This document provides a short introduction to cluster analysis and discusses assessing the quality of clusterings. It begins with an overview of cluster analysis and benchmarking quality. It then discusses cluster quality statistics and examples. Finally, it considers assessing the quality of a clustering and different cluster analysis methods.

1. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Assessing the quality of a clustering
Christian Hennig
Christian Hennig Assessing the quality of a clustering

2. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
1. A short introduction to cluster analysis
Cluster analysis is about finding groups in data.
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Christian Hennig Assessing the quality of a clustering

3. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
1.1 Cluster analysis methods
1.1.1 k-means (Fix & Hodges 1951)
n
i=1
xi − ¯xC(i)
2
= min!
Christian Hennig Assessing the quality of a clustering

4. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
1.1 Cluster analysis methods
1.1.1 k-means (Fix & Hodges 1951)
n
i=1
xi − ¯xC(i)
2
= min!
represents all objects by centroid,
“compact” clusters.
Christian Hennig Assessing the quality of a clustering

5. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
1.1 Cluster analysis methods
1.1.1 k-means (Fix & Hodges 1951)
n
i=1
xi − ¯xC(i)
2
= min!
represents all objects by centroid,
“compact” clusters.
Version: Don’t square, other centroids than mean (“pam”).
Christian Hennig Assessing the quality of a clustering

6. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
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MDS 1
MDS2
Christian Hennig Assessing the quality of a clustering

7. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
1.1.2 Gaussian mixture model (Pearson 1894)
f(x) =
k
j=1
πjϕaj ,Σj
(x).
Clusters are described by Gaussian distributions.
Elliptical clusters, flexible size and shape.
Christian Hennig Assessing the quality of a clustering

8. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
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−0.4 −0.2 0.0 0.2 0.4
−0.4−0.20.00.20.4
MDS 1
MDS2
Christian Hennig Assessing the quality of a clustering

9. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
1.1.3 Classical hierarchical methods
Operate on dissimilarity matrices;
compute dissimilarity measure for every pair of observations.
Can use Euclidean distance,
but also tailor-made distances for other data formats.
Christian Hennig Assessing the quality of a clustering

10. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
1.1.3 Classical hierarchical methods
Operate on dissimilarity matrices;
compute dissimilarity measure for every pair of observations.
Can use Euclidean distance,
but also tailor-made distances for other data formats.
“Cluster”: a collection of similar objects,
dissimilar to the others.
Christian Hennig Assessing the quality of a clustering

11. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
Genetic data: 236 Tetragonula bees, 13 allele pairs
[,1] [,2] [,3] [,4] [,5] [,6] (...)
[1,] "NO" "AA" "PP" "HH" "EH" "FF"
[2,] "EO" "AA" "PP" "HH" "GH" "FF"
[3,] "NQ" "AA" "PT" "HH" "GF" "EF"
[4,] "OO" "AA" "PP" "GH" "GH" "EF"
[5,] "OO" "AA" "PP" "GH" "GH" "EF"
[6,] "LN" "AA" "PP" "HH" "EG" "FE"
(...)
Compute “shared allele distance”.
Christian Hennig Assessing the quality of a clustering

12. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
[,1] [,2] [,3] [,4] [,5]
[1,] 0.00 0.21 0.33 0.29 0.25
[2,] 0.21 0.00 0.33 0.25 0.21
[3,] 0.33 0.33 0.00 0.29 0.33 (...)
[4,] 0.29 0.25 0.29 0.00 0.08
[5,] 0.25 0.21 0.33 0.08 0.00
(...)
Dataset seen before is a
Euclidean approximation (“MDS”) of this.
Christian Hennig Assessing the quality of a clustering

13. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
1.1.3 Classical hierarchical methods
Operate on dissimilarities and produce hierarchical trees
(originally motivated by biological classification).
Differ in definition of “dissimilarity between clusters”.
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0.00.20.40.6
Cluster Dendrogram
hclust (*, "single")
as.dist(tai$distmat)
Height
Christian Hennig Assessing the quality of a clustering

14. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
Single Linkage: (Florek and Perkal 1951)
˜d(A, B) = min
a∈A,b∈B
d(a, b)
Complete Linkage:
˜d(A, B) = max
a∈A,b∈B
d(a, b)
Average Linkage:
˜d(A, B) = avea∈A,b∈Bd(a, b)
These can deliver quite different clusterings.
(Complete L. very compact,
Single L. separated but maybe widespread)
Christian Hennig Assessing the quality of a clustering

15. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
1.1.4 Spectral clustering (Shi and Malik 2000)
Dissimilarity-based nonlinear dimension reduction
for k-means.
Christian Hennig Assessing the quality of a clustering

16. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
1
1
1111
1
1 1
1
1
1
111
11
1
1
1
11
1
1
1
1
1
1
11
1
1
1
1
1
62
6
6 6
2
6
6
6
6
6
66
66
6
6
6
66
6 6
6
6
6
2
2
2
2
6
2
2
2
2
2
2
2
2
2
2
2
55
55
2
2
3
3
3
3
3
3
35 33
3
7
7
7
7
7
7
7
7 7
7
7
7
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44
4
4 4
44
4
4
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4
4
4
44
44
4
4
4
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4
4
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4
4
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4
4
4
4
4
4
444
4
44 4
4
44
4
4444
4
444
4
4
5
5
55
7
7
7
7
777
777
7
7
7
7
7
7
7
77 7
7
7
7
7
7
77
7
7
7
7
7
7
7
7
7
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7
7
7
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7
7
7
7
7
7
7
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7
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7
7
7
77
7
7
7
7
7
7
−0.4 −0.2 0.0 0.2 0.4
−0.4−0.20.00.20.4
ctai$points[,1]
ctai$points[,2]
Christian Hennig Assessing the quality of a clustering

17. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
1.1.5 Density-based methods
such as “DBSCAN” (Ester et al. 1996),
joins observations with all neighbouring points,
and neighbourhoods if they share enough points.
Christian Hennig Assessing the quality of a clustering

18. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
1
1
1111
1
1 1
1
1
1
111
11
1
1
1
11
1
1
1
1
1
1
11
1
1
1
1
1
2N
2
2 2
N
2
2
2
2
2
22
22
2
2
2
22
2 2
2
2
2
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
NN
NN
N
N
3
3
3
3
3
3
3N 33
3
N
N
4
4
4
N
4
4 4
N
N
4
4
55
5
5 5
55
5
5
55
5
5
5
55
55
5
5
5
55
5
5
5
5
55
N
5
5
55
5 5
5
5
5
5
5
5
555
5
55 5
5
55
5
5555
5
555
N
5
N
N
NN
6
6
6
6
666
666
6
N
6
6
6
6
6
66 6
6
6
6
6
6
66
6
6
6
6
6
6
6
6
6
6
6
6
6
6 6
6
6
6
6
6
6
7
7
7
7
N
7
7
77
7
7
7
7
7
7
−0.4 −0.2 0.0 0.2 0.4
−0.4−0.20.00.20.4
ctai$points[,1]
ctai$points[,2]
Christian Hennig Assessing the quality of a clustering

19. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Cluster analysis methods
1.1.6 Further issues in cluster analysis
Number of clusters
Cluster validation
Dissimilarity definition
Choice of method
Christian Hennig Assessing the quality of a clustering

20. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
2. Benchmarking and measurement of quality
Which clustering is better?
(Old faithful geyser data)
−2 −1 0 1 2
−2−101
mclust
waiting
duration
−2 −1 0 1 2
−2−101
pam
waiting
duration
Christian Hennig Assessing the quality of a clustering

21. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Which clustering is better?
−10 0 10 20 30
010203040
xy[,1]
xy[,2]
−10 0 10 20 30
010203040
xy[,1]
xy[,2]
Christian Hennig Assessing the quality of a clustering

22. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Benchmarking approaches:
Real datasets with known classes
Simulated datasets from mixture distributions
Real datasets without known classes
With known truth can compute misclassification rates.
Christian Hennig Assessing the quality of a clustering

23. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Disadvantages of benchmarking with known truth
In datasets with known classes
clustering is not of real scientific interest.
Deviate systematically from real clustering problems.
Christian Hennig Assessing the quality of a clustering

24. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Disadvantages of benchmarking with known truth
In datasets with known classes
clustering is not of real scientific interest.
Deviate systematically from real clustering problems.
The fact that we know certain true classes
doesn’t preclude other legitimate/”true” clusterings.
Christian Hennig Assessing the quality of a clustering

25. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Disadvantages of benchmarking with known truth
In datasets with known classes
clustering is not of real scientific interest.
Deviate systematically from real clustering problems.
The fact that we know certain true classes
doesn’t preclude other legitimate/”true” clusterings.
Classes in supervised classification problems
may not qualify as data analytic clusters.
Christian Hennig Assessing the quality of a clustering

26. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Disadvantages of benchmarking with known truth
In datasets with known classes
clustering is not of real scientific interest.
Deviate systematically from real clustering problems.
The fact that we know certain true classes
doesn’t preclude other legitimate/”true” clusterings.
Classes in supervised classification problems
may not qualify as data analytic clusters.
So there could be better truths than the known one.
Christian Hennig Assessing the quality of a clustering

27. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
How true are the true given classes?
(Hennig and Liao 2013, social stratification data)
Christian Hennig Assessing the quality of a clustering

28. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
7 standard occupation classes such as “manual workers”,
“managerials and professionals”, “not working”
Christian Hennig Assessing the quality of a clustering

29. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
These are not “data analytic clusters”.
Christian Hennig Assessing the quality of a clustering

30. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Mixture components aren’t always “data analytic clusters”
either.
55
3
54
3
54
5
5
4
4
5
4
3
5
4
5
5
4
3
534
4
4
4
4
4
4
5
54
5
4 5
4
4
4
5
5
5
4
54
4
5
5
4
5
4
5
5
5
4
4
4
4
5
4
5
5
55
5
4
4 4
4
4
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5
5
4
3
4
1
4
4
3
4
4
5
4
1
4 54
5
5
3
5
54
5
1
4
4
4
4
4
5
4
3
4
5
4
55
5
4
5
4
5
3
5
4 4
4
5
3
5
4 5
34
4
5
5
5
4
3
4
5
5
55
4 5
4
54
4
5
4
4
4
4
4
5
4
5
3
5
5
3
4
5 5
4 5
4 5
5
54
4
4
4 4
5
5
4
54
5
5
4
4
5
4
5
5
5 5
4
4
4
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5
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4
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2
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3
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5
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5
4
4
4
4
5
4
5
5
5
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5
4
5
54
4 4 5
4
5
5
4
5
5
5
5
4
4 5
3
5
5 54
5
4
4 4
4
35
5
5
4
5
4
4
5
3
4
5
5
4
4
4
5
4
4
4 5
5
4
54
5
44
5
4 5
3
4
4
3
3
4
4
55
4 5
4
4
4
5
5
4
5
5
555
4
5
4
4
5
5
3
5
4
4
5
5
4
4
5
3
4
55
4 54
4
4
45
3
4
5
3
5
5
4
4
3
4
2 5
4
54
4
4
4
4
4
4
2 5
4
4
4
5
5
4
4
5
5
5
4
5
5
5
55
4 5
5
5
4 54
5
4
4
554
4 551
4
5
4
5
2
33
4
4
45
54
4
5
1
5
44
4
4
4
4
54
4
3
4
4
4
4
5
3
5 554
4
44
5
4
4
5
5
4
5
4
4
5
4
4
5
5
5
2
3
5
34
5
4
5
3
4
4
5
5
1
4
5
4
4 5 4 54 54
5
4
4
4
4
55
4 5
5
54
4 5
4
5
5
5
2
4
3
4
4
4
5
5
5 5
5
4
4
5
4 4
3
5
0 20 40 60 80 100
−20−10010203040
xdata$x[,1]
xdata$x[,2]
Christian Hennig Assessing the quality of a clustering

31. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Using a known truth is useful and fair enough
but also want to evaluate clusterings
on data for which truth is not known.
Christian Hennig Assessing the quality of a clustering

32. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
There is a range of cluster validation indexes
measuring clustering quality, such as
Average silhouette width (ASW)
(Kaufman and Rouseeuw 1990)
sw(i, C) = b(i,C)−a(i,C)
max(a(i,C),b(i,C)),
a(i, C) =
1
|Cj| − 1
x∈Cj
d(xi, x), b(i, C) = min
xi ∈Cl
1
|Cl|
x∈Cl
d(xi, x).
Maximum average sw ⇒ good C.
Christian Hennig Assessing the quality of a clustering

33. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
“One size fits it all”-approach.
Christian Hennig Assessing the quality of a clustering

34. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
“One size fits it all”-approach.
Homogeneity will normally dominate here:
−10 0 10 20 30
010203040
xy[,1]
xy[,2]
−10 0 10 20 30
010203040
xy[,1]
xy[,2]
Christian Hennig Assessing the quality of a clustering

35. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
My general philosophy
There are various different aims of clustering.
Measure them separately to characterise
what a method does best,
instead of producing a single ranking.
Christian Hennig Assessing the quality of a clustering

36. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Typical clustering aims
Between-cluster separation
Christian Hennig Assessing the quality of a clustering

37. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Typical clustering aims
Between-cluster separation
Within-cluster homogeneity (low distances)
Christian Hennig Assessing the quality of a clustering

38. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Typical clustering aims
Between-cluster separation
Within-cluster homogeneity (low distances)
Within-cluster homogeneous distributional shape
Christian Hennig Assessing the quality of a clustering

39. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Typical clustering aims
Between-cluster separation
Within-cluster homogeneity (low distances)
Within-cluster homogeneous distributional shape
Good representation of data by centroids
Christian Hennig Assessing the quality of a clustering

40. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Typical clustering aims
Between-cluster separation
Within-cluster homogeneity (low distances)
Within-cluster homogeneous distributional shape
Good representation of data by centroids
Little loss of information
from original distance between objects.
Christian Hennig Assessing the quality of a clustering

41. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Typical clustering aims
Between-cluster separation
Within-cluster homogeneity (low distances)
Within-cluster homogeneous distributional shape
Good representation of data by centroids
Little loss of information
from original distance between objects.
Clusters are regions of high density
without within-cluster gaps
Christian Hennig Assessing the quality of a clustering

42. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Typical clustering aims
Between-cluster separation
Within-cluster homogeneity (low distances)
Within-cluster homogeneous distributional shape
Good representation of data by centroids
Little loss of information
from original distance between objects.
Clusters are regions of high density
without within-cluster gaps
Uniform cluster sizes
Christian Hennig Assessing the quality of a clustering

43. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
Typical clustering aims
Between-cluster separation
Within-cluster homogeneity (low distances)
Within-cluster homogeneous distributional shape
Good representation of data by centroids
Little loss of information
from original distance between objects.
Clusters are regions of high density
without within-cluster gaps
Uniform cluster sizes
Stability
Christian Hennig Assessing the quality of a clustering

44. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
These may be in conflict with each other.
−10 0 10 20 30
010203040
xy[,1]
xy[,2]
−10 0 10 20 30
010203040
xy[,1]
xy[,2]
Christian Hennig Assessing the quality of a clustering

45. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
E.g., pattern recognition in images
requires separation,
Christian Hennig Assessing the quality of a clustering

46. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
E.g., pattern recognition in images
requires separation,
clustering for information reduction requires
good representation by centroids,
Christian Hennig Assessing the quality of a clustering

47. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
E.g., pattern recognition in images
requires separation,
clustering for information reduction requires
good representation by centroids,
groups in social network analysis shouldn’t have
large within-cluster gaps,
Christian Hennig Assessing the quality of a clustering

48. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Which clustering is better?
Benchmarking approaches
Cluster validation indexes
My general philosophy
Typical clustering aims
E.g., pattern recognition in images
requires separation,
clustering for information reduction requires
good representation by centroids,
groups in social network analysis shouldn’t have
large within-cluster gaps,
underlying “true” classes (biological species)
may cause homogeneous distributional shapes.
Christian Hennig Assessing the quality of a clustering

49. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
3. Cluster quality statistics
Measuring between-cluster separation
∃ several ways measuring separation (as for other aims).
Straightforward: min distance between any two clusters,
or distance between centroids (e.g., k-means).
Christian Hennig Assessing the quality of a clustering

50. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
−2 −1 0 1 2
−2−101
waiting
duration
Christian Hennig Assessing the quality of a clustering

51. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
−2 −1 0 1 2
−2−101
waiting
duration
M
M
Christian Hennig Assessing the quality of a clustering

52. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
Measuring between-cluster separation
∃ several ways measuring separation (as for other aims).
Straightforward: min distance between any two clusters,
or distance between centroids (e.g., k-means).
These measure quite different concepts of separation.
(min distance relies on only two points;
centroid distance ignores what goes on at border.)
Christian Hennig Assessing the quality of a clustering

53. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
p-separation index:
More stable version of “min distance”:
Average distance to nearest point in different cluster for
p = 10% “border” points in any cluster.
−2 −1 0 1 2
−2−101
waiting
duration
X
X
X
X
X
X
X
XX
X
X XX
X
X
X
X
X
X
X
X
X X
X
X
X X
X
X
X
X
Christian Hennig Assessing the quality of a clustering

54. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
Measuring “density mountains vs. valleys”
Index that measures whether clusters correspond
to “density mountains”,
and whether “valleys” are between clusters.
Note: This is current research and may be revised.
Christian Hennig Assessing the quality of a clustering

55. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
Two aspects:
(a) Density goes down from mode;
no gaps and valleys within clusters.
(b) Cluster borders are valleys;
they don’t run through mountains.
Estimate density by weighted count of close points
(“kernel density”).
0.00.51.01.52.0
x
k(x)
10% quantile of within−cluster distances
Christian Hennig Assessing the quality of a clustering

56. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
−2 −1 0 1 2
−2−101
sinlink g= 2
waiting
duration
Christian Hennig Assessing the quality of a clustering

57. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
Start from cluster modes
−0.5 0.0 0.5 1.0 1.5 2.0
−1.8−1.6−1.4−1.2−1.0−0.8
sinlink g= 2 Step 1
waiting
duration
X
Christian Hennig Assessing the quality of a clustering

58. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
Connect closest point to cluster
−0.5 0.0 0.5 1.0 1.5 2.0
−1.8−1.6−1.4−1.2−1.0−0.8
sinlink g= 2 Step 3
waiting
duration
X
Christian Hennig Assessing the quality of a clustering

59. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
As long as density goes down, no penalty
−0.5 0.0 0.5 1.0 1.5 2.0
−1.8−1.6−1.4−1.2−1.0−0.8
sinlink g= 2 Step 6
waiting
duration
X
Christian Hennig Assessing the quality of a clustering

60. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
Penalty for density increase
−2 −1 0 1 2
−2−101
sinlink g= 2 Step 98
waiting
duration
X
Christian Hennig Assessing the quality of a clustering

61. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
−2 −1 0 1 2
−2−101
sinlink g= 2 Step 99
waiting
duration
X
Christian Hennig Assessing the quality of a clustering

62. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
−2 −1 0 1 2
−2−101
sinlink g= 2 Step 100
waiting
duration
X
Christian Hennig Assessing the quality of a clustering

63. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
−2 −1 0 1 2
−2−101
sinlink g= 2 Step 101
waiting
duration
X
Christian Hennig Assessing the quality of a clustering

64. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
−2 −1 0 1 2
−2−101
sinlink g= 2 Step 102
waiting
duration
X
Christian Hennig Assessing the quality of a clustering

65. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
−2 −1 0 1 2
−2−101
sinlink g= 2 Step 103
waiting
duration
X
Christian Hennig Assessing the quality of a clustering

66. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
−2 −1 0 1 2
−2−101
sinlink g= 2 Step 104
waiting
duration
X
Christian Hennig Assessing the quality of a clustering

67. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
−2 −1 0 1 2
−2−101
sinlink g= 2 Step 105
waiting
duration
X
Christian Hennig Assessing the quality of a clustering

68. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
−2 −1 0 1 2
−2−101
sinlink g= 2 Step 106
waiting
duration
X
Christian Hennig Assessing the quality of a clustering

69. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
−2 −1 0 1 2
−2−101
sinlink g= 2 Step 107
waiting
duration
X
Christian Hennig Assessing the quality of a clustering

70. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
−2 −1 0 1 2
−2−101
sinlink g= 2 Step 108
waiting
duration
X
Christian Hennig Assessing the quality of a clustering

71. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
−2 −1 0 1 2
−2−101
sinlink g= 2 Step 297
waiting
duration
X
Christian Hennig Assessing the quality of a clustering

72. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
Add penalty density∗density from other clusters
−2 −1 0 1 2
−2−101
specc g= 3
waiting
duration
P P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
P
PP
P
P
Christian Hennig Assessing the quality of a clustering

73. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
Other statistics
Within-cluster average distance
Christian Hennig Assessing the quality of a clustering

74. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
Other statistics
Within-cluster average distance
Within-cluster similarity measure
to normal/uniform
Christian Hennig Assessing the quality of a clustering

75. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
Other statistics
Within-cluster average distance
Within-cluster similarity measure
to normal/uniform
Within-cluster (squared) distance to centroid
Christian Hennig Assessing the quality of a clustering

76. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
Other statistics
Within-cluster average distance
Within-cluster similarity measure
to normal/uniform
Within-cluster (squared) distance to centroid
ρ(distance, cluster induced distance) (Hubert’s Γ)
Christian Hennig Assessing the quality of a clustering

77. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
Other statistics
Within-cluster average distance
Within-cluster similarity measure
to normal/uniform
Within-cluster (squared) distance to centroid
ρ(distance, cluster induced distance) (Hubert’s Γ)
Entropy of cluster sizes
Christian Hennig Assessing the quality of a clustering

78. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
Other statistics
Within-cluster average distance
Within-cluster similarity measure
to normal/uniform
Within-cluster (squared) distance to centroid
ρ(distance, cluster induced distance) (Hubert’s Γ)
Entropy of cluster sizes
Average largest within-cluster gap
Christian Hennig Assessing the quality of a clustering

79. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
Other statistics
Within-cluster average distance
Within-cluster similarity measure
to normal/uniform
Within-cluster (squared) distance to centroid
ρ(distance, cluster induced distance) (Hubert’s Γ)
Entropy of cluster sizes
Average largest within-cluster gap
Variation of clusterings on bootstrapped data
Christian Hennig Assessing the quality of a clustering

80. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Measuring between-cluster separation
Measuring “density mountains vs. valleys”
Other statistics
Other statistics
Within-cluster average distance
Within-cluster similarity measure
to normal/uniform
Within-cluster (squared) distance to centroid
ρ(distance, cluster induced distance) (Hubert’s Γ)
Entropy of cluster sizes
Average largest within-cluster gap
Variation of clusterings on bootstrapped data
Standardise all indexes to [0, 1] so that
“large is good”.
Christian Hennig Assessing the quality of a clustering

81. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
4. Examples
−10 0 10 20 30
010203040
xy[,1]
xy[,2]
−10 0 10 20 30
010203040
xy[,1]
xy[,2]
3-means mclust-3
ave within 0.811 0.643
sep index 0.163 0.306
density index 0.977 0.978
within gap 0.927 0.949
Christian Hennig Assessing the quality of a clustering

82. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
−2 −1 0 1 2
−2−101
mclust
waiting
duration
−2 −1 0 1 2
−2−101
pam
waitingduration
−2 −1 0 1 2
−2−101
spectral
waiting
duration
−2 −1 0 1 2
−2−101
ave.linkage
waiting
duration
−2 −1 0 1 2
−2−101
single linkage
waiting
duration
−2 −1 0 1 2
−2−101
pdfCluster (3)
waiting
duration
Christian Hennig Assessing the quality of a clustering

83. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
mclust pam spect ave.l sing.l pdf3
ave within 0.71 0.95 0.82 0.90 0.04 0.98
sep index 0.98 0.30 0.94 0.60 0.99 0.78
density 0.99 0.44 0.70 0.63 0.59 0.99
gap 0.14 0.46 0.46 0.46 0.99 0.48
gamma 0.81 0.91 0.92 0.96 0.06 0.98
normality 0.69 0.44 0.45 0.48 0.11 0.52
Note: These values are quantile-standardised,
implementation of this in fpc is still to come.
Christian Hennig Assessing the quality of a clustering

84. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
0.00.20.40.60.81.0
Number of clusters
dindex
2 3 4 5
kmeans
kmeans
kmeans
kmeans
avelink
avelink
avelink avelink
sinlink
sinlink
sinlink
sinlink
comlink
comlink
comlink
comlink
mclust
mclust
mclust mclust
pam
pam
pam
pam
specc
specc
specc
specc
pdfclus
Quantile−calibrated density index
Christian Hennig Assessing the quality of a clustering

85. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
6. Discussion
Clustering quality is multidimensional.
Christian Hennig Assessing the quality of a clustering

86. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
6. Discussion
Clustering quality is multidimensional.
Provide multidimensional evaluation,
characterising a method’s behaviour.
Christian Hennig Assessing the quality of a clustering

87. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
6. Discussion
Clustering quality is multidimensional.
Provide multidimensional evaluation,
characterising a method’s behaviour.
Can aggregate criteria by weighted mean
given well justified weights.
Christian Hennig Assessing the quality of a clustering

88. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
6. Discussion
Clustering quality is multidimensional.
Provide multidimensional evaluation,
characterising a method’s behaviour.
Can aggregate criteria by weighted mean
given well justified weights.
Benchmarking without known truth
and comparison of clusterings in practice.
Christian Hennig Assessing the quality of a clustering

89. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
6. Discussion
Clustering quality is multidimensional.
Provide multidimensional evaluation,
characterising a method’s behaviour.
Can aggregate criteria by weighted mean
given well justified weights.
Benchmarking without known truth
and comparison of clusterings in practice.
Required: standardisation to compare
different indexes and numbers of clusters.
Christian Hennig Assessing the quality of a clustering

90. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Much of this is implemented in R-package fpc,
more will be.
Christian Hennig Assessing the quality of a clustering

91. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
Much of this is implemented in R-package fpc,
more will be.
Soon to come:
IFCS Cluster Benchmarking Repository
(Iven Van Mechelen, Nema Dean, Isabelle Guyon,
Anne-Laure Boulesteix, Doug Steinley, Friedrich Leisch,
Christian Hennig, Rainer Dangl)
This work is supported by EPSRC Grant EP/K033972/1.
Christian Hennig Assessing the quality of a clustering

92. A short introduction to cluster analysis
Benchmarking and measurement of quality
Cluster quality statistics
Examples
Discussion
A bit of marketing:
Christian Hennig Assessing the quality of a clustering