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use distribution;
use random;
#[derive(Clone, Copy)]
pub struct Binomial {
n: usize,
p: f64,
q: f64,
np: f64,
nq: f64,
npq: f64,
}
impl Binomial {
pub fn new(n: usize, p: f64) -> Binomial {
should!(0.0 < p && p < 1.0);
let q = 1.0 - p;
let np = n as f64 * p;
let nq = n as f64 * q;
Binomial { n: n, p: p, q: q, np: np, nq: nq, npq: np * q }
}
pub fn with_failure(n: usize, q: f64) -> Binomial {
should!(0.0 < q && q < 1.0);
let p = 1.0 - q;
let np = n as f64 * p;
let nq = n as f64 * q;
Binomial { n: n, p: p, q: q, np: np, nq: nq, npq: np * q }
}
#[inline(always)]
pub fn n(&self) -> usize { self.n }
#[inline(always)]
pub fn p(&self) -> f64 { self.p }
#[inline(always)]
pub fn q(&self) -> f64 { self.q }
}
impl distribution::Distribution for Binomial {
type Value = usize;
fn cdf(&self, x: f64) -> f64 {
use special::{inc_beta, ln_beta};
if x < 0.0 {
return 0.0;
}
let x = x as usize;
if x == 0 {
return self.q.powi(self.n as i32);
}
if x >= self.n {
return 1.0;
}
let (p, q) = ((self.n - x) as f64, (x + 1) as f64);
inc_beta(self.q, p, q, ln_beta(p, q))
}
}
impl distribution::Discrete for Binomial {
fn pmf(&self, x: usize) -> f64 {
use std::f64::consts::PI;
if self.p == 0.0 {
return if x == 0 { 1.0 } else { 0.0 };
}
if self.p == 1.0 {
return if x == self.n { 1.0 } else { 0.0 };
}
let n = self.n as f64;
if x == 0 {
(n * self.q.ln()).exp()
} else if x == self.n {
(n * self.p.ln()).exp()
} else {
let x = x as f64;
let n_m_x = n - x;
let ln_c = stirlerr(n) - stirlerr(x) - stirlerr(n_m_x)
- ln_d0(x, self.np) - ln_d0(n_m_x, self.nq);
ln_c.exp() * (n / (2.0 * PI * x * (n_m_x))).sqrt()
}
}
}
impl distribution::Entropy for Binomial {
fn entropy(&self) -> f64 {
use distribution::Discrete;
use std::f64::consts::PI;
if self.n > 10000 && self.npq > 80.0 {
0.5 * ((2.0 * PI * self.npq).ln() + 1.0)
} else {
-(0..(self.n + 1)).fold(0.0, |sum, i| sum + self.pmf(i) * self.pmf(i).ln())
}
}
}
impl distribution::Inverse for Binomial {
fn inv_cdf(&self, p: f64) -> usize {
use distribution::{Distribution, Discrete, Modes};
should!(0.0 <= p && p <= 1.0);
let u = p;
macro_rules! buttom_up_sum(
($prod_term: expr) => ({
let mut k = 1;
let mut a = self.q.powi(self.n as i32);
let mut sum = a - u;
while sum < 0.0 {
a *= $prod_term(k);
sum += a;
k += 1;
}
k - 1
});
);
macro_rules! top_down_sum(
($prod_term: expr) => ({
let mut k = 1;
let mut a = self.p.powi(self.n as i32);
let mut sum = (1.0 - u) - a;
while sum >= 0.0 {
a *= $prod_term(k);
sum -= a;
k += 1;
}
self.n - k + 1
});
);
if u == 1.0 {
self.n
} else if u == 0.0 {
0
} else if self.n < 1000 {
if u <= self.cdf((self.n / 2) as f64) {
buttom_up_sum!(|k| self.p / self.q * ((self.n - k + 1) as f64 / k as f64))
} else {
top_down_sum!(|k| self.q / self.p * ((self.n - k + 1) as f64 / k as f64))
}
} else if self.npq > 80.0 {
approximate_by_normal(self.p, self.np, self.npq, u).floor() as usize
} else {
let modes = self.modes();
let mut m = modes[0];
loop {
let next = (u - self.cdf(m as f64)) / self.pmf(m);
if -0.5 < next && next < 0.5 {
break;
}
m = (m as isize + next.round() as isize) as usize;
}
m
}
}
}
impl distribution::Kurtosis for Binomial {
#[inline]
fn kurtosis(&self) -> f64 {
(1.0 - 6.0 * self.p * self.q) / self.npq
}
}
impl distribution::Mean for Binomial {
#[inline]
fn mean(&self) -> f64 { self.np }
}
impl distribution::Median for Binomial {
fn median(&self) -> f64 {
use distribution::Inverse;
use std::f64::consts::LN_2;
if self.np.fract() == 0.0 {
self.np
} else if self.p == 0.5 && self.n % 2 != 0 {
self.np
} else if self.p <= 1.0 - LN_2 || self.p >= LN_2 ||
(self.np.round() - self.np).abs() <= self.p.min(self.q) {
self.np.round()
} else if self.n > 1000 && self.npq > 80.0 {
self.np.floor()
} else {
self.inv_cdf(0.5) as f64
}
}
}
impl distribution::Modes for Binomial {
fn modes(&self) -> Vec<usize> {
let r = self.p * (self.n + 1) as f64;
if r == 0.0 {
vec![0]
} else if self.p == 1.0 {
vec![self.n]
} else if r.fract() != 0.0 {
vec![r.floor() as usize]
} else {
vec![r as usize - 1, r as usize]
}
}
}
impl distribution::Sample for Binomial {
#[inline]
fn sample<S>(&self, source: &mut S) -> usize where S: random::Source {
use distribution::Inverse;
self.inv_cdf(source.read::<f64>())
}
}
impl distribution::Skewness for Binomial {
#[inline]
fn skewness(&self) -> f64 {
(1.0 - 2.0 * self.p) / (self.npq).sqrt()
}
}
impl distribution::Variance for Binomial {
#[inline]
fn variance(&self) -> f64 { self.npq }
}
fn approximate_by_normal(p: f64, np: f64, v: f64, u: f64) -> f64 {
use distribution::gaussian;
let w = gaussian::inv_cdf(u);
let w2 = w * w;
let w3 = w2 * w;
let w4 = w3 * w;
let w5 = w4 * w;
let w6 = w5 * w;
let sd = v.sqrt();
let sd_em1 = sd.recip();
let sd_em2 = v.recip();
let sd_em3 = sd_em1 * sd_em2;
let sd_em4 = sd_em2 * sd_em2;
let p2 = p * p;
let p3 = p2 * p;
let p4 = p2 * p2;
np +
sd * w +
(p + 1.0) / 3.0 -
(2.0 * p - 1.0) * w2 / 6.0 +
sd_em1 * w3 * (2.0 * p2 - 2.0 * p - 1.0) / 72.0 -
w * (7.0 * p2 - 7.0 * p + 1.0) / 36.0 +
sd_em2 * (2.0 * p - 1.0) * (p + 1.0) * (p - 2.0) * (3.0 * w4 + 7.0 * w2 - 16.0 / 1620.0) +
sd_em3 * (
w5 * (4.0 * p4 - 8.0 * p3 - 48.0 * p2 + 52.0 * p - 23.0) / 17280.0 +
w3 * (256.0 * p4 - 512.0 * p3 - 147.0 * p2 + 403.0 * p - 137.0) / 38880.0 -
w * (433.0 * p4 - 866.0 * p3 - 921.0 * p2 + 1354.0 * p - 671.0) / 38880.0
) +
sd_em4 * (
w6 * (2.0 * p - 1.0) * (p2 - p + 1.0) * (p2 - p + 19.0) / 34020.0 +
w4 * (2.0 * p - 1.0) * (9.0 * p4 - 18.0 * p3 - 35.0 * p2 + 44.0 * p - 25.0) / 15120.0 +
w2 * (2.0 * p - 1.0) * (
923.0 * p4 - 1846.0 * p3 + 5271.0 * p2 - 4348.0 * p + 5189.0
) / 408240.0 -
4.0 * (2.0 * p - 1.0) * (p + 1.0) * (p - 2.0) * (23.0 * p2 - 23.0 * p + 2.0) / 25515.0
)
}
fn stirlerr(n: f64) -> f64 {
const S0: f64 = 1.0 / 12.0;
const S1: f64 = 1.0 / 360.0;
const S2: f64 = 1.0 / 1260.0;
const S3: f64 = 1.0 / 1680.0;
const S4: f64 = 1.0 / 1188.0;
const SFE: [f64; 16] = [
0.000000000000000000e+00, 8.106146679532725822e-02,
4.134069595540929409e-02, 2.767792568499833915e-02,
2.079067210376509311e-02, 1.664469118982119216e-02,
1.387612882307074800e-02, 1.189670994589177010e-02,
1.041126526197209650e-02, 9.255462182712732918e-03,
8.330563433362871256e-03, 7.757367548795184079e-03,
6.942840107209529866e-03, 6.408994188004207068e-03,
5.951370112758847736e-03, 5.554733551962801371e-03,
];
if n < 16.0 {
return SFE[n as usize];
}
let nn = n * n;
if n > 500.0 {
(S0 - S1 / nn) / n
} else if n > 80.0 {
(S0 - (S1 - S2 / nn) / nn) / n
} else if n > 35.0 {
(S0 - (S1 - (S2 - S3 / nn) / nn) / nn) / n
} else {
(S0 - (S1 - (S2 - (S3 - S4 / nn) / nn) / nn) / nn) / n
}
}
fn ln_d0(x: f64, np: f64) -> f64 {
if (x - np).abs() < 0.1 * (x + np) {
let mut s = (x - np).powi(2) / (x + np);
let v = (x - np) / (x + np);
let mut ej = 2.0 * x * v;
let mut j = 1;
loop {
ej *= v * v;
let s1 = s + ej / (2 * j + 1) as f64;
if s1 == s {
return s1;
}
s = s1;
j += 1;
}
}
x * (x / np).ln() + np - x
}
#[cfg(test)]
mod tests {
use assert;
use prelude::*;
macro_rules! new {
($n:expr, $p:expr) => (Binomial::new($n, $p));
}
#[test]
fn cdf() {
let d = new!(16, 0.75);
let p = vec![
0.000000000000000e+00, 2.328306436538699e-10, 2.628657966852194e-07,
3.810715861618527e-05, 1.644465373829007e-03, 2.712995628826319e-02,
1.896545726340262e-01, 5.950128899421541e-01, 9.365235602017492e-01,
1.000000000000000e+00,
];
let x = (-1..9).map(|i| d.cdf(2.0 * i as f64)).collect::<Vec<_>>();
assert::close(&x, &p, 1e-14);
let x = (-1..9).map(|i| d.cdf(2.0 * i as f64 + 0.5)).collect::<Vec<_>>();
assert::close(&x, &p, 1e-14);
}
#[test]
fn pmf() {
let d = new!(16, 0.25);
let p = vec![
1.002259575761855e-02, 1.336346101015806e-01, 2.251990651711821e-01,
1.100973207503558e-01, 1.966023584827779e-02, 1.359226182103156e-03,
3.432389348745344e-05, 2.514570951461788e-07, 2.328306436538698e-10,
];
assert::close(&(0..9).map(|i| d.pmf(2 * i)).collect::<Vec<_>>(), &p, 1e-14);
}
#[test]
fn entropy() {
assert_eq!(new!(16, 0.25).entropy(), 1.9588018945068573);
assert_eq!(new!(10_000_000, 0.5).entropy(), 8.784839178123887);
}
#[test]
fn inv_cdf() {
let d = Binomial::new(250, 0.55);
assert_eq!(d.inv_cdf(0.1), 127);
assert_eq!(d.inv_cdf(0.025), 122);
let x = 1298;
let d = new!(2500, 0.55);
assert_eq!(d.inv_cdf(d.cdf(x as f64)), x);
assert_eq!(new!(1001, 0.25).inv_cdf(0.5), 250);
assert_eq!(new!(1500, 0.15).inv_cdf(0.2), 213);
assert_eq!(new!(1_000_000, 2.5e-5).inv_cdf(0.9995), 42);
assert_eq!(new!(1_000_000_000, 6.66e-9).inv_cdf(0.8), 8);
}
#[test]
fn kurtosis() {
assert_eq!(new!(16, 0.25).kurtosis(), -0.041666666666666664);
}
#[test]
fn mean() {
assert_eq!(new!(16, 0.25).mean(), 4.0);
}
#[test]
fn median() {
assert_eq!(new!(16, 0.25).median(), 4.0);
assert_eq!(new!(3, 0.5).median(), 1.5);
assert_eq!(new!(1000, 0.015).median(), 15.0);
assert_eq!(new!(39, 0.1).median(), 4.0);
}
#[test]
fn modes() {
assert_eq!(new!(16, 0.25).modes(), vec![4]);
assert_eq!(new!(3, 0.5).modes(), vec![1, 2]);
assert_eq!(new!(1000, 0.015).modes(), vec![15]);
assert_eq!(new!(39, 0.1).modes(), vec![3, 4]);
}
#[test]
fn skewness() {
assert_eq!(new!(16, 0.25).skewness(), 0.2886751345948129);
}
#[test]
fn variance() {
assert_eq!(new!(16, 0.25).variance(), 3.0);
}
}