GammaBase.php

Go to the documentation of this file.

<?php

namespace PhpOffice\PhpSpreadsheet\Calculation\Statistical\Distributions;

use PhpOffice\PhpSpreadsheet\Calculation\Functions;

abstract class GammaBase
{
    private const LOG_GAMMA_X_MAX_VALUE = 2.55e305;

    private const EPS = 2.22e-16;

    private const MAX_VALUE = 1.2e308;

    private const SQRT2PI = 2.5066282746310005024157652848110452530069867406099;

    private const MAX_ITERATIONS = 256;

    protected static function calculateDistribution(float $value, float $a, float $b, bool $cumulative)
    {
        if ($cumulative) {
            return self::incompleteGamma($a, $value / $b) / self::gammaValue($a);
        }

        return (1 / ($b ** $a * self::gammaValue($a))) * $value ** ($a - 1) * exp(0 - ($value / $b));
    }

    protected static function calculateInverse(float $probability, float $alpha, float $beta)
    {
        $xLo = 0;
        $xHi = $alpha * $beta * 5;

        $dx = 1024;
        $x = $xNew = 1;
        $i = 0;

        while ((abs($dx) > Functions::PRECISION) && (++$i <= self::MAX_ITERATIONS)) {
            // Apply Newton-Raphson step
            $result = self::calculateDistribution($x, $alpha, $beta, true);
            $error = $result - $probability;

            if ($error == 0.0) {
                $dx = 0;
            } elseif ($error < 0.0) {
                $xLo = $x;
            } else {
                $xHi = $x;
            }

            $pdf = self::calculateDistribution($x, $alpha, $beta, false);
            // Avoid division by zero
            if ($pdf !== 0.0) {
                $dx = $error / $pdf;
                $xNew = $x - $dx;
            }

            // If the NR fails to converge (which for example may be the
            // case if the initial guess is too rough) we apply a bisection
            // step to determine a more narrow interval around the root.
            if (($xNew < $xLo) || ($xNew > $xHi) || ($pdf == 0.0)) {
                $xNew = ($xLo + $xHi) / 2;
                $dx = $xNew - $x;
            }
            $x = $xNew;
        }

        if ($i === self::MAX_ITERATIONS) {
            return Functions::NA();
        }

        return $x;
    }

    //
    //    Implementation of the incomplete Gamma function
    //
    public static function incompleteGamma(float $a, float $x): float
    {
        static $max = 32;
        $summer = 0;
        for ($n = 0; $n <= $max; ++$n) {
            $divisor = $a;
            for ($i = 1; $i <= $n; ++$i) {
                $divisor *= ($a + $i);
            }
            $summer += ($x ** $n / $divisor);
        }

        return $x ** $a * exp(0 - $x) * $summer;
    }

    //
    //    Implementation of the Gamma function
    //
    public static function gammaValue(float $value): float
    {
        if ($value == 0.0) {
            return 0;
        }

        static $p0 = 1.000000000190015;
        static $p = [
            1 => 76.18009172947146,
            2 => -86.50532032941677,
            3 => 24.01409824083091,
            4 => -1.231739572450155,
            5 => 1.208650973866179e-3,
            6 => -5.395239384953e-6,
        ];

        $y = $x = $value;
        $tmp = $x + 5.5;
        $tmp -= ($x + 0.5) * log($tmp);

        $summer = $p0;
        for ($j = 1; $j <= 6; ++$j) {
            $summer += ($p[$j] / ++$y);
        }

        return exp(0 - $tmp + log(self::SQRT2PI * $summer / $x));
    }

    // Log Gamma related constants
    private const  LG_D1 = -0.5772156649015328605195174;

    private const LG_D2 = 0.4227843350984671393993777;

    private const LG_D4 = 1.791759469228055000094023;

    private const LG_P1 = [
        4.945235359296727046734888,
        201.8112620856775083915565,
        2290.838373831346393026739,
        11319.67205903380828685045,
        28557.24635671635335736389,
        38484.96228443793359990269,
        26377.48787624195437963534,
        7225.813979700288197698961,
    ];

    private const LG_P2 = [
        4.974607845568932035012064,
        542.4138599891070494101986,
        15506.93864978364947665077,
        184793.2904445632425417223,
        1088204.76946882876749847,
        3338152.967987029735917223,
        5106661.678927352456275255,
        3074109.054850539556250927,
    ];

    private const LG_P4 = [
        14745.02166059939948905062,
        2426813.369486704502836312,
        121475557.4045093227939592,
        2663432449.630976949898078,
        29403789566.34553899906876,
        170266573776.5398868392998,
        492612579337.743088758812,
        560625185622.3951465078242,
    ];

    private const LG_Q1 = [
        67.48212550303777196073036,
        1113.332393857199323513008,
        7738.757056935398733233834,
        27639.87074403340708898585,
        54993.10206226157329794414,
        61611.22180066002127833352,
        36351.27591501940507276287,
        8785.536302431013170870835,
    ];

    private const LG_Q2 = [
        183.0328399370592604055942,
        7765.049321445005871323047,
        133190.3827966074194402448,
        1136705.821321969608938755,
        5267964.117437946917577538,
        13467014.54311101692290052,
        17827365.30353274213975932,
        9533095.591844353613395747,
    ];

    private const LG_Q4 = [
        2690.530175870899333379843,
        639388.5654300092398984238,
        41355999.30241388052042842,
        1120872109.61614794137657,
        14886137286.78813811542398,
        101680358627.2438228077304,
        341747634550.7377132798597,
        446315818741.9713286462081,
    ];

    private const LG_C = [
        -0.001910444077728,
        8.4171387781295e-4,
        -5.952379913043012e-4,
        7.93650793500350248e-4,
        -0.002777777777777681622553,
        0.08333333333333333331554247,
        0.0057083835261,
    ];

    // Rough estimate of the fourth root of logGamma_xBig
    private const LG_FRTBIG = 2.25e76;

    private const PNT68 = 0.6796875;

    // Function cache for logGamma
    private static $logGammaCacheResult = 0.0;

    private static $logGammaCacheX = 0.0;

    public static function logGamma(float $x): float
    {
        if ($x == self::$logGammaCacheX) {
            return self::$logGammaCacheResult;
        }

        $y = $x;
        if ($y > 0.0 && $y <= self::LOG_GAMMA_X_MAX_VALUE) {
            if ($y <= self::EPS) {
                $res = -log($y);
            } elseif ($y <= 1.5) {
                $res = self::logGamma1($y);
            } elseif ($y <= 4.0) {
                $res = self::logGamma2($y);
            } elseif ($y <= 12.0) {
                $res = self::logGamma3($y);
            } else {
                $res = self::logGamma4($y);
            }
        } else {
            // --------------------------
            //    Return for bad arguments
            // --------------------------
            $res = self::MAX_VALUE;
        }

        // ------------------------------
        //    Final adjustments and return
        // ------------------------------
        self::$logGammaCacheX = $x;
        self::$logGammaCacheResult = $res;

        return $res;
    }

    private static function logGamma1(float $y)
    {
        // ---------------------
        //    EPS .LT. X .LE. 1.5
        // ---------------------
        if ($y < self::PNT68) {
            $corr = -log($y);
            $xm1 = $y;
        } else {
            $corr = 0.0;
            $xm1 = $y - 1.0;
        }

        $xden = 1.0;
        $xnum = 0.0;
        if ($y <= 0.5 || $y >= self::PNT68) {
            for ($i = 0; $i < 8; ++$i) {
                $xnum = $xnum * $xm1 + self::LG_P1[$i];
                $xden = $xden * $xm1 + self::LG_Q1[$i];
            }

            return $corr + $xm1 * (self::LG_D1 + $xm1 * ($xnum / $xden));
        }

        $xm2 = $y - 1.0;
        for ($i = 0; $i < 8; ++$i) {
            $xnum = $xnum * $xm2 + self::LG_P2[$i];
            $xden = $xden * $xm2 + self::LG_Q2[$i];
        }

        return $corr + $xm2 * (self::LG_D2 + $xm2 * ($xnum / $xden));
    }

    private static function logGamma2(float $y)
    {
        // ---------------------
        //    1.5 .LT. X .LE. 4.0
        // ---------------------
        $xm2 = $y - 2.0;
        $xden = 1.0;
        $xnum = 0.0;
        for ($i = 0; $i < 8; ++$i) {
            $xnum = $xnum * $xm2 + self::LG_P2[$i];
            $xden = $xden * $xm2 + self::LG_Q2[$i];
        }

        return $xm2 * (self::LG_D2 + $xm2 * ($xnum / $xden));
    }

    protected static function logGamma3(float $y)
    {
        // ----------------------
        //    4.0 .LT. X .LE. 12.0
        // ----------------------
        $xm4 = $y - 4.0;
        $xden = -1.0;
        $xnum = 0.0;
        for ($i = 0; $i < 8; ++$i) {
            $xnum = $xnum * $xm4 + self::LG_P4[$i];
            $xden = $xden * $xm4 + self::LG_Q4[$i];
        }

        return self::LG_D4 + $xm4 * ($xnum / $xden);
    }

    protected static function logGamma4(float $y)
    {
        // ---------------------------------
        //    Evaluate for argument .GE. 12.0
        // ---------------------------------
        $res = 0.0;
        if ($y <= self::LG_FRTBIG) {
            $res = self::LG_C[6];
            $ysq = $y * $y;
            for ($i = 0; $i < 6; ++$i) {
                $res = $res / $ysq + self::LG_C[$i];
            }
            $res /= $y;
            $corr = log($y);
            $res = $res + log(self::SQRT2PI) - 0.5 * $corr;
            $res += $y * ($corr - 1.0);
        }

        return $res;
    }
}