Slides for a workshop on clustering methods with R. R codes are available at http://rpubs.com/mrkm_a/ClusteringWithR

1. Clustering Methods with R
Akira Murakami
Department of English Language and Applied Linguistics
University of Birmingham
a.murakami@bham.ac.uk

2. Cluster Analysis
• Cluster analysis finds groups in data.
• Objects in the same cluster are similar to each other.
• Objects in different clusters are dissimilar.
• A variety of algorithms have been proposed.
• Saying “I ran a cluster analysis” does not mean much.
• Used in data mining or as a statistical analysis.
• Unsupervised machine learning technique.
2

3. Cluster Analysis in SLA
• In SLA, clustering has been applied to identify the typology of
learners’
• motivational profiles (Csizér & Dörnyei, 2005),
• ability/aptitude profiles (Rysiewicz, 2008),
• developmental profiles based on international posture, L2
willingness to communicate, and frequency of communication
in L2 (Yashima & Zenuk-Nishide, 2008),
• cognitive and achievement profiles based on L1 achievement,
intelligence, L2 aptitude, and L2 proficiency (Sparks, Patton,
& Ganschow, 2012).
3

4. Similarity Measure
• Cluster analysis groups the observations that are
“similar”. But how do we measure similarity?
• Let’s suppose that we are interested in clustering L1
groups according to their accuracy of different
linguistic features (i.e., accuracy profile of L1
groups).
• As the measure of accuracy, we use an index that
takes the value between 0 and 1, such as the TLU
score.
4

5. ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mathematical Distance
5

6. ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
L1 Korean
Mathematical Distance
6

7. ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
L1 Korean
L1 German
Mathematical Distance
7

8. ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
L1 Korean
L1 German
Distance = 0.2
Mathematical Distance
8

9. ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨ ￨
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
L1 Korean
L1 German
Distance = 0.2
L1 Japanese
Distance = 0.1
Mathematical Distance
9

10. (Dis)Similarity Matrix
10
L1 Korean L1 German L1 Japanese
L1 Korean 0.0
L1 German 0.2 0.0
L1 Japanese 0.1 0.3 0.0

11. Distance Measures
• Things are simple in 1D, but get more complicated in 2D or above.
• Different measures of distance
• Euclidean distance
• Manhattan distance
• Maximum distance
• Mahalanobis distance
• Hamming distance
• etc
11

12. Distance Measures
• Things are simple in 1D, but get more complicated in 2D or above.
• Different measures of distance
• Euclidean distance
• Manhattan distance
• Maximum distance
• Mahalanobis distance
• Hamming distance
• etc
12

13. Article Accuracy
Pasttense−edAccuracy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Euclidean Distance
13

14. Article Accuracy
Pasttense−edAccuracy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
L1 German
(0.8, 0.6)
Euclidean Distance
14

15. Article Accuracy
Pasttense−edAccuracy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
L1 German
(0.8, 0.6)
L1 Korean
(0.4, 0.8)
Euclidean Distance
15

16. Article Accuracy
Pasttense−edAccuracy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
L1 German
(0.8, 0.6)
L1 Korean
(0.4, 0.8)
(0.4−0.8)2
+(0.8−0.6)2
Euclidean Distance
16

17. Article Accuracy
Pasttense−edAccuracy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
L1 German
(0.8, 0.6)
L1 Korean
(0.4, 0.8)
0.45
Euclidean Distance
17

18. Article Accuracy
Pasttense−edAccuracy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
L1 German
(0.8, 0.6)
L1 Korean
(0.4, 0.8)
0.45
L1 Japanese (0.6, 0.5)
Euclidean Distance
18

19. Article Accuracy
Pasttense−edAccuracy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
L1 German
(0.8, 0.6)
L1 Korean
(0.4, 0.8)
0.45
L1 Japanese (0.6, 0.5)
0.36
0.22
Euclidean Distance
19

20. (Dis)Similarity Matrix
20
L1 Korean L1 German L1 Japanese
L1 Korean 0.00
L1 German 0.45 0.00
L1 Japanese 0.36 0.22 0.00

21. 0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
Plural−sAccuracy
L1 German (0.3, 0.6, 0.9)
L1 Korean (0.6, 0.9, 0.6)
L1 Japanese (0.9, 0.4, 0.5)
Euclidean Distance (3D)
21

22. 0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
Plural−sAccuracy
L1 German (0.3, 0.6, 0.9)
L1 Korean (0.6, 0.9, 0.6)
L1 Japanese (0.9, 0.4, 0.5)
0.75
0.52
0.59
Euclidean Distance (3D)
22

23. (Dis)Similarity Matrix
23
L1 Korean L1 German L1 Japanese
L1 Korean 0.00
L1 German 0.52 0.00
L1 Japanese 0.59 0.75 0.00

24. Distance Measures
• Things are simple in 1D, but get more complicated in 2D or above.
• Different measures of distance
• Euclidean distance
• Manhattan distance
• Maximum distance
• Mahalanobis distance
• Hamming distance
• etc
24

25. Article Accuracy
Pasttense−edAccuracy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
L1 German
(0.8, 0.6)
L1 Korean
(0.4, 0.8)
Manhattan Distance
25

26. Article Accuracy
Pasttense−edAccuracy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
L1 German
(0.8, 0.6)
L1 Korean
(0.4, 0.8)
Manhattan Distance
26

27. Article Accuracy
Pasttense−edAccuracy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
L1 German
(0.8, 0.6)
L1 Korean
(0.4, 0.8)
0.4
0.2
Manhattan Distance
27

28. Article Accuracy
Pasttense−edAccuracy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
L1 German
(0.8, 0.6)
L1 Korean
(0.4, 0.8)
0.4
0.2
Manhattan Distance
28
→ Distance = 0.4 + 0.2 = 0.6

29. Article Accuracy
Pasttense−edAccuracy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
(0.1, 0.4)
(0.9, 0.3)
(0.6, 0.9)
Manhattan Distance
29

30. Article Accuracy
Pasttense−edAccuracy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
(0.1, 0.4)
(0.9, 0.3)
(0.6, 0.9)
0.5
0.5
0.71
0.1
0.8
0.81
Manhattan Distance
30

31. Article Accuracy
Pasttense−edAccuracy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
(0.1, 0.4)
(0.9, 0.3)
(0.6, 0.9)
0.5
0.5
0.71
0.1
0.8
0.81
Manhattan Distance
31
Euclidean: 0.71
Manhattan: 0.5 + 0.5 = 1.00
Euclidean: 0.81
Manhattan: 0.1 + 0.8 = 0.90

32. dist()
• In R, dist function is used to obtain dissimilarity
matrices.
• Practicals
32

33. Clustering Methods
• Now that we know the concept of similarity, we
move on to the clustering of objects based on the
similarity.
• A number of methods have been proposed for
clustering. We will look at the following two:
• agglomerative hierarchical cluster analysis
• k-means
33

34. Clustering Methods
• Now that we know the concept of similarity, we
move on to the clustering of objects based on the
similarity.
• A number of methods have been proposed for
clustering. We will look at the following two:
• agglomerative hierarchical cluster analysis
• k-means
34

35. Agglomerative Hierarchical Cluster Analysis
• In agglomerative hierarchical clustering,
observations are clustered in a bottom-up manner.
1. Each observation forms an independent cluster
at the beginning.
2. The two clusters that are most similar are
clustered together.
3. 2 is repeated until all the observations are
clustered in a single cluster.
35

36. Linkage Criteria
• How do we calculate the similarity between clusters
that each includes multiple observations?
• Ward’s criterion (Ward’s method)
• complete-linkage
• single-linkage
• etc.
36

37. Linkage Criteria
• How do we calculate the similarity between clusters
that each includes multiple observations?
• Ward’s criterion (Ward’s method)
• complete-linkage
• single-linkage
• etc.
37

38. Ward’s Method
• Ward’s method leads to the smallest within-cluster
variance.
• At each iteration, two clusters are merged so that it
yields the smallest increase of the sum of squared
errors.
• Sum of Squared Errors (SSE): the sum of the
squared difference between the mean of the cluster
and individual data points.
38

39. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
Ward’s Method
39

40. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
Ward’s Method
40

41. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
x
mean (0.3, 0.6)
Ward’s Method
41

42. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
x
mean (0.3, 0.6)
0.22
0.22
Ward’s Method
42

43. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
x
mean (0.3, 0.6)
0.05
0.05
Ward’s Method
43

44. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
x
mean (0.3, 0.6)
0.05
0.05
Ward’s Method
44→ 0.05 + 0.05 = 0.10

45. • This procedure is repeated for all of the pairs.
45

46. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
x
Ward’s Method
46

47. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
x
x
Ward’s Method
47

48. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
x
x
(0.3, 0.3)
(0.6, 0.8)
Ward’s Method
48

49. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
x
x
(0.3, 0.3)
(0.6, 0.8)
( 0.1
2
+0.1
2
)
2
= 0.02
0.2
2
= 0.04
Ward’s Method
49

50. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
x
x
(0.3, 0.3)
(0.6, 0.8)
( 0.1
2
+0.1
2
)
2
= 0.02
0.2
2
= 0.04
Ward’s Method
SSE = 0.02 + 0.02 + 0.04 + 0.04 = 0.12

51. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
x (0.45, 0.55)
Ward’s Method

52. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
x (0.45, 0.55)
0.12
0.08
0.06
0.18
Ward’s Method
SSE = 0.12 + 0.08 + 0.06 + 0.18 = 0.46

53. ΔSSE
• SSE before the merger: 0.12
• SSE after the merger: 0.46
• Difference (ΔSSE): 0.46 - 0.12 = 0.34
53

54. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
x
x
Ward’s Method
54

55. Dendrogram
55
1
2
5
3
4
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Cluster Dendrogram
hclust (*, "ward.D2")
dd.dist
Height

56. 56
Practicals

57. Linkage Criteria
• How do we know the similarity between clusters
that each includes multiple observations?
• Ward’s criterion (Ward’s method)
• complete-linkage
• single-linkage
• etc.
57

58. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
Complete Linkage
58

59. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
Complete Linkage
59

60. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
0.7
Complete Linkage
60

61. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
0.4
Single Linkage
61

62. Potential Pitfall of Hierarchical Clustering
• It assumes hierarchical structure in the clustering.
• Let us say that our data included two L1 groups over three
proficiency levels.
• If we group the data into two clusters, the best split may be
between the two L1 groups.
• If we group them into three clusters, the best groups may be by
proficiency groups.
• In this case, three-cluster solution is not nested within two-
cluster solution, and hierarchical clustering may fail to identify
the two clusters.
62

63. 63
k-means Clustering

64. k-means Clustering
• K-means clustering does not assume a hierarchical
structure of clusters.
• i.e., no parent/child clusters
• Analysts need to specify the number of clusters.
64

65. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1 (0.4, 0.2)
2 (0.2, 0.4)
3 (0.4, 0.8) 4 (0.8, 0.8)
5 (0.9, 0.4)
k-means Clustering
65

66. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
x
x
1
2
3 4
5
(Centroid 1)
(Centroid 2)
k-means Clustering
66

67. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
x
x
1
2
3 4
5
(Centroid 1)
(Centroid 2)
0.28
0.60
0.45 0.72
0.72
0.64
0.70
k-means Clustering
67

68. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1
2
3 4
5
x
x
Centroid 1
Centroid 2
k-means Clustering
68

69. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1
2
3 4
5
x
x
Centroid 1
Centroid 2
0.40
0.41
0.50
0.22
0.45
0.22
0.28
0.42
0.21
0.63
k-means Clustering
69

70. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Article Accuracy
Pasttense−edAccuracy
1
2
3 4
5
x
x
Centroid 1
Centroid 2
k-means Clustering
70

71. k-Means Clustering
• The optimal number of clusters depends on the intended use.
• There is no “correct” or “wrong” choice in the number of
clusters.
• NP hard
• The algorithm only approximates solutions.
• Randomness is involved in the solution. You get different
solutions every time you run it.
• It assumes convex clusters.
71

72. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
y1 Concave
72

73. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
y1
x
x
xx
x
x
x
x
x
xx x
x
x
x
x x
x
x xx
x
xx
x
x
x
xx
x
x
xx
x
x
x
xx
x
x
x
x
x
x
x
x
x
x
xx
x
x
x
x
x
x
x
x
x
x
x
x
xx
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
xx
x
xx
x
x
x
x
x
x
x
x
x
x
x 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
xx
x
xx
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x x
x
x
x
x
x
x
x
xx
x
x
x
x
xx
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Concave
73

74. 74
Practicals

75. Within-Learner Centering
• The mean accuracy value of each learner was subtracted from all the
data points of the learner.
• For example, let's suppose the mean sentence length (MSL) of
Learner A over 10 writings was
• {4.0, 4.2, 4.4, 4.6, 4.8, 5.0, 5.2, 5.4, 5.6, 5.8}
and that of Learner B was
• {8.0, 8.2, 8.4, 8.6, 8.8, 9.0, 9.2, 9.4, 9.6, 9.8}
• The difference in MSL is identical in the two learners (+0.2 per writing).
• But the absolute MSL is widely different.
75

76. Within-Learner Centering
• The mean value of Learner A (4.9) is subtracted from all the data
points of Learner A:
• → {-0.90, -0.70, -0.50, -0.30, -0.10, 0.10, 0.30, 0.50, 0.70,
0.90}.
• Similarly, the mean value of Learner B (8.90) is subtracted from
all the data points of Learner B:
• → {-0.90, -0.70, -0.50, -0.30, -0.10, 0.10, 0.30, 0.50, 0.70,
0.90}.
• It is guaranteed that these two learners are clustered into the
same group as they have exactly the same set of values.
76

77. 77
Cluster Validation

78. Cluster Validation/Evaluation
• We got clusters and explored them, but how do we
know how good the clusters are, or whether they
indeed capture signal and not just noise?
• Are the clusters ‘real’?
• Is it the difference in the true learning curve that
the earlier clustering captured or is it just the
random noise?
78

79. Two Types of Validation
• External Validation
• Internal Validation
79

80. External Validation
• If there is a a systematic pattern between clusters
and some external criteria, such as the proficiency
or L1 of learners, then what the cluster analysis
captured is unlikely to be just noise.
80

81. Internal Validation
• Measures of goodness of clusters
• silhouette width
• Davies–Bouldin index
• Dunn index
• etc.
81

82. Internal Validation
• Measures of goodness of clusters
• silhouette width
• Davies–Bouldin index
• Dunn index
• etc.
82

83. Silhouette Width
• Intuitively, the silhouette value is large if within-
cluster dissimilarity is small (i.e., learners within
each cluster have similar developmental
trajectories) and between-cluster dissimilarity is
large (i.e., learners in different clusters have
different learning curves).
• The silhouette is given to each data point (i.e.,
learner), and all the silhouette values are averaged
to measure the cluster distinctiveness of a cluster
analysis.
83

84. • Let’s say there are three clusters, A through C.
• Let’s further say that i is a member of Cluster A.
• Let a(i) be the average distance between that learner and all the
other learners that belong to the same cluster.
• We also calculate the average distances
1. between the learner and all the other learners that belong to
Cluster B
2. between the learner and all the other learners that belong to
Cluster C
• Let b(i) be the smaller of the two above (1-2).
• s(i) = (b(i) - a(i)) / max(a(i), b(i))
84
Silhouette Width

85. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
y1
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x x
x
Silhouette Width
85

86. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
y1
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x x
x
Silhouette Width
86

87. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
y1
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x x
x
Silhouette Width
87
→ Average = 0.022 (the value of a(i))

88. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
y1
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x x
x
Silhouette Width
88
→ Average = 0.191

89. 0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
y1
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x x
x
Silhouette Width
89
→ Average = 0.240

90. Silhouette Width
• a(i) = 0.022
• b(i) = 0.191 (the smaller of the other two)
• s(i) = (b(i) - a(i)) / max(a(i), b(i))
• s(i) = (0.191 - 0.022) / 0.191 = 0.882
• This is repeated for all the data points.
• Goodness of clustering: mean silhouette width across
all the data points.
90

91. Bootstrapping
• Now that we have a measure of how good our
clustering is, the next question is whether it is good
enough to be considered non-random.
• We can address this question through the technique
called bootstrapping.
• The idea is similar to the usual hypothesis-testing
procedure.
• We obtain the null distribution of the silhouette value
and see where our value falls.
91

92. • More specific procedure is as follows:
1. For each learner, we sample 30 writings (with replacement).
2. We run a k-means cluster analysis with the data obtained in
1 and calculate the mean silhouette value.
3. 1 and 2 are repeated e.g., 10,000 times, resulting in 10,000
mean silhouette values which we consider as the null
distribution.
4. We examine whether the 95% range of 3 includes our
observed mean silhouette value.
92
Bootstrapping

93. • The idea here is that we practically randomize the order
of the writings within individual learners and follow the
same procedure as our main analysis.
• Since the order of writings is random, there should not
be any systematic pattern of development observed.
• The clusters obtained in this manner thus captures noise
alone. We calculate the mean silhouette value on the
noise-only, random clusters, and obtain its distribution by
repeating the whole procedure a large number of times.
93
Bootstrapping

94. 94
langtest.jp

95. langtest.jp
95
http://langtest.jp

96. Paper Introducing langtest.jp
96
http://applij.oxfordjournals.org/content/early/2015/06/24/applin.amv025.abstract

97. langtest.jp
97

98. 98
Demo