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#![allow(clippy::manual_range_contains)]
#![allow(unstable_name_collisions)]
#[allow(unused_imports)]
use primitive::Primitive;
use Gamma;
/// Beta functions.
pub trait Beta {
/// Compute the regularized incomplete beta function.
///
/// The code is based on a [C implementation][1] by John Burkardt. The
/// original algorithm was published in Applied Statistics and is known as
/// [Algorithm AS 63][2] and [Algorithm AS 109][3].
///
/// [1]: http://people.sc.fsu.edu/~jburkardt/c_src/asa109/asa109.html
/// [2]: http://www.jstor.org/stable/2346797
/// [3]: http://www.jstor.org/stable/2346887
fn inc_beta(self, p: Self, q: Self, ln_beta: Self) -> Self;
/// Compute the inverse of the regularized incomplete beta function.
///
/// The code is based on a [C implementation][1] by John Burkardt. The
/// original algorithm was published in Applied Statistics and is known as
/// [Algorithm AS 64][2] and [Algorithm AS 109][3].
///
/// [1]: http://people.sc.fsu.edu/~jburkardt/c_src/asa109/asa109.html
/// [2]: http://www.jstor.org/stable/2346798
/// [3]: http://www.jstor.org/stable/2346887
fn inv_inc_beta(self, p: Self, q: Self, ln_beta: Self) -> Self;
/// Compute the natural logarithm of the beta function.
fn ln_beta(self, other: Self) -> Self;
}
#[rustfmt::skip]
macro_rules! implement { ($kind:ident) => { impl Beta for $kind {
fn inc_beta(self, mut p: Self, mut q: Self, ln_beta: Self) -> Self {
// Algorithm AS 63
// http://www.jstor.org/stable/2346797
//
// The function uses the method discussed by Soper (1921). If p is not
// less than (p + q)x and the integral part of q + (1 - x)(p + q) is a
// positive integer, say s, reductions are made up to s times “by parts”
// using the recurrence relation
//
// Γ(p + q)
// I(x, p, q) = ------------- x^p (1 - x)^(q - 1) + I(x, p + 1, q - 1)
// Γ(p + 1) Γ(q)
//
// and then reductions are continued by “raising p” with the recurrence
// relation
//
// Γ(p + q)
// I(x, p + s, q - s) = --------------------- x^(p + s) (1 - x)^(q - s)
// Γ(p + s + 1) Γ(q - s)
//
// + I(x, p + s + 1, q - s)
//
// If s is not a positive integer, reductions are made only by “raising
// p.” The process of reduction is terminated when the relative
// contribution to the integral is not greater than the value of ACU. If
// p is less than (p + q)x, I(1 - x, q, p) is first calculated by the
// above procedure and then I(x, p, q) is obtained from the relation
//
// I(x, p, q) = 1 - I(1 - x, p, q).
//
// Soper (1921) demonstrated that the expansion of I(x, p, q) by “parts”
// and “raising p” method as described above converges more rapidly than
// any other series expansions.
const ACU: $kind = 0.1e-14;
let x = self;
debug_assert!(x >= 0.0 && x <= 1.0 && p > 0.0 && q > 0.0);
if x == 0.0 {
return 0.0;
}
if x == 1.0 {
return 1.0;
}
let mut psq = p + q;
let pbase;
let qbase;
let mut temp;
let flip = p < psq * x;
if flip {
pbase = 1.0 - x;
qbase = x;
temp = q;
q = p;
p = temp;
} else {
pbase = x;
qbase = 1.0 - x;
}
let mut term = 1.0;
let mut ai = 1.0;
let mut rx;
let mut ns = (q + qbase * psq) as isize;
if ns == 0 {
rx = pbase;
} else {
rx = pbase / qbase;
}
let mut a = 1.0;
temp = q - ai;
loop {
term = term * temp * rx / (p + ai);
a += term;
temp = if term < 0.0 { -term } else { term };
if temp <= ACU && temp <= ACU * a {
break;
}
ai += 1.0;
ns -= 1;
if 0 < ns {
temp = q - ai;
} else if ns == 0 {
temp = q - ai;
rx = pbase;
} else {
temp = psq;
psq += 1.0;
}
}
// Remark AS R19 and Algorithm AS 109
// http://www.jstor.org/stable/2346887
a = a * (p * pbase.ln() + (q - 1.0) * qbase.ln() - ln_beta).exp() / p;
if flip {
1.0 - a
} else {
a
}
}
fn inv_inc_beta(self, mut p: Self, mut q: Self, ln_beta: Self) -> Self {
// Algorithm AS 64
// http://www.jstor.org/stable/2346798
//
// An approximation x₀ to x if found from (cf. Scheffé and Tukey, 1944)
//
// 1 + x₀ 4p + 2q - 2
// ------ = -----------
// 1 - x₀ χ²(α)
//
// where χ²(α) is the upper α point of the χ² distribution with 2q
// degrees of freedom and is obtained from Wilson and Hilferty’s
// approximation (cf. Wilson and Hilferty, 1931)
//
// χ²(α) = 2q (1 - 1 / (9q) + y(α) sqrt(1 / (9q)))^3,
//
// y(α) being Hastings’ approximation (cf. Hastings, 1955) for the upper
// α point of the standard normal distribution. If χ²(α) < 0, then
//
// x₀ = 1 - ((1 - α)q B(p, q))^(1 / q).
//
// Again if (4p + 2q - 2) / χ²(α) does not exceed 1, x₀ is obtained from
//
// x₀ = (αp B(p, q))^(1 / p).
//
// The final solution is obtained by the Newton–Raphson method from the
// relation
//
// f(x[i - 1])
// x[i] = x[i - 1] - ------------
// f'(x[i - 1])
//
// where
//
// f(x) = I(x, p, q) - α.
// Remark AS R83
// http://www.jstor.org/stable/2347779
const SAE: i32 = -30;
const FPU: $kind = 1e-30; // 10^SAE
let mut a = self;
debug_assert!(a >= 0.0 && a <= 1.0 && p > 0.0 && q > 0.0);
if a == 0.0 {
return 0.0;
}
if a == 1.0 {
return 1.0;
}
let mut x;
let mut y;
let flip = 0.5 < a;
if flip {
x = p;
p = q;
q = x;
a = 1.0 - a;
}
x = (-(a * a).ln()).sqrt();
y = x - (2.30753 + 0.27061 * x) / (1.0 + (0.99229 + 0.04481 * x) * x);
if 1.0 < p && 1.0 < q {
// Remark AS R19 and Algorithm AS 109
// http://www.jstor.org/stable/2346887
//
// For p and q > 1, the approximation given by Carter (1947), which
// improves the Fisher–Cochran formula, is generally better. For
// other values of p and q en empirical investigation has shown that
// the approximation given in AS 64 is adequate.
let r = (y * y - 3.0) / 6.0;
let s = 1.0 / (2.0 * p - 1.0);
let t = 1.0 / (2.0 * q - 1.0);
let h = 2.0 / (s + t);
let w = y * (h + r).sqrt() / h - (t - s) * (r + 5.0 / 6.0 - 2.0 / (3.0 * h));
x = p / (p + q * (2.0 * w).exp());
} else {
let mut t = 1.0 / (9.0 * q);
t = 2.0 * q * (1.0 - t + y * t.sqrt()).powf(3.0);
if t <= 0.0 {
x = 1.0 - ((((1.0 - a) * q).ln() + ln_beta) / q).exp();
} else {
t = 2.0 * (2.0 * p + q - 1.0) / t;
if t <= 1.0 {
x = (((a * p).ln() + ln_beta) / p).exp();
} else {
x = 1.0 - 2.0 / (t + 1.0);
}
}
}
if x < 0.0001 {
x = 0.0001;
} else if 0.9999 < x {
x = 0.9999;
}
// Remark AS R83
// http://www.jstor.org/stable/2347779
let e = (-5.0 / p / p - 1.0 / a.powf(0.2) - 13.0) as i32;
let acu = if e > SAE { $kind::powi(10.0, e) } else { FPU };
let mut tx;
let mut yprev = 0.0;
let mut sq = 1.0;
let mut prev = 1.0;
'outer: loop {
// Remark AS R19 and Algorithm AS 109
// http://www.jstor.org/stable/2346887
y = x.inc_beta(p, q, ln_beta);
y = (y - a) * (ln_beta + (1.0 - p) * x.ln() + (1.0 - q) * (1.0 - x).ln()).exp();
// Remark AS R83
// http://www.jstor.org/stable/2347779
if y * yprev <= 0.0 {
prev = if sq > FPU { sq } else { FPU };
}
// Remark AS R19 and Algorithm AS 109
// http://www.jstor.org/stable/2346887
let mut g = 1.0;
loop {
loop {
let adj = g * y;
sq = adj * adj;
if sq < prev {
tx = x - adj;
if 0.0 <= tx && tx <= 1.0 {
break;
}
}
g /= 3.0;
}
if prev <= acu || y * y <= acu {
x = tx;
break 'outer;
}
if tx != 0.0 && tx != 1.0 {
break;
}
g /= 3.0;
}
if tx == x {
break;
}
x = tx;
yprev = y;
}
if flip {
1.0 - x
} else {
x
}
}
fn ln_beta(self, other: Self) -> Self {
debug_assert!(self > 0.0 && other > 0.0);
self.ln_gamma().0 + other.ln_gamma().0 - (self + other).ln_gamma().0
}
}}}
implement!(f32);
implement!(f64);
#[cfg(test)]
mod tests {
use alloc::{vec, vec::Vec};
use assert;
use super::Beta;
#[test]
fn inc_beta_small() {
let (p, q) = (0.1, 0.2);
let ln_beta = p.ln_beta(q);
let x = vec![
0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60, 0.65,
0.70, 0.75, 0.80, 0.85, 0.90, 0.95, 1.00,
];
let a = vec![
0.000000000000000e+00,
5.095391215346399e-01,
5.482400859052436e-01,
5.732625733722232e-01,
5.925346573554778e-01,
6.086596697678208e-01,
6.228433547203172e-01,
6.357578563479236e-01,
6.478288604374864e-01,
6.593557133297501e-01,
6.705707961028990e-01,
6.816739425887479e-01,
6.928567823206671e-01,
7.043251807250750e-01,
7.163269829958610e-01,
7.291961263917867e-01,
7.434379555965913e-01,
7.599272566076309e-01,
7.804880320024465e-01,
8.104335200313719e-01,
1.000000000000000e+00,
];
let y = x
.iter()
.map(|&x| x.inc_beta(p, q, ln_beta))
.collect::<Vec<_>>();
assert::close(&y, &a, 1e-14);
}
#[test]
fn inc_beta_large() {
let (p, q) = (2.0, 3.0);
let ln_beta = p.ln_beta(q);
let x = vec![
0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60, 0.65,
0.70, 0.75, 0.80, 0.85, 0.90, 0.95, 1.00,
];
let a = vec![
0.000000000000000e+00,
1.401875000000000e-02,
5.230000000000002e-02,
1.095187500000000e-01,
1.807999999999999e-01,
2.617187500000001e-01,
3.483000000000000e-01,
4.370187500000001e-01,
5.248000000000003e-01,
6.090187500000001e-01,
6.875000000000000e-01,
7.585187500000001e-01,
8.208000000000000e-01,
8.735187499999999e-01,
9.163000000000000e-01,
9.492187500000000e-01,
9.728000000000000e-01,
9.880187500000001e-01,
9.963000000000000e-01,
9.995187500000000e-01,
1.000000000000000e+00,
];
let y = x
.iter()
.map(|&x| x.inc_beta(p, q, ln_beta))
.collect::<Vec<_>>();
assert::close(&y, &a, 1e-14);
}
#[test]
fn inv_inc_beta_small() {
let (p, q) = (0.2, 0.3);
let ln_beta = p.ln_beta(q);
let a = vec![
0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60, 0.65,
0.70, 0.75, 0.80, 0.85, 0.90, 0.95, 1.00,
];
let x = vec![
0.000000000000000e+00,
2.793072851850660e-06,
8.937381711316164e-05,
6.784491773826951e-04,
2.855345858289119e-03,
8.684107512129325e-03,
2.144658503798324e-02,
4.568556852983932e-02,
8.683942933344659e-02,
1.502095712585510e-01,
2.391350361479824e-01,
3.527066234122371e-01,
4.840600731467657e-01,
6.206841200371190e-01,
7.474718280552188e-01,
8.514539745840592e-01,
9.257428898178934e-01,
9.707021084050310e-01,
9.923134416335146e-01,
9.992341305241808e-01,
1.000000000000000e+00,
];
let y = a
.iter()
.map(|&a| a.inv_inc_beta(p, q, ln_beta))
.collect::<Vec<_>>();
assert::close(&y, &x, 1e-14);
}
#[test]
fn inv_inc_beta_large() {
let (p, q) = (1.0, 2.0);
let ln_beta = p.ln_beta(q);
let a = vec![
0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60, 0.65,
0.70, 0.75, 0.80, 0.85, 0.90, 0.95, 1.00,
];
let x = vec![
0.000000000000000e+00,
0.025320565519104e+00,
0.051316701949486e+00,
0.078045554270711e+00,
0.105572809000084e+00,
0.133974596215561e+00,
0.163339973465924e+00,
0.193774225170145e+00,
0.225403330758517e+00,
0.258380151290432e+00,
0.292893218813452e+00,
0.329179606750063e+00,
0.367544467966324e+00,
0.408392021690038e+00,
0.452277442494834e+00,
0.500000000000000e+00,
0.552786404500042e+00,
0.612701665379257e+00,
0.683772233983162e+00,
0.776393202250021e+00,
1.000000000000000e+00,
];
let y = a
.iter()
.map(|&a| a.inv_inc_beta(p, q, ln_beta))
.collect::<Vec<_>>();
assert::close(&y, &x, 1e-14);
}
#[test]
fn ln_beta() {
let x = vec![(0.25, 0.5), (0.5, 0.75), (0.75, 1.0), (1.0, 1.25)];
let y = vec![
1.6571065161914820,
0.8739177307778084,
0.2876820724517809,
-0.2231435513142098,
];
let z = x.iter().map(|&(p, q)| p.ln_beta(q)).collect::<Vec<_>>();
assert::close(&z, &y, 1e-14);
}
}