Receiver Operating Characteristics: ROC

ROC space is the two-dimensional space spanned by True Positive Rate and False Positive Rate.

AUC: Area under Curve

1. TPR = TP Rate
2. FPR = FP Rate

The ROC curve is defined by the relation $f(TPR, FPR)$. Area under the ROC curve is

$$ \int TPR(FPR) d(FPR) \sim \sum_i TPR_i *\Delta FPR. $$

If AUC = 1, we have TP Rate = 1 for all FP Rate. This is the best performance a model could have.

How to Calculate ROC Curve

Not every model has an AUC. To get the AUC curve, we need a hyperparameter to be tuned to get different TP Rate and FP Rate.

In logistic regression, a threshold $T$ is predetermined to decide which label to use in classifications.

By tuning the threshold $T$, we get different TP Rate $TPR$ and FP Rate $FPR$, i.e., $TPR(T)$ and $FPR(T)$. The parametric relations between $TPR(T)$ and $FPR(T)$ forms the ROC curve.