# Formula for the omission and the commission errors

I'm confused by the formula for the commission-error and the omission-error as it was stated a bit differently in a paper I've read, compared to the one I'm giving below (maybe the authors changed that because of the context of a change detection, not a casual classification).

Are these the correct formula for a given class?

$$commisionError = \frac{FP}{FP + TP} = \frac{FP}{totalPredicted}$$ $$omissionError = \frac{FN}{FN + TP} = \frac{FN}{totalReference}$$

Where:

• FP: The false positive.
• TP: The true positive.
• FN: The false negative.
• TN: The true negative.

If we had only two classes, should these two errors be calculated for the two classes, or is there a way to infer the errors of the second class from those of the first one?

I'm asking because it's clear that for a two-classes case we have:

$$FP_{class1} = FN_{class2}$$

$$FN_{class1} = FP_{class2}$$

## Arithmetic

We have four numbers: TP, FP, TN, FN that sum to $M$, total number of samples: $$TP + FP + TN + FN = M$$

$M$ is usually known. So if you know any 3 of those 4 numbers, you can calculate the forth one.

Those four numbers are base of many scores (aka coefficients, rates, values), see this wikipedia article. By juggling with above formula you can find many relations between these scores.

## Formulas

commisionError and omissionError are not very common names. You ask if those are correct. If this is the definition, they are correct by definition. It is really hard to answer, as you say:

Are these the correct formula for a given class?

There is no given class, because using TP, TN, ... implies that there are exactly two classes. So you may want clarify what you mean by given class.

• Yes, I was asking in the context of two classes, as I wasn't sure if the errors mentioned above should be calculated for both classes, and whether the formulas were correct at all. I recently came across the FPR and the FNR which were calculated in a way that $FPR_{class1} = FNR_{class2}$ and vice versa. Commented Feb 8, 2018 at 18:48
• If you have only two classes, than FPR = FP/(FP+TP) and FNR = FN/ (FN+TP). No class subindex. If there were more classes, situation gets more complicated
– hans
Commented Feb 12, 2018 at 21:38
• The formulas for the FPR and the FNR you gave above are different from the ones I've found online. Do you have a reference? As for the omission and commission errors, I got them from this book. The problem actually with all these performance indicators is that sometimes you find different formulas for the same indicator, and it's one reason why I was confused. Commented Feb 14, 2018 at 14:57
• @Hakim I just discovered that I didn't answer to your comment. You are right, my FPR formula is not correct (maybe I did a typo). Right formula: FPR = FP/(FP+TN) if it still matters... Sorry for my mistake. source
– hans
Commented Aug 9, 2018 at 9:56