n {
<span class="kw">continue</span>;
} <span class="kw">else </span>{
v <span class="kw-2">*</span>= (u - p3) * lambda_r;
}
}
<span class="comment">// Step 5: Acceptance/rejection comparison.
// Step 5.0: Test for appropriate method of evaluating f(y).
</span><span class="kw">let </span>k = (y - m).abs();
<span class="kw">if </span>!(k &gt; SQUEEZE_THRESHOLD &amp;&amp; (k <span class="kw">as </span>f64) &lt; <span class="number">0.5 </span>* npq - <span class="number">1.</span>) {
<span class="comment">// Step 5.1: Evaluate f(y) via the recursive relationship. Start the
// search from the mode.
</span><span class="kw">let </span>s = p / q;
<span class="kw">let </span>a = s * (n + <span class="number">1.</span>);
<span class="kw">let </span><span class="kw-2">mut </span>f = <span class="number">1.0</span>;
<span class="kw">match </span>m.cmp(<span class="kw-2">&amp;</span>y) {
Ordering::Less =&gt; {
<span class="kw">let </span><span class="kw-2">mut </span>i = m;
<span class="kw">loop </span>{
i += <span class="number">1</span>;
f <span class="kw-2">*</span>= a / (i <span class="kw">as </span>f64) - s;
<span class="kw">if </span>i == y {
<span class="kw">break</span>;
}
}
},
Ordering::Greater =&gt; {
<span class="kw">let </span><span class="kw-2">mut </span>i = y;
<span class="kw">loop </span>{
i += <span class="number">1</span>;
f /= a / (i <span class="kw">as </span>f64) - s;
<span class="kw">if </span>i == m {
<span class="kw">break</span>;
}
}
},
Ordering::Equal =&gt; {},
}
<span class="kw">if </span>v &gt; f {
<span class="kw">continue</span>;
} <span class="kw">else </span>{
<span class="kw">break</span>;
}
}
<span class="comment">// Step 5.2: Squeezing. Check the value of ln(v) against upper and
// lower bound of ln(f(y)).
</span><span class="kw">let </span>k = k <span class="kw">as </span>f64;
<span class="kw">let </span>rho = (k / npq) * ((k * (k / <span class="number">3. </span>+ <span class="number">0.625</span>) + <span class="number">1. </span>/ <span class="number">6.</span>) / npq + <span class="number">0.5</span>);
<span class="kw">let </span>t = -<span class="number">0.5 </span>* k * k / npq;
<span class="kw">let </span>alpha = v.ln();
<span class="kw">if </span>alpha &lt; t - rho {
<span class="kw">break</span>;
}
<span class="kw">if </span>alpha &gt; t + rho {
<span class="kw">continue</span>;
}
<span class="comment">// Step 5.3: Final acceptance/rejection test.
</span><span class="kw">let </span>x1 = (y + <span class="number">1</span>) <span class="kw">as </span>f64;
<span class="kw">let </span>f1 = (m + <span class="number">1</span>) <span class="kw">as </span>f64;
<span class="kw">let </span>z = (f64_to_i64(n) + <span class="number">1 </span>- m) <span class="kw">as </span>f64;
<span class="kw">let </span>w = (f64_to_i64(n) - y + <span class="number">1</span>) <span class="kw">as </span>f64;
<span class="kw">fn </span>stirling(a: f64) -&gt; f64 {
<span class="kw">let </span>a2 = a * a;
(<span class="number">13860. </span>- (<span class="number">462. </span>- (<span class="number">132. </span>- (<span class="number">99. </span>- <span class="number">140. </span>/ a2) / a2) / a2) / a2) / a / <span class="number">166320.
</span>}
<span class="kw">if </span>alpha
&gt; x_m * (f1 / x1).ln()
+ (n - (m <span class="kw">as </span>f64) + <span class="number">0.5</span>) * (z / w).ln()
+ ((y - m) <span class="kw">as </span>f64) * (w * p / (x1 * q)).ln()
<span class="comment">// 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.
</span>+ stirling(f1)
+ stirling(z)
- stirling(x1)
- stirling(w)
{
<span class="kw">continue</span>;
}
<span class="kw">break</span>;
}
<span class="macro">assert!</span>(y &gt;= <span class="number">0</span>);
result = y <span class="kw">as </span>u64;
}
<span class="comment">// Invert the result for p &lt; 0.5.
</span><span class="kw">if </span>p != <span class="self">self</span>.p {
<span class="self">self</span>.n - result
} <span class="kw">else </span>{
result
}
}
}
<span class="attribute">#[cfg(test)]
</span><span class="kw">mod </span>test {
<span class="kw">use </span><span class="kw">super</span>::Binomial;
<span class="kw">use </span><span class="kw">crate</span>::Distribution;
<span class="kw">use </span>rand::Rng;
<span class="kw">fn </span>test_binomial_mean_and_variance&lt;R: Rng&gt;(n: u64, p: f64, rng: <span class="kw-2">&amp;mut </span>R) {
<span class="kw">let </span>binomial = Binomial::new(n, p).unwrap();
<span class="kw">let </span>expected_mean = n <span class="kw">as </span>f64 * p;
<span class="kw">let </span>expected_variance = n <span class="kw">as </span>f64 * p * (<span class="number">1.0 </span>- p);
<span class="kw">let </span><span class="kw-2">mut </span>results = [<span class="number">0.0</span>; <span class="number">1000</span>];
<span class="kw">for </span>i <span class="kw">in </span>results.iter_mut() {
<span class="kw-2">*</span>i = binomial.sample(rng) <span class="kw">as </span>f64;
}
<span class="kw">let </span>mean = results.iter().sum::&lt;f64&gt;() / results.len() <span class="kw">as </span>f64;
<span class="macro">assert!</span>((mean <span class="kw">as </span>f64 - expected_mean).abs() &lt; expected_mean / <span class="number">50.0</span>);
<span class="kw">let </span>variance =
results.iter().map(|x| (x - mean) * (x - mean)).sum::&lt;f64&gt;() / results.len() <span class="kw">as </span>f64;
<span class="macro">assert!</span>((variance - expected_variance).abs() &lt; expected_variance / <span class="number">10.0</span>);
}
<span class="attribute">#[test]
</span><span class="kw">fn </span>test_binomial() {
<span class="kw">let </span><span class="kw-2">mut </span>rng = <span class="kw">crate</span>::test::rng(<span class="number">351</span>);
test_binomial_mean_and_variance(<span class="number">150</span>, <span class="number">0.1</span>, <span class="kw-2">&amp;mut </span>rng);
test_binomial_mean_and_variance(<span class="number">70</span>, <span class="number">0.6</span>, <span class="kw-2">&amp;mut </span>rng);
test_binomial_mean_and_variance(<span class="number">40</span>, <span class="number">0.5</span>, <span class="kw-2">&amp;mut </span>rng);
test_binomial_mean_and_variance(<span class="number">20</span>, <span class="number">0.7</span>, <span class="kw-2">&amp;mut </span>rng);
test_binomial_mean_and_variance(<span class="number">20</span>, <span class="number">0.5</span>, <span class="kw-2">&amp;mut </span>rng);
}
<span class="attribute">#[test]
</span><span class="kw">fn </span>test_binomial_end_points() {
<span class="kw">let </span><span class="kw-2">mut </span>rng = <span class="kw">crate</span>::test::rng(<span class="number">352</span>);
<span class="macro">assert_eq!</span>(rng.sample(Binomial::new(<span class="number">20</span>, <span class="number">0.0</span>).unwrap()), <span class="number">0</span>);
<span class="macro">assert_eq!</span>(rng.sample(Binomial::new(<span class="number">20</span>, <span class="number">1.0</span>).unwrap()), <span class="number">20</span>);
}
<span class="attribute">#[test]
#[should_panic]
</span><span class="kw">fn </span>test_binomial_invalid_lambda_neg() {
Binomial::new(<span class="number">20</span>, -<span class="number">10.0</span>).unwrap();
}
}
</code></pre></div>
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