Similarity Measure – IOU – (Jaccard index)

Similarity Measure

The Jaccard index/coefficient, aka Intersection over Union is used in computer vision as one of the similarity measures for object detection on images.

Formula

The Jaccard similarity coefficient,aka Intersection over Union (IOU) score:

$J(A,B) = {{|A \cap B|}\over{|A \cup B|}} = {{|A \cap B|}\over{|A| + |B| - |A \cap B|}}.$

Binary classification Formula

The same score above can be described as binary classification as well to get an ML algorithm to use it:

$J = {M_{11} \over M_{01} + M_{10} + M_{11}}.$

And can also be noted in writing as:

true positive / (true positive + false positive + false negative)

M11 is always the intersection- aka true positive
M01 is the false positive
M10 is the false negative
M00 is not used in the formula as it is the complement

Code Example in R

 Calc_Jaccard_Index <- function(vector1, vector2){
         Jaccard_Index<(sum(which(vector1==1)%in%which(vector2==1)) /
                       (sum(which(vector1==0)%in%which(vector2==1)) +
                       sum(which(vector1==1)%in%which(vector2==0)) +
                       sum(which(vector1==1)%in%which(vector2==1))))
        return(Jaccard_Index)
}