/**
* Sum of squares
*
* ∑⟮xᵢ⟯²
*
* @param array $numbers
*
* @return number
*/
public static function sumOfSquares(array $numbers)
{
if (empty($numbers)) {
return null;
}
$∑⟮xᵢ⟯² = array_sum(Map\Single::square($numbers));
return $∑⟮xᵢ⟯²;
}
/**
* ρ - Spearman's rank correlation coefficient (Spearman's rho)
*
* https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient
*
* 6 ∑ dᵢ²
* ρ = 1 - ---------
* n(n² − 1)
*
* Where
* dᵢ: the difference between the two ranks of each observation
*
* @param array $X values for random variable X
* @param array $Y values for random variable Y
*
* @return number
*
* @throws BadDataException if both random variables do not have the same number of elements
*/
public static function spearmansRho(array $X, array $Y)
{
if (count($X) !== count($Y)) {
throw new Exception\BadDataException('Both random variables must have the same number of elements');
}
$n = count($X);
// Sorted Xs and Ys
$Xs = $X;
$Ys = $Y;
rsort($Xs);
rsort($Ys);
// Determine ranks of each X and Y
// Some items might show up multiple times, so record each successive rank.
$rg⟮X⟯ = [];
$rg⟮Y⟯ = [];
foreach ($Xs as $rank => $xᵢ) {
if (!isset($rg⟮X⟯[$xᵢ])) {
$rg⟮X⟯[$xᵢ] = [];
}
$rg⟮X⟯[$xᵢ][] = $rank;
}
foreach ($Ys as $rank => $yᵢ) {
if (!isset($rg⟮Y⟯[$yᵢ])) {
$rg⟮Y⟯[$yᵢ] = [];
}
$rg⟮Y⟯[$yᵢ][] = $rank;
}
// Determine average rank of each X and Y
// Rank will not change if value only shows up once.
// Average is for when values show up multiple times.
$rg⟮X⟯ = array_map(function ($x) {
return array_sum($x) / count($x);
}, $rg⟮X⟯);
$rg⟮Y⟯ = array_map(function ($y) {
return array_sum($y) / count($y);
}, $rg⟮Y⟯);
// Difference between the two ranks of each observation
$d = array_map(function ($x, $y) use($rg⟮X⟯, $rg⟮Y⟯) {
return abs($rg⟮X⟯[$x] - $rg⟮Y⟯[$y]);
}, $X, $Y);
// Numerator: 6 ∑ dᵢ²
$d² = Map\Single::square($d);
$∑d² = array_sum($d²);
// Denominator: n(n² − 1)
$n⟮n² − 1⟯ = $n * ($n ** 2 - 1);
/*
* 6 ∑ dᵢ²
* ρ = 1 - ---------
* n(n² − 1)
*/
return 1 - 6 * $∑d² / $n⟮n² − 1⟯;
}
/**
* SSreg - The Sum Squares of the regression (Explained sum of squares)
*
* The sum of the squares of the deviations of the predicted values from
* the mean value of a response variable, in a standard regression model.
* https://en.wikipedia.org/wiki/Explained_sum_of_squares
*
* SSreg = ∑(ŷᵢ - ȳ)²
* When a constant is fit to the regression, the average of y = average of ŷ.
*
* In the case where the constant is not fit, we use the sum of squares of the predicted value
* SSreg = ∑ŷᵢ²
*
* @return number
*/
public function sumOfSquaresRegression()
{
if ($this->fit_constant == 1) {
return RandomVariable::sumOfSquaresDeviations($this->Yhat());
}
return array_sum(Single::square($this->reg_Yhat));
}
/**
* l²-norm (|x|₂)
* Also known as Euclidean norm, Euclidean length, L² distance, ℓ² distance
* Used to normalize a vector.
*
* http://mathworld.wolfram.com/L2-Norm.html
* https://en.wikipedia.org/wiki/Norm_(mathematics)#Euclidean_norm
* ______
* |x|₂ = √∑|xᵢ|²
*
* @return number
*/
public function l2Norm()
{
return sqrt(array_sum(Map\Single::square($this->A)));
}