Christian Hennig- Assessing the quality of a clustering
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
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
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
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
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
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11
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1
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1
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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
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5
5
5
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5
5
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N
5
5
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5 5
5
5
5
5
5
5
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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
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6
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7
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77
7
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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
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