https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function
This function looks at the values of x, a, and b, and determines which algorithm is best to calculate
the value of Iₓ(a, b)
http://www.boost.org/doc/libs/1_35_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/sf_beta/ibeta_function.html
https://github.com/boostorg/math/blob/develop/include/boost/math/special_functions/beta.hpp
| public static regularizedIncompleteBeta ( $x, $a, $b ) : number | ||
| $x | Upper limit of the integration 0 ≦ x ≦ 1 | |
| $a | Shape parameter a > 0 | |
| $b | Shape parameter b > 0 | |
| Результат | number | |
/**
* Cumulative distribution function
* Calculate the cumulative t value up to a point, left tail.
*
* cdf = 1 - ½Iₓ₍t₎(ν/2, ½)
*
* ν
* where x(t) = ------
* t² + ν
*
* Iₓ₍t₎(ν/2, ½) is the regularized incomplete beta function
*
* @param number $t t score
* @param int $ν degrees of freedom > 0
*/
public static function CDF($t, int $ν)
{
Support::checkLimits(self::LIMITS, ['t' => $t, 'ν' => $ν]);
if ($t == 0) {
return 0.5;
}
$x⟮t⟯ = $ν / ($t ** 2 + $ν);
$ν／2 = $ν / 2;
$½ = 0.5;
$Iₓ = Special::regularizedIncompleteBeta($x⟮t⟯, $ν／2, $½);
if ($t < 0) {
return $½ * $Iₓ;
}
// $t ≥ 0
return 1 - $½ * $Iₓ;
}
/**
* F used within CDF
* _ _
* 1 ∞ | / 1 ν \ / ν \ |
* Fᵥ,ᵤ(x) = Φ(-μ) + - ∑ | pⱼIy | j + -, - | + qⱼIy | j + 1, - | |
* 2 ʲ⁼⁰ |_ \ 2 2 / \ 2 / _|
*
* where
* Φ = cumulative distribution function of the standard normal distribution
* Iy(a,b) = regularized incomplete beta function
*
* x²
* y = ------
* x² + ν
*
* 1 / μ² \ / μ² \ʲ
* pⱼ = -- exp| - - | | - |
* j! \ 2 / \ 2 /
*
* μ / μ² \ / μ² \ʲ
* qⱼ = ------------ exp| - - | | - |
* √2Γ(j + 3/2) \ 2 / \ 2 /
*
* @param number $x
* @param int $ν Degrees of freedom
* @param number $μ Noncentrality parameter
*
* @return number
*/
private static function F($x, int $ν, $μ)
{
Support::checkLimits(self::LIMITS, ['x' => $x, 'ν' => $ν, 'μ' => $μ]);
$Φ = StandardNormal::CDF(-$μ);
$y = $x ** 2 / ($x ** 2 + $ν);
$sum = $Φ;
$tol = 1.0E-8;
$j = 0;
do {
$exp = exp(-1 * $μ ** 2 / 2) * ($μ ** 2 / 2) ** $j;
$pⱼ = 1 / Combinatorics::factorial($j) * $exp;
$qⱼ = $μ / sqrt(2) / Special::gamma($j + 3 / 2) * $exp;
$I1 = Special::regularizedIncompleteBeta($y, $j + 1 / 2, $ν / 2);
$I2 = Special::regularizedIncompleteBeta($y, $j + 1, $ν / 2);
$delta = $pⱼ * $I1 + $qⱼ * $I2;
$sum += $delta / 2;
$j += 1;
} while ($delta / $sum > $tol || $j < 10);
return $sum;
}
/**
* Cumulative distribution function
*
* cdf = Iₓ(α,β)
*
* @param number $α shape parameter α > 0
* @param number $β shape parameter β > 0
* @param number $x x ∈ (0,1)
*
* @return float
*/
public static function CDF($x, $α, $β)
{
Support::checkLimits(self::LIMITS, ['x' => $x, 'α' => $α, 'β' => $β]);
return Special::regularizedIncompleteBeta($x, $α, $β);
}
/**
* Cumulative distribution function
*
* / d₁ d₂ \
* I | --, -- |
* ᵈ¹ˣ \ 2 2 /
* ------
* ᵈ¹ˣ⁺ᵈ²
*
* Where I is the regularized incomplete beta function.
*
* @param number $x percentile ≥ 0
* @param int $d₁ degree of freedom v1 > 0
* @param int $d₂ degree of freedom v2 > 0
*
* @return number
*/
public static function CDF($x, int $d₁, int $d₂)
{
Support::checkLimits(self::LIMITS, ['x' => $x, 'd₁' => $d₁, 'd₂' => $d₂]);
$ᵈ¹ˣ／d₁x＋d₂ = $d₁ * $x / ($d₁ * $x + $d₂);
return Special::regularizedIncompleteBeta($ᵈ¹ˣ／d₁x＋d₂, $d₁ / 2, $d₂ / 2);
}
