Distance measure for asymmetric binary attributes in data mining

How to calculate proximity measure for asymmetric binary attributes?

In this tutorial, we will learn about the proximity measure for asymmetric binary attributes

Contingency table for binary data

Here in this example, consider 1 for positive/True  and 0 for negative/False.

Object 2
Object 1   1 / True / Positive 0 / False / Negative Sum
1 / True / Positive A B A + B
0 / False / Negative C D C + D
Sum A + C B + D  

In table 1 we can consider the following facts.

A represents that object 1 is True and object 2 is also True.

B represents that object 1 is True and object 2 is also False.

C represents that object 1 is False and object 2 is also True.

D represents that object 1 is False and object 2 is also False.

Name Fever Cough Test 1 Test 2 Test 3 Test 4
Asad Negative Yes  Negative  Positive  Negative  Negative
Bilal Negative  Yes  Negative Positive  Positive  Negative
Tahir Positive Yes  Negative  Negative  Negative  Negative

In table 2, Asad, Bilal and Tahir are objects. Negative values represents False and Positive represents Negative.

dissimilarity of binary variables

In the results, we can see the following facts;

The distance between object 1 and 2 is 0.67. Asad is object 1 and Tahir is in object 2 and the distance between both is 0.67.

Less distance is between Asad and Bilal. It means that Asad and Bilal are more similar to each other as compared to other objects.