Receiver operating characteristic (ROC) curve

ROC curve is a plot of true positive rate (true positive / total actual positive) vs. false positive rate (false positive / total actual negative, FP / (TN + FP) ). It shows the performance of a classifier as a trade off between selectivity (specificity, true negative rate TN / (TN + FP)) and sensitivity (recall rate, true positive rate).

false positive rate = 1 - true negative rate

Choosing the operating point

Let 

alpha = cost of false positive 

beta = cost of missing a positive (false negative)

p = proportion of positive cases

The average expected cost of classification at point x, y in the ROC space is 

C = (1-p) alpha x + p beta (1-y)

The error rate can be obtained by setting the misclassification costs equal to each other and unity. So

error rate = (1 - p) x + p (1 - y)

This equation means that points on the ROC space with equal error rate are straight lines. 

EER - euqal error rate is at the point where false positive rate = false negative rate.

References:

http://www0.cs.ucl.ac.uk/staff/ucacbbl/roc/

http://www.cs.ubc.ca/~murphyk/Teaching/CS340-Fall07/ROC.pdf

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