Choosing an optimal k value in KNN determines the number of neighbors we look at when we assign a value to any new observation.

For a very low value of k (suppose k=1), the model overfits on the training data, which leads to a high error rate on the validation set. On the other hand, for a high value of k, the model performs poorly on both train and validation set. When k increases, validation error decreases and then starts increasing in a “U” shape. An optimal value of k can be determined from the elbow curve of the validation error.

The value of k in the KNN algorithm is related to the error rate of the model. A small value of k could lead to overfitting as well as a big value of k can lead to underfitting. Overfitting imply that the model is well on the training data but has poor performance when new data is coming. Underfitting refers to a model that is not good on the training data and also cannot be generalized to predict new data.

Scikit-learn is a very popular Machine Learning library in Python which provides a KNeighborsClassifier object which performs the KNN classification. The n_neighbors parameter passed to the KNeighborsClassifier object sets the desired k value that checks the k closest neighbors for each unclassified point.

The object provides a .fit() method which takes in training data and a .predict() method which returns the classification of a set of data points.

The Euclidean Distance between two points can be computed, knowing the coordinates of those points.

On a 2-D plane, the distance between two points p and q is the square-root of the sum of the squares of the difference between their x and y components. Remember the Pythagorean Theorem: a^2 + b^2 = c^2 ?

We can write a function to compute this distance. Let’s assume that points are represented by tuples of the form (x_coord, y_coord). Also remember that computing the square-root of some value n can be done in a couple of ways: math.sqrt(n), using the math library, or n ** 0.5 (n raised to the power of 1/2).

def distance(p1, p2):
  x_diff_squared = (p1[0] - p2[0]) ** 2
  y_diff_squared = (p1[1] - p2[1]) ** 2
  return (x_diff_squared + y_diff_squared) ** 0.5

distance( (0, 0), (3, 4) )      # => 5.0