Lets say you have a dataset for binary classification. Each data point x has n-dimensions, and the target value is y = 1 or -1 for the two classes.

The perceptron

K-nearest neighbors (KNN) is one of the simplest Machine Learning algorithms. It is a supervised learning algorithm which can be used for both classification and regression.

Understanding the classification algorithm (illustration)

Let us understand this algorithm with a classification problem. For simplicity of visualization, we'll assume that our input data is 2 dimensional. We have two possible classes - green and red.

Lets plot out training data in feature space.

There is no explicit training phase in KNN! In other words, for classifying new data points, we'll directly use our dataset (in some sense, the dataset is the model).

To classify a new data point, we find the k points in the training data closest to it, and make a prediction based on whichever class is most common among these k points (i.e. we simulate a vote). Here closest is defined by a suitable distance metric such as euclidean distance. Other distance metrics are discussed below.

For example, if we want to classify blue point as shown in following figure, we consider k nearest data points and we assign the class which has the majority.

If k = 3, we get two data points with green class and one data point with red class. Hence, we'll predict green class for the new point.

Here's another example, let us change the position of new point (blue point) as shown below.

If we take k = 5 then we get four neighbors with red class and one neighbor with green class. Hence, new point will be classified as red point.

KNN as regression algorithm

In case of regression (when target variable is a real value), we take the average of the K nearest neighbors.

Tuning the hyper-parameter K

A small value of k means that noise will have a higher influence on the result and large value will make the algorithm computationally expensive. Usually, we perform cross-validation to find out best k value (or to choose the value of k that best suits our accuracy / speed trade-off). If you don't want to try multiple values of k, a rule of thumb is to set k equal to the square root of total number of data points. For more on choosing best value of k, refer this stackoverflow thread.

Distance metrics

There are various options available for distance metric such as euclidian or manhattan distance. The most commonly used metric is euclidian distance.

Minkowski is the generalization of Euclidian and Manhattan distance.

Note that you'll want to do some pre-processing on the input data (for example, make sure each dimension has 0 mean and unit variance) so that the distance metrics above are meaningful.

Scikit-learn implementation

## load the dataset 
from sklearn.datasets import load_iris
dataset = load_iris()
X = dataset.data
y = dataset.target
 
## precessing
# standardize the data to make sure each feature contributes equally 
# to the distance
from sklearn.preprocessing import StandardScaler
ss = StandardScaler()
X_processed = ss.fit_transform(X)
 
## split the dataset into train and test set
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X_processed, y, test_size=0.3, random_state=42)
 
## fit n nearest neighbor model 
from sklearn.neighbors import KNeighborsClassifier
model = KNeighborsClassifier(n_neighbors = 5, metric="minkowski", p=2) 
# p=2 for euclidian distance
model.fit(X_train, y_train)
# output: 
# KNeighborsClassifier(algorithm='auto', leaf_size=30, metric='minkowski',
#          metric_params=None, n_jobs=1, n_neighbors=5, p=2,
#          weights='uniform')
 
## evaluate 
model.score(X_test, y_test)
# output: 1.0

Parametric and non-parametric models

In a parametric model, we continuously update a fixed number of parameters to learn a function which can classify new data point without requiring the training data (for example, logistic regression). In a non-parametric model, the number of parameters grows with the size of training data. This is what happens in KNN.