# Normalizing error when data passes through zero

Given a time-dependent measure of error, $e(t)$, and a time-dependent function to normalize it with, $f(t)$, one would report the normalized error as $$E(t) = \left|\frac{e(t)}{f(t)}\right|.$$ However, if $f(t)$ passes through zero at certain times, $E(t)$ will be very large near those instances, despite the fact that the actual error could be very small.

If one hopes to normalize an error measurement with a function that passes through zero, is there a standard convention to normalized error?

Modifying the denominator to avoid its zeros often involves adding constant factor which is equivalent to using an absolute error test near $f(t) = 0$. That is,
$$E(t) = \frac{\left|e(t)\right|}{\left|f(t)\right| + C}$$ Which is equivalent to normalizing $e(t)$ by $C$ near $f(t) = 0$.
There's no standard convention for this nor the choice of $C$. There are an infinite number of ways to normalize your error, so it's better to just state what you're doing when you're presenting your results.
You could also compute the error with respect to a typical size of $f(t)$. For example, the average of its magnitude, or the maximum of the magnitude. This avoids the problem you encounter.