K Nearest Neighbours (K-NN)

Machine Learning Techniques

Dr. Ashish Tendulkar

IIT Madras

K-nearest neighbour or KNN is a supervised learning algorithm that can be used for both regression and classification tasks.

It is an instance based learning technique.

There is no explicit model, but KNN compares a new example with existing training examples, obtains \(k\) nearest neighbours and assigns an output label based on the labels of \(k\) nearest neighbors.

The examples are compared with a variety of distance metrics such as Euclidean and Manhattan distance.

There are two important questions or hyper parameters:
- How many neighbours (k) to choose?
- Which distance metric should be used for comparing examples?

Overview

Key insight

An example is labeled by the company it keeps.

An example of 3-NN classifier

This is an example of binary classification problem with 3-NN classifier. In this example, nearest neighbours are calculated based on Euclidean distance.

It has examples from two classes: Red and Green.

Now there are two new points \(P_1\) and \(P_2\), for which labels are unknown or yet to be predicted.

Each point looks at its 3 neighbours and computes the class that is represented by 2 or 3 neighbours. Hence, the point is labeled with the majority class in its neighbourhood.

In the figure, for \(P_1\), 2 out of 3 neighbours are red, therefore, it is predicted to be in class Red.

For \(P_2\), 2 out of 3 neighbours are green, therefore, it is predicted to be in class Green.

Distance Metric

Following two metrics are used quite often:

Euclidean distance

Manhatten distance

Manhatten Distance:

Euclidean distance :

Distance between two points \(\mathbf{x}_1\) and \(\mathbf{x}_2\) represented with \(m\) features is calculated as follows:

\delta(\mathbf{x}^{(1)}, \mathbf{x}^{(2)}) = \mid x_1^{(1)} - x_1^{(2)} \mid + \mid x_2^{(1)} - x_2^{(2)} \mid+ \dots+ \mid x_m^{(1)} - x_m^{(2)} \mid

\delta(\mathbf{x}^{(1)}, \mathbf{x}^{(2)}) = \sqrt{(x_1^{(1)}-x_1^{(2)})^2+(x_2^{(1)}-x_2^{(2)})^2+\dots + (x_m^{(1)} -x_m^{(2)})^2 }

Euclidean distance

\begin{aligned} \delta(\mathbf{x}^{(1)}, \mathbf{x}^{(2)}) &= \sqrt{(x_1^{(1)}-x_1^{(2)})^2+(x_2^{(1)}-x_2^{(2)})^2+\dots + (x_m^{(1)} -x_m^{(2)})^2 } \\ \end{aligned}

\begin{aligned} &= \left( \sum_{j=1}^m \left(x_j^{(1)}-x_j^{(2)} \right)^2 \right)^{\frac{1}{2}} \\ \end{aligned}

\begin{aligned} &= \left( \left(\mathbf{x}^{(1)}-\mathbf{x}^{(2)} \right)^T \left(\mathbf{x}^{(1)}-\mathbf{x}^{(2)} \right) \right)^{\frac{1}{2}} \\ \end{aligned}

Writing this compactly

This can be rewritten in vectorized format as follows

Manhattan distance

\begin{aligned} \delta(\mathbf{x}^{(1)}, \mathbf{x}^{(2)}) &= \mid x_1^{(1)}-x_1^{(2)} \mid + \mid x_2^{(1)}-x_2^{(2)} \mid +\dots + \mid x_m^{(1)}-x_m^{(2)} \mid \\ \end{aligned}

\begin{aligned} &= \sum_{j=1}^m \mid x_j^{(1)}-x_j^{(2)} \mid \\ \end{aligned}

\begin{aligned} &= \mathbf{1}_{1 \times m}^T \mid \mathbf{x}^{(1)}-\mathbf{x}^{(2)} \mid_{m \times 1} \\ \end{aligned}

Writing this compactly as follows

The vectorized form is as follows:

Model

Classification

Regression

For classification task, the \(k\) neighbours take part in voting. The class that receives highest number of votes is the predicted class.

For regression task, the output/prediction is calculated as average of the outputs/labels of \(k\) neighbours.

\hat{y} = \dfrac{1}{k} \sum\limits_{i=1}^{k}y_i

Visualization

Let us apply KNN technique and visualise how a new example is assigned a label. For this example value of \(k\) is set to be 3.

Visualize data and the new example

Find nearest 3 neighbours

Label the new data point with majority class out of 3.

Visualization

Decision Boundary

Finding best value of \(k\)

On the other hand, if \(k\) is too large, then our model will be biased. The model will tend to ignore the underlying trend. In this case, the model will underfit.

If \(k\) is too small e.g. 1 or 2, then our model is sensitive to noise. The model will try to adjust to small changes in variance. In this case, the model will overfit. The decision boundary will be very jagged.