Mathematical explanation of K-Nearest Neighbour

KNN stands for K-nearest neighbour, it’s one of the Supervised learning algorithm mostly used for classification of data on the basis how it’s neighbour are classified. KNN stores all available cases and classifies new cases based on a similarity measure. K in KNN is a parameter that refers to the number of the nearest neighbours to include in the majority voting process.

How do we choose K?

Sqrt(n), where n is a total number of data points(if in case n is even we have to make the value  odd by adding 1 or subtracting 1 that helps in select better)

When to use KNN?

We can use KNN when Dataset is labelled and noise-free and it’s must be small because KNN is a “Lazy learner”. Let’s understand KNN algorithm with the help of an example

NAME	AGE	GENDER	CLASS OF SPORTS
Ajay	32	0	Football
Mark	40	0	Neither
Sara	16	1	Cricket
Zaira	34	1	Cricket
Sachin	55	0	Neither
Rahul	40	0	Cricket
Pooja	20	1	Neither
Smith	15	0	Cricket
Laxmi	55	1	Football
Michael	15	0	Football

Here male is denoted with numeric value 0 and female with 1. Let’s find in which class of people Angelina will lie whose k factor is 3 and age is 5. So we have to find out the distance using 

  d=√((x2-x1)²+(y2-y1)²) to find the distance between any two points.

So let’s find out the distance between Ajay and Angelina using formula  

d=√((age2-age1)²+(gender2-gender1)²)

d=√((5-32)²+(1-0)²)

d=√729+1

d=27.02

Similarly, we find out all distance one by one.

Distance between Angelina and	Distance
Ajay	27.02
Mark	35.01
Sara	11.00
Zaira	29.00
Sachin	50.01
Rahul	35.01
Pooja	15.00
Smith	10.05
Laxmi	50.00
Michael	10.05

So the value of k factor is 3 for Angelina. And the closest to 3 is 9,10,10.5 that is closest to Angelina are Zaira, Smith and Michael.

                                       Zaira         9           cricket

                                      Michael      10         cricket    

                                      smith          10.5      football

so according to KNN algorithm, Angelina will be in the class of people who like cricket. So this is how KNN algorithm works.