continue impl. of distributions/stochastics/statistics

Math/Statistic/Correlation.php

@@ -48,7 +48,7 @@ final class Correlation
      */
     public static function bravaisPersonCorrelationCoefficient(array $x, array $y) : float
     {
-        return MeasureOfDispersion::empiricalCovariance($x, $y) / (MeasureOfDispersion::standardDeviation($x) * MeasureOfDispersion::standardDeviation($y));
+        return MeasureOfDispersion::empiricalCovariance($x, $y) / (MeasureOfDispersion::standardDeviationSample($x) * MeasureOfDispersion::standardDeviationSample($y));
     }
 
     /**

Math/Statistic/MeasureOfDispersion.php

@@ -79,11 +79,11 @@ final class MeasureOfDispersion
             throw new ZeroDivisionException();
         }
 
-        return self::standardDeviation($values) / $mean;
+        return self::standardDeviationSample($values) / $mean;
     }
 
     /**
-     * Calculage standard deviation.
+     * Calculate standard deviation of sample.
      *
      * Example: ([4, 5, 9, 1, 3])
      *
@@ -96,7 +96,7 @@ final class MeasureOfDispersion
      *
      * @since 1.0.0
      */
-    public static function standardDeviation(array $values, float $mean = null) : float
+    public static function standardDeviationSample(array $values, float $mean = null) : float
     {
         $mean = $mean !== null ? $mean : Average::arithmeticMean($values);
         $sum  = 0.0;
@@ -108,6 +108,32 @@ final class MeasureOfDispersion
         return \sqrt($sum / (\count($values) - 1));
     }
 
+    /**
+     * Calculate standard deviation of entire population
+     *
+     * Example: ([4, 5, 9, 1, 3])
+     *
+     * @latex \sigma = \sqrt{\sigma^{2}} = \sqrt{Var(X)}
+     *
+     * @param array<int, int|float> $values Values
+     * @param float                 $mean   Mean
+     *
+     * @return float
+     *
+     * @since 1.0.0
+     */
+    public static function standardDeviationPopulation(array $values, float $mean = null) : float
+    {
+        $mean = $mean !== null ? $mean : Average::arithmeticMean($values);
+        $sum  = 0.0;
+
+        foreach ($values as $value) {
+            $sum += ($value - $mean) ** 2;
+        }
+
+        return \sqrt($sum / \count($values));
+    }
+
     /**
      * Calculage sample variance.
      *

Math/Stochastic/Distribution/HypergeometricDistribution.php

@@ -139,4 +139,24 @@ final class HypergeometricDistribution
         return (($N - 1) * $N ** 2 * ($N * ($N + 1) - 6 * $K * ($N - $K) - 6 * $n * ($N - $n)) + 6 * $n * $K * ($N - $K) * ($N - $n) * (5 * $N - 6))
             / ($n * $K * ($N - $K) * ($N - $n) * ($N - 2) * ($N - 3));
     }
+
+    /**
+     * Hypergeometric-Distribution
+     *
+     * @param int $sampleSuccesses     Amount of sample successes
+     * @param int $samples             Sample size
+     * @param int $populationSuccesses Amount of population successes
+     * @param int $population          Population size
+     *
+     * @return float
+     *
+     * @since 1.0.0
+     */
+    public static function dist(int $sampleSuccesses, int $samples, int $populationSuccesses, int $population) : float
+    {
+        // Each multiplication calculates the total amount of possible group combinations based on a total amount of items.
+        return (int) (\round(Functions::fact($populationSuccesses) / Functions::fact($populationSuccesses - $sampleSuccesses)) / Functions::fact($sampleSuccesses)
+            * \round(Functions::fact($population - $populationSuccesses) / Functions::fact($population - $populationSuccesses - ($samples - $sampleSuccesses))) / Functions::fact($samples - $sampleSuccesses)
+            * \round(Functions::fact($population) / Functions::fact($population - $samples)) / Functions::fact($samples));
+    }
 }

Math/Stochastic/Distribution/LogNormalDistribution.php

@@ -173,4 +173,20 @@ final class LogNormalDistribution
             [0, 1 / (2 * $sigma ** 2)],
         ];
     }
+
+    /**
+     * Log-Normal-Distribution
+     *
+     * @param float $value             Value
+     * @param float $mean              Mean
+     * @param float $standardDeviation Standard deviation
+     *
+     * @return float
+     *
+     * @since 1.0.0
+     */
+    public static function dist(float $value, float $mean, float $standardDeviation) : float
+    {
+        return NormalDistribution::dist((\log($value) - $mean) / $standardDeviation, 0.0, 1.0, true);
+    }
 }

Math/Stochastic/Distribution/NormalDistribution.php

@@ -14,6 +14,8 @@ declare(strict_types=1);
 
 namespace phpOMS\Math\Stochastic\Distribution;
 
+use phpOMS\Math\Functions\Functions;
+
 /**
  * Normal distribution.
  *
@@ -109,6 +111,8 @@ final class NormalDistribution
      *
      * @return float
      *
+     * @todo: compare with Functions::getErf($x);
+     *
      * @since 1.0.0
      */
     private static function erf(float $x) : float
@@ -245,4 +249,23 @@ final class NormalDistribution
     {
         return 0;
     }
+
+    /**
+     * Normal-Distribution
+     *
+     * @param float $value             Value
+     * @param float $mean              Mean
+     * @param float $standardDeviation Standard deviation
+     * @param bool  $isCumulative      Cumulative
+     *
+     * @return float
+     *
+     * @since 1.0.0
+     */
+    public static function dist(float $value, float $mean, float $standardDeviation, bool $isCumulative = true) : float
+    {
+        return $isCumulative
+            ? 0.5 * (1 + Functions::getErf(($value - $mean) / $standardDeviation * \sqrt(2)))
+            : 1 / (\sqrt(2 * \M_PI) * $standardDeviation) * \exp (-\pow($value - $mean, 2) / (2 * $standardDeviation * $standardDeviation));
+    }
 }

Math/Stochastic/Distribution/PoissonDistribution.php

@@ -191,4 +191,31 @@ final class PoissonDistribution
     {
         return \pow($lambda, -1);
     }
+
+    /**
+     * Poisson-Distribution
+     *
+     * @param float $value        Value
+     * @param float $mean         Mean
+     * @param bool  $isCumulative Cumulative
+     *
+     * @return float
+     *
+     * @since 1.0.0
+     */
+    public static function dist(float $value, float $mean, bool $isCumulative = true) : float
+    {
+        if (!$isCumulative) {
+            return \exp(-$mean) * \pow($mean, $value) / Functions::fact((int) \floor($value));
+        }
+
+        $sum   = 0.0;
+        $limit = \floor($value);
+
+        for ($i = 0; $i <= $limit; ++$i) {
+            $sum += \pow($mean, $i) / Functions::fact($i);
+        }
+
+        return \exp(-$mean) * $sum;
+    }
 }

Math/Stochastic/Distribution/TDistribution.php

@@ -159,4 +159,57 @@ final class TDistribution
     {
         return $nu < 5 && $nu > 2 ? \PHP_FLOAT_MAX : 6 / ($nu - 4);
     }
+
+    /**
+     * T-Distribution
+     *
+     * @param float $value   Value
+     * @param int   $degrees Degrees of freedom
+     * @param int   $tails   Tails (1 or 2)
+     *
+     * @return float
+     *
+     * @since 1.0.0
+     */
+    public static function dist(float $value, int $degrees, int $tails = 2) : float
+    {
+        if ($value < 0.0 || $degrees < 1 || $tails < 1 || $tails > 2) {
+            return 0.0;
+        }
+
+        /**
+         * "AS 3" by B E Cooper of the Atlas Computer Laboratory
+         * Ellis Horwood Ltd.; W. Sussex, England
+         */
+        $term  = $degrees;
+        $theta = \atan2($value, \sqrt($term));
+        $cos   = \cos($theta);
+        $sin   = \sin($theta);
+        $sum   = 0.0;
+
+        if ($degrees % 2 === 1) {
+            $i = 3;
+            $term = $cos;
+        } else {
+            $i = 2;
+            $term = 1;
+        }
+
+        $sum = $term;
+        while ($i < $degrees) {
+            $term *= $cos ** 2 * ($i - 1) / $i;
+            $sum  += $term;
+            $i    += 2;
+        }
+
+        $sum *= $sin;
+
+        if ($degrees % 2 === 1) {
+            $sum = 2 / M_PI * ($sum + $theta);
+        }
+
+        $t = 0.5 * (1 + $sum);
+
+        return $tails === 1 ? 1 - \abs($t) : 1 - \abs(1 - $t - $t);
+    }
 }

Math/Stochastic/Distribution/WeibullDistribution.php

@@ -101,4 +101,23 @@ final class WeibullDistribution
 
         return $gamma * (1 - 1 / $k) + \log($lambda / $k) + 1;
     }
+
+    /**
+     * Weibull-Distribution
+     *
+     * @param float $value        Value
+     * @param float $alpha        Alpha
+     * @param float $beta         Beta
+     * @param bool  $isCumulative Cumulative
+     *
+     * @return float
+     *
+     * @since 1.0.0
+     */
+    public static function dist(float $value, float $alpha, float $beta, bool $isCumulative = true) : float
+    {
+        return $isCumulative
+            ? 1 - \exp(-\pow($value / $beta, $alpha))
+            : $alpha / \pow($beta, $alpha) * \pow($value, $alpha - 1) * \exp(-\pow($value / $beta, $alpha));
+    }
 }

Math/Stochastic/Distribution/ZTest.php

@@ -14,6 +14,9 @@ declare(strict_types=1);
 
 namespace phpOMS\Math\Stochastic\Distribution;
 
+use phpOMS\Math\Statistic\Average;
+use phpOMS\Math\Statistic\MeasureOfDispersion;
+
 /**
  * ZTest
  *
@@ -38,26 +41,20 @@ final class ZTest
     /**
      * Test hypthesis.
      *
-     * @param float $dataset      Value observed
+     * @param array      $data  Data
-     * @param float $expected     Expected value
+     * @param float      $alpha Alpha / Observed dataset size
-     * @param float $total        Observed dataset size
+     * @param null|float $sigma Sigma / Significance
-     * @param float $significance Significance
      *
-     * @return bool
+     * @return float
      *
      * @since 1.0.0
      */
-    public static function testHypothesis(float $dataset, float $expected, float $total, float $significance = 0.95) : bool
+    public static function testHypothesis(array $data, float $alpha, float $sigma = null) : float
     {
-        $z = ($dataset - $expected) / \sqrt($expected * (1 - $expected) / $total);
+        if ($sigma === null) {
-
+            return MeasureOfDispersion::standardDeviationSample($data);
-        $zSignificance = 0.0;
-        foreach (self::TABLE as $key => $value) {
-            if ($significance === $value) {
-                $zSignificance = (float) $key;
-            }
         }
 
-        return $z > -$key && $z < $key;
+        return 1 - NormalDistribution::dist((Average::arithmeticMean($data) - $alpha) / ($sigma / \sqrt(\count($data))), 0.0, 1.0, true);
     }
 }

tests/Math/Statistic/MeasureOfDispersionTest.php

@@ -36,9 +36,9 @@ class MeasureOfDispersionTest extends \PHPUnit\Framework\TestCase
      * @testdox The standard deviation is correctly calculated
      * @group framework
      */
-    public function testStandardDeviation() : void
+    public function testStandardDeviationSample() : void
     {
-        self::assertEqualsWithDelta(2.160246, MeasureOfDispersion::standardDeviation([1, 2, 3, 4, 5, 6, 7]), 0.01);
+        self::assertEqualsWithDelta(2.160246, MeasureOfDispersion::standardDeviationSample([1, 2, 3, 4, 5, 6, 7]), 0.01);
     }
 
     /**