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Posted on July 27, 2006 @ 06:34:55 PM by Paul Meagher
Here is a stochoastic proof that the Euclidean metric obeys the triangle inequality.
<?php
include "Array_Distance.php";
function randomIntegers($size, $lo, $hi) {
$int_vector = array();
for ($i=0; $i<$size; $i++)
$int_vector[] = mt_rand($lo, $hi);
return $int_vector;
}
$num_experiments = 10;
$num_dimensions = 2;
$num_rule_matches = 0;
$num_rule_failures = 0;
?>
<h1>Triangle Inequality Experiments using the Euclidean Metric</h1>
<pre>
d(x, z) <= d(x, y) + d(y, z) (triangle inequality)
</pre>
<?php
for($e=0; $e<$num_experiments; $e++) {
$X = randomIntegers($num_dimensions, 0, 9);
$Y = randomIntegers($num_dimensions, 0, 9);
$Z = randomIntegers($num_dimensions, 0, 9);
$d = new Array_Distance($X, $Z);
$dXZ = $d->euclidean();
$d = new Array_Distance($X, $Y);
$dXY = $d->euclidean();
$d = new Array_Distance($Y, $Z);
$dYZ = $d->euclidean();
if ($dYZ <= ($dXY + $dYZ)) {
echo "Experiment #".($e+1)." obeys triangle inequality: ";
echo $dYZ."<=(".$dXY." + ".$dYZ.")<br/>";
$num_rule_matches++;
} else {
echo "Does not obey triangle inequality<br />";
$num_rule_failures++;
}
}
?>
<h1>Results</h1>
<?php
echo "<b>Num Experiments: </b> $num_experiments <br />";
echo "<b>Num Rule Matches: </b> $num_rule_matches <br />";
echo "<b>Num Rule Failures: </b> $num_rule_failures <br />";
echo "<b>Rule Success Rate: </b> ".(($num_rule_matches/$num_experiments)*100)."%<br />";
?>
You can see the ouput of this script here.
I wonder what would happen if I substituted the minkowski($exp) method for the euclidean() method in the above test_triangle_inequality.php script?
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