public static function kendallsTau(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);
// Match X and Y pairs and sort by X rank
$xy = array_map(function ($x, $y) {
return [$x, $y];
}, $X, $Y);
usort($xy, function ($a, $b) {
return $a[0] <=> $b[0];
});
// Initialize counters
$nc = 0;
// concordant pairs
$nd = 0;
// discordant pairs
$ties_x = 0;
// ties xᵢ = xⱼ
$ties_y = 0;
// ties yᵢ = yⱼ
$ties_xy = 0;
// ties xᵢ = xⱼ and yᵢ = yⱼ
// Tally concordant, discordant, and tied pairs
for ($i = 0; $i < $n; $i++) {
for ($j = $i + 1; $j < $n; $j++) {
// xᵢ = xⱼ and yᵢ = yⱼ -- neither concordant or discordant
if ($xy[$i][self::X] == $xy[$j][self::X] && $xy[$i][self::Y] == $xy[$j][self::Y]) {
$ties_xy++;
// xᵢ = xⱼ -- neither concordant or discordant
} elseif ($xy[$i][self::X] == $xy[$j][self::X]) {
$ties_x++;
// yᵢ = yⱼ -- neither concordant or discordant
} elseif ($xy[$i][self::Y] == $xy[$j][self::Y]) {
$ties_y++;
// xᵢ < xⱼ and yᵢ < yⱼ -- concordant
} elseif ($xy[$i][self::X] < $xy[$j][self::X] && $xy[$i][self::Y] < $xy[$j][self::Y]) {
$nc++;
// xᵢ > xⱼ and yᵢ < yⱼ or xᵢ < xⱼ and yᵢ > yⱼ -- discordant
} else {
$nd++;
}
}
}
// Numerator: (number of concordant pairs) - (number of discordant pairs)
$⟮nc − nd⟯ = $nc - $nd;
/* tau-a (no rank ties):
*
* nc - nd
* τ = ----------
* n(n - 1)/2
*/
if ($ties_x == 0 && $ties_y == 0) {
return $⟮nc − nd⟯ / ($n * ($n - 1) / 2);
}
/* tau-b (rank ties exist):
*
* nc - nd
* τ = -----------------------------
* √(nc + nd + X₀)(nc + nd + Y₀)
*/
return $⟮nc − nd⟯ / sqrt(($nc + $nd + $ties_x) * ($nc + $nd + $ties_y));
}
