PHP Trait MathPHP\Statistics\Regression\Methods\WeightedLeastSquares

Datei anzeigen Open project: markrogoyski/math-php

Public Methods

Method Description
leastSquares ( array $ys, array $xs, array $ws, integer $order = 1 ) : Matrix Weighted linear least squares fitting using Matrix algebra (Polynomial).

Method Details

leastSquares() public method

Generalizing from a straight line (first degree polynomial) to a kᵗʰ degree polynomial: y = a₀ + a₁x + ⋯ + akxᵏ Leads to equations in matrix form: [n Σxᵢ ⋯ Σxᵢᵏ ] [a₀] [Σyᵢ ] [Σxᵢ Σxᵢ² ⋯ Σxᵢᵏ⁺¹] [a₁] [Σxᵢyᵢ ] [ ⋮ ⋮ ⋱ ⋮ ] [ ⋮ ] = [ ⋮ ] [Σxᵢᵏ Σxᵢᵏ⁺¹ ⋯ Σxᵢ²ᵏ ] [ak] [Σxᵢᵏyᵢ] This is a Vandermonde matrix: [1 x₁ ⋯ x₁ᵏ] [a₀] [y₁] [1 x₂ ⋯ x₂ᵏ] [a₁] [y₂] [⋮ ⋮ ⋱ ⋮ ] [ ⋮ ] = [ ⋮] [1 xn ⋯ xnᵏ] [ak] [yn] Can write as equation: y = Xa Solve by premultiplying by transpose Xᵀ: XᵀWy = XᵀWXa Invert to yield vector solution: a = (XᵀWX)⁻¹XᵀWy (http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html) For reference, the traditional way to do least squares: _ _ __ x y - xy _ _ m = _________ b = y - mx _ __ (x)² - x²
public leastSquares ( array $ys, array $xs, array $ws, integer $order = 1 ) : Matrix
$ys array y values
$xs array x values
$ws array weight values
$order integer
return MathPHP\LinearAlgebra\Matrix [[m], [b]]