What is a model in the mathematical sense
Error as a measure of quality
Error is a function (commonly called a loss function) that compares the model’s prediction with the true value and returns a number that shows how wrong we were. The smaller this number, the better the model. For example, the simplest error is the difference between prediction and reality: $ŷ - y$.
In practice, we often use the squared error (Squared Error or SE), because it is always non‑negative and penalizes large mistakes more strongly: $(ŷ - y)^2$.
Example of use:
<?php
// Simple error: difference between prediction and true value
// ŷ - y. Positive means we overestimated, negative means we underestimated.
function error(float $yTrue, float $yPredicted): float {
return $yPredicted - $yTrue;
}
// Squared error: (ŷ - y)^2
// Always non‑negative and penalizes large mistakes more strongly than small ones.
function squaredError(float $yTrue, float $yPredicted): float {
return ($yPredicted - $yTrue) ** 2;
}