The k-Nearest Neighbors (k-NN) algorithm is a simple yet powerful classification and regression technique used in machine learning. It’s a non-parametric method, meaning it doesn’t make any assumptions about the underlying data distribution. Instead, it makes predictions based on the similarity of input features to the training data.

The algorithm requires a dataset with labeled examples. Each example consists of features (independent variables) and a corresponding label (dependent variable). For example, in a dataset of fruits, features could include color, size, and weight, with labels indicating the type of fruit.

To predict the label of a new data point, the algorithm calculates the distance between that point and all other points in the dataset. Common distance metrics include Euclidean distance, Manhattan distance, and Minkowski distance. Euclidean distance is the most commonly used metric and is calculated as the square root of the sum of squared differences between corresponding elements of two vectors.

The parameter ‘k’ represents the number of nearest neighbors to consider when making a prediction. It’s crucial to choose an appropriate value for ‘k’ to balance bias and variance. A small ‘k’ may lead to overfitting, while a large ‘k’ may lead to underfitting. The optimal value of ‘k’ can be determined through techniques like cross-validation.

Once the distances are calculated, the algorithm identifies the ‘k’ nearest neighbors to the new data point based on the calculated distance metrics.

For classification tasks, the algorithm assigns the majority class among the ‘k’ nearest neighbors to the new data point. In regression tasks, it calculates the weighted average of the labels of the ‘k’ nearest neighbors, where weights can be inversely proportional to the distance.

Finally, the algorithm assigns the predicted label to the new data point based on the majority class or the calculated average.

Let’s consider a simple example of classifying fruits based on two features: sweetness and acidity. We have a dataset with three types of fruits: apples, oranges, and bananas. Each fruit is represented by its sweetness, acidity, and label:

Now, suppose we want to classify a new fruit with sweetness=5 and acidity=4 using the k-NN algorithm with k=3.

Compute the Euclidean distance between the new fruit and each fruit in the dataset:

Distance to Apple 1: sqrt((5–8)^2 + (4–3)^2) = sqrt(9 + 1) = sqrt(10) ≈ 3.16

Distance to Apple 2: sqrt((5–6)^2 + (4–2)^2) = sqrt(1 + 4) = sqrt(5) ≈ 2.24

Distance to Orange 1: sqrt((5–2)^2 + (4–8)^2) = sqrt(9 + 16) = sqrt(25) = 5

Distance to Orange 2: sqrt((5–7)^2 + (4–5)^2) = sqrt(4 + 1) = sqrt(5) ≈ 2.24

Distance to Banana 1: sqrt((5–4)^2 + (4–9)^2) = sqrt(1 + 25) = sqrt(26) ≈ 5.1

Distance to Banana 2: sqrt((5–2)^2 + (4–3)^2) = sqrt(9 + 1) = sqrt(10) ≈ 3.16

Select the 3 nearest neighbors based on the calculated distances. In this case, the nearest neighbors are Apple 2, Orange 2, and Apple 1.

Voting: Since k=3, we consider the majority class among the nearest neighbors. Here, two out of three nearest neighbors are apples, so we predict the new fruit as an apple.

Thus, according to the k-NN algorithm, the new fruit with sweetness=5 and acidity=4 is predicted to be an apple.

The k-Nearest Neighbors algorithm is a versatile and intuitive method for classification and regression tasks. However, it’s important to preprocess data, choose an appropriate distance metric, and select the right value for ‘k’ to ensure accurate predictions. Additionally, while k-NN is easy to understand and implement, it may not be suitable for large datasets due to its computational inefficiency.