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From the Fuzzy C-Means point of view: Every data point belongs to all the cluster which is given by their degree of membership to each cluster.

How do I view it in the case of Hierarchical Clustering? How will the membership matrix be like at every level of the hierarchy?

At the parent node(single cluster), should the membership value of all data points to that parent node be 1? But how about at the leave node? When a leaf node contains only a single data point, then by definition of compactness of a cluster, the leaf node containing only a single data point is the best clustering, and hence shouldn't the membership be 1 in such a case?

Jerry
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  • Hierarchical agglomerative clustering is [not](https://stats.stackexchange.com/a/63549/3277) a hierarchical taxonomy; rather, it is a _story_ of collecting objects. Each next step clusters depend on the clusters formed on the previous step. If on step i a decision is not crisp but is fuzzy - how that should be incorporated on step i+1? on step i+k? I have no idea. Have you? – ttnphns May 19 '20 at 14:53
  • Say we have N data points: At parent(1 cluster), membership matrix(U) will be of 1xN, i.e., every data point to that 1 cluster will be 1. Suppose at the next lower level we have 2 clusters: then U will be 2xN. Similarly at ith level having k clusters(considering the leaf nodes as the number of clusters) U will be kXN. – Jerry May 19 '20 at 15:09
  • So, I think that the LEVEL which corresponds to the actual number of clusters in the dataset will give an optimum separation and compactness of the clusters obtained. HOWEVER, how should the U at i-th level correspond to the (i-1)th level is my concern. – Jerry May 19 '20 at 15:11
  • To me, *fuzzy hierarchical clustering* is an oxymoron. By definition, hierarchies are tree-like structures based on strict, non-fuzzy partitions which don't allow a degree or grade of membership likelihood. You would have to use a different clustering method to arrive at an estimate like that. –  May 19 '20 at 15:58
  • Can we not think it in these terms? Root Node: Fuzzy C means of 1 cluster. i-th level: Fuzzy C means of k clusters – Jerry May 19 '20 at 16:04

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