README

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.

Features

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

Setup

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

{
  "require": {
      "markrogoyski/math-php": "2.*"
  }
}

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:2.*

Minimum Requirements

• PHP 7.2

Note: For PHP 7.0 and 7.1, use v1.0 (markrogoyski/math-php:1.*)

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]

// Linear equation of one variable: ax + b = 0
[$a, $b] = [2, 4]; // 2x + 4 = 0
$x       = Algebra::linear($a, $b);

// Quadratic equation: ax² + bx + c = 0
[$a, $b, $c] = [1, 2, -8]; // x² + 2x - 8
[$x₁, $x₂]   = Algebra::quadratic($a, $b, $c);

// Discriminant: Δ = b² - 4ac
[$a, $b, $c] = [2, 3, 4]; // 3² - 4(2)(4)
$Δ           = Algebra::discriminant($a, $b, $c);

// Cubic equation: z³ + a₂z² + a₁z + a₀ = 0
[$a₃, $a₂, $a₁, $a₀] = [2, 9, 3, -4]; // 2x³ + 9x² + 3x -4
[$x₁, $x₂, $x₃]      = Algebra::cubic($a₃, $a₂, $a₁, $a₀);

// Quartic equation: a₄z⁴ + a₃z³ + a₂z² + a₁z + a₀ = 0
[$a₄, $a₃, $a₂, $a₁, $a₀] = [1, -10, 35, -50, 24]; // z⁴ - 10z³ + 35z² - 50z + 24 = 0
[$z₁, $z₂, $z₃, $z₄]      = Algebra::quartic($a₄, $a₃, $a₂, $a₁, $a₀);

Arithmetic

use MathPHP\Arithmetic;

$√x  = Arithmetic::isqrt(8);     // 2 Integer square root
$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

// Modulo (Differs from PHP remainder (%) operator for negative numbers)
$dividend = 12;
$divisor  = 5;
$modulo   = Arithmetic::modulo($dividend, $divisor);  // 2
$modulo   = Arithmetic::modulo(-$dividend, $divisor); // 3

Expression - Polynomial

use MathPHP\Expression\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]

// Companion matrix
$companion = $polynomial->companionMatrix();

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
$rate      = Finance::rate($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];
$profitability_index = Finance::profitabilityIndex($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 - Special Functions

use MathPHP\Functions\Special;

// Gamma function Γ(z)
$z = 4;
$Γ = Special::gamma($z);
$Γ = Special::gammaLanczos($z);   // Lanczos approximation
$Γ = Special::gammaStirling($z);  // Stirling approximation
$l = Special::logGamma($z);
$c = Special::logGammaCorr($z);   // Log gamma correction

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

// Beta function
[$x, $y] = [1, 2];
$β  = Special::beta($x, $y);
$lβ = Special::logBeta($x, $y);

// Incomplete beta functions
[$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
$𝐳    = [1, 2, 3, 4, 1, 2, 3];
$σ⟮𝐳⟯ⱼ = Special::softmax($𝐳);

// Log of the error term in the Stirling-De Moivre factorial series
$err = Special::stirlingError($n);

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
[$mᵢ, $mⱼ, $k] = [1, 2, 5];
$R = $A->rowInterchange($mᵢ, $mⱼ);
$R = $A->rowExclude($mᵢ);             // Exclude row $mᵢ
$R = $A->rowMultiply($mᵢ, $k);        // Multiply row mᵢ by k
$R = $A->rowDivide($mᵢ, $k);          // Divide row mᵢ by k
$R = $A->rowAdd($mᵢ, $mⱼ, $k);        // Add k * row mᵢ to row mⱼ
$R = $A->rowAddScalar($mᵢ, $k);       // Add k to each item of row mᵢ
$R = $A->rowAddVector($mᵢ, $V);       // Add Vector V to row mᵢ
$R = $A->rowSubtract($mᵢ, $mⱼ, $k);   // Subtract k * row mᵢ from row mⱼ
$R = $A->rowSubtractScalar($mᵢ, $k);  // Subtract k from each item of row mᵢ

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

// SVD (Singular Value Decomposition)
$SVD = $A->svd();
$U   = $A->U;  // m x m orthogonal matrix
$V   = $A->V;  // n x n orthogonal matrix
$S   = $A->S;  // m x n diagonal matrix of singular values
$D   = $A->D;  // Vector of diagonal elements from S

// 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

// Walk maps a function to all values without mutation or returning a value
$A->walk($func);

// 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]);

// Create from row or column vector
$A = MatrixFactory::createFromRowVector([1, 2, 3]);    // 1 × n matrix consisting of a single row of n elements
$A = MatrixFactory::createFromColumnVector([1, 2, 3]); // m × 1 matrix consisting of a single column of m elements

// Specialized matrices
[$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
$max               = $A->max();
$min               = $A->min();
$A⋅B               = $A->dotProduct($B);                      // same as innerProduct
$A⋅B               = $A->innerProduct($B);                    // same as dotProduct
$A⊥⋅B              = $A->perpDotProduct($B);
$radAngle          = $A->angleBetween($B);                    // angle in radians
$degAngle          = $A->angleBetween($B, $inDegrees = true); // angle in degrees
$taxicabDistance   = $A->l1Distance($B);                      // same as minkowskiDistance($B, 1)
$euclidDistance    = $A->l2Distance($B);                      // same as minkowskiDistance($B, 2)
$minkowskiDistance = $A->minkowskiDistance($B, $p = 2);

// 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);
[$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;

[$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        = $complex->sqrt();    // positive square root
[$z₁, $z₂] = $complex->roots();
$c⁻¹       = $complex->inverse();
$−c        = $complex->negate();
[$r, $θ]   = $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 - Quaternion

Use MathPHP\Number\Quaternion;

$r = 4;
$i = 1;
$j = 2;
$k = 3;

$quaternion = new Quaternion($r, $i, $j, $k);

// Get individual parts
[$r, $i, $j, $k] = [$quaternion->r, $quaternion->i, $quaternion->j, $quaternion->k];

// Unary functions
$conjugate    = $quaternion->complexConjugate();
$│q│          = $quaternion->abs();  // absolute value (magnitude)
$quaternion⁻¹ = $quaternion->inverse();
$−q           = $quaternion->negate();

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

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

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();
$inverse    = $rational->inverse();

// Binary functions
$sum            = $rational->add($rational);
$diff           = $rational->subtract($rational);
$product        = $rational->multiply($rational);
$quotient       = $rational->divide($rational);
$exponentiation = $rational->pow(2);

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

Number Theory - Integers

use MathPHP\NumberTheory\Integer;

$n = 225;

// Prime numbers
$bool    = Integer::isPrime($n);
$factors = Integer::primeFactorization($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);
[$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 * $x + 1;
};
[$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*$x + 2;
};
[$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

// Regular Grid Interpolation
// Returns a scalar

// Points defining the regular grid
$xs = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9];
$ys = [10, 11, 12, 13, 14, 15, 16, 17, 18, 19];
$zs = [110, 111, 112, 113, 114, 115, 116, 117, 118, 119];

// Data on the regular grid in n dimensions
$data = [];
$func = function ($x, $y, $z) {
    return 2 * $x + 3 * $y - $z;
};
foreach ($xs as $i => $x) {
    foreach ($ys as $j => $y) {
        foreach ($zs as $k => $z) {
            $data[$i][$j][$k] = $func($x, $y, $z);
        }
    }
}

// Constructing a RegularGridInterpolator
$rgi = new Interpolation\RegularGridInterpolator([$xs, $ys, $zs], $data, 'linear');  // 'nearest' method also available

// Interpolating coordinates on the regular grid
$coordinates   = [2.21, 12.1, 115.9];
$interpolation = $rgi($coordinates);  // -75.18

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]];

$target = 0;
[$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]];

$target = 0;
[$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];

$target = 1;
[$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 * $x + 1;
};
[$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 * $x + 1;
};
[$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 * $x + 1;
};
[$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 * $x + 1;
};
[$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 * $x + 1;
};
[$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 * $x + 1;
};
[$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;

[$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();

// Zipf distribution
$k    = 2;   // rank
$s    = 3;   // exponent
$N    = 10;  // number of elements
$zipf = new Discrete\Zipf($s, $N);
$pmf  = $zipf->pmf($k);
$cdf  = $zipf->cdf($k);
$μ    = $zipf->mean();
$mode = $zipf->mode();

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);

// Hypergeometric distribution
$quantities   = [5, 10, 15];   // Suppose there are 5 black, 10 white, and 15 red marbles in an urn.
$choices      = [2, 2, 2];     // If six marbles are chosen without replacement, the probability that exactly two of each color are chosen is:
$distribution = new Multivariate\Hypergeometric($quantities);
$probability  = $distribution->pmf($choices);    // 0.0795756

// 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];

Sample Data

use MathPHP\SampleData;

// Famous sample data sets to experiment with

// Motor Trend Car Road Tests (mtcars)
$mtCars      = new SampleData\MtCars();
$rawData     = $mtCars->getData();                     // [[21, 6, 160, ... ], [30.4, 4, 71.1, ... ], ... ]
$labeledData = $mtCars->getLabeledData();              // ['Mazda RX4' => ['mpg' => 21, 'cyl' => 6, 'disp' => 160, ... ], 'Honda Civic' => [ ... ], ...]
$modelData   = $mtCars->getModelData('Ferrari Dino');  // ['mpg' => 19.7, 'cyl' => 6, 'disp' => 145, ... ]
$mpgs        = $mtCars->getMpg();                      // ['Mazda RX4' => 21, 'Honda civic' => 30.4, ... ]
// Getters for Mpg, Cyl, Disp, Hp, Drat, Wt, Qsec, Vs, Am, Gear, Carb

// Edgar Anderson's Iris Data (iris)
$iris         = new SampleData\Iris();
$rawData      = $iris->getData();         // [[5.1, 3.5, 1.4, 0.2, 'setosa'], [4.9, 3.0, 1.4, 0.2, 'setosa'], ... ]
$labeledData  = $iris->getLabeledData();  // [['sepalLength' => 5.11, 'sepalWidth' => 3.5, 'petalLength' => 1.4, 'petalWidth' => 0.2, 'species' => 'setosa'], ... ]
$petalLengths = $iris->getSepalLength();  // [5.1, 4.9, 4.7, ... ]
// Getters for SepalLength, SepalWidth, PetalLength, PetalWidth, Species

// The Effect of Vitamin C on Tooth Growth in Guinea Pigs (ToothGrowth)
$toothGrowth = new SampleData\ToothGrowth();
$rawData     = $toothGrowth->getData();         // [[4.2, 'VC', 0.5], [11.5, 'VC', '0.5], ... ]
$labeledData = $toothGrowth->getLabeledData();  // [['len' => 4.2, 'supp' => 'VC', 'dose' => 0.5], ... ]
$lengths     = $toothGrowth->getLen();          // [4.2, 11.5, ... ]
// Getters for Len, Supp, Dose

// Results from an Experiment on Plant Growth (PlantGrowth)
$plantGrowth = new SampleData\PlantGrowth();
$rawData     = $plantGrowth->getData();         // [[4.17, 'ctrl'], [5.58, 'ctrl'], ... ]
$labeledData = $plantGrowth->getLabeledData();  // [['weight' => 4.17, 'group' => 'ctrl'], ['weight' => 5.58, 'group' => 'ctrl'], ... ]
$weights     = $plantGrowth->getWeight();       // [4.17, 5.58, ... ]
// Getters for Weight, Group

// Violent Crime Rates by US State (USArrests)
$usArrests   = new SampleData\UsArrests();
$rawData     = $usArrests->rawData();              // [[13.2, 236, 58, 21.2], [10.0, 263, 48, 44.5], ... ]
$labeledData = $usArrests->getLabeledData();       // ['Alabama' => ['murder' => 13.2, 'assault' => 236, 'urbanPop' => 58, 'rape' => 21.2], ... ]
$stateData   = $usArrests->getStateData('Texas');  // ['murder' => 12.7, 'assault' => 201, 'urbanPop' => 80, 'rape' => 25.5]
$murders     = $usArrests->getMurders();           // ['Alabama' => 13.2, 'Alaska' => 10.1, ... ]
// Getters for Murder, Assault, UrbanPop, Rape

// Data from Cereals (cereal)
$cereal  = new SampleData\Cereal();
$cereals = $cereal->getCereals();    // ['B1', 'B2', 'B3', 'M1', 'M2', ... ]
$X       = $cereal->getXData();      // [[0.002682755, 0.003370673, 0.004085942, ... ], [0.002781597, 0.003474863, 0.004191472, ... ], ... ]
$Y       = $cereal->getYData();      // [[18373, 41.61500, 6.565000, ... ], [18536, 41.40500, 6.545000, ... ], ... ]
$Ysc     = $cereal->getYscData();    // [[-0.1005049, 0.6265746, -1.1716630, ... ], [0.9233889, 0.1882929, -1.3185289, ... ], ... ]
// Labeled data: getLabeledXData(), getLabeledYData(), getLabeledYscData()

// Data from People (people)
$people      = new SampleData\People();
$rawData     = $people->getData();         // [198, 92, -1, ... ], [184, 84, -1, ... ], ... ]
$labeledData = $people->getLabeledData();  // ['Lars' => ['height' => 198, 'weight' => 92, 'hairLength' => -1, ... ]]
$names       = $people->getNames();
// Getters for names, height, weight, hairLength, shoeSize, age, income, beer, wine, sex, swim, region, iq

Search

use MathPHP\Search;

// Search lists of numbers to find specific indexes

$list = [1, 2, 3, 4, 5];

$index   = Search::sorted($list, 2);   // Find the array index where an item should be inserted to maintain sorted order
$index   = Search::argMax($list);      // Find the array index of the maximum value
$index   = Search::nanArgMax($list);   // Find the array index of the maximum value, ignoring NANs
$index   = Search::argMin($list);      // Find the array index of the minimum value
$index   = Search::nanArgMin($list);   // Find the array index of the minimum value, ignoring NANs
$indices = Search::nonZero($list);     // Find the array indices of the scalar values that are non-zero

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)
$heptagons = Advanced::heptagonalNumber($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
$harmonic = NonInteger::harmonic($n);
// [1, 3/2, 11/6, 25/12] - Indexed from 1

// Generalized harmonic sequence
$m           = 2;  // exponent
$generalized = NonInteger::generalizedHarmonic($n, $m);
// [1, 5 / 4, 49 / 36, 205 / 144] - Indexed from 1

// Hyperharmonic sequence
$r             = 2;  // depth of recursion
$hyperharmonic = NonInteger::hyperharmonic($n, $r);
// [1, 5/2, 26/6, 77/12] - 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;  // 25 percent of observations trimmed from each end of distribution
$truncated_mean = Average::truncatedMean($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
[$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);