Origin: this data is a modified version of the Optical Recognition of Handwritten Digits Dataset from the

UCI repository. It contains pre$-$processed black and white images of the digits $5$ and $6.$ Each feature

indicates how many pixels are black in a patch of $4$ x $4$ pixels.

Format: there is one row per image and one column per feature. The class labels are $5$ and $6.$ The file

train$\_$targets.csv includes the targets for the data in train$\_$inputs.csv$.$

Implement k$-$nearest neighbours. You are required to personally implement the core functionality of

KNN$,$ which includes calculating distances between data points and determining predictions based on a

majority vote. Refrain from using external libraries for these specific tasks of the KNN algorithm.

However, you may utilize libraries for auxiliary tasks such as data loading, cross$-$validation, and other

supportive functions.

Feel free to run the Jupyter notebook on any machine or Google Colab. Google Colab is a free cloud

environment provided by Google that allows you to run Jupyter notebooks very easily. Python and all

necessary libraries are already installed.

Once you are done, run the Jupyter notebook entirely and save the following results:

$1.$ A graph that shows the average accuracy based on $10-$fold cross validation when varying the

number of neighbours from $1$ to $30.$

$2.$ The best number of neighbours found by $10-$fold cross validation and its cross$-$validation

accuracy.

$3.$ The test accuracy based on the best number of neighbours