SVM: achieving ROC curve by varying misclassification costs

Question

Assume that we have SVM model for binary classification with objective function as follows: $$ min(\frac{1}{2}\omega\cdot\omega +C^{+}\sum_{i|y_{i}=+1}^{n}\xi_{i}\quad+C^{-}\sum_{i|y_{i}=-1}^{n}\xi_{i}) $$ After many times of running optimization with different misclassification costs($C^{+}$ and $C^{-}$) for testing part of one dataset, I achieved a ROC(?) representation of the model:

Then, I decided to create logarithmic trendline for this set of points achieving:

After reading some questions, inter alia here, I came to a conclusion, that it is not a typical way to obtain a ROC curve for SVM. The questions are:

1. Will I lie if I name it a (approximated) ROC curve for this model?
2. Was this a proper way to achieve ROC curve?

Answer 1

1. Will I lie if I name it a (approximated) ROC curve for this model?

Yes, this is definitely not a ROC curve. In addition what you are approximating is not the ROC curve.

2. Was this a proper way to achieve ROC curve?

No, what you did is has definitely nothing to do with a ROC curve.

A ROC curve is a curve that shows the monotone relationship between FPR and TPR as the decision threshold is varied. All the remaining variables like model parameters, the dataset on which FPR and TPR are computed, or meta parameters like (mis-)classification costs must remain constant. It is very easy to calculate exactly and doesn't require any approximation. Check out the question Understanding ROC curve on this site to see how to understand and properly build a ROC curve.