how to Merge the two closest clusters.?

How to Merge the Closest Clusters?

To merge the two closest clusters in hierarchical clustering, you need to determine which pair of clusters has the smallest distance according to your chosen linkage method (e.g., single-linkage, complete-linkage, average-linkage). Once you identify the pair with the smallest distance, you merge them into a new cluster.

Here's a step-by-step guide:

Example Data: Let's consider the clusters after an initial iteration:

$\text{Clusters}=\{�\},\{�\},\{�\},\{�\},\{�\},\{�\},\{�\}$

Step 1: Pairwise Euclidean Distance Calculation: Calculate the Euclidean distance between each pair of clusters using the average-linkage method.

$\text{Distance}(��)=\frac{1}{\mid �\mid \cdot \mid �\mid }{\sum }_{�\in �}{\sum }_{�\in �}\text{EuclideanDistance}(�,�)$

Example Calculation: $\text{Distance}(\{�\},\{�\})=\frac{1}{1\cdot 1}\cdot \text{EuclideanDistance}(�,�)$ $\text{Distance}(\{�\},\{�\})=\frac{1}{1\cdot 1}\cdot \text{EuclideanDistance}(�,�)$ $\text{Distance}(\{�\},\{�\})=\frac{1}{1\cdot 1}\cdot \text{EuclideanDistance}(�,�)$ $\text{Distance}(\{�\},\{�\})=\frac{1}{1\cdot 1}\cdot \text{EuclideanDistance}(�,�)$

Repeat this process for all pairs of clusters.

Step 2: Identify Closest Clusters: Find the pair of clusters with the smallest distance.

Example: Suppose the smallest distance is between clusters {A} and {C}.

Step 3: Merge Clusters: Merge the two closest clusters into a new cluster.

Updated Clusters: $\text{Clusters}=\{��\},\{�\},\{�\},\{�\},\{�\},\{�\}$

Now, {AC} represents the merged cluster. Repeat the process of pairwise distance calculation and merging until only one cluster remains.

In practice, hierarchical clustering algorithms handle these details automatically, and you can use libraries like scipy and scikit-learn in Python to perform hierarchical clustering efficiently. The linkage method used will determine how distances are calculated between clusters.