In machine learning the Area Under the Receiver Operating Characteristic Curve ($AUC$) can be illustrated in a plot of the True Positive Rate ($TPR$) against the False Positive Rate ($FPR$). Formally, it can also be expressed as the expected value of the $TPR$, given a uniform distribution of $FPR$:

\begin{align} AUC = \int_{0}^{1}TPR\ dFPR \end{align}

This can also be expressed as as a function of $TPR$ and a decision threshold $k$ (assuming that first, a scoring function estimates the probability of belonging to the positiv class and an indicator function then assigns the class based on a decision threshold $k$): \begin{align} \label{eq:auc} AUC = \int_{0}^{1}TPR\ dFPR = \int_{-\infty}^{\infty} TPR_{k}\ f_{o}(k)dk \end{align} where $f_{o}(k)$ is the probability that the scoring function $F(X)$ produces a score of exactly $k$ in the instances that have a negative class label (see Hand 2009).

I am wondering what is the value of integrating the $FPR$ over the $FPR$:

\begin{align} \int_{-\infty}^{\infty}FPR\ f_{0}(k)dk = \int_{-\infty}^{\infty}FPR\ dFPR \end{align}

Dalessandro et al. (2014) suggest that

\begin{align} \int_{-\infty}^{\infty}FPR\ f_{0}(k)dk = \int_{-\infty}^{\infty}FPR\ dFPR = 0.5 \end{align}

but this is not very intuitive to me and causes contradictory results in my further proceedings.


HAND, D. J. (2009): “Measuring classifier performance: a coherent alternative to the area under the ROC curve”, Machine Learning, 77, 103–123.

DALESSANDRO, B., C. PERLICH, AND T. RAEDER (2014): “Bigger is Better, but at What Cost? Estimating the Economic Value of Incremental Data Assets”, Big data, 2, 87–96.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.