How to build ROC curve (or AUC) of classification model from confusion matrix only

Question

I've been looking into using ROC curves as a evaluation tool of a multi-class classification. The only data I have about this classification is in form of 7-by-7 confusion matrix.

Visualisation of the ROC curves is not important, result in form of AUC is sufficient.

Is this possible?

Edit: Simplified the question, omitted unnecessary presumption I have made.

Answer 1

You cannot construct a ROC curve from the confusion matrix alone. A confusion matrix represents a single point in the ROC space, and you need all possible confusion matrices at all thresholds to build a full curve and compute the AUC.

This holds true for multi-class ROC analysis.

Answer 2

You can compute a ROC curve with a confusion matrix. You'll need to compute the true positive rate and the false positive rate. This will give you a point on the ``ROC'' space. The two other points are always $(0,0)$ and $(1,1)$, as varying the threshold positive $0$ to $1$ will always give you these points :

• A threshold of $0$ will consider everything as positive so the $tpr$ is $1$ and the $fpr$ is $1$ ;
• A threshold of $1$ will consider everything as negative so the $tpr$ is $0$ and the $fpr$ is $0$.

As a reminder :

• True positive rate: $tpr = tp / p$
• False positive rate: $fpr = fp / n$

Where:

• $tp$ : true positive ;
• $p$ : all positives values (false and true) ;
• $fp$ : false positives ;
• $n$ : all negatives values (false and true).