AHC stands for Agglomerative Hierarchical Clustering, which is a hierarchical clustering algorithm that creates a tree of clusters by merging smaller clusters into larger ones. The algorithm does not require specifying the number of clusters in advance, unlike k-means or GMM. Instead, it allows choosing any number of clusters by cutting the tree at the right level.

## Working

**Step 1:** Treat each data point as a singleton cluster and calculate the distance matrix between all pairs of clusters.

**Step 2:** Find the pair of clusters with the smallest distance and merge them into a new cluster. Update the distance matrix by removing the rows and columns of the merged clusters and adding a row and column for the new cluster.

**Step 3:** Repeat step 2 until there is only one cluster left, which is the root of the tree.

## Pseudocode

```
# Input: data points X, distance measure D
# Output: cluster tree T
# Step 1: Initialize n singleton clusters and n X n distance matrix
C = singleton_clusters(X)
M = distance_matrix(C, D)
# Initialize cluster tree
T = empty_tree()
# Loop until there is only one cluster left
while C.size > 1:
# Step 2: Find the pair of clusters with the smallest distance and merge them
i, j = argmin(M)
new_cluster = merge(C[i], C[j])
# Update the distance matrix
M = update_matrix(M, i, j, new_cluster, D)
# Update the cluster list
C = update_list(C, i, j, new_cluster)
# Step 3: Add the merged cluster to the tree
T = add_node(T, new_cluster)
# Return the final cluster tree
return T
```

## Advantages

- It can capture the hierarchy and structure of the data, unlike k-means or GMM which produce flat clusters.
- It can handle clusters with different shapes and sizes, unlike k-means which assumes spherical clusters.
- It can choose any number of clusters by cutting the tree at the right level, unlike k-means or GMM which need k as an input.

## Disadvantages

- It is computationally expensive, as it requires calculating and updating a large distance matrix at each iteration.
- It is sensitive to noise and outliers, as they can affect the distance measure and the merging process.
- It is difficult to handle different cluster densities, as it may merge clusters too early or too late depending on the distance measure.

## References and Further Reading

- Hierarchical Clustering | Machine Learning | Google for Developers
- Agglomerative Hierarchical Clustering - GeeksforGeeks
- Hierarchical clustering - Wikipedia

This will be a multipart series which will follow up with more clustering algorithms with their working, pseudocode, advantages and disadvantages.

Please stay tuned for more such content.

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