ROC space is the two-dimensional space spanned by True Positive Rate and False Positive Rate.
AUC: Area under Curve
- TPR = TP Rate
- 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.
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