Hierarchical Clustering
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1 Hierarchical Clustering Produces a set of nested clusters organized as a hierarchical tree Can be visualized as a dendrogram A tree like diagram that records the sequences of merges or splits /8/00 8
2 Strengths of Hierarchical Clustering Do not have to assume any particular number of clusters Any desired number of clusters can be obtained by cutting the dendogram at the proper level They may correspond to meaningful taxonomies Example in biological sciences (eg, animal kingdom, phylogeny reconstruction, ) /8/00 9
3 Hierarchical Clustering Two main types of hierarchical clustering Agglomerative: Start with the points as individual clusters At each step, merge the closest pair of clusters until only one cluster (or k clusters) left Divisive: Start with one, all-inclusive cluster At each step, split a cluster until each cluster contains a point (or there are k clusters) Traditional hierarchical algorithms use a similarity or distance matrix Merge or split one cluster at a time /8/00 0
4 Agglomerative Clustering Algorithm More popular hierarchical clustering technique Basic algorithm is straightforward Compute the proximity matrix Let each data point be a cluster Repeat Merge the two closest clusters Update the proximity matrix Until only a single cluster remains Key operation is the computation of the proximity of two clusters Different approaches to defining the distance between clusters distinguish the different algorithms /8/00
5 Starting Situation Start with clusters of individual points and a proximity matrix p p p p p p p p p p Proximity Matrix /8/00
6 Intermediate Situation After some merging steps, we have some clusters C C C C C C C C C C C C Proximity Matrix C C C /8/00
7 Intermediate Situation We want to merge the two closest clusters (C and C) and update the proximity matrix C C C C C C C C C C C C Proximity Matrix C C C /8/00
8 After Merging The question is How do we update the proximity matrix? C C C U C C C C U C C C????? C? C? Proximity Matrix C C U C /8/00
9 How to Define Inter-Cluster Similarity p Similarity? p p p p p p p p p MIN MAX Group Average Proximity Matrix Distance Between Centroids Other methods driven by an objective function Ward s Method uses squared error /8/00
10 How to Define Inter-Cluster Similarity p p p p p p p p p p MIN MAX Group Average Proximity Matrix Distance Between Centroids Other methods driven by an objective function Ward s Method uses squared error /8/00 7
11 How to Define Inter-Cluster Similarity p p p p p p p p p p MIN MAX Group Average Proximity Matrix Distance Between Centroids Other methods driven by an objective function Ward s Method uses squared error /8/00 8
12 How to Define Inter-Cluster Similarity p p p p p p p p p p MIN MAX Group Average Proximity Matrix Distance Between Centroids Other methods driven by an objective function Ward s Method uses squared error /8/00 9
13 How to Define Inter-Cluster Similarity p p p p p p p p p p MIN MAX Group Average Proximity Matrix Distance Between Centroids Other methods driven by an objective function Ward s Method uses squared error /8/00 0
14 Cluster Similarity: MIN or Single Link Similarity of two clusters is based on the two most similar (closest) points in the different clusters Determined by one pair of points, ie, by one link in the proximity graph I I I I I I,00 0,90 0,0 0, 0,0 I 0,90,00 0,70 0,0 0,0 I 0,0 0,70,00 0,0 0,0 I 0, 0,0 0,0,00 0,80 I 0,0 0,0 0,0 0,80,00 /8/00
15 Hierarchical Clustering: MIN Nested Clusters Dendrogram /8/00
16 Strength of MIN Original Points Two Clusters Can handle non-elliptical shapes /8/00
17 Limitations of MIN Original Points Two Clusters Sensitive to noise and outliers Particular case: chain effect /8/00
18 Cluster Similarity: MAX or Complete Linkage Similarity of two clusters is based on the two least similar (most distant) points in the different clusters Determined by all pairs of points in the two clusters I I I I I I,00 0,90 0,0 0, 0,0 I 0,90,00 0,70 0,0 0,0 I 0,0 0,70,00 0,0 0,0 I 0, 0,0 0,0,00 0,80 I 0,0 0,0 0,0 0,80,00 /8/00
19 Hierarchical Clustering: MAX Nested Clusters 0 Dendrogram /8/00
20 Strength of MAX Original Points Two Clusters Less susceptible to noise and outliers /8/00 7
21 Limitations of MAX Original Points Two Clusters Tends to break large clusters Biased towards globular clusters /8/00 8
22 Cluster Similarity: Group Average Proximity of two clusters is the average of pairwise proximity between points in the two clusters Need to use average connectivity for scalability since total proximity favors large clusters I I I I I I,00 0,90 0,0 0, 0,0 I 0,90,00 0,70 0,0 0,0 I 0,0 0,70,00 0,0 0,0 I 0, 0,0 0,0,00 0,80 I 0,0 0,0 0,0 0,80,00 /8/00 9
23 Hierarchical Clustering: Group Average Nested Clusters Dendrogram /8/00 70
24 Hierarchical Clustering: Group Average Compromise between Single and Complete Link Strengths Less susceptible to noise and outliers Limitations Biased towards globular clusters /8/00 7
25 Cluster Similarity: Ward s Method Similarity of two clusters is based on the increase in squared error when two clusters are merged Similar to group average if distance between points is distance squared Less susceptible to noise and outliers Biased towards globular clusters Hierarchical analogue of K-means Can be used to initialize K-means /8/00 7
26 Hierarchical Clustering: Comparison MIN MAX Ward s Method Group Average /8/00 7
27 Hierarchical Clustering: Time and Space requirements O(N) space since it uses the proximity matrix N is the number of points O(N) time in many cases There are N steps and at each step the proximity matrix (size: O(N)) must be updated and searched Complexity can be reduced to O(N log(n) ) time for some approaches /8/00 7
28 Hierarchical Clustering: Problems and Limitations Once a decision is made to combine two clusters, it cannot be undone No objective function is directly minimized Different schemes have problems with one or more of the following: Sensitivity to noise and outliers Difficulty handling different sized clusters and convex shapes Breaking large clusters /8/00 7
29 Single-link HAC vs Graphs Cut edges with D>0 D =0 D=0 0 0 Clusters = connected components /8/00 7
30 Single-link HAC vs Graphs Cut edges with D> Cut edges with D> /8/00 77
31 Complete-link HAC vs Graphs Clusters ~ cliques constraint: earlier cliques have precedence Cut edges with D> Cut edges with D> /8/00 78
32 MST: Divisive Hierarchical Clustering Build MST (Minimum Spanning Tree) Start with a tree that consists of any point In successive steps, look for the closest pair of points (p, q) such that one point (p) is in the current tree but the other (q) is not Add q to the tree and put an edge between p and q /8/00 79
33 MST: Divisive Hierarchical Clustering X /8/00 80
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Clustering Basic concepts Road map K-means algorithm Representation of clusters Hierarchical clustering Distance functions Data standardization Handling mixed attributes Which clustering algorithm to use?
Olmo S. Zavala Romero. Clustering Hierarchical Distance Group Dist. K-means. Center of Atmospheric Sciences, UNAM.
Center of Atmospheric Sciences, UNAM November 16, 2016 Cluster Analisis Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster)