# ROC Curve for unbounded scores

Say I have a classifier that assigns a score to an image based on whether it has a cat in it. The higher the score, the more likely there's a cat in it. But for this classifier, the value of the score is unbounded, and could be any positive number, in principle. Is there a well-defined way to create a ROC curve for this classifier, if all the images have yes/no labels? Just as a traditional ROC curve involves all thresholds between 0 and 1, could a modified ROC curve involve all decision thresholds between 0 and the highest score?

I could normalize the scores to [0,1] by dividing them by the highest score in the set, but I want to avoid that if possible.

• Typically a model like this would use some kind of function to squeeze the values into $[0,1]$ e.g. logistic regression. However, I see no reason why you couldn’t slide your threshold up and down the real like and calculate the sensitivity and specificity to plot.
– Dave
Sep 30 '20 at 15:08
• ROC curves only care about the ranking/the ordering of your images. It doesn't matter how you normalize your scores, the ROC won't change. Sep 30 '20 at 15:25
• "traditional ROC curve involves all thresholds between 0 and 1" Where did you get this information from? It is plain wrong! Sep 30 '20 at 17:06
• @LaksanNathan care to write it as an answer? Sep 30 '20 at 17:07

Yes, the two rates will always be between $$0$$ and $$1$$. But there is no reason whatsoever that the threshold needs to be in some specified interval, like $$[0,1]$$, or any other fixed $$[a,b]$$.
Just start with your threshold at the lowest score you have, at which you will have neither FPs nor TPs, so this is the point at $$(0,0)$$ of the ROC curve. Then increase the threshold. The FPRs and TPRs will change; plot them. When you end up at the highest score you have, you will have the FPR and the TPR both equal to $$1$$ and plot the $$(1,1)$$ point of the ROC curve.