How to Calculate pairwise Euclidean distances between clusters.? 

To calculate pairwise Euclidean distances between clusters, you need to consider the distances between the data points in different clusters. The distance between two clusters can be computed using various linkage methods, such as single-linkage, complete-linkage, or average-linkage. Let's focus on the average-linkage method for simplicity.

Here's a step-by-step guide to calculating pairwise Euclidean distances between clusters using the average-linkage method:

Example Data: Let's use a set of data points and their coordinates:

$�(1,2),�(2,3),�(2,2),�(3,3),�(8,7),�(9,8),�(10,7)$

Step 1: Initialization:

• Start with each data point as a separate cluster.

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

Step 2: Pairwise Euclidean Distance Calculation:

• Calculate the Euclidean distance between each pair of clusters using the average-linkage method.

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

Where $�$ and $�$ are clusters, and $\text{EuclideanDistance}(�,�)$ is the Euclidean distance between data points $�$ and $�$.

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

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

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

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

Repeat this process for all pairs of clusters.

Step 3: Merge Clusters:

• Merge the two clusters with the smallest distance.

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

Repeat Steps 2-3 until only one cluster remains.

In practice, hierarchical clustering algorithms, such as those implemented in Python libraries like scipy and scikit-learn, handle the details of pairwise distance calculations and clustering efficiently.