The document describes research on defining set-valued prototypes through consensus analysis. It begins with definitions of prototypes and an overview of consensus analysis and measuring consensus between partitions. It then discusses partitioning methods like k-means clustering and fuzzy c-means, and how they can be used to generate multiple partitions. The document outlines how simulated and real data examples will be used to test and apply the consensus analysis approach to derive set-valued prototypes.

1. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Set-valued prototypes
through Consensus Analysis
M. Fordellone1 F. Palumbo2
1Department of Statistical Sciences
University of Padua (Italy)
email: fordellone@stat.unipd.it
2Department of Political Sciences
University of Naples (Italy)
email: fpalumbo@unina.it
IFCS Conference
July 6th 2015, Bologna (Italy)
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

2. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Outline
1 Prototypes definition
What is a prototype?
2 Consensus Analysis
Consensus clustering
Consensus measurement
3 Partitioning methods
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
4 Simulated data examples
Eight experimental contexts
5 Application on real data
I.P.I.P. test
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

3. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
What is a prototype?
Outline
1 Prototypes definition
What is a prototype?
2 Consensus Analysis
Consensus clustering
Consensus measurement
3 Partitioning methods
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
4 Simulated data examples
Eight experimental contexts
5 Application on real data
I.P.I.P. test
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

4. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
What is a prototype?
What is a prototype?
According to Rosch (1975, 1999), prototypes are the elements that
better than others represent a category.
Smith and Medin (1981) refer to the concept of category as the
highest order of genera that cannot be defined by a mere listing of
properties shared by all elements.
A prototype is not necessarily a real element of the category, it
can be observed or unobserved (abstract) entity (Medin, D. L. and
Schaffer, M. M., 1978).
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

5. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Consensus clustering
Consensus measurement
Outline
1 Prototypes definition
What is a prototype?
2 Consensus Analysis
Consensus clustering
Consensus measurement
3 Partitioning methods
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
4 Simulated data examples
Eight experimental contexts
5 Application on real data
I.P.I.P. test
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

6. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Consensus clustering
Consensus measurement
Consensus concept
Finding and measuring the agreement between two or more parti-
tions of the same data set is of substantial interest in cluster analysis.
This particular case of consensus analysis is also known as consensus
clustering.
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

7. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Consensus clustering
Consensus measurement
Comparing partitions
Let X be a N×J data matrix, and T and V two partitions of X, then
nrc (r = 1, . . . , R; c = 1, . . . , C) represents the number of objects
assigned to the classes tr and vc, with respect to the two partitioning
criteria. Consensus between the partitions T and V is evaluated
starting from the entries of the cross-classifying contingency table.
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

8. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Consensus clustering
Consensus measurement
Comparing partitions
Let X be a N×J data matrix, and T and V two partitions of X, then
nrc (r = 1, . . . , R; c = 1, . . . , C) represents the number of objects
assigned to the classes tr and vc, with respect to the two partitioning
criteria. Consensus between the partitions T and V is evaluated
starting from the entries of the cross-classifying contingency table.
Table : Contingency table
Partition V
v1 v2 · · · vC
Partition T
t1 n11 n12 · · · n1C n1·
t2 n21 n22 · · · n2C n2·
...
...
...
...
...
...
tR nR1 nR2 · · · nRC nR·
n·1 n·2 · · · n·C n
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

9. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Consensus clustering
Consensus measurement
Outline
1 Prototypes definition
What is a prototype?
2 Consensus Analysis
Consensus clustering
Consensus measurement
3 Partitioning methods
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
4 Simulated data examples
Eight experimental contexts
5 Application on real data
I.P.I.P. test
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

10. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Consensus clustering
Consensus measurement
Measure of Consensus
Number of ways that n units can pair:
S = n
2 = n(n−1)
2
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

11. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Consensus clustering
Consensus measurement
Measure of Consensus
Number of ways that n units can pair:
S = n
2 = n(n−1)
2
Total number of Agreements:
A = n
2 + R
r=1
C
c=1 n2
rc − 1
2
R
r=1 n2
r· + C
c=1 n2
·c
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

12. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Consensus clustering
Consensus measurement
Measure of Consensus
Number of ways that n units can pair:
S = n
2 = n(n−1)
2
Total number of Agreements:
A = n
2 + R
r=1
C
c=1 n2
rc − 1
2
R
r=1 n2
r· + C
c=1 n2
·c
Total number of Disagreements:
D = 1
2
R
r=1 n2
r· + C
c=1 n2
·c − R
r=1
C
c=1 n2
rc
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

13. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Consensus clustering
Consensus measurement
Measure of Consensus
Number of ways that n units can pair:
S = n
2 = n(n−1)
2
Total number of Agreements:
A = n
2 + R
r=1
C
c=1 n2
rc − 1
2
R
r=1 n2
r· + C
c=1 n2
·c
Total number of Disagreements:
D = 1
2
R
r=1 n2
r· + C
c=1 n2
·c − R
r=1
C
c=1 n2
rc
Table : Measures of Consensus
Authors Measure Range
Rand (1971) A/S ∈ [0, 1]
Arabie et al. (1973) D/S ∈ [0, 1]
Hubert (1977) (A − D)/S ∈ [0, 1]
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

14. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
Outline
1 Prototypes definition
What is a prototype?
2 Consensus Analysis
Consensus clustering
Consensus measurement
3 Partitioning methods
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
4 Simulated data examples
Eight experimental contexts
5 Application on real data
I.P.I.P. test
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

15. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
k-Means method
K-Means method is developed by Queen (1967). He suggests the
name k-Means for describing an algorithm that assigns each unit
to the group having the nearest centroid (mean). The iterative
procedure consists in four principal steps:
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

16. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
k-Means method
K-Means method is developed by Queen (1967). He suggests the
name k-Means for describing an algorithm that assigns each unit
to the group having the nearest centroid (mean). The iterative
procedure consists in four principal steps:
1 Randomly select K group centers;
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

17. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
k-Means method
K-Means method is developed by Queen (1967). He suggests the
name k-Means for describing an algorithm that assigns each unit
to the group having the nearest centroid (mean). The iterative
procedure consists in four principal steps:
1 Randomly select K group centers;
2 Calculate the distance between each data point and group
centers;
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

18. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
k-Means method
K-Means method is developed by Queen (1967). He suggests the
name k-Means for describing an algorithm that assigns each unit
to the group having the nearest centroid (mean). The iterative
procedure consists in four principal steps:
1 Randomly select K group centers;
2 Calculate the distance between each data point and group
centers;
3 Assign the data point to the group whose distance from the
group center is minimum among all the group centers;
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

19. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
k-Means method
K-Means method is developed by Queen (1967). He suggests the
name k-Means for describing an algorithm that assigns each unit
to the group having the nearest centroid (mean). The iterative
procedure consists in four principal steps:
1 Randomly select K group centers;
2 Calculate the distance between each data point and group
centers;
3 Assign the data point to the group whose distance from the
group center is minimum among all the group centers;
4 Recalculate the new group centers.
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

20. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
k-Means method
K-Means method is developed by Queen (1967). He suggests the
name k-Means for describing an algorithm that assigns each unit
to the group having the nearest centroid (mean). The iterative
procedure consists in four principal steps:
1 Randomly select K group centers;
2 Calculate the distance between each data point and group
centers;
3 Assign the data point to the group whose distance from the
group center is minimum among all the group centers;
4 Recalculate the new group centers.
The procedure repeats from step 2 until no more assignments take
place.
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

21. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
k-Means method
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

22. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
k-Means method
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

23. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
k-Means method
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

24. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
Outline
1 Prototypes definition
What is a prototype?
2 Consensus Analysis
Consensus clustering
Consensus measurement
3 Partitioning methods
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
4 Simulated data examples
Eight experimental contexts
5 Application on real data
I.P.I.P. test
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

25. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
Fuzzy clustering
In fuzzy clustering data elements can belong to more than one group,
in according to a measure of association given by a set of member-
ship levels.
The memberships, ∈ [0, 1], indicate the strength of the association
between each data element and each group.
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

26. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
Fuzzy clustering
In fuzzy clustering data elements can belong to more than one group,
in according to a measure of association given by a set of member-
ship levels.
The memberships, ∈ [0, 1], indicate the strength of the association
between each data element and each group.
In our case the units with the max membership degree can be uni-
vocally assigned to the corresponding group.
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

27. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
Outline
1 Prototypes definition
What is a prototype?
2 Consensus Analysis
Consensus clustering
Consensus measurement
3 Partitioning methods
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
4 Simulated data examples
Eight experimental contexts
5 Application on real data
I.P.I.P. test
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

28. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
FCM and AA
Fuzzy c-means (Bezdek et al., 1984) and Archetypal Analysis (Cutler
and Breiman, 1994) can be seen as a fuzzy approach of the k-Means,
under different constraints.
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

29. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
FCM and AA
Fuzzy c-means (Bezdek et al., 1984) and Archetypal Analysis (Cutler
and Breiman, 1994) can be seen as a fuzzy approach of the k-Means,
under different constraints.
Fuzzy c-Means minimizes the sum of distances between each point
and a set of K centers; Archetypal Analysis minimizes the sum of
distances between each point and a set of K archetypes as defined
by a convex combination of extreme points.
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

30. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

31. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
FCM and AA
Fuzzy c-Means
W =
n
i=1
K
k=1
γ2
ik xi − ck
2
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

32. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
FCM and AA
Fuzzy c-Means
W =
n
i=1
K
k=1
γ2
ik xi − ck
2
γik is the membership level of
the i-th unit and of the k-th
group
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

33. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
FCM and AA
Fuzzy c-Means
W =
n
i=1
K
k=1
γ2
ik xi − ck
2
γik is the membership level of
the i-th unit and of the k-th
group
ck is the center of the k-th
group
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

34. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
FCM and AA
Fuzzy c-Means
W =
n
i=1
K
k=1
γ2
ik xi − ck
2
γik is the membership level of
the i-th unit and of the k-th
group
ck is the center of the k-th
group
Constraints:
K
k=1 γik = 1;
γik ≥ 0.
∀k ∈ 1, 2, . . . , K
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

35. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
FCM and AA
Fuzzy c-Means
W =
n
i=1
K
k=1
γ2
ik xi − ck
2
γik is the membership level of
the i-th unit and of the k-th
group
ck is the center of the k-th
group
Constraints:
K
k=1 γik = 1;
γik ≥ 0.
∀k ∈ 1, 2, . . . , K
Archetypal Analysis
J =
n
i=1
K
k=1
xi − δikak
2
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

36. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
FCM and AA
Fuzzy c-Means
W =
n
i=1
K
k=1
γ2
ik xi − ck
2
γik is the membership level of
the i-th unit and of the k-th
group
ck is the center of the k-th
group
Constraints:
K
k=1 γik = 1;
γik ≥ 0.
∀k ∈ 1, 2, . . . , K
Archetypal Analysis
J =
n
i=1
K
k=1
xi − δikak
2
δik is the membership level of
the i-th unit and of the k-th
group
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

37. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
FCM and AA
Fuzzy c-Means
W =
n
i=1
K
k=1
γ2
ik xi − ck
2
γik is the membership level of
the i-th unit and of the k-th
group
ck is the center of the k-th
group
Constraints:
K
k=1 γik = 1;
γik ≥ 0.
∀k ∈ 1, 2, . . . , K
Archetypal Analysis
J =
n
i=1
K
k=1
xi − δikak
2
δik is the membership level of
the i-th unit and of the k-th
group
ak = n
i=1 xi βik is the
archetype of the k-th group
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

38. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
FCM and AA
Fuzzy c-Means
W =
n
i=1
K
k=1
γ2
ik xi − ck
2
γik is the membership level of
the i-th unit and of the k-th
group
ck is the center of the k-th
group
Constraints:
K
k=1 γik = 1;
γik ≥ 0.
∀k ∈ 1, 2, . . . , K
Archetypal Analysis
J =
n
i=1
K
k=1
xi − δikak
2
δik is the membership level of
the i-th unit and of the k-th
group
ak = n
i=1 xi βik is the
archetype of the k-th group
Constraints:
K
k=1 δik = 1; δik ≥ 0;
K
k=1 βik = 1; βik ≥ 0.
∀k ∈ 1, 2, . . . , K
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

39. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Outline
1 Prototypes definition
What is a prototype?
2 Consensus Analysis
Consensus clustering
Consensus measurement
3 Partitioning methods
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
4 Simulated data examples
Eight experimental contexts
5 Application on real data
I.P.I.P. test
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

40. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data
Three groups of units in different experimental contexts have been
generated by a multivariate Gaussian distribution with eight dimen-
sions (four variables are white noise).
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

41. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data
Three groups of units in different experimental contexts have been
generated by a multivariate Gaussian distribution with eight dimen-
sions (four variables are white noise).
Table : Experimental contexts
Size Correlation Kurtosis
Case 1 900 0.2 − 0.4 β = 3
Case 2 300 0.2 − 0.4 β = 3
Case 3 900 0.2 − 0.4 β < 3
Case 4 300 0.2 − 0.4 β < 3
Case 5 900 0.6 − 0.8 β = 3
Case 6 300 0.6 − 0.8 β = 3
Case 7 900 0.6 − 0.8 β < 3
Case 8 300 0.6 − 0.8 β < 3
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

42. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 1
K-means groups
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

43. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 2
K-means groups
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

44. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 3
K-means groups
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

45. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 4
K-means groups
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

46. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 5
K-means groups
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

47. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 6
K-means groups
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

48. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 7
K-means groups
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

49. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 8
K-means groups
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

50. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 1
Memberships FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

51. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 2
Memberships FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

52. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 3
Memberships FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

53. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 4
Memberships FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

54. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 5
Memberships FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

55. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 6
Memberships FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

56. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 7
Memberships FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

57. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 8
Memberships FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

58. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 1
Consensus Analysis between FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

59. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 2
Consensus Analysis between FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

60. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 3
Consensus Analysis between FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

61. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 4
Consensus Analysis between FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

62. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 5
Consensus Analysis between FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

63. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 6
Consensus Analysis between FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

64. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 7
Consensus Analysis between FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

65. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 8
Consensus Analysis between FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

66. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 1
Consensus groups FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

67. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 2
Consensus groups FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

68. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 3
Consensus groups FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

69. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 4
Consensus groups FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

70. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 5
Consensus groups FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

71. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 6
Consensus groups FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

72. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 7
Consensus groups FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

73. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Case 8
Consensus groups FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

74. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Eight experimental contexts
Simulated data: Summary
Table : Results of Consensus Analysis and definition of the prototypes
Experimental Conditions Prototyping Results Consensus Measuring
N Corr. Kurt. K Size Rand Arabie Hubert
900 0.2 − 0.4 β = 3 3 900 (100.0%) 1.000 0.000 1.000
300 0.2 − 0.4 β = 3 3 300 (100.0%) 1.000 0.000 1.000
900 0.2 − 0.4 β < 3 3 625 (69.4%) 0.725 0.275 0.449
300 0.2 − 0.4 β < 3 3 185 (61.7%) 0.683 0.317 0.365
900 0.6 − 0.8 β = 3 3 599 (66.6%) 0.753 0.247 0.506
300 0.6 − 0.8 β = 3 3 202 (67.3%) 0.758 0.242 0.517
900 0.6 − 0.8 β < 3 3 533 (59.2%) 0.698 0.302 0.397
300 0.6 − 0.8 β < 3 3 189 (63.0%) 0.720 0.280 0.439
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

75. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
Outline
1 Prototypes definition
What is a prototype?
2 Consensus Analysis
Consensus clustering
Consensus measurement
3 Partitioning methods
k-Means
Fuzzy criterion
Fuzzy c-Means (FCM) and Archetypal Analysis (AA)
4 Simulated data examples
Eight experimental contexts
5 Application on real data
I.P.I.P. test
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

76. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
About data
Web Site: http://personality-testing.info/ rawdata/
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

77. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
About data
Four different scales were used as part of an experiment DISC per-
sonality test. The scales are from the International Personality Item
Pool (http://ipip.ori.org/newCPIKey.htm).
The scales used are:
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

78. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
About data
Four different scales were used as part of an experiment DISC per-
sonality test. The scales are from the International Personality Item
Pool (http://ipip.ori.org/newCPIKey.htm).
The scales used are:
Assertiveness, is the quality of being self-assured and
confident without being aggressive
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

79. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
About data
Four different scales were used as part of an experiment DISC per-
sonality test. The scales are from the International Personality Item
Pool (http://ipip.ori.org/newCPIKey.htm).
The scales used are:
Assertiveness, is the quality of being self-assured and
confident without being aggressive
Social confidence, is generally described as a state of being
certain
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

80. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
About data
Four different scales were used as part of an experiment DISC per-
sonality test. The scales are from the International Personality Item
Pool (http://ipip.ori.org/newCPIKey.htm).
The scales used are:
Assertiveness, is the quality of being self-assured and
confident without being aggressive
Social confidence, is generally described as a state of being
certain
Adventurousness, is represented by the activities with some
potential for physical danger
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

81. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
About data
Four different scales were used as part of an experiment DISC per-
sonality test. The scales are from the International Personality Item
Pool (http://ipip.ori.org/newCPIKey.htm).
The scales used are:
Assertiveness, is the quality of being self-assured and
confident without being aggressive
Social confidence, is generally described as a state of being
certain
Adventurousness, is represented by the activities with some
potential for physical danger
Dominance, is conceptualized as a measure of individual
differences in levels of group-based discrimination
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

82. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
About data
Dataset consists in 40 items (10 for each scale) and 898 individuals.
The items were rated on a 5 point scale where:
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

83. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
About data
Dataset consists in 40 items (10 for each scale) and 898 individuals.
The items were rated on a 5 point scale where:
1=Strongly disagree,
2=Disagree,
3=Neither agree not disagree,
4=Agree,
5=Strongly agree.
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

84. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
About data
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

85. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
Principal Component Analysis
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

86. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
Scree-plots FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

87. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
K-means groups
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

88. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
Memberships FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

89. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
Consensus Analysis between FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

90. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
Consensus groups FCM and AA
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

91. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
I.P.I.P. test
I.P.I.P. test
Description of prototypes
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

92. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
Conclusions
The results of the applications confirm the following hypothesis:
When the groups are well defined, avoiding any overlapping,
the consensus analysis between the two different partitioning
methods underlined the presence of the groups;
The simulation has been useful to study which are the causes
that can deeply affect the consensus among the two
approaches: firstly correlation between variables, secondly
presence of multivariate outliers (different kurtosis levels).
We believe that the prototypes definitions through the consensus
approach is more reliable in comparison to the classical approaches:
the finding of the groups in respect to the consensus-criterion, guar-
antees more homogeneous prototypes.
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

93. Prototypes definition
Consensus Analysis
Partitioning methods
Simulated data examples
Application on real data
Conclusions
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis

94. Appendix Bibliography
M. Fordellone, F. Palumbo Set-valued prototypes through Consensus Analysis