getMessage(); } if ($stdDev < 0) { return Functions::NAN(); } if ($cumulative) { return 0.5 * (1 + Engineering\Erf::erfValue(($value - $mean) / ($stdDev * sqrt(2)))); } return (1 / (self::SQRT2PI * $stdDev)) * exp(0 - (($value - $mean) ** 2 / (2 * ($stdDev * $stdDev)))); } /** * NORMINV. * * Returns the inverse of the normal cumulative distribution for the specified mean and standard deviation. * * @param mixed $probability Float probability for which we want the value * Or can be an array of values * @param mixed $mean Mean Value as a float * Or can be an array of values * @param mixed $stdDev Standard Deviation as a float * Or can be an array of values * * @return array|float|string The result, or a string containing an error * If an array of numbers is passed as an argument, then the returned result will also be an array * with the same dimensions */ public static function inverse($probability, $mean, $stdDev) { if (is_array($probability) || is_array($mean) || is_array($stdDev)) { return self::evaluateArrayArguments([self::class, __FUNCTION__], $probability, $mean, $stdDev); } try { $probability = DistributionValidations::validateProbability($probability); $mean = DistributionValidations::validateFloat($mean); $stdDev = DistributionValidations::validateFloat($stdDev); } catch (Exception $e) { return $e->getMessage(); } if ($stdDev < 0) { return Functions::NAN(); } return (self::inverseNcdf($probability) * $stdDev) + $mean; } /* * inverse_ncdf.php * ------------------- * begin : Friday, January 16, 2004 * copyright : (C) 2004 Michael Nickerson * email : nickersonm@yahoo.com * */ private static function inverseNcdf($p) { // Inverse ncdf approximation by Peter J. Acklam, implementation adapted to // PHP by Michael Nickerson, using Dr. Thomas Ziegler's C implementation as // a guide. http://home.online.no/~pjacklam/notes/invnorm/index.html // I have not checked the accuracy of this implementation. Be aware that PHP // will truncate the coeficcients to 14 digits. // You have permission to use and distribute this function freely for // whatever purpose you want, but please show common courtesy and give credit // where credit is due. // Input paramater is $p - probability - where 0 < p < 1. // Coefficients in rational approximations static $a = [ 1 => -3.969683028665376e+01, 2 => 2.209460984245205e+02, 3 => -2.759285104469687e+02, 4 => 1.383577518672690e+02, 5 => -3.066479806614716e+01, 6 => 2.506628277459239e+00, ]; static $b = [ 1 => -5.447609879822406e+01, 2 => 1.615858368580409e+02, 3 => -1.556989798598866e+02, 4 => 6.680131188771972e+01, 5 => -1.328068155288572e+01, ]; static $c = [ 1 => -7.784894002430293e-03, 2 => -3.223964580411365e-01, 3 => -2.400758277161838e+00, 4 => -2.549732539343734e+00, 5 => 4.374664141464968e+00, 6 => 2.938163982698783e+00, ]; static $d = [ 1 => 7.784695709041462e-03, 2 => 3.224671290700398e-01, 3 => 2.445134137142996e+00, 4 => 3.754408661907416e+00, ]; // Define lower and upper region break-points. $p_low = 0.02425; //Use lower region approx. below this $p_high = 1 - $p_low; //Use upper region approx. above this if (0 < $p && $p < $p_low) { // Rational approximation for lower region. $q = sqrt(-2 * log($p)); return ((((($c[1] * $q + $c[2]) * $q + $c[3]) * $q + $c[4]) * $q + $c[5]) * $q + $c[6]) / (((($d[1] * $q + $d[2]) * $q + $d[3]) * $q + $d[4]) * $q + 1); } elseif ($p_high < $p && $p < 1) { // Rational approximation for upper region. $q = sqrt(-2 * log(1 - $p)); return -((((($c[1] * $q + $c[2]) * $q + $c[3]) * $q + $c[4]) * $q + $c[5]) * $q + $c[6]) / (((($d[1] * $q + $d[2]) * $q + $d[3]) * $q + $d[4]) * $q + 1); } // Rational approximation for central region. $q = $p - 0.5; $r = $q * $q; return ((((($a[1] * $r + $a[2]) * $r + $a[3]) * $r + $a[4]) * $r + $a[5]) * $r + $a[6]) * $q / ((((($b[1] * $r + $b[2]) * $r + $b[3]) * $r + $b[4]) * $r + $b[5]) * $r + 1); } }