/**
* Log normal distribution - cumulative distribution function
*
* https://en.wikipedia.org/wiki/Log-normal_distribution
*
* 1 1 / ln x - μ \
* cdf = - + - erf | -------- |
* 2 2 \ √2σ /
*
* @param number $x > 0
* @param number $μ location parameter
* @param number $σ scale parameter > 0
* @return number
*/
public static function CDF($x, $μ, $σ)
{
Support::checkLimits(self::LIMITS, ['x' => $x, 'μ' => $μ, 'σ' => $σ]);
$π = \M_PI;
$⟮ln x − μ⟯ = log($x) - $μ;
$√2σ = sqrt(2) * $σ;
return 1 / 2 + 1 / 2 * Special::erf($⟮ln x − μ⟯ / $√2σ);
}
/**
* Cumulative distribution function
*
* Cumulative t value up to a point, left tail.
*
* / k x \
* γ | - , - |
* \ 2 2 /
* CDF = -------------
* / k \
* Γ | - |
* \ 2 /
*
* @param number $x Chi-square critical value (CV) > 0
* @param int $k degrees of freedom > 0
*
* @return number cumulative probability
*/
public static function CDF($x, int $k)
{
Support::checkLimits(self::LIMITS, ['x' => $x, 'k' => $k]);
// Numerator
$γ⟮k／2、x／2⟯ = Special::γ($k / 2, $x / 2);
// Demoninator
$Γ⟮k／2⟯ = Special::Γ($k / 2);
return $γ⟮k／2、x／2⟯ / $Γ⟮k／2⟯;
}
/**
* Verify the input arguments are valid for correct use of the bisection
* method. If the tolerance is less than zero, an Exception will be thrown.
* If f($a) and f($b) have the same sign, we cannot use the intermediate
* value theorem to guarantee a root is between $a and $b. This exposes the
* risk of an endless loop, so we throw an Exception. If $a = $b, then clearly
* we cannot run our loop as $a and $b will themselves be the midpoint, so we
* throw an Exception.
*
* @param Callable $function f(x) callback function
* @param number $a The start of the interval which contains a root
* @param number $b The end of the interval which contains a root
* @param number $tol Tolerance; How close to the actual solution we would like.
*
* @throws Exception if $tol (the tolerance) is negative
* @throws Exception if $a = $b
* @throws Exception if f($a) and f($b) share the same sign
*/
private static function validate(callable $function, $a, $b, $tol)
{
Validation::tolerance($tol);
Validation::interval($a, $b);
$f⟮a⟯ = $function($a);
$f⟮b⟯ = $function($b);
if (Special::sgn($f⟮a⟯) === Special::sgn($f⟮b⟯)) {
throw new Exception\BadDataException('Input function has the same sign at the start and end of the interval. Choose start and end points such that the function evaluated at those points has a different sign (one positive, one negative).');
}
}
/**
* 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;
}
/**
* Mean of the distribution
* _
* /ν Γ((ν - 1)/2)
* E[T] = μ / - ------------ if ν > 1
* √ 2 Γ(ν/2)
*
* = Does not exist if ν ≤ 1
*
* @param int $ν Degrees of freedom
* @param number $μ Noncentrality parameter
*
* @return number
*/
public static function mean(int $ν, $μ)
{
Support::checkLimits(self::LIMITS, ['ν' => $ν, 'μ' => $μ]);
if ($ν == 1) {
return \NAN;
}
return $μ * sqrt($ν / 2) * Special::gamma(($ν - 1) / 2) / Special::gamma($ν / 2);
}
/**
* Mean of the distribution
*
* μ = λΓ(1 + 1/k)
*
* @param number $k shape parameter
* @param number $λ scale parameter
*
* @return number
*/
public static function mean($k, $λ)
{
Support::checkLimits(self::LIMITS, ['k' => $k, 'λ' => $λ]);
return $λ * Special::gamma(1 + 1 / $k);
}
/**
* Cumulative distribution function
* Probability of being below X.
* Area under the normal distribution from -∞ to X.
* _ _
* 1 | / x - μ \ |
* cdf(x) = - | 1 + erf| ----- | |
* 2 |_ \ σ√2 / _|
*
* @param number $x upper bound
* @param number $μ mean
* @param number $σ standard deviation
*
* @return float cdf(x) below
*/
public static function CDF($x, $μ, $σ) : float
{
Support::checkLimits(self::LIMITS, ['x' => $x, 'μ' => $μ, 'σ' => $σ]);
return 1 / 2 * (1 + Special::erf(($x - $μ) / ($σ * sqrt(2))));
}
/**
* 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);
}