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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?

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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.

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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.

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