**π‘ Problem Formulation:** Imagine you have a collection of points in a two-dimensional space and you want to divide them into ‘k’ distinct groups based on their proximity to each other. For instance, given the set of points [(1,2), (1,4), (3,5), (6,8), (7,7)] and k=2, the desired output could be either {(1,2), (1,4), (3,5)} and {(6,8), (7,7)} or any other optimal grouping based on the chosen method.

## Method 1: K-Means Clustering

The K-Means clustering algorithm is aimed at partitioning n observations into k clusters in which each observation belongs to the cluster with the nearest mean. It is a popular method for vector quantization and is particularly well-suited for creating groups of points.

Here’s an example:

from sklearn.cluster import KMeans import numpy as np points = np.array([(1,2), (1,4), (3,5), (6,8), (7,7)]) kmeans = KMeans(n_clusters=2, random_state=0).fit(points) print(kmeans.labels_)

Output:

[1 1 1 0 0]

This snippet uses the `sklearn.cluster.KMeans`

class to perform clustering. It initializes the KMeans algorithm with two clusters and a fixed random state for reproducibility. The `fit`

method computes the centroids and assigns each point in ‘points’ to one of the two clusters, as shown in the output labels.

## Method 2: DBSCAN

Density-Based Spatial Clustering of Applications with Noise (DBSCAN) is an algorithm that groups together points that are closely packed together, marking as outliers points that lie alone in low-density regions. It doesnβt require one to specify the number of clusters in advance.

Here’s an example:

from sklearn.cluster import DBSCAN points = [(1,2), (1,4), (3,5), (6,8), (7,7)] clustering = DBSCAN(eps=3, min_samples=2).fit(points) print(clustering.labels_)

Output:

[ 0 0 0 1 1]

This snippet uses the `sklearn.cluster.DBSCAN`

class with an epsilon value of 3 and a minimum sample count of 2 to perform clustering. DBSCAN successfully identifies dense regions of points and assigns cluster labels accordingly.

## Method 3: Agglomerative Hierarchical Clustering

Agglomerative Hierarchical Clustering starts with each point as a separate cluster and iteratively combines the closest pairs of clusters until all points have been merged into a single remaining cluster. This method is useful when the number of clusters is not predetermined.

Here’s an example:

from sklearn.cluster import AgglomerativeClustering points = [(1,2), (1,4), (3,5), (6,8), (7,7)] clustering = AgglomerativeClustering(n_clusters=2).fit(points) print(clustering.labels_)

Output:

[1 1 1 0 0]

This code uses the `sklearn.cluster.AgglomerativeClustering`

class to perform hierarchical clustering, specifying that we want to end up with 2 clusters. It fits the model to the points and outputs the labels that indicate cluster membership.

## Method 4: Mean Shift Clustering

Mean Shift Clustering aims at discovering blobs in a smooth density of samples. It is a centroid-based algorithm, which works by updating candidates for centroids to be the mean of the points within a given region. These candidates are then filtered in a post-processing stage to eliminate near-duplicates.

Here’s an example:

from sklearn.cluster import MeanShift points = [(1,2), (1,4), (3,5), (6,8), (7,7)] clustering = MeanShift().fit(points) print(clustering.labels_)

Output:

[1 1 1 0 0]

The example uses the Mean Shift algorithm implemented in `sklearn.cluster.MeanShift`

. After fitting the points, it outputs labels indicating which points belong to which cluster. This method requires no specification of the number of clusters.

## Bonus One-Liner Method 5: SciPy Hierarchical Clustering

For a hierarchical clustering approach that doesn’t require scikit-learn, SciPy library provides a method that computes a hierarchy of clusters and returns the labels accordingly.

Here’s an example:

from scipy.cluster.hierarchy import fcluster, linkage points = [(1,2), (1,4), (3,5), (6,8), (7,7)] Z = linkage(points, 'ward') labels = fcluster(Z, 2, criterion='maxclust') print(labels)

Output:

[2 2 2 1 1]

This snippet uses SciPy’s `linkage`

and `fcluster`

functions to perform agglomerative hierarchical clustering with Ward’s method and then assigns points to clusters, specifying a maximum number of clusters.

## Summary/Discussion

**Method 1:**K-Means Clustering. Fast and efficient for large datasets. Requires specifying the number of clusters and may produce different results if the algorithm runs with different initializations.**Method 2:**DBSCAN. Capable of finding arbitrarily shaped clusters and robust to outliers. Determining the optimal parameters can be difficult, and it doesnβt perform well with varying cluster densities.**Method 3:**Agglomerative Hierarchical Clustering. Provides a tree of clusters which is informative. Computationally expensive and not suitable for large datasets. Number of clusters needs to be specified after the model is run.**Method 4:**Mean Shift Clustering. Automatically determines the number of clusters. Computation can be expensive and it may not work well in high dimensional spaces.**Bonus One-Liner Method 5:**SciPy Hierarchical Clustering. A simple one-liner for hierarchical clustering with SciPy. It’s a method good for visualizing dendrogram but can be computationally intensive.