# Reconstruct ROC curve from AUC and EER

I have the values for AUC (Area under curve) and EER (Equal error rate), presented in a paper I'm reading. Is it possible to reconstruct the ROC curve from those AUC and EER values?

Short answer is no.

Longer answer is that it is not possible since AUC and ERR are just two aggregate information about the ROC curve, the original informatio is lost.

Another illustrative way to see why it is not possible could be the following. Supposing the ROC curve is a continuous function. We know it is not, but it does not hurt and makes things simpler. So, what do we know about that function (ROC curve continuous version)?

• f is defined as $f : [0,1] \to [0,1]$
• f is monotone and increasing function $\frac{f(x)-f(y)}{x-y} >= 0$ or otherwise it's derivative is positive $f'(x)>=0$
• AUC equals definite integral $AUC = \int_0^1 f(x)dx$
• EER is basically some concept related with fixed points $f(ERR) = 1-ERR$ (see the image below to understand that)

That being established, how many functions can you find to meet those criterias? I do not know how to prove, but I bet there is an infinity of them).

• You don’t necessarily have to prove that there are infinitely many, just that there are two curves which satisfy these constraints, since if there are more than one curve, it’s not unwieldy determined. (But there is only one curve for is=1.0.)
– Sycorax
Commented Jun 13, 2018 at 13:42
• *not uniquely determined; only one curve for AUC=1.0 -- I shouldn't type comments on my phone.
– Sycorax
Commented Jun 13, 2018 at 15:48