Question

The following dataset contains eight items representing colored points on the x-y plane:

х y color
1 1 red
1 3 green
2 5 blue
3 5 green
4 1 blue
4 4 red
5 3 blue
5 4 green
Using this data as the training set, run the k-nearest-neighbors classification algorithm (k=1) using square distance and choose the color for a new item with x = 3 and y = 3.

Answer 1

According to the question, the following datasheet contains eight items representing the colored points o the x-y plane.

We have to use this data as training set to run the k-nearest classification algorithm to decide most likely color for a new item with x = 3 and y = 3.

The distance between the points is actual distance on x-y plane, called as Eucledian distance.

We will make a data table, by calculating distance from (3, 3) of each point. By using formula :

Distance,
`$d = √((x_2-x_1)^2+(y_2-y_1)^2)$`

x y Color Distance from point (3, 3)

1 1 Red
`$√((3-1)^2+(3-1)^2) = 2.82$`

1 3 Green
`$√((3-1)^2+(3-3)^2) = 2$`

2 5 Blue
`$√((3-2)^2+(3-5)^2) = 2.23$`

3 5 Green
`$√((3-3)^2+(3-5)^2) = 2$`

4 1 Blue
`$√((3-4)^2+(3-1)^2) = 2.23$`

4 4 Red
`$√((3-4)^2+(3-4)^2) = 1.41$`

5 3 Blue
`$√((3-5)^2+(3-3)^2) = 2$`

5 4 Green
`$√((3-5)^2+(3-4)^2) = 2.23$`

Now, we will do sorting of colors with distance in ascending order.

We get, [ Red, Green, Green, Blue, Blue, Blue, Green, Red]

Now if we run the algorithm with k = 1, then we pick only 1 color having the shortest distance that will be assigned to the given point.

Therefore the color is RED.

If we run the algorithm with k = 4, we will pick up
`$4 \text{ colors}$` with shortest distance which are
`$\text{red, green, green, blue}$`. Since, now we know, Green has the greatest frequency among 4, hence the answer is Green.