math-php

MathPHP - Powerful Modern Math Library for PHP

The only library you need to integrate mathematical functions into your applications. It is a self-contained library in pure PHP with no external dependencies.

It is actively under development with development (0.y.z) releases.

Features

• Algebra
• Arithmetic
• Finance
• Functions
  • Map
  • Polynomial
  • Special Functions
• Information Theory
  • Entropy
• Linear Algebra
  • Matrix
  • Vector
• Numbers
  • Arbitrary Integer
  • Complex
  • Rational
• Number Theory
  • Integers
• Numerical Analysis
  • Interpolation
  • Numerical Differentiation
  • Numerical Integration
  • Root Finding
• Probability
  • Combinatorics
  • Distributions
    • Continuous
    • Discrete
    • Multivariate
    • Tables
• Sequences
  • Basic
  • Advanced
  • NonInteger
• Set Theory
• Statistics
  • ANOVA
  • Averages
  • Circular
  • Correlation
  • Descriptive
  • Distance and Divergence
  • Distributions
  • Effect Size
  • Experiments
  • Kernel Density Estimation
  • Multivariate
    • PCA (Principal Component Analysis)
  • Outlier
  • Random Variables
  • Regressions
  • Significance Testing
• Trigonometry

Setup

Add the library to your composer.json file in your project:

{
  "require": {
      "markrogoyski/math-php": "0.*"
  }
}

Use composer to install the library:

$ php composer.phar install

Composer will install MathPHP inside your vendor folder. Then you can add the following to your
.php files to use the library with Autoloading.

require_once(__DIR__ . '/vendor/autoload.php');

Alternatively, use composer on the command line to require and install MathPHP:

$ php composer.phar require markrogoyski/math-php:0.*

Minimum Requirements

• PHP 7

Usage

Algebra

use MathPHP\Algebra;

// Greatest common divisor (GCD)
$gcd = Algebra::gcd(8, 12);

// Extended greatest common divisor - gcd(a, b) = a*a' + b*b'
$gcd = Algebra::extendedGcd(12, 8); // returns array [gcd, a', b']

// Least common multiple (LCM)
$lcm = Algebra::lcm(5, 2);

// Factors of an integer
$factors = Algebra::factors(12); // returns [1, 2, 3, 4, 6, 12]

// Quadradic equation
list($a, $b, $c) = [1, 2, -8]; // x² + 2x - 8
list($x₁, $x₂)   = Algebra::quadradic($a, $b, $c);

// Cubic equation
list($a₃, $a₂, $a₁, $a₀) = [2, 9, 3, -4]; // 2x³ + 9x² + 3x -4
list($x₁, $x₂, $x₃)      = Algebra::cubic($a₃, $a₂, $a₁, $a₀);

// Quartic equation
list($a₄, $a₃, $a₂, $a₁, $a₀) = [1, -10, 35, -50, 24]; // z⁴ - 10z³ + 35z² - 50z + 24 = 0
list($z₁, $z₂, $z₃, $z₄)      = Algebra::quartic($a₄, $a₃, $a₂, $a₁, $a₀);

Arithmetic

use MathPHP\Arithmetic;

$³√x = Arithmetic::cubeRoot(-8); // -2
$ⁿ√x = Arithmetic::root(81, 4);  // nᵗʰ root (4ᵗʰ): 3

// Sum of digits
$digit_sum    = Arithmetic::digitSum(99);    // 18
$digital_root = Arithmetic::digitalRoot(99); // 9

// Equality of numbers within a tolerance
$x = 0.00000003458;
$y = 0.00000003455;
$ε = 0.0000000001;
$almostEqual = Arithmetic::almostEqual($x, $y, $ε); // true

// Copy sign
$magnitude = 5;
$sign      = -3;
$signed_magnitude = Arithmetic::copySign($magnitude, $sign); // -5

Finance

use MathPHP\Finance;

// Financial payment for a loan or annuity with compound interest
$rate          = 0.035 / 12; // 3.5% interest paid at the end of every month
$periods       = 30 * 12;    // 30-year mortgage
$present_value = 265000;     // Mortgage note of $265,000.00
$future_value  = 0;
$beginning     = false;      // Adjust the payment to the beginning or end of the period
$pmt           = Finance::pmt($rate, $periods, $present_value, $future_value, $beginning);

// Interest on a financial payment for a loan or annuity with compound interest.
$period = 1; // First payment period
$ipmt   = Finance::ipmt($rate, $period, $periods, $present_value, $future_value, $beginning);

// Principle on a financial payment for a loan or annuity with compound interest
$ppmt = Finance::ppmt($rate, $period, $periods, $present_value, $future_value = 0, $beginning);

// Number of payment periods of an annuity.
$periods = Finance::periods($rate, $payment, $present_value, $future_value, $beginning);

// Annual Equivalent Rate (AER) of an annual percentage rate (APR)
$nominal = 0.035; // APR 3.5% interest
$periods = 12;    // Compounded monthly
$aer     = Finance::aer($nominal, $periods);

// Annual nominal rate of an annual effective rate (AER)
$nomial = Finance::nominal($aer, $periods);

// Future value for a loan or annuity with compound interest
$payment = 1189.97;
$fv      = Finance::fv($rate, $periods, $payment, $present_value, $beginning)

// Present value for a loan or annuity with compound interest
$pv = Finance::pv($rate, $periods, $payment, $future_value, $beginning)

// Net present value of cash flows
$values = [-1000, 100, 200, 300, 400];
$npv    = Finance::npv($rate, $values);

// Interest rate per period of an annuity
$beginning = false; // Adjust the payment to the beginning or end of the period
$rate      = rate($periods, $payment, $present_value, $future_value, $beginning);

// Internal rate of return
$values = [-100, 50, 40, 30];
$irr    = Finance:irr($values); // Rate of return of an initial investment of $100 with returns of $50, $40, and $30

// Modified internal rate of return
$finance_rate      = 0.05; // 5% financing
$reinvestment_rate = 0.10; // reinvested at 10%
$mirr              = Finance:mirr($values, $finance_rate); // rate of return of an initial investment of $100 at 5% financing with returns of $50, $40, and $30 reinvested at 10%

// Discounted payback of an investment
$values  = [-1000, 100, 200, 300, 400, 500];
$rate    = 0.1;
$payback = Finance::payback($values, $rate); // The payback period of an investment with a $1,000 investment and future returns of $100, $200, $300, $400, $500 and a discount rate of 0.10

// Profitability index
$values              = [-100, 50, 50, 50];
$profitability_index = profitabilityIndex($values, $rate); // The profitability index of an initial $100 investment with future returns of $50, $50, $50 with a 10% discount rate

Functions - Map - Single Array

use MathPHP\Functions\Map;

$x = [1, 2, 3, 4];

$sums        = Map\Single::add($x, 2);      // [3, 4, 5, 6]
$differences = Map\Single::subtract($x, 1); // [0, 1, 2, 3]
$products    = Map\Single::multiply($x, 5); // [5, 10, 15, 20]
$quotients   = Map\Single::divide($x, 2);   // [0.5, 1, 1.5, 2]
$x²          = Map\Single::square($x);      // [1, 4, 9, 16]
$x³          = Map\Single::cube($x);        // [1, 8, 27, 64]
$x⁴          = Map\Single::pow($x, 4);      // [1, 16, 81, 256]
$√x          = Map\Single::sqrt($x);        // [1, 1.414, 1.732, 2]
$∣x∣         = Map\Single::abs($x);         // [1, 2, 3, 4]
$maxes       = Map\Single::max($x, 3);      // [3, 3, 3, 4]
$mins        = Map\Single::min($x, 3);      // [1, 2, 3, 3]
$reciprocals = Map\Single::reciprocal($x);  // [1, 1/2, 1/3, 1/4]

Functions - Map - Multiple Arrays

use MathPHP\Functions\Map;

$x = [10, 10, 10, 10];
$y = [1,   2,  5, 10];

// Map function against elements of two or more arrays, item by item (by item ...)
$sums        = Map\Multi::add($x, $y);      // [11, 12, 15, 20]
$differences = Map\Multi::subtract($x, $y); // [9, 8, 5, 0]
$products    = Map\Multi::multiply($x, $y); // [10, 20, 50, 100]
$quotients   = Map\Multi::divide($x, $y);   // [10, 5, 2, 1]
$maxes       = Map\Multi::max($x, $y);      // [10, 10, 10, 10]
$mins        = Map\Multi::mins($x, $y);     // [1, 2, 5, 10]

// All functions work on multiple arrays; not limited to just two
$x    = [10, 10, 10, 10];
$y    = [1,   2,  5, 10];
$z    = [4,   5,  6,  7];
$sums = Map\Multi::add($x, $y, $z); // [15, 17, 21, 27]

Functions - Polynomial

use MathPHP\Functions\Polynomial;

// Polynomial x² + 2x + 3
$coefficients = [1, 2, 3]
$polynomial   = new Polynomial($coefficients);

// Evaluate for x = 3
$x = 3;
$y = $polynomial($x);  // 18: 3² + 2*3 + 3

// Calculus
$derivative = $polynomial->differentiate();  // Polynomial 2x + 2
$integral   = $polynomial->integrate();      // Polynomial ⅓x³ + x² + 3x

// Arithmetic
$sum        = $polynomial->add($polynomial);       // Polynomial 2x² + 4x + 6
$sum        = $polynomial->add(2);                 // Polynomial x² + 2x + 5
$difference = $polynomial->subtract($polynomial);  // Polynomial 0
$difference = $polynomial->subtract(2);            // Polynomial x² + 2x + 1
$product    = $polynomial->multiply($polynomial);  // Polynomial x⁴ + 4x³ + 10x² + 12x + 9
$product    = $polynomial->multiply(2);            // Polynomial 2x² + 4x + 6
$negated    = $polynomial->negate();               // Polynomial -x² - 2x - 3

// Data
$degree       = $polynomial->getDegree();        // 2
$coefficients = $polynomial->getCoefficients();  // [1, 2, 3]

// String representation
print($polynomial);  // x² + 2x + 3

// Roots
$polynomial = new Polynomial([1, -3, -4]);
$roots      = $polynomial->roots();         // [-1, 4]

Functions - Special Functions

use MathPHP\Functions\Special;

// Gamma function Γ(z)
$z = 4;
$Γ = Special::gamma($z);          // Uses gamma definition for integers and half integers; uses Lanczos approximation for real numbers
$Γ = Special::gammaLanczos($z);   // Lanczos approximation
$Γ = Special::gammaStirling($z);  // Stirling approximation

// Incomplete gamma functions - γ(s,t), Γ(s,x)
list($x, $s) = [1, 2];
$γ = Special::lowerIncompleteGamma($x, $s); // same as γ
$γ = Special::γ($x, $s);                    // same as lowerIncompleteGamma
$Γ = Special::upperIncompleteGamma($x, $s);

// Beta function
list($x, $y) = [1, 2];
$β = Special::beta($x, $y); // same as β
$β = Special::β($x, $y);    // same as beta

// Incomplete beta functions
list($x, $a, $b) = [0.4, 2, 3];
$B  = Special::incompleteBeta($x, $a, $b);
$Iₓ = Special::regularizedIncompleteBeta($x, $a, $b);

// Multivariate beta function
$αs = [1, 2, 3];
$β  = Special::multivariateBeta($αs);

// Error function (Gauss error function)
$error = Special::errorFunction(2);              // same as erf
$error = Special::erf(2);                        // same as errorFunction
$error = Special::complementaryErrorFunction(2); // same as erfc
$error = Special::erfc(2);                       // same as complementaryErrorFunction

// Hypergeometric functions
$pFq = Special::generalizedHypergeometric($p, $q, $a, $b, $c, $z);
$₁F₁ = Special::confluentHypergeometric($a, $b, $z);
$₂F₁ = Special::hypergeometric($a, $b, $c, $z);

// Sign function (also known as signum or sgn)
$x    = 4;
$sign = Special::signum($x); // same as sgn
$sign = Special::sgn($x);    // same as signum

// Logistic function (logistic sigmoid function)
$x₀ = 2; // x-value of the sigmoid's midpoint
$L  = 3; // the curve's maximum value
$k  = 4; // the steepness of the curve
$x  = 5;
$logistic = Special::logistic($x₀, $L, $k, $x);

// Sigmoid function
$t = 2;
$sigmoid = Special::sigmoid($t);

// Softmax function
$?    = [1, 2, 3, 4, 1, 2, 3];
$σ⟮?⟯ⱼ = Special::softmax($?);

Information Theory - Entropy

use MathPHP\InformationTheory\Entropy;

// Probability distributions
$p = [0.2, 0.5, 0.3];
$q = [0.1, 0.4, 0.5];

// Shannon entropy
$bits  = Entropy::shannonEntropy($p);         // log₂
$nats  = Entropy::shannonNatEntropy($p);      // ln
$harts = Entropy::shannonHartleyEntropy($p);  // log₁₀

// Cross entropy
$H⟮p、q⟯ = Entropy::crossEntropy($p, $q);       // log₂

// Joint entropy
$P⟮x、y⟯ = [1/2, 1/4, 1/4, 0];
H⟮x、y⟯ = Entropy::jointEntropy($P⟮x、y⟯);        // log₂

// Rényi entropy
$α    = 0.5;
$Hₐ⟮X⟯ = Entropy::renyiEntropy($p, $α);         // log₂

// Perplexity
$perplexity = Entropy::perplexity($p);         // log₂

Linear Algebra - Matrix

use MathPHP\LinearAlgebra\Matrix;
use MathPHP\LinearAlgebra\MatrixFactory;

// Create an m × n matrix from an array of arrays
$matrix = [
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9],
];
$A = MatrixFactory::create($matrix);

// Basic matrix data
$array = $A->getMatrix();  // Original array of arrays
$rows  = $A->getM();       // number of rows
$cols  = $A->getN();       // number of columns

// Basic matrix element getters (zero-based indexing)
$row = $A->getRow(2);
$col = $A->getColumn(2);
$Aᵢⱼ = $A->get(2, 2);
$Aᵢⱼ = $A[2][2];

// Row operations
list($mᵢ, $mⱼ, $k) = [1, 2, 5];
$R = $A->rowInterchange($mᵢ, $mⱼ);
$R = $A->rowMultiply($mᵢ, $k);     // Multiply row mᵢ by k
$R = $A->rowAdd($mᵢ, $mⱼ, $k);     // Add k * row mᵢ to row mⱼ
$R = $A->rowExclude($mᵢ);          // Exclude row $mᵢ

// Column operations
list($nᵢ, $nⱼ, $k) = [1, 2, 5];
$R = $A->columnInterchange($nᵢ, $nⱼ);
$R = $A->columnMultiply($nᵢ, $k);     // Multiply column nᵢ by k
$R = $A->columnAdd($nᵢ, $nⱼ, $k);     // Add k * column nᵢ to column nⱼ
$R = $A->columnExclude($nᵢ);          // Exclude column $nᵢ

// Matrix augmentations - return a new Matrix
$⟮A∣B⟯ = $A->augment($B);        // Augment on the right - standard augmentation
$⟮A∣I⟯ = $A->augmentIdentity();  // Augment with the identity matrix
$⟮A∣B⟯ = $A->augmentBelow($B);
$⟮A∣B⟯ = $A->augmentAbove($B);
$⟮B∣A⟯ = $A->augmentLeft($B);

// Matrix arithmetic operations - return a new Matrix
$A＋B = $A->add($B);
$A⊕B  = $A->directSum($B);
$A⊕B  = $A->kroneckerSum($B);
$A−B  = $A->subtract($B);
$AB   = $A->multiply($B);
$２A  = $A->scalarMultiply(2);
$A／2 = $A->scalarDivide(2);
$−A   = $A->negate();
$A∘B  = $A->hadamardProduct($B);
$A⊗B  = $A->kroneckerProduct($B);

// Matrix operations - return a new Matrix
$Aᵀ 　 = $A->transpose();
$D  　 = $A->diagonal();
$A⁻¹   = $A->inverse();
$Mᵢⱼ   = $A->minorMatrix($mᵢ, $nⱼ);        // Square matrix with row mᵢ and column nⱼ removed
$Mk    = $A->leadingPrincipalMinor($k);    // kᵗʰ-order leading principal minor
$CM    = $A->cofactorMatrix();
$B     = $A->meanDeviation();              // optional parameter to specify data direction (variables in 'rows' or 'columns')
$S     = $A->covarianceMatrix();           // optional parameter to specify data direction (variables in 'rows' or 'columns')
$adj⟮A⟯ = $A->adjugate();
$Mᵢⱼ   = $A->submatrix($mᵢ, $nᵢ, $mⱼ, $nⱼ) // Submatrix of A from row mᵢ, column nᵢ to row mⱼ, column nⱼ
$H     = $A->householder();

// Matrix value operations - return a value
$tr⟮A⟯   = $A->trace();
$, A, = $a->det();              // Determinant
$Mᵢⱼ    = $A->minor($mᵢ, $nⱼ);    // First minor
$Cᵢⱼ    = $A->cofactor($mᵢ, $nⱼ);
$rank⟮A⟯ = $A->rank();

// Matrix vector operations - return a new Vector
$AB = $A->vectorMultiply($X₁);
$M  = $A->rowSums();
$M  = $A->columnSums();
$M  = $A->rowMeans();
$M  = $A->columnMeans();

// Matrix norms - return a value
$‖A‖₁ = $A->oneNorm();
$‖A‖F = $A->frobeniusNorm(); // Hilbert–Schmidt norm
$‖A‖∞ = $A->infinityNorm();
$max   = $A->maxNorm();

// Matrix reductions
$ref  = $A->ref();   // Matrix in row echelon form
$rref = $A->rref();  // Matrix in reduced row echelon form

// Matrix decompositions
// LU decomposition
$LU = $A->luDecomposition();
$L  = $LU->L;  // lower triangular matrix
$U  = $LU->U;  // upper triangular matrix
$P  = $LU-P;   // permutation matrix

// QR decomposition
$QR = $A->qrDecomposition();
$Q  = $QR->Q;  // orthogonal matrix
$R  = $QR->R;  // upper triangular matrix

// Crout decomposition
$LU = $A->croutDecomposition();
$L  = $LU->L;  // lower triangular matrix
$U  = $LU->U;  // normalized upper triangular matrix

// Cholesky decomposition
$LLᵀ = $A->choleskyDecomposition();
$L   = $LLᵀ->L;   // lower triangular matrix
$LT  = $LLᵀ->LT;  // transpose of lower triangular matrix

// Eigenvalues and eigenvectors
$eigenvalues   = $A->eigenvalues();   // array of eigenvalues
$eigenvecetors = $A->eigenvectors();  // Matrix of eigenvectors

// Solve a linear system of equations: Ax = b
$b = new Vector(1, 2, 3);
$x = $A->solve($b);

// Map a function over each element
$func = function($x) {
    return $x * 2;
};
$R = $A->map($func);  // using closure
$R = $A->map('abs');  // using callable

// Map a function over each row
$array = $A->mapRows('array_reverse');  // using callable returns matrix-like array of arrays
$array = $A->mapRows('array__sum');     // using callable returns array of aggregate calculations

// Matrix comparisons
$bool = $A->isEqual($B);

// Matrix properties - return a bool
$bool = $A->isSquare();
$bool = $A->isSymmetric();
$bool = $A->isSkewSymmetric();
$bool = $A->isSingular();
$bool = $A->isNonsingular();           // Same as isInvertible
$bool = $A->isInvertible();            // Same as isNonsingular
$bool = $A->isPositiveDefinite();
$bool = $A->isPositiveSemidefinite();
$bool = $A->isNegativeDefinite();
$bool = $A->isNegativeSemidefinite();
$bool = $A->isLowerTriangular();
$bool = $A->isUpperTriangular();
$bool = $A->isTriangular();
$bool = $A->isDiagonal();
$bool = $A->isRectangularDiagonal();
$bool = $A->isUpperBidiagonal();
$bool = $A->isLowerBidiagonal();
$bool = $A->isBidiagonal();
$bool = $A->isTridiagonal();
$bool = $A->isUpperHessenberg();
$bool = $A->isLowerHessenberg();
$bool = $A->isOrthogonal();
$bool = $A->isNormal();
$bool = $A->isIdempotent();
$bool = $A->isNilpotent();
$bool = $A->isInvolutory();
$bool = $A->isSignature();
$bool = $A->isRef();
$bool = $A->isRref();

// Other representations of matrix data
$vectors = $A->asVectors();                 // array of column vectors
$D       = $A->getDiagonalElements();       // array of the diagonal elements
$d       = $A->getSuperdiagonalElements();  // array of the superdiagonal elements
$d       = $A->getSubdiagonalElements();    // array of the subdiagonal elements

// String representation - Print a matrix
print($A);
/*
 [1, 2, 3]
 [2, 3, 4]
 [3, 4, 5]
 */

// PHP Predefined Interfaces
$json = json_encode($A); // JsonSerializable
$Aᵢⱼ  = $A[$mᵢ][$nⱼ];    // ArrayAccess

Linear Algebra - Matrix Construction (Factory)

$matrix = [
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9],
];

// Matrix factory creates most appropriate matrix
$A = MatrixFactory::create($matrix);

// Matrix factory can create a matrix from an array of column vectors
use MathPHP\LinearAlgebra\Vector;
$X₁ = new Vector([1, 4, 7]);
$X₂ = new Vector([2, 5, 8]);
$X₃ = new Vector([3, 6, 9]);
$A  = MatrixFactory::createFromVectors([$X₁, $X₂, $X₃]);

// Specialized matrices
list($m, $n, $k, $angle, $size) = [4, 4, 2, 3.14159, 2];
$identity_matrix                = MatrixFactory::identity($n);                   // Ones on the main diagonal
$zero_matrix                    = MatrixFactory::zero($m, $n);                   // All zeros
$ones_matrix                    = MatrixFactory::one($m, $n);                    // All ones
$eye_matrix                     = MatrixFactory::eye($m, $n, $k);                // Ones (or other value) on the k-th diagonal
$exchange_matrix                = MatrixFactory::exchange($n);                   // Ones on the reverse diagonal
$downshift_permutation_matrix   = MatrixFactory::downshiftPermutation($n);       // Permutation matrix that pushes the components of a vector down one notch with wraparound
$upshift_permutation_matrix     = MatrixFactory::upshiftPermutation($n);         // Permutation matrix that pushes the components of a vector up one notch with wraparound
$diagonal_matrix                = MatrixFactory::diagonal([1, 2, 3]);            // 3 x 3 diagonal matrix with zeros above and below the diagonal
$hilbert_matrix                 = MatrixFactory::hilbert($n);                    // Square matrix with entries being the unit fractions
$vandermonde_matrix             = MatrixFactory::vandermonde([1, 2, 3], 4);      // 4 x 3 Vandermonde matrix
$random_matrix                  = MatrixFactory::random($m, $n);                 // m x n matrix of random integers
$givens_matrix                  = MatrixFactory::givens($m, $n, $angle, $size);  // givens rotation matrix

Linear Algebra - Vector

use MathPHP\LinearAlgebra\Vector;

// Vector
$A = new Vector([1, 2]);
$B = new Vector([2, 4]);

// Basic vector data
$array = $A->getVector();
$n     = $A->getN();           // number of elements
$M     = $A->asColumnMatrix(); // Vector as an nx1 matrix
$M     = $A->asRowMatrix();    // Vector as a 1xn matrix

// Basic vector elements (zero-based indexing)
$item = $A->get(1);

// Vector numeric operations - return a value
$sum  = $A->sum();
$│A│  = $A->length();           // same as l2Norm
$A⋅B  = $A->dotProduct($B);     // same as innerProduct
$A⋅B  = $A->innerProduct($B);   // same as dotProduct
$A⊥⋅B = $A->perpDotProduct($B);

// Vector arithmetic operations - return a Vector
$A＋B  = $A->add($B);
$A−B   = $A->subtract($B);
$A×B   = $A->multiply($B);
$A／B  = $A->divide($B);
$kA    = $A->scalarMultiply($k);
$A／k  = $A->scalarDivide($k);

// Vector operations - return a Vector or Matrix
$A⨂B  = $A->outerProduct($B);  // Same as direct product
$AB    = $A->directProduct($B); // Same as outer product
$AxB   = $A->crossProduct($B);
$A⨂B   = $A->kroneckerProduct($B);
$Â     = $A->normalize();
$A⊥    = $A->perpendicular();
$projᵇA = $A->projection($B);   // projection of A onto B
$perpᵇA = $A->perp($B);         // perpendicular of A on B

// Vector norms - return a value
$l₁norm = $A->l1Norm();
$l²norm = $A->l2Norm();
$pnorm  = $A->pNorm();
$max    = $A->maxNorm();

// String representation
print($A);  // [1, 2]

// PHP standard interfaces
$n    = count($A);                // Countable
$json = json_encode($A);          // JsonSerializable
$Aᵢ   = $A[$i];                   // ArrayAccess
foreach ($A as $element) { ... }  // Iterator

Number - Arbitrary Length Integers

use MathPHP\Number;
use MathPHP\Functions;

// Create arbitrary-length big integers from int or string
$bigInt = new Number\ArbitraryInteger('876937869482938749389832');

// Unary functions
$−bigInt  = $bigInt->negate();
$√bigInt  = $bigInt->isqrt();       // Integer square root
$│bitInt│ = $bigInt->abs();         // Absolute value
$bigInt！  = $bigInt->fact();
$bool     = $bigInt->isPositive();

// Binary functions
$sum                  = $bigInt->add($bigInt);
$difference           = $bigInt->subtract($bigInt);
$product              = $bigInt->multiply($bigInt);
$quotient             = $bigInt->intdiv($divisor);
$mod                  = $bigInt->mod($divisor);
list($quotient, $mod) = $bigInt->fullIntdiv($divisor);
$pow                  = $bigInt->pow($exponent);
$shifted              = $bigInt->leftShift(2);

// Comparison functions
$bool = $bigInt->equals($bigInt);
$bool = $bigInt->greaterThan($bigInt);
$bool = $bigInt->lessThan($bigInt);

// Conversions
$int    = $bigInt->toInt();
$float  = $bigInt->toFloat();
$binary = $bigInt->toBinary();
$string = (string) $bigInt;

// Functions
$ackermann    = Functions\ArbitraryInteger::ackermann($bigInt);
$randomBigInt = Functions\ArbitaryInteger::rand($intNumberOfBytes);

Number - Complex Numbers

use MathPHP\Number\Complex;

list($r, $i) = [2, 4];
$complex     = new Complex($r, $i);

// Accessors
$r = $complex->r;
$i = $complex->i;

// Unary functions
$conjugate     = $complex->complexConjugate();
$│c│           = $complex->abs();     // absolute value (modulus)
$arg⟮c⟯         = $complex->arg();     // argument (phase)
$√c            = $complex->sqrt();    // positive square root
list($z₁, $z₂) = $complex->roots();
$c⁻¹           = $complex->inverse();
$−c            = $complex->negate();
$polar         = $complex->polarForm();

// Binary functions
$c＋c = $complex->add($complex);
$c−c  = $complex->subtract($complex);
$c×c  = $complex->multiply($complex);
$c／c = $complex->divide($complex);

// Other functions
$bool   = $complex->equals($complex);
$string = (string) $complex;

Number - Rational Numbers

use MathPHP\Number\Rational;

$whole       = 0;
$numerator   = 2;
$denominator = 3;

$rational = new Rational($whole, $numerator, $denominator);  // ²/₃

// Get individual parts
$whole       = $rational->getWholePart();
$numerator   = $rational->getNumerator();
$denominator = $rational->getDenominator();

// Unary functions
$│rational│ = $rational->abs();

// Binary functions
$sum      = $rational->add($rational);
$diff     = $rational->subtract($rational);
$product  = $rational->multiply($rational);
$quotient = $rational->divide($rational);

// Other functions
$bool   = $rational->equals($rational);
$float  = $rational->toFloat();
$string = (string) $rational;

Number Theory - Integers

use MathPHP\NumberTheory\Integer;

$n = 225;

// Prime factorization
$factors = Integer::primeFactorization($n);

// Divisor function
$int  = Integer::numberOfDivisors($n);
$int  = Integer::sumOfDivisors($n);

// Aliquot sums
$int  = Integer::aliquotSum($n);        // sum-of-divisors - n
$bool = Integer::isPerfectNumber($n);   // n = aliquot sum
$bool = Integer::isDeficientNumber($n); // n > aliquot sum
$bool = Integer::isAbundantNumber($n);  // n < aliquot sum

// Totients
$int  = Integer::totient($n);        // Jordan's totient k=1 (Euler's totient)
$int  = Integer::totient($n, 2);     // Jordan's totient k=2
$int  = Integer::cototient($n);      // Cototient
$int  = Integer::reducedTotient($n); // Carmichael's function

// Möbius function
$int  = Integer::mobius($n);

// Radical/squarefree kernel
$int  = Integer::radical($n);

// Squarefree
$bool = Integer::isSquarefree($n);

// Refactorable number
$bool = Integer::isRefactorableNumber($n);

// Sphenic number
$bool = Integer::isSphenicNumber($n);

// Perfect powers
$bool        = Integer::isPerfectPower($n);
list($m, $k) = Integer::perfectPower($n);

// Coprime
$bool = Integer::coprime(4, 35);

// Even and odd
$bool = Integer::isEven($n);
$bool = Integer::isOdd($n);

Numerical Analysis - Interpolation

use MathPHP\NumericalAnalysis\Interpolation;

// Interpolation is a method of constructing new data points with the range
// of a discrete set of known data points.
// Each integration method can take input in two ways:
//  1) As a set of points (inputs and outputs of a function)
//  2) As a callback function, and the number of function evaluations to
//     perform on an interval between a start and end point.

// Input as a set of points
$points = [[0, 1], [1, 4], [2, 9], [3, 16]];

// Input as a callback function
$f⟮x⟯ = function ($x) {
    return $x**2 + 2 * $x + 1;
};
list($start, $end, $n) = [0, 3, 4];

// Lagrange Polynomial
// Returns a function p(x) of x
$p = Interpolation\LagrangePolynomial::interpolate($points);                // input as a set of points
$p = Interpolation\LagrangePolynomial::interpolate($f⟮x⟯, $start, $end, $n); // input as a callback function

$p(0) // 1
$p(3) // 16

// Nevilles Method
// More accurate than Lagrange Polynomial Interpolation given the same input
// Returns the evaluation of the interpolating polynomial at the $target point
$target = 2;
$result = Interpolation\NevillesMethod::interpolate($target, $points);                // input as a set of points
$result = Interpolation\NevillesMethod::interpolate($target, $f⟮x⟯, $start, $end, $n); // input as a callback function

// Newton Polynomial (Forward)
// Returns a function p(x) of x
$p = Interpolation\NewtonPolynomialForward::interpolate($points);                // input as a set of points
$p = Interpolation\NewtonPolynomialForward::interpolate($f⟮x⟯, $start, $end, $n); // input as a callback function

$p(0) // 1
$p(3) // 16

// Natural Cubic Spline
// Returns a piecewise polynomial p(x)
$p = Interpolation\NaturalCubicSpline::interpolate($points);                // input as a set of points
$p = Interpolation\NaturalCubicSpline::interpolate($f⟮x⟯, $start, $end, $n); // input as a callback function

$p(0) // 1
$p(3) // 16

// Clamped Cubic Spline
// Returns a piecewise polynomial p(x)

// Input as a set of points
$points = [[0, 1, 0], [1, 4, -1], [2, 9, 4], [3, 16, 0]];

// Input as a callback function
$f⟮x⟯ = function ($x) {
    return $x**2 + 2 * $x + 1;
};
$f’⟮x⟯ = function ($x) {
    return 2*$x + 2;
};
list($start, $end, $n) = [0, 3, 4];

$p = Interpolation\ClampedCubicSpline::interpolate($points);                // input as a set of points
$p = Interpolation\ClampedCubicSpline::interpolate($f⟮x⟯, $f’⟮x⟯, $start, $end, $n); // input as a callback function

$p(0) // 1
$p(3) // 16

Numerical Analysis - Numerical Differentiation

use MathPHP\NumericalAnalysis\NumericalDifferentiation;

// Numerical Differentiation approximates the derivative of a function.
// Each Differentiation method can take input in two ways:
//  1) As a set of points (inputs and outputs of a function)
//  2) As a callback function, and the number of function evaluations to
//     perform on an interval between a start and end point.

// Input as a callback function
$f⟮x⟯ = function ($x) {
    return $x**2 + 2 * $x + 1;
};

// Three Point Formula
// Returns an approximation for the derivative of our input at our target

// Input as a set of points
$points = [[0, 1], [1, 4], [2, 9]];

$target = 0;
list($start, $end, $n) = [0, 2, 3];
$derivative = NumericalDifferentiation\ThreePointFormula::differentiate($target, $points);                // input as a set of points
$derivative = NumericalDifferentiation\ThreePointFormula::differentiate($target, $f⟮x⟯, $start, $end, $n); // input as a callback function

// Five Point Formula
// Returns an approximation for the derivative of our input at our target

// Input as a set of points
$points = [[0, 1], [1, 4], [2, 9], [3, 16], [4, 25]];

$target = 0;
list($start, $end, $n) = [0, 4, 5];
$derivative = NumericalDifferentiation\FivePointFormula::differentiate($target, $points);                // input as a set of points
$derivative = NumericalDifferentiation\FivePointFormula::differentiate($target, $f⟮x⟯, $start, $end, $n); // input as a callback function

// Second Derivative Midpoint Formula
// Returns an approximation for the second derivative of our input at our target

// Input as a set of points
$points = [[0, 1], [1, 4], [2, 9];

$target = 1;
list($start, $end, $n) = [0, 2, 3];
$derivative = NumericalDifferentiation\SecondDerivativeMidpointFormula::differentiate($target, $points);                // input as a set of points
$derivative = NumericalDifferentiation\SecondDerivativeMidpointFormula::differentiate($target, $f⟮x⟯, $start, $end, $n); // input as a callback function

Numerical Analysis - Numerical Integration

use MathPHP\NumericalAnalysis\NumericalIntegration;

// Numerical integration approximates the definite integral of a function.
// Each integration method can take input in two ways:
//  1) As a set of points (inputs and outputs of a function)
//  2) As a callback function, and the number of function evaluations to
//     perform on an interval between a start and end point.

// Trapezoidal Rule (closed Newton-Cotes formula)
$points = [[0, 1], [1, 4], [2, 9], [3, 16]];
$∫f⟮x⟯dx = NumericalIntegration\TrapezoidalRule::approximate($points); // input as a set of points

$f⟮x⟯ = function ($x) {
    return $x**2 + 2 * $x + 1;
};
list($start, $end, $n) = [0, 3, 4];
$∫f⟮x⟯dx = NumericalIntegration\TrapezoidalRule::approximate($f⟮x⟯, $start, $end, $n); // input as a callback function

// Simpsons Rule (closed Newton-Cotes formula)
$points = [[0, 1], [1, 4], [2, 9], [3, 16], [4,3]];
$∫f⟮x⟯dx = NumericalIntegration\SimpsonsRule::approximate($points); // input as a set of points

$f⟮x⟯ = function ($x) {
    return $x**2 + 2 * $x + 1;
};
list($start, $end, $n) = [0, 3, 5];
$∫f⟮x⟯dx = NumericalIntegration\SimpsonsRule::approximate($f⟮x⟯, $start, $end, $n); // input as a callback function

// Simpsons 3/8 Rule (closed Newton-Cotes formula)
$points = [[0, 1], [1, 4], [2, 9], [3, 16]];
$∫f⟮x⟯dx = NumericalIntegration\SimpsonsThreeEighthsRule::approximate($points); // input as a set of points

$f⟮x⟯ = function ($x) {
    return $x**2 + 2 * $x + 1;
};
list($start, $end, $n) = [0, 3, 5];
$∫f⟮x⟯dx = NumericalIntegration\SimpsonsThreeEighthsRule::approximate($f⟮x⟯, $start, $end, $n); // input as a callback function

// Booles Rule (closed Newton-Cotes formula)
$points = [[0, 1], [1, 4], [2, 9], [3, 16], [4, 25]];
$∫f⟮x⟯dx = NumericalIntegration\BoolesRule::approximate($points); // input as a set of points

$f⟮x⟯ = function ($x) {
    return $x**3 + 2 * $x + 1;
};
list($start, $end, $n) = [0, 4, 5];
$∫f⟮x⟯dx = NumericalIntegration\BoolesRuleRule::approximate($f⟮x⟯, $start, $end, $n); // input as a callback function

// Rectangle Method (open Newton-Cotes formula)
$points = [[0, 1], [1, 4], [2, 9], [3, 16]];
$∫f⟮x⟯dx = NumericalIntegration\RectangleMethod::approximate($points); // input as a set of points

$f⟮x⟯ = function ($x) {
    return $x**2 + 2 * $x + 1;
};
list($start, $end, $n) = [0, 3, 4];
$∫f⟮x⟯dx = NumericalIntegration\RectangleMethod::approximate($f⟮x⟯, $start, $end, $n); // input as a callback function

// Midpoint Rule (open Newton-Cotes formula)
$points = [[0, 1], [1, 4], [2, 9], [3, 16]];
$∫f⟮x⟯dx = NumericalIntegration\MidpointRule::approximate($points); // input as a set of points

$f⟮x⟯ = function ($x) {
    return $x**2 + 2 * $x + 1;
};
list($start, $end, $n) = [0, 3, 4];
$∫f⟮x⟯dx = NumericalIntegration\MidpointRule::approximate($f⟮x⟯, $start, $end, $n); // input as a callback function

Numerical Analysis - Root Finding

use MathPHP\NumericalAnalysis\RootFinding;

// Root-finding methods solve for a root of a polynomial.

// f(x) = x⁴ + 8x³ -13x² -92x + 96
$f⟮x⟯ = function($x) {
    return $x**4 + 8 * $x**3 - 13 * $x**2 - 92 * $x + 96;
};

// Newton's Method
$args     = [-4.1];  // Parameters to pass to callback function (initial guess, other parameters)
$target   = 0;       // Value of f(x) we a trying to solve for
$tol      = 0.00001; // Tolerance; how close to the actual solution we would like
$position = 0;       // Which element in the $args array will be changed; also serves as initial guess. Defaults to 0.
$x        = RootFinding\NewtonsMethod::solve($f⟮x⟯, $args, $target, $tol, $position); // Solve for x where f(x) = $target

// Secant Method
$p₀  = -1;      // First initial approximation
$p₁  = 2;       // Second initial approximation
$tol = 0.00001; // Tolerance; how close to the actual solution we would like
$x   = RootFinding\SecantMethod::solve($f⟮x⟯, $p₀, $p₁, $tol); // Solve for x where f(x) = 0

// Bisection Method
$a   = 2;       // The start of the interval which contains a root
$b   = 5;       // The end of the interval which contains a root
$tol = 0.00001; // Tolerance; how close to the actual solution we would like
$x   = RootFinding\BisectionMethod::solve($f⟮x⟯, $a, $b, $tol); // Solve for x where f(x) = 0

// Fixed-Point Iteration
// f(x) = x⁴ + 8x³ -13x² -92x + 96
// Rewrite f(x) = 0 as (x⁴ + 8x³ -13x² + 96)/92 = x
// Thus, g(x) = (x⁴ + 8x³ -13x² + 96)/92
$g⟮x⟯ = function($x) {
    return ($x**4 + 8 * $x**3 - 13 * $x**2 + 96)/92;
};
$a   = 0;       // The start of the interval which contains a root
$b   = 2;       // The end of the interval which contains a root
$p   = 0;       // The initial guess for our root
$tol = 0.00001; // Tolerance; how close to the actual solution we would like
$x   = RootFinding\FixedPointIteration::solve($g⟮x⟯, $a, $b, $p, $tol); // Solve for x where f(x) = 0

Probability - Combinatorics

use MathPHP\Probability\Combinatorics;

list($n, $x, $k) = [10, 3, 4];

// Factorials
$n！  = Combinatorics::factorial($n);
$n‼︎   = Combinatorics::doubleFactorial($n);
$x⁽ⁿ⁾ = Combinatorics::risingFactorial($x, $n);
$x₍ᵢ₎ = Combinatorics::fallingFactorial($x, $n);
$！n  = Combinatorics::subfactorial($n);

// Permutations
$nPn = Combinatorics::permutations($n);     // Permutations of n things, taken n at a time (same as factorial)
$nPk = Combinatorics::permutations($n, $k); // Permutations of n things, taking only k of them

// Combinations
$nCk  = Combinatorics::combinations($n, $k);                            // n choose k without repetition
$nC′k = Combinatorics::combinations($n, $k, Combinatorics::REPETITION); // n choose k with repetition (REPETITION const = true)

// Central binomial coefficient
$cbc = Combinatorics::centralBinomialCoefficient($n);

// Catalan number
$Cn = Combinatorics::catalanNumber($n);

// Lah number
$L⟮n、k⟯ = Combinatorics::lahNumber($n, $k)

// Multinomial coefficient
$groups    = [5, 2, 3];
$divisions = Combinatorics::multinomial($groups);

Probability - Continuous Distributions

use MathPHP\Probability\Distribution\Continuous;

$p = 0.1;

// Beta distribution
$α      = 1; // shape parameter
$β      = 1; // shape parameter
$x      = 2;
$beta   = new Continuous\Beta($α, $β);
$pdf    = $beta->pdf($x);
$cdf    = $beta->cdf($x);
$icdf   = $beta->inverse($p);
$μ      = $beta->mean();
$median = $beta->median();
$mode   = $beta->mode();
$σ²     = $beta->variance();

// Cauchy distribution
$x₀     = 2; // location parameter
$γ      = 3; // scale parameter
$x      = 1;
$cauchy = new Continuous\Cauchy(x₀, γ);
$pdf    = $cauchy->pdf(x);
$cdf    = $cauchy->cdf(x);
$icdf   = $cauchy->inverse($p);
$μ      = $cauchy->mean();
$median = $cauchy->median();
$mode   = $cauchy->mode();

// χ²-distribution (Chi-Squared)
$k      = 2; // degrees of freedom
$x      = 1;
$χ²     = new Continuous\ChiSquared($k);
$pdf    = $χ²->pdf($x);
$cdf    = $χ²->cdf($x);
$μ      = $χ²->mean($x);
$median = $χ²->median();
$mode   = $χ²->mode();
$σ²     = $χ²->variance();

// Dirac delta distribution
$x     = 1;
$dirac = new Continuous\DiracDelta();
$pdf   = $dirac->pdf($x);
$cdf   = $dirac->cdf($x);
$icdf  = $dirac->inverse($p);
$μ     = $dirac->mean();

// Exponential distribution
$λ           = 1; // rate parameter
$x           = 2;
$exponential = new Continuous\Exponential($λ);
$pdf         = $exponential->pdf($x);
$cdf         = $exponential->cdf($x);
$icdf        = $exponential->inverse($p);
$μ           = $exponential->mean();
$median      = $exponential->median();
$σ²          = $exponential->variance();

// F-distribution
$d₁   = 3; // degree of freedom v1
$d₂   = 4; // degree of freedom v2
$x    = 2;
$f    = new Continuous\F($d₁, $d₂);
$pdf  = $f->pdf($x);
$cdf  = $f->cdf($x);
$μ    = $f->mean();
$mode = $f->mode();
$σ²   = $f->variance();

// Gamma distribution
$k      = 2; // shape parameter
$θ      = 3; // scale parameter
$x      = 4;
$gamma  = new Continuous\Gamma($k, $θ);
$pdf    = $gamma->pdf($x);
$cdf    = $gamma->cdf($x);
$μ      = $gamma->mean();
$median = $gamma->median();
$mode   = $gamma->mode();
$σ²     = $gamma->variance();

// Laplace distribution
$μ       = 1;   // location parameter
$b       = 1.5; // scale parameter (diversity)
$x       = 1;
$laplace = new Continuous\Laplace($μ, $b);
$pdf     = $laplace->pdf($x);
$cdf     = $laplace->cdf($x);
$icdf    = $laplace->inverse($p);
$μ       = $laplace->mean();
$median  = $laplace->median();
$mode    = $laplace->mode();
$σ²      = $laplace->variance();

// Logistic distribution
$μ        = 2;   // location parameter
$s        = 1.5; // scale parameter
$x        = 3;
$logistic = new Continuous\Logistic($μ, $s);
$pdf      = $logistic->pdf($x);
$cdf      = $logistic->cdf($x);
$icdf     = $logistic->inverse($p);
$μ        = $logistic->mean();
$median   = $logistic->median();
$mode     = $logistic->mode();
$σ²       = $logisitic->variance();

// Log-logistic distribution (Fisk distribution)
$α           = 1; // scale parameter
$β           = 1; // shape parameter
$x           = 2;
$logLogistic = new Continuous\LogLogistic($α, $β);
$pdf         = $logLogistic->pdf($x);
$cdf         = $logLogistic->cdf($x);
$icdf        = $logLogistic->inverse($p);
$μ           = $logLogistic->mean();
$median      = $logLogistic->median();
$mode        = $logLogistic->mode();
$σ²          = $logLogistic->variance();

// Log-normal distribution
$μ         = 6;   // scale parameter
$σ         = 2;   // location parameter
$x         = 4.3;
$logNormal = new Continuous\LogNormal($μ, $σ);
$pdf       = $logNormal->pdf($x);
$cdf       = $logNormal->cdf($x);
$icdf      = $logNormal->inverse($p);
$μ         = $logNormal->mean();
$median    = $logNormal->median();
$mode      = $logNormal->mode();
$σ²        = $logNormal->variance();

// Noncentral T distribution
$ν            = 50; // degrees of freedom
$μ            = 10; // noncentrality parameter
$x            = 8;
$noncenetralT = new Continuous\NoncentralT($ν, $μ);
$pdf          = $noncenetralT->pdf($x);
$cdf          = $noncenetralT->cdf($x);
$μ            = $noncenetralT->mean();

// Normal distribution
$σ      = 1;
$μ      = 0;
$x      = 2;
$normal = new Continuous\Normal($μ, $σ);
$pdf    = $normal->pdf($x);
$cdf    = $normal->cdf($x);
$icdf   = $normal->inverse($p);
$μ      = $normal->mean();
$median = $normal->median();
$mode   = $normal->mode();
$σ²     = $normal->variance();

// Pareto distribution
$a      = 1; // shape parameter
$b      = 1; // scale parameter
$x      = 2;
$pareto = new Continuous\Pareto($a, $b);
$pdf    = $pareto->pdf($x);
$cdf    = $pareto->cdf($x);
$icdf   = $pareto->inverse($p);
$μ      = $pareto->mean();
$median = $pareto->median();
$mode   = $pareto->mode();
$σ²     = $pareto->variance();

// Standard normal distribution
$z              = 2;
$standardNormal = new Continuous\StandardNormal();
$pdf            = $standardNormal->pdf($z);
$cdf            = $standardNormal->cdf($z);
$icdf           = $standardNormal->inverse($p);
$μ              = $standardNormal->mean();
$median         = $standardNormal->median();
$mode           = $standardNormal->mode();
$σ²             = $standardNormal->variance();

// Student's t-distribution
$ν        = 3;   // degrees of freedom
$p        = 0.4; // proportion of area
$x        = 2;
$studentT = new Continuous\StudentT::pdf($ν);
$pdf      = $studentT->pdf($x);
$cdf      = $studentT->cdf($x);
$t        = $studentT->inverse2Tails($p);  // t such that the area greater than t and the area beneath -t is p
$μ        = $studentT->mean();
$median   = $studentT->median();
$mode     = $studentT->mode();
$σ²       = $studentT->variance();

// Uniform distribution
$a       = 1; // lower boundary of the distribution
$b       = 4; // upper boundary of the distribution
$x       = 2;
$uniform = new Continuous\Uniform($a, $b);
$pdf     = $uniform->pdf($x);
$cdf     = $uniform->cdf($x);
$μ       = $uniform->mean();
$median  = $uniform->median();
$mode    = $uniform->mode();
$σ²      = $uniform->variance();

// Weibull distribution
$k       = 1; // shape parameter
$λ       = 2; // scale parameter
$x       = 2;
$weibull = new Continuous\Weibull($k, $λ);
$pdf     = $weibull->pdf($x);
$cdf     = $weibull->cdf($x);
$icdf    = $weibull->inverse($p);
$μ       = $weibull->mean();
$median  = $weibull->median();
$mode    = $weibull->mode();

// Other CDFs - All continuous distributions - Replace {$distribution} with desired distribution.
$between = $distribution->between($x₁, $x₂);  // Probability of being between two points, x₁ and x₂
$outside = $distribution->outside($x₁, $x);   // Probability of being between below x₁ and above x₂
$above   = $distribution->above($x);          // Probability of being above x to ∞

// Random Number Generator
$random  = $distribution->rand();  // A random number with a given distribution

Probability - Discrete Distributions

use MathPHP\Probability\Distribution\Discrete;

// Bernoulli distribution (special case of binomial where n = 1)
$p         = 0.3;
$k         = 0;
$bernoulli = new Discrete\Bernoulli($p);
$pmf       = $bernoulli->pmf($k);
$cdf       = $bernoulli->cdf($k);
$μ         = $bernoulli->mean();
$median    = $bernoulli->median();
$mode      = $bernoulli->mode();
$σ²        = $bernoulli->variance();

// Binomial distribution
$n        = 2;   // number of events
$p        = 0.5; // probability of success
$r        = 1;   // number of successful events
$binomial = new Discrete\Binomial($n, $p);
$pmf      = $binomial->pmf($r);
$cdf      = $binomial->cdf($r);
$μ        = $binomial->mean();
$σ²       = $binomial->variance();

// Categorical distribution
$k             = 3;                                    // number of categories
$probabilities = ['a' => 0.3, 'b' => 0.2, 'c' => 0.5]; // probabilities for categorices a, b, and c
$categorical   = new Discrete\Categorical($k, $probabilities);
$pmf_a         = $categorical->pmf('a');
$mode          = $categorical->mode();

// Geometric distribution (failures before the first success)
$p         = 0.5; // success probability
$k         = 2;   // number of trials
$geometric = new Discrete\Geometric($p);
$pmf       = $geometric->pmf($k);
$cdf       = $geometric->cdf($k);
$μ         = $geometric->mean();
$median    = $geometric->median();
$mode      = $geometric->mode();
$σ²        = $geometric->variance();

// Hypergeometric distribution
$N        = 50; // population size
$K        = 5;  // number of success states in the population
$n        = 10; // number of draws
$k        = 4;  // number of observed successes
$hypergeo = new Discrete\Hypergeometric($N, $K, $n);
$pmf      = $hypergeo->pmf($k);
$cdf      = $hypergeo->cdf($k);
$μ        = $hypergeo->mean();
$mode     = $hypergeo->mode();
$σ²       = $hypergeo->variance();

// Negative binomial distribution (Pascal)
$r                = 1;   // number of failures until the experiment is stopped
$P                = 0.5; // probability of success on an individual trial
$x                = 2;   // number of successes
$negativeBinomial = new Discrete\NegativeBinomial($r, $p);
$pmf              = $negativeBinomial->pmf($x);
$cdf              = $negativeBinomial->cdf($x);
$μ                = $negativeBinomial->mean();
$mode             = $negativeBinomial->mode();
$σ²               = $negativeBinomial->variance();

// Pascal distribution (Negative binomial)
$r      = 1;   // number of failures until the experiment is stopped
$P      = 0.5; // probability of success on an individual trial
$x      = 2;   // number of successes
$pascal = new Discrete\Pascal($r, $p);
$pmf    = $pascal->pmf($x);
$cdf    = $pascal->cdf($x);
$μ      = $pascal->mean();
$mode   = $pascal->mode();
$σ²     = $pascal->variance();

// Poisson distribution
$λ       = 2; // average number of successful events per interval
$k       = 3; // events in the interval
$poisson = new Discrete\Poisson($λ);
$pmf     = $poisson->pmf($k);
$cdf     = $poisson->cdf($k);
$μ       = $poisson->mean();
$median  = $poisson->median();
$mode    = $poisson->mode();
$σ²      = $poisson->variance();

// Shifted geometric distribution (probability to get one success)
$p                = 0.5; // success probability
$k                = 2;   // number of trials
$shiftedGeometric = new Discrete\ShiftedGeometric($p);
$pmf              = $shiftedGeometric->pmf($k);
$cdf              = $shiftedGeometric->cdf($k);
$μ                = $shiftedGeometric->mean();
$median           = $shiftedGeometric->median();
$mode             = $shiftedGeometric->mode();
$σ²               = $shiftedGeometric->variance();

// Uniform distribution
$a       = 1; // lower boundary of the distribution
$b       = 4; // upper boundary of the distribution
$k       = 2; // percentile
$uniform = new Discrete\Uniform($a, $b);
$pmf     = $uniform->pmf();
$cdf     = $uniform->cdf($k);
$μ       = $uniform->mean();
$median  = $uniform->median();
$σ²      = $uniform->variance();

Probability - Multivariate Distributions

use MathPHP\Probability\Distribution\Multivariate;

// Dirichlet distribution
$αs        = [1, 2, 3];
$xs        = [0.07255081, 0.27811903, 0.64933016];
$dirichlet = new Multivariate\Dirichlet($αs);
$pdf       = $dirichlet->pdf($xs);

// Normal distribution
$μ      = [1, 1.1];
$∑      = MatrixFactory::create([
    [1, 0],
    [0, 1],
]);
$X      = [0.7, 1.4];
$normal = new Multivariate\Normal($μ, $∑);
$pdf    = $normal->pdf($X);

// Multinomial distribution
$frequencies   = [7, 2, 3];
$probabilities = [0.40, 0.35, 0.25];
$multinomial   = new Multivariate\Multinomial($probabilities);
$pmf           = $multinomial->pmf($frequencies);

Probability - Distribution Tables

use MathPHP\Probability\Distribution\Table;

// Provided solely for completeness' sake.
// It is statistics tradition to provide these tables.
// MathPHP has dynamic distribution CDF functions you can use instead.

// Standard Normal Table (Z Table)
$table       = Table\StandardNormal::Z_SCORES;
$probability = $table[1.5][0];                 // Value for Z of 1.50

// t Distribution Tables
$table   = Table\TDistribution::ONE_SIDED_CONFIDENCE_LEVEL;
$table   = Table\TDistribution::TWO_SIDED_CONFIDENCE_LEVEL;
$ν       = 5;  // degrees of freedom
$cl      = 99; // confidence level
$t       = $table[$ν][$cl];

// t Distribution Tables
$table = Table\TDistribution::ONE_SIDED_ALPHA;
$table = Table\TDistribution::TWO_SIDED_ALPHA;
$ν     = 5;     // degrees of freedom
$α     = 0.001; // alpha value
$t     = $table[$ν][$α];

// χ² Distribution Table
$table = Table\ChiSquared::CHI_SQUARED_SCORES;
$df    = 2;    // degrees of freedom
$p     = 0.05; // P value
$χ²    = $table[$df][$p];

Sequences - Basic

use MathPHP\Sequence\Basic;

$n = 5; // Number of elements in the sequence

// Arithmetic progression
$d           = 2;  // Difference between the elements of the sequence
$a₁          = 1;  // Starting number for the sequence
$progression = Basic::arithmeticProgression($n, $d, $a₁);
// [1, 3, 5, 7, 9] - Indexed from 1

// Geometric progression (arⁿ⁻¹)
$a           = 2; // Scalar value
$r           = 3; // Common ratio
$progression = Basic::geometricProgression($n, $a, $r);
// [2(3)⁰, 2(3)¹, 2(3)², 2(3)³] = [2, 6, 18, 54] - Indexed from 1

// Square numbers (n²)
$squares = Basic::squareNumber($n);
// [0², 1², 2², 3², 4²] = [0, 1, 4, 9, 16] - Indexed from 0

// Cubic numbers (n³)
$cubes = Basic::cubicNumber($n);
// [0³, 1³, 2³, 3³, 4³] = [0, 1, 8, 27, 64] - Indexed from 0

// Powers of 2 (2ⁿ)
$po2 = Basic::powersOfTwo($n);
// [2⁰, 2¹, 2², 2³, 2⁴] = [1,  2,  4,  8,  16] - Indexed from 0

// Powers of 10 (10ⁿ)
$po10 = Basic::powersOfTen($n);
// [10⁰, 10¹, 10², 10³,  10⁴] = [1, 10, 100, 1000, 10000] - Indexed from 0

// Factorial (n!)
$fact = Basic::factorial($n);
// [0!, 1!, 2!, 3!, 4!] = [1,  1,  2,  6,  24] - Indexed from 0

// Digit sum
$digit_sum = Basic::digitSum($n);
// [0, 1, 2, 3, 4] - Indexed from 0

// Digital root
$digit_root = Basic::digitalRoot($n);
// [0, 1, 2, 3, 4] - Indexed from 0

Sequences - Advanced

use MathPHP\Sequence\Advanced;

$n = 6; // Number of elements in the sequence

// Fibonacci (Fᵢ = Fᵢ₋₁ + Fᵢ₋₂)
$fib = Advanced::fibonacci($n);
// [0, 1, 1, 2, 3, 5] - Indexed from 0

// Lucas numbers
$lucas = Advanced::lucasNumber($n);
// [2, 1, 3, 4, 7, 11] - Indexed from 0

// Pell numbers
$pell = Advanced::pellNumber($n);
// [0, 1, 2, 5, 12, 29] - Indexed from 0

// Triangular numbers (figurate number)
$triangles = Advanced::triangularNumber($n);
// [1, 3, 6, 10, 15, 21] - Indexed from 1

// Pentagonal numbers (figurate number)
$pentagons = Advanced::pentagonalNumber($n);
// [1, 5, 12, 22, 35, 51] - Indexed from 1

// Hexagonal numbers (figurate number)
$hexagons = Advanced::hexagonalNumber($n);
// [1, 6, 15, 28, 45, 66] - Indexed from 1

// Heptagonal numbers (figurate number)
$hexagons = Advanced::heptagonalNumber($n);
// [1, 4, 7, 13, 18, 27] - Indexed from 1

// Look-and-say sequence (describe the previous term!)
$look_and_say = Advanced::lookAndSay($n);
// ['1', '11', '21', '1211', '111221', '312211'] - Indexed from 1

// Lazy caterer's sequence (central polygonal numbers)
$lazy_caterer = Advanced::lazyCaterers($n);
// [1, 2, 4, 7, 11, 16] - Indexed from 0

// Magic squares series (magic constants; magic sums)
$magic_squares = Advanced::magicSquares($n);
// [0, 1, 5, 15, 34, 65] - Indexed from 0

// Perfect numbers
$perfect_numbers = Advanced::perfectNumbers($n);
// [6, 28, 496, 8128, 33550336, 8589869056] - Indexed from 0

// Perfect powers sequence
$perfect_powers = Advanced::perfectPowers($n);
// [4, 8, 9, 16, 25, 27] - Indexed from 0

// Not perfect powers sequence
$not_perfect_powers = Advanced::notPerfectPowers($n);
// [2, 3, 5, 6, 7, 10] - Indexed from 0

// Prime numbers up to n (n is not the number of elements in the sequence)
$primes = Advanced::primesUpTo(30);
// [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] - Indexed from 0

Sequences - Non-Integer

use MathPHP\Sequence\NonInteger;

$n = 4; // Number of elements in the sequence

// Harmonic sequence
$harmonic_sequence = NonInteger::harmonic($n);
// [1, 3/2, 11/6, 25/12] - Indexed from 1

// Hyperharmonic sequence
$p = 2;
$hyperharmonic_sequence = NonInteger::hyperharmonic($n, $p);
// [1, 5/4, 49/36, 205/144] - Indexed from 1

Set Theory

use MathPHP\SetTheory\Set;
use MathPHP\SetTheory\ImmutableSet;

// Sets and immutable sets
$A = new Set([1, 2, 3]);          // Can add and remove members
$B = new ImmutableSet([3, 4, 5]); // Cannot modify set once created

// Basic set data
$set         = $A->asArray();
$cardinality = $A->length();
$bool        = $A->isEmpty();

// Set membership
$true = $A->isMember(2);
$true = $A->isNotMember(8);

// Add and remove members
$A->add(4);
$A->add(new Set(['a', 'b']));
$A->addMulti([5, 6, 7]);
$A->remove(7);
$A->removeMulti([5, 6]);
$A->clear();

// Set properties against other sets - return boolean
$bool = $A->isDisjoint($B);
$bool = $A->isSubset($B);         // A ⊆ B
$bool = $A->isProperSubset($B);   // A ⊆ B & A ≠ B
$bool = $A->isSuperset($B);       // A ⊇ B
$bool = $A->isProperSuperset($B); // A ⊇ B & A ≠ B

// Set operations with other sets - return a new Set
$A∪B  = $A->union($B);
$A∩B  = $A->intersect($B);
$A＼B = $A->difference($B);          // relative complement
$AΔB  = $A->symmetricDifference($B);
$A×B  = $A->cartesianProduct($B);

// Other set operations
$P⟮A⟯ = $A->powerSet();
$C   = $A->copy();

// Print a set
print($A); // Set{1, 2, 3, 4, Set{a, b}}

// PHP Interfaces
$n = count($A);                 // Countable
foreach ($A as $member) { ... } // Iterator

// Fluent interface
$A->add(5)->add(6)->remove(4)->addMulti([7, 8, 9]);

Statistics - ANOVA

use MathPHP\Statistics\ANOVA;

// One-way ANOVA
$sample1 = [1, 2, 3];
$sample2 = [3, 4, 5];
$sample3 = [5, 6, 7];
   ⋮            ⋮

$anova = ANOVA::oneWay($sample1, $sample2, $sample3);
print_r($anova);
/* Array (
    [ANOVA] => Array (             // ANOVA hypothesis test summary data
            [treatment] => Array (
                    [SS] => 24     // Sum of squares (between)
                    [df] => 2      // Degrees of freedom
                    [MS] => 12     // Mean squares
                    [F]  => 12     // Test statistic
                    [P]  => 0.008  // P value
                )
            [error] => Array (
                    [SS] => 6      // Sum of squares (within)
                    [df] => 6      // Degrees of freedom
                    [MS] => 1      // Mean squares
                )
            [total] => Array (
                    [SS] => 30     // Sum of squares (total)
                    [df] => 8      // Degrees of freedom
                )
        )
    [total_summary] => Array (     // Total summary data
            [n]        => 9
            [sum]      => 36
            [mean]     => 4
            [SS]       => 174
            [variance] => 3.75
            [sd]       => 1.9364916731037
            [sem]      => 0.6454972243679
        )
    [data_summary] => Array (      // Data summary (each input sample)
            [0] => Array ([n] => 3 [sum] => 6  [mean] => 2 [SS] => 14  [variance] => 1 [sd] => 1 [sem] => 0.57735026918963)
            [1] => Array ([n] => 3 [sum] => 12 [mean] => 4 [SS] => 50  [variance] => 1 [sd] => 1 [sem] => 0.57735026918963)
            [2] => Array ([n] => 3 [sum] => 18 [mean] => 6 [SS] => 110 [variance] => 1 [sd] => 1 [sem] => 0.57735026918963)
        )
) */

// Two-way ANOVA
/*, Factor B₁, Factor B₂, Factor B₃, ⋯
Factor A₁, 4, 6, 8, 6, 6, 9, 8, 9, 13, ⋯
Factor A₂, 4, 8, 9, 7, 10, 13, 12, 14, 16, ⋯
    ⋮           ⋮           ⋮           ⋮         */
$factorA₁ = [
  [4, 6, 8],    // Factor B₁
  [6, 6, 9],    // Factor B₂
  [8, 9, 13],   // Factor B₃
];
$factorA₂ = [
  [4, 8, 9],    // Factor B₁
  [7, 10, 13],  // Factor B₂
  [12, 14, 16], // Factor B₃
];
       ⋮

$anova = ANOVA::twoWay($factorA₁, $factorA₂);
print_r($anova);
/* Array (
    [ANOVA] => Array (          // ANOVA hypothesis test summary data
            [factorA] => Array (
                    [SS] => 32                 // Sum of squares
                    [df] => 1                  // Degrees of freedom
                    [MS] => 32                 // Mean squares
                    [F]  => 5.6470588235294    // Test statistic
                    [P]  => 0.034994350619895  // P value
                )
            [factorB] => Array (
                    [SS] => 93                 // Sum of squares
                    [df] => 2                  // Degrees of freedom
                    [MS] => 46.5               // Mean squares
                    [F]  => 8.2058823529412    // Test statistic
                    [P]  => 0.0056767297582031 // P value
                )
            [interaction] => Array (
                    [SS] => 7                  // Sum of squares
                    [df] => 2                  // Degrees of freedom
                    [MS] => 3.5                // Mean squares
                    [F]  => 0.61764705882353   // Test statistic
                    [P]  => 0.5555023440712    // P value
                )
            [error] => Array (
                    [SS] => 68                 // Sum of squares (within)
                    [df] => 12                 // Degrees of freedom
                    [MS] => 5.6666666666667    // Mean squares
                )
            [total] => Array (
                    [SS] => 200                // Sum of squares (total)
                    [df] => 17                 // Degrees of freedom
                )
        )
    [total_summary] => Array (    // Total summary data
            [n]        => 18
            [sum]      => 162
            [mean]     => 9
            [SS]       => 1658
            [variance] => 11.764705882353
            [sd]       => 3.4299717028502
            [sem]      => 0.80845208345444
        )
    [summary_factorA]     => Array ( ... )   // Summary data of factor A
    [summary_factorB]     => Array ( ... )   // Summary data of factor B
    [summary_interaction] => Array ( ... )   // Summary data of interactions of factors A and B
) */

Statistics - Averages

use MathPHP\Statistics\Average;

$numbers = [13, 18, 13, 14, 13, 16, 14, 21, 13];

// Mean, median, mode
$mean   = Average::mean($numbers);
$median = Average::median($numbers);
$mode   = Average::mode($numbers); // Returns an array — may be multimodal

// Weighted mean
$weights       = [12, 1, 23, 6, 12, 26, 21, 12, 1];
$weighted_mean = Average::weightedMean($numbers, $weights)

// Other means of a list of numbers
$geometric_mean      = Average::geometricMean($numbers);
$harmonic_mean       = Average::harmonicMean($numbers);
$contraharmonic_mean = Average::contraharmonicMean($numbers);
$quadratic_mean      = Average::quadraticMean($numbers);  // same as rootMeanSquare
$root_mean_square    = Average::rootMeanSquare($numbers); // same as quadraticMean
$trimean             = Average::trimean($numbers);
$interquartile_mean  = Average::interquartileMean($numbers); // same as iqm
$interquartile_mean  = Average::iqm($numbers);               // same as interquartileMean
$cubic_mean          = Average::cubicMean($numbers);

// Truncated mean (trimmed mean)
$trim_percent   = 25;
$truncated_mean = Average::truncatedMean($numbers, $trim_percent);

// Generalized mean (power mean)
$p                = 2;
$generalized_mean = Average::generalizedMean($numbers, $p); // same as powerMean
$power_mean       = Average::powerMean($numbers, $p);       // same as generalizedMean

// Lehmer mean
$p           = 3;
$lehmer_mean = Average::lehmerMean($numbers, $p);

// Moving averages
$n       = 3;
$weights = [3, 2, 1];
$SMA     = Average::simpleMovingAverage($numbers, $n);             // 3 n-point moving average
$CMA     = Average::cumulativeMovingAverage($numbers);
$WMA     = Average::weightedMovingAverage($numbers, $n, $weights);
$EPA     = Average::exponentialMovingAverage($numbers, $n);

// Means of two numbers
list($x, $y) = [24, 6];
$agm           = Average::arithmeticGeometricMean($x, $y); // same as agm
$agm           = Average::agm($x, $y);                     // same as arithmeticGeometricMean
$log_mean      = Average::logarithmicMean($x, $y);
$heronian_mean = Average::heronianMean($x, $y);
$identric_mean = Average::identricMean($x, $y);

// Averages report
$averages = Average::describe($numbers);
print_r($averages);
/* Array (
    [mean]                => 15
    [median]              => 14
    [mode]                => Array ( [0] => 13 )
    [geometric_mean]      => 14.789726414533
    [harmonic_mean]       => 14.605077399381
    [contraharmonic_mean] => 15.474074074074
    [quadratic_mean]      => 15.235193176035
    [trimean]             => 14.5
    [iqm]                 => 14
    [cubic_mean]          => 15.492307432707
) */

Statistics - Circular

use MathPHP\Statistics\Circular;

$angles = [1.51269877, 1.07723915, 0.81992282];

$θ = Circular::mean($angles);
$R = Circular::resultantLength($angles);
$ρ = Circular::meanResultantLength($angles);
$V = Circular::variance($angles);
$ν = Circular::standardDeviation($angles);

// Descriptive circular statistics report
$stats = Circular::describe($angles);
print_r($stats);
/* Array (
    [n]                     => 3
    [mean]                  => 1.1354043006436
    [resultant_length]      => 2.8786207547493
    [mean_resultant_length] => 0.9595402515831
    [variance]              => 0.040459748416901
    [sd]                    => 0.28740568481722
); */

Statistics - Correlation

use MathPHP\Statistics\Correlation;

$X = [1, 2, 3, 4, 5];
$Y = [2, 3, 4, 4, 6];

// Covariance
$σxy = Correlation::covariance($X, $Y);  // Has optional parameter to set population (defaults to sample covariance)

// Weighted covariance
$w    = [2, 3, 1, 1, 5];
$σxyw = Correlation::weightedCovariance($X, $Y, $w);

// r - Pearson product-moment correlation coefficient (Pearson's r)
$r = Correlation::r($X, $Y);  // Has optional parameter to set population (defaults to sample correlation coefficient)

// Weighted correlation coefficient
$rw = Correlation::weightedCorrelationCoefficient($X, $Y, $w);

// R² - Coefficient of determination
$R² = Correlation::r2($X, $Y);  // Has optional parameter to set population (defaults to sample coefficient of determination)

// τ - Kendall rank correlation coefficient (Kendall's tau)
$τ = Correlation::kendallsTau($X, $Y);

// ρ - Spearman's rank correlation coefficient (Spearman's rho)
$ρ = Correlation::spearmansRho($X, $Y);

// Descriptive correlation report
$stats = Correlation::describe($X, $Y);
print_r($stats);
/* Array (
    [cov] => 2.25
    [r]   => 0.95940322360025
    [r2]  => 0.92045454545455
    [tau] => 0.94868329805051
    [rho] => 0.975
) */

// Confidence ellipse - create an ellipse surrounding the data at a specified standard deviation
$sd           = 1;
$num_points   = 11; // Optional argument specifying number of points of the ellipse
$ellipse_data = Correlation::confidenceEllipse($X, $Y, $sd, $num_points);

Statistics - Descriptive

use MathPHP\Statistics\Descriptive;

$numbers = [13, 18, 13, 14, 13, 16, 14, 21, 13];

// Range and midrange
$range    = Descriptive::range($numbers);
$midrange = Descriptive::midrange($numbers);

// Variance (population and sample)
$σ² = Descriptive::populationVariance($numbers); // n degrees of freedom
$S² = Descriptive::sampleVariance($numbers);     // n - 1 degrees of freedom

// Variance (Custom degrees of freedom)
$df = 5;                                    // degrees of freedom
$S² = Descriptive::variance($numbers, $df); // can specify custom degrees of freedom

// Weighted sample variance
$weights = [0.1, 0.2, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1];
$σ²w     = Descriptive::weightedSampleVariance($numbers, $weights, $biased = false);

// Standard deviation (For a sample; uses sample variance)
$σ = Descriptive::sd($numbers);                // same as standardDeviation;
$σ = Descriptive::standardDeviation($numbers); // same as sd;

// SD+ (Standard deviation for a population; uses population variance)
$SD＋ = Descriptive::sd($numbers, Descriptive::POPULATION); // POPULATION constant = true
$SD＋ = Descriptive::standardDeviation($numbers, true);     // same as sd with POPULATION constant

// Coefficient of variation (cᵥ)
$cᵥ = Descriptive::coefficientOfVariation($numbers);

// MAD - mean/median absolute deviations
$mean_mad   = Descriptive::meanAbsoluteDeviation($numbers);
$median_mad = Descriptive::medianAbsoluteDeviation($numbers);

// Quartiles (inclusive and exclusive methods)
// [0% => 13, Q1 => 13, Q2 => 14, Q3 => 17, 100% => 21, IQR => 4]
$quartiles = Descriptive::quartiles($numbers);          // Has optional parameter to specify method. Default is Exclusive
$quartiles = Descriptive::quartilesExclusive($numbers);
$quartiles = Descriptive::quartilesInclusive($numbers);

// IQR - Interquartile range
$IQR = Descriptive::interquartileRange($numbers); // Same as IQR; has optional parameter to specify quartile method.
$IQR = Descriptive::iqr($numbers);                // Same as interquartileRange; has optional parameter to specify quartile method.

// Percentiles
$twentieth_percentile    = Descriptive::percentile($numbers, 20);
$ninety_fifth_percentile = Descriptive::percentile($numbers, 95);

// Midhinge
$midhinge = Descriptive::midhinge($numbers);

// Describe a list of numbers - descriptive stats report
$stats = Descriptive::describe($numbers); // Has optional parameter to set population or sample calculations
print_r($stats);
/* Array (
    [n]          => 9
    [min]        => 13
    [max]        => 21
    [mean]       => 15
    [median]     => 14
    [mode]       => Array ( [0] => 13 )
    [range]      => 8
    [midrange]   => 17
    [variance]   => 8
    [sd]         => 2.8284271247462
    [cv]         => 0.18856180831641
    [mean_mad]   => 2.2222222222222
    [median_mad] => 1
    [quartiles]  => Array (
            [0%]   => 13
            [Q1]   => 13
            [Q2]   => 14
            [Q3]   => 17
            [100%] => 21
            [IQR]  => 4
        )
    [midhinge]   => 15
    [skewness]   => 1.4915533665654
    [ses]        => 0.71713716560064
    [kurtosis]   => 0.1728515625
    [sek]        => 1.3997084244475
    [sem]        => 0.94280904158206
    [ci_95]      => Array (
            [ci]          => 1.8478680091392
            [lower_bound] => 13.152131990861
            [upper_bound] => 16.847868009139
        )
    [ci_99]      => Array (
            [ci]          => 2.4285158135783
            [lower_bound] => 12.571484186422
            [upper_bound] => 17.428515813578
        )
) */

// Five number summary - five most important sample percentiles
$summary = Descriptive::fiveNumberSummary($numbers);
// [min, Q1, median, Q3, max]

Statistics - Distance and Divergence

use MathPHP\Statistics\Distance;

// Probability distributions
$p = [0.2, 0.5, 0.3];
$q = [0.1, 0.4, 0.5];

// Distances
$DB⟮p、q⟯ = Distance::bhattacharyyaDistance($p, $q);
$H⟮p、q⟯  = Distance::hellingerDistance($p, $q);

// Mahalanobis distance
$x    = new Matrix([[6], [5]]);
$data = new Matrix([
    [4, 4, 5, 2, 3, 6, 9, 7, 4, 5],
    [3, 7, 5, 7, 9, 5, 6, 2, 2, 7],
]);
$otherData = new Matrix([
    [4, 4, 5, 2, 3, 6, 9, 7, 4, 5],
    [3, 7, 5, 7, 9, 5, 6, 2, 2, 7],
]);
$y = new Matrix([[2], [2]]);
$D = Distance::Mahalanobis($x, $data);          // Mahalanobis distance from x to the centroid of the data.
$D = Distance::Mahalanobis($x, $data, $y);      // Mahalanobis distance between $x and $y using the data.
$D = Distance::Mahalanobis($data, $otherData);  // Mahalanobis distance between the centroids of two sets of data.

// Divergences
$Dkl⟮P‖Q⟯ = Distance::kullbackLeiblerDivergence($p, $q);
$JSD⟮P‖Q⟯ = Distance::jensenShannonDivergence($p, $q);

Statistics - Distributions

use MathPHP\Statistics\Distribution;

$grades = ['A', 'A', 'B', 'B', 'B', 'B', 'C', 'C', 'D', 'F'];

// Frequency distributions (frequency and relative frequency)
$frequencies          = Distribution::frequency($grades);         // [ A => 2,   B => 4,   C => 2,   D => 1,   F => 1   ]
$relative_frequencies = Distribution::relativeFrequency($grades); // [ A => 0.2, B => 0.4, C => 0.2, D => 0.1, F => 0.1 ]

// Cumulative frequency distributions (cumulative and cumulative relative)
$cumulative_frequencies          = Distribution::cumulativeFrequency($grades);         // [ A => 2,   B => 6,   C => 8,   D => 9,   F => 10  ]
$cumulative_relative_frequencies = Distribution::cumulativeRelativeFrequency($grades); // [ A => 0.2, B => 0.6, C => 0.8, D => 0.9, F => 1   ]

// Stem and leaf plot
// Return value is array where keys are the stems, values are the leaves
$values             = [44, 46, 47, 49, 63, 64, 66, 68, 68, 72, 72, 75, 76, 81, 84, 88, 106];
$stem_and_leaf_plot = Distribution::stemAndLeafPlot($values);
// [4 => [4, 6, 7, 9], 5 => [], 6 => [3, 4, 6, 8, 8], 7 => [2, 2, 5, 6], 8 => [1, 4, 8], 9 => [], 10 => [6]]

// Optional second parameter will print stem and leaf plot to STDOUT
Distribution::stemAndLeafPlot($values, Distribution::PRINT);
/*
 4, 4 6 7 9
 5, 6, 3 4 6 8 8
 7, 2 2 5 6
 8, 1 4 8
 9, 10, 6
*/

Statistics - Effect Size

use MathPHP\Statistics\EffectSize;

$SSt = 24;  // Sum of squares treatment
$SSE = 300; // Sum of squares error
$SST = 600; // Sum of squares total
$dft = 1;   // Degrees of freedom treatment
$MSE = 18;  // Mean squares error

// η² - Eta-squared
$η²  = EffectSize::etaSquared($SSt, $SST);
$η²p = EffectSize::partialEtaSquared($SSt, $SSE);

// ω² - Omega-squared
$ω² = EffectSize::omegaSquared($SSt, $dft, $SST, $MSE);

// Cohen's ƒ²
$ƒ² = EffectSize::cohensF($η²);
$ƒ² = EffectSize::cohensF($ω²);
$ƒ² = EffectSize::cohensF($R²);

// Cohen's q
list($r₁, $r₂) = [0.1, 0.2];
$q = EffectSize::cohensQ($r₁, $r₂);

// Cohen's d
list($μ₁, $σ₁) = [6.7, 1.2];
list($μ₂, $σ₂) = [6, 1];
$d = EffectSize::cohensD($μ₁, $μ₂, $σ₁, $σ₂);

// Hedges' g
list($μ₁, $σ₁, $n₁) = [6.7, 1.2, 15];
list($μ₂, $σ₂, $n₂) = [6, 1, 15];
$g = EffectSize::hedgesG($μ₁, $μ₂, $σ₁, $σ₂, $n₁, $n₂);

// Glass' Δ
$Δ = EffectSize::glassDelta($μ₁, $μ₂, $σ₂);

Statistics - Experiments

use MathPHP\Statistics\Experiment;

$a = 28;   // Exposed and event present
$b = 129;  // Exposed and event absent
$c = 4;    // Non-exposed and event present
$d = 133;  // Non-exposed and event absent

// Risk ratio (relative risk) - RR
$RR = Experiment::riskRatio($a, $b, $c, $d);
// ['RR' => 6.1083, 'ci_lower_bound' => 2.1976, 'ci_upper_bound' => 16.9784, 'p' => 0.0005]

// Odds ratio (OR)
$OR = Experiment::oddsRatio($a, $b, $c, $d);
// ['OR' => 7.2171, 'ci_lower_bound' => 2.4624, 'ci_upper_bound' => 21.1522, 'p' => 0.0003]

// Likelihood ratios (positive and negative)
$LL = Experiment::likelihoodRatio($a, $b, $c, $d);
// ['LL+' => 7.4444, 'LL-' => 0.3626]

$sensitivity = 0.67;
$specificity = 0.91;
$LL          = Experiment::likelihoodRatioSS($sensitivity, $specificity);

Statistics - Kernel Density Estimation

use MathPHP\Statistics\KernelDensityEstimation

$data = [-2.76, -1.09, -0.5, -0.15, 0.22, 0.69, 1.34, 1.75];
$x    = 0.5;

// Density estimator with default bandwidth (normal distribution approximation) and kernel function (standard normal)
$kde     = new KernelDensityEstimation($data);
$density = $kde->evaluate($x)

// Custom bandwidth
$h = 0.1;
$kde->setBandwidth($h);

// Library of built-in kernel functions
$kde->setKernelFunction(KernelDensityEstimation::STANDARD_NORMAL);
$kde->setKernelFunction(KernelDensityEstimation::NORMAL);
$kde->setKernelFunction(KernelDensityEstimation::UNIFORM);
$kde->setKernelFunction(KernelDensityEstimation::TRIANGULAR);
$kde->setKernelFunction(KernelDensityEstimation::EPANECHNIKOV);
$kde->setKernelFunction(KernelDensityEstimation::TRICUBE);

// Set custom kernel function (user-provided callable)
$kernel = function ($x) {
  if (abs($x) > 1) {
      return 0;
  } else {
      return 70 / 81 * ((1 - abs($x) ** 3) ** 3);
  }
};
$kde->setKernelFunction($kernel);

// All customization optionally can be done in the constructor
$kde = new KernelDesnsityEstimation($data, $h, $kernel);

Statistics - Multivariate - Principal Component Analysis

use MathPHP\Statistics\Multivariate\PCA;
use MathPHP\LinearAlgebra\MatrixFactory;

// Given
$matrix = MatrixFactory::create($data);  // observations of possibly correlated variables
$center = true;                          // do mean centering of data
$scale  = true;                          // do standardization of data

// Build a principal component analysis model to explore
$model = new PCA($matrix, $center, $scale);

// Scores and loadings of the PCA model
$scores      = $model->getScores();       // Matrix of transformed standardized data with the loadings matrix
$loadings    = $model->getLoadings();     // Matrix of unit eigenvectors of the correlation matrix
$eigenvalues = $model->getEigenvalues();  // Vector of eigenvalues of components

// Residuals, limits, critical values and more
$R²         = $model->getR2();           // array of R² values
$cumR²      = $model->getCumR2();        // array of cummulative R² values
$Q          = $model->getQResiduals();   // Matrix of Q residuals
$T²         = $model->getT2Distances();  // Matrix of T² distances
$T²Critical = $model->getCriticalT2();   // array of critical limits of T²
$QCritical  = $model->getCriticalQ();    // array of critical limits of Q

Statistics - Outlier

use MathPHP\Statistics\Outlier;

$data = [199.31, 199.53, 200.19, 200.82, 201.92, 201.95, 202.18, 245.57];
$n    = 8;    // size of data
$?    = 0.05; // significance level

// Grubb's test - two sided test
$grubbsStatistic = Outlier::grubbsStatistic($data, Outlier::TWO_SIDED);
$criticalValue   = Outlier::grubbsCriticalValue($?, $n, Outlier::TWO_SIDED);

// Grubbs' test - one sided test of minimum value
$grubbsStatistic = Outlier::grubbsStatistic($data, Outlier::ONE_SIDED_LOWER);
$criticalValue   = Outlier::grubbsCriticalValue($?, $n, Outlier::ONE_SIDED);

// Grubbs' test - one sided test of maximum value
$grubbsStatistic = Outlier::grubbsStatistic($data, Outlier::ONE_SIDED_UPPER);
$criticalValue   = Outlier::grubbsCriticalValue($?, $n, Outlier::ONE_SIDED);

Statistics - Random Variables

use MathPHP\Statistics\RandomVariable;

$X = [1, 2, 3, 4];
$Y = [2, 3, 4, 5];

// Central moment (nth moment)
$second_central_moment = RandomVariable::centralMoment($X, 2);
$third_central_moment  = RandomVariable::centralMoment($X, 3);

// Skewness (population, sample, and alternative general method)
$skewness = RandomVariable::skewness($X);            // Optional type parameter to choose skewness type calculation. Defaults to sample skewness (similar to Excel's SKEW).
$skewness = RandomVariable::sampleSkewness($X);      // Same as RandomVariable::skewness($X, RandomVariable::SAMPLE_SKEWNESS) - Similar to Excel's SKEW, SAS and SPSS, R (e1071) skewness type 2
$skewness = RandomVariable::populationSkewness($X);  // Same as RandomVariable::skewness($X, RandomVariable::POPULATION_SKEWNESS) - Similar to Excel's SKEW.P, classic textbook definition, R (e1071) skewness type 1
$skewness = RandomVariable::alternativeSkewness($X); // Same as RandomVariable::skewness($X, RandomVariable::ALTERNATIVE_SKEWNESS) - Alternative, classic definition of skewness
$SES      = RandomVariable::ses(count($X));          // standard error of skewness

// Kurtosis (excess)
$kurtosis    = RandomVariable::kurtosis($X);
$platykurtic = RandomVariable::isPlatykurtic($X); // true if kurtosis is less than zero
$leptokurtic = RandomVariable::isLeptokurtic($X); // true if kurtosis is greater than zero
$mesokurtic  = RandomVariable::isMesokurtic($X);  // true if kurtosis is zero
$SEK         = RandomVariable::sek(count($X));    // standard error of kurtosis

// Standard error of the mean (SEM)
$sem = RandomVariable::standardErrorOfTheMean($X); // same as sem
$sem = RandomVariable::sem($X);                    // same as standardErrorOfTheMean

// Confidence interval
$μ  = 90; // sample mean
$n  = 9;  // sample size
$σ  = 36; // standard deviation
$cl = 99; // confidence level
$ci = RandomVariable::confidenceInterval($μ, $n, $σ, $cl); // Array( [ci] => 30.91, [lower_bound] => 59.09, [upper_bound] => 120.91 )

Statistics - Regressions

use MathPHP\Statistics\Regression;

$points = [[1,2], [2,3], [4,5], [5,7], [6,8]];

// Simple linear regression (least squares method)
$regression = new Regression\Linear($points);
$parameters = $regression->getParameters();          // [m => 1.2209302325581, b => 0.6046511627907]
$equation   = $regression->getEquation();            // y = 1.2209302325581x + 0.6046511627907
$y          = $regression->evaluate(5);              // Evaluate for y at x = 5 using regression equation
$ci         = $regression->ci(5, 0.5);               // Confidence interval for x = 5 with p-value of 0.5
$pi         = $regression->pi(5, 0.5);               // Prediction interval for x = 5 with p-value of 0.5; Optional number of trials parameter.
$Ŷ          = $regression->yHat();
$r          = $regression->r();                      // same as correlationCoefficient
$r²         = $regression->r2();                     // same as coefficientOfDetermination
$se         = $regression->standardErrors();         // [m => se(m), b => se(b)]
$t          = $regression->tValues();                // [m => t, b => t]
$p          = $regression->tProbability();           // [m => p, b => p]
$F          = $regression->fStatistic();
$p          = $regression->fProbability();
$h          = $regression->leverages();
$e          = $regression->residuals();
$D          = $regression->cooksD();
$DFFITS     = $regression->dffits();
$SStot      = $regression->sumOfSquaresTotal();
$SSreg      = $regression->sumOfSquaresRegression();
$SSres      = $regression->sumOfSquaresResidual();
$MSR        = $regression->meanSquareRegression();
$MSE        = $regression->meanSquareResidual();
$MSTO       = $regression->meanSquareTotal();
$error      = $regression->errorSd();                // Standard error of the residuals
$V          = $regression->regressionVariance();
$n          = $regression->getSampleSize();          // 5
$points     = $regression->getPoints();              // [[1,2], [2,3], [4,5], [5,7], [6,8]]
$xs         = $regression->getXs();                  // [1, 2, 4, 5, 6]
$ys         = $regression->getYs();                  // [2, 3, 5, 7, 8]
$ν          = $regression->degreesOfFreedom();

// Linear regression through a fixed point (least squares method)
$force_point = [0,0];
$regression  = new Regression\LinearThroughPoint($points, $force_point);
$parameters  = $regression->getParameters();
$equation    = $regression->getEquation();
$y           = $regression->evaluate(5);
$Ŷ           = $regression->yHat();
$r           = $regression->r();
$r²          = $regression->r2();
 ⋮                     ⋮

// Theil–Sen estimator (Sen's slope estimator, Kendall–Theil robust line)
$regression  = new Regression\TheilSen($points);
$parameters  = $regression->getParameters();
$equation    = $regression->getEquation();
$y           = $regression->evaluate(5);
 ⋮                     ⋮

// Use Lineweaver-Burk linearization to fit data to the Michaelis–Menten model: y = (V * x) / (K + x)
$regression  = new Regression\LineweaverBurk($points);
$parameters  = $regression->getParameters();  // [V, K]
$equation    = $regression->getEquation();    // y = Vx / (K + x)
$y           = $regression->evaluate(5);
 ⋮                     ⋮

// Use Hanes-Woolf linearization to fit data to the Michaelis–Menten model: y = (V * x) / (K + x)
$regression  = new Regression\HanesWoolf($points);
$parameters  = $regression->getParameters();  // [V, K]
$equation    = $regression->getEquation();    // y = Vx / (K + x)
$y           = $regression->evaluate(5);
 ⋮                     ⋮

// Power law regression - power curve (least squares fitting)
$regression = new Regression\PowerLaw($points);
$parameters = $regression->getParameters();   // [a => 56.483375436574, b => 0.26415375648621]
$equation   = $regression->getEquation();     // y = 56.483375436574x^0.26415375648621
$y          = $regression->evaluate(5);
 ⋮                     ⋮

// LOESS - Locally Weighted Scatterplot Smoothing (Local regression)
$α          = 1/3;                         // Smoothness parameter
$λ          = 1;                           // Order of the polynomial fit
$regression = new Regression\LOESS($points, $α, $λ);
$y          = $regression->evaluate(5);
$Ŷ          = $regression->yHat();
 ⋮                     ⋮

Statistics - Significance Testing

use MathPHP\Statistics\Significance;

// Z test - One sample (z and p values)
$Hₐ = 20;   // Alternate hypothesis (M Sample mean)
$n  = 200;  // Sample size
$H₀ = 19.2; // Null hypothesis (μ Population mean)
$σ  = 6;    // SD of population (Standard error of the mean)
$z  = Significance:zTest($Hₐ, $n, $H₀, $σ);           // Same as zTestOneSample
$z  = Significance:zTestOneSample($Hₐ, $n, $H₀, $σ);  // Same as zTest
/* [
  'z'  => 1.88562, // Z score
  'p1' => 0.02938, // one-tailed p value
  'p2' => 0.0593,  // two-tailed p value
] */

// Z test - Two samples (z and p values)
$μ₁ = 27;   // Sample mean of population 1
$μ₂ = 33;   // Sample mean of population 2
$n₁ = 75;   // Sample size of population 1
$n₂ = 50;   // Sample size of population 2
$σ₁ = 14.1; // Standard deviation of sample mean 1
$σ₂ = 9.5;  // Standard deviation of sample mean 2
$z  = Significance::zTestTwoSample($μ₁, $μ₂, $n₁, $n₂, $σ₁, $σ₂);
/* [
  'z'  => -2.36868418147285,  // z score
  'p1' => 0.00893,            // one-tailed p value
  'p2' => 0.0179,             // two-tailed p value
] */

// Z score
$M = 8; // Sample mean
$μ = 7; // Population mean
$σ = 1; // Population SD
$z = Significance::zScore($M, $μ, $σ);

// T test - One sample (from sample data)
$a     = [3, 4, 4, 5, 5, 5, 6, 6, 7, 8]; // Data set
$H₀    = 300;                            // Null hypothesis (μ₀ Population mean)
$tTest = Significance::tTest($a, $H₀)
print_r($tTest);
/* Array (
    [t]    => 0.42320736951516  // t score
    [df]   => 9                 // degrees of freedom
    [p1]   => 0.34103867713806  // one-tailed p value
    [p2]   => 0.68207735427613  // two-tailed p value
    [mean] => 5.3               // sample mean
    [sd]   => 1.4944341180973   // standard deviation
) */

// T test - One sample (from summary data)
$Hₐ    = 280; // Alternate hypothesis (M Sample mean)
$s     = 50;  // Standard deviation of sample
$n     = 15;  // Sample size
$H₀    = 300; // Null hypothesis (μ₀ Population mean)
$tTest = Significance::tTestOneSampleFromSummaryData($Hₐ, $s, $n, $H₀);
print_r($tTest);
/* Array (
    [t]    => -1.549193338483    // t score
    [df]   => 14                 // degreees of freedom
    [p1]   => 0.071820000122611  // one-tailed p value
    [p2]   => 0.14364000024522   // two-tailed p value
    [mean] => 280                // sample mean
    [sd]   => 50                 // standard deviation
) */

// T test - Two samples (from sample data)
$x₁    = [27.5, 21.0, 19.0, 23.6, 17.0, 17.9, 16.9, 20.1, 21.9, 22.6, 23.1, 19.6, 19.0, 21.7, 21.4];
$x₂    = [27.1, 22.0, 20.8, 23.4, 23.4, 23.5, 25.8, 22.0, 24.8, 20.2, 21.9, 22.1, 22.9, 20.5, 24.4];
$tTest = Significance::tTest($x₁, $x₂);
print_r($tTest);
/* Array (
    [t]     => -2.4553600286929   // t score
    [df]    => 24.988527070145    // degrees of freedom
    [p1]    => 0.010688914613979  // one-tailed p value
    [p2]    => 0.021377829227958  // two-tailed p value
    [mean1] => 20.82              // mean of sample x₁
    [mean2] => 22.98667           // mean of sample x₂
    [sd1]   => 2.804894           // standard deviation of x₁
    [sd2]   => 1.952605           // standard deviation of x₂
) */

// T test - Two samples (from summary data)
$μ₁    = 42.14; // Sample mean of population 1
$μ₂    = 43.23; // Sample mean of population 2
$n₁    = 10;    // Sample size of population 1
$n₂    = 10;    // Sample size of population 2
$σ₁    = 0.683; // Standard deviation of sample mean 1
$σ₂    = 0.750; // Standard deviation of sample mean 2
$tTest = Significance::tTestTwoSampleFromSummaryData($μ₁, $μ₂, $n₁, $n₂, $σ₁, $σ₂);
print_r($tTest);
/* Array (
   [t] => -3.3972305988708     // t score
   [df] => 17.847298548027     // degrees of freedom
   [p1] => 0.0016211251126198  // one-tailed p value
   [p2] => 0.0032422502252396  // two-tailed p value
   [mean1] => 42.14
   [mean2] => 43.23
   [sd1] => 0.6834553
   [sd2] => 0.7498889
] */

// T score
$Hₐ = 280; // Alternate hypothesis (M Sample mean)
$s  = 50;  // SD of sample
$n  = 15;  // Sample size
$H₀ = 300; // Null hypothesis (μ₀ Population mean)
$t  = Significance::tScore($Hₐ, $s, $n, $H);

// χ² test (chi-squared goodness of fit test)
$observed = [4, 6, 17, 16, 8, 9];
$expected = [10, 10, 10, 10, 10, 10];
$χ²       = Significance::chiSquaredTest($observed, $expected);
// ['chi-square' => 14.2, 'p' => 0.014388]

Trigonometry

use MathPHP\Trigonometry;

$n      = 9;
$points = Trigonometry::unitCircle($n); // Produce n number of points along the unit circle

Unit Tests

Beyond 100% code coverage!

MathPHP has thousands of unit tests testing individual functions directly with numerous data inputs to achieve 100% test coverage.
MathPHP unit tests also test mathematical axioms which indirectly test the same functions in multiple different ways ensuring that those math properties all work out according to the axioms.

$ cd tests
$ phpunit

Standards

MathPHP conforms to the following standards:

• PSR-1 - Basic coding standard (http://www.php-fig.org/psr/psr-1/)
• PSR-4 - Autoloader (http://www.php-fig.org/psr/psr-4/)
• PSR-12 - Extended coding style guide (http://www.php-fig.org/psr/psr-12/)

License

MathPHP is licensed under the MIT License.