# Clarification of algorithm for agglomerative hierarchical clustering

I am given the following algorithm for agglomerative hierarchical clustering:

Consider $$\mathbb{X} = [ \mathbf{X}_1 \dots \mathbf{X}_n ]$$. Fix $$\Delta$$ and $$\mathcal{L}$$ (where $$\Delta$$ and $$\mathcal{L}$$ are the distance function and linkage, respectively). Allocate $$\mathbb{X}$$ to $$K$$ clusters. Suppose $$\mathbb{X}$$ are in $$\kappa$$ clusters, $$\kappa \le n$$. To assign $$\mathbb{X}$$ to $$K < \kappa$$ clusters, put $$\nu = \kappa$$ and consider the $$\nu$$ clusters $$\mathcal{C}_1 \dots \mathcal{C}_\nu$$.
1. Calculate pairwise linkages: For $$\mathcal{C}_k, \mathcal{C}_\mathscr{l}$$, calculate $$\mathcal{L}_{k, \mathscr{l}}$$.
2. Find the smallest linkage: Call it $$\mathcal{L}_{(1)}$$.
3. Marge the closest clusters: If $$\mathcal{C}_\alpha$$ and $$\mathcal{C}_\beta$$ such that $$\mathcal{L}_{\alpha, \beta} = \mathcal{L}_{(1)}$$, then merge $$\mathcal{C}_\alpha$$ and $$\mathcal{C}_\beta$$ into one new cluster.
4. Rename the remaining clusters: Consider the collection $$\mathcal{C}_1 \dots \mathcal{C}_{\nu - 1}$$ that is derived from replacing $$\mathcal{C}_\alpha$$ and $$\mathcal{C}_\beta$$ by their merged cluster. Now put $$\nu = \nu - 1$$.
5. Update $$\nu$$: If $$\nu > K$$ steps, then repeat steps 1 to 4.

I don't understand this part:

Allocate $$\mathbb{X}$$ to $$K$$ clusters. Suppose $$\mathbb{X}$$ are in $$\kappa$$ clusters, $$\kappa \le n$$.

If we allocate $$\mathbb{X}$$ to $$K$$ clusters, then how does it then make sense to say that $$\mathbb{X}$$ are in $$\kappa$$ clusters? What is meant here by "allocate"?

• I think this just means that $K$ is the target number of clusters where the procedure stops. Oct 10 '20 at 9:02
• @Lewian So "allocate" means that it is the number of clusters that we are setting the algorithm to cluster the data in? Oct 10 '20 at 9:04
• I'd think so. That would make sense. I also think the bit you are having trouble with is badly written, and ultimately I cannot guarantee what is meant by something that is badly written. Oct 10 '20 at 10:16
• @Lewian I agree. Anyway, thank you for the help. Feel free to post an answer, and I will accept it. Oct 10 '20 at 10:17