1. Learning the threshold in Hierarchical Agglomerative Clustering
Kristine Daniels
Christophe Giraud-Carrier
Speaker: Ngai Wang Kay

2. Hierarchical clustering
Threshold
Dendrogram d3 d2 d1 d2 d1 d3

3. Distance metric
Single-link distance metric – the minimum of the simple distances (e.g. Euclidean distances) between the objects in the two clusters.

4. Distance metric
Complete-link distance metric – the maximum of the simple distances between the objects in the two clusters.

5. Threshold determination
Some applications may just want a set of clusters for a particular threshold instead of a dendrogram.
A more efficient clustering algorithm may be developed for such case.
There are many possible thresholds.
So, it is hard to determine the threshold that gives an accurate clustering result (based on a measure against the correct clusters).

6. Threshold determination
Suppose C1, …, Cn are the correct clusters and H1, …, Hm are the computed clusters.
A F-measure is used to determine the accuracy of the computed clusters as follows:

7. Threshold determination
where N is the dataset size

8. Semi-supervised algorithm
Select a random subset S of the dataset.
Label the correct clusters of the data in S.
Cluster S using the previous algorithm.
Compute the F-measure value for each threshold in the dendrogram.
Find the threshold with the highest F-measure value.
Cluster the dataset using this threshold.

9. Sample set
Preliminary experiments show that a sample set of size 50 gives reasonable clustering results.
The time complexity of the hierarchical clustering is usually O(N2) or higher in simple distance computations and numerical comparisons.
So, learning the threshold may be a very small cost in comparison to that of clustering the dataset.

10. Experimental results
Experiments are conducted by complete-link clustering on various real datasets in the UCI repository (
They are originally collected for the classification problem.
The class labels of the data are used as the cluster labels in these experiments.

11. Experimental results Dataset Size # Classes Breast-Wisconsin 699 2 Car
1728 4
Diabetes
768
Glass
214
Hepatitis
155
Ionosphere
351
Kr-vs-Kp
3196
Tic-Tac-Toe
958
Vehicle
946

12. Experimental results Dataset Target threshold Learned threshold
Breast-Wisconsin
13.17
11.91
Car
7.35
6.68
Diabetes
8.84
11.61
Glass
9.39
8.06
Hepatitis
17.12
14.50
Ionosphere
24.81
24.00
Kr-vs-Kp
50.37
Tic-Tac-Toe
7.52
7.45
Vehicle
13.09
6.11

13. Experimental results
Because of the nature of the data, there may be many good threshold values.
So, large differences between the target and learned thresholds do not have to yield large differences between the corresponding F-measure values.

14. Experimental results Dataset F-measure (Target / Learned) # Clusters
Breast-Wisconsin
0.97 / 0.97
2 / 2
Car
0.90 / 0.64
2 / 5
Diabetes
0.71 / 0.65
13 / 4
Glass
0.82 / 0.82
11 / 13
Hepatitis
0.77 / 0.77
1 / 2
Ionosphere
0.69 / 0.66
Kr-vs-Kp
0.67 / 0.67
1 / 4
Tic-Tac-Toe
0.69 / 0.58
Vehicle
0.46 / 0.31
3 / 36

15. Experimental results
The Vehicle dataset shows a huge difference in the number of clusters but a moderate difference in the F-measure.
The Car dataset suffers from a serious loss of the F-measure, but the difference in the number of clusters is small.
These anomalies may be explained, in part, by the sparseness of the data, the skewness of the underlying class distributions, and the cluster labels are based on the classification labels.

16. Experimental results
The Diabetes dataset achieves a F-measure value close to optimal with fewer clusters when using the learned threshold.
In summary, the learned threshold achieves clustering results close to the optimal ones at a fraction of the computational cost of clustering the whole dataset.

17. Conclusion
Hierarchical clustering does not produce a single clustering result but a dendrogram, a series of nested clusters based on distance thresholds.
This leads to the open problem of choosing the preferred threshold.
An efficient semi-supervised algorithm is proposed to obtain such threshold.
Experimental results show the clustering results obtained using the learned threshold are close to optimal.