/**
* Evaluate for x
* Use the smoothness parameter α to determine the subset of data to consider for
* local regression. Perform a weighted least squares regression and evaluate x.
*
* @param number $x
*
* @return number
*/
public function evaluate($x)
{
$α = $this->α;
$λ = $this->λ;
$n = $this->n;
// The number of points considered in the local regression
$Δx = Single::abs(Single::subtract($this->xs, $x));
$αᵗʰΔx = Average::kthSmallest($Δx, $this->number_of_points - 1);
$arg = Single::min(Single::divide($Δx, $αᵗʰΔx * max($α, 1)), 1);
// Kernel function: tricube = (1-arg³)³
$tricube = Single::cube(Single::multiply(Single::subtract(Single::cube($arg), 1), -1));
$weights = $tricube;
// Local Regression Parameters
$parameters = $this->leastSquares($this->ys, $this->xs, $weights, $λ);
$X = new VandermondeMatrix([$x], $λ + 1);
return $X->multiply($parameters)[0][0];
}
/**
* Infinity norm (‖A‖∞)
* Maximum absolute row sum of the matrix
*
* @return number
*/
public function infinityNorm()
{
$m = $this->m;
$‖A‖∞ = array_sum(Map\Single::abs($this->A[0]));
for ($i = 1; $i < $m; $i++) {
$‖A‖∞ = max($‖A‖∞, array_sum(Map\Single::abs($this->A[$i])));
}
return $‖A‖∞;
}
/**
* Max norm (infinity norm) (|x|∞)
*
* |x|∞ = max |x|
*
* @return number
*/
public function maxNorm()
{
return max(Map\Single::abs($this->A));
}