Editing Receiver operating characteristic (section)

==Curves in ROC space==
[[File:ROC curves.svg|thumb|right|300px]]
In binary classification, the class prediction for each instance is often made based on a [[Continuous probability distribution|continuous random variable]] <math> X </math>, which is a "score" computed for the instance (e.g. the estimated probability in logistic regression). Given a threshold parameter <math> T </math>, the instance is classified as "positive" if <math> X>T </math>, and "negative" otherwise. <math> X </math> follows a probability density <math> f_1 (x) </math> if the instance actually belongs to class "positive", and <math> f_0 (x) </math> if otherwise. Therefore, the true positive rate is given by <math> \mbox{TPR}(T)= \int_{T}^\infty f_1(x) \, dx </math> and the false positive rate is given by <math> \mbox{FPR}(T)= \int_{T}^\infty f_0(x) \, dx </math>. 
The ROC curve plots parametrically <math>\mbox{TPR}(T)</math> versus <math>\mbox{FPR}(T)</math> with <math>T</math> as the varying parameter.

For example, imagine that the blood protein levels in diseased people and healthy people are [[normal distribution|normally distributed]] with means of 2 [[gram|g]]/[[decilitre|dL]] and 1 g/dL respectively. A medical test might measure the level of a certain protein in a blood sample and classify any number above a certain threshold as indicating disease. The experimenter can adjust the threshold (green vertical line in the figure), which will in turn change the false positive rate. Increasing the threshold would result in fewer false positives (and more false negatives), corresponding to a leftward movement on the curve. The actual shape of the curve is determined by how much overlap the two distributions have.