On Normalizing Fuzzy Coincidence Matrices to Compare Fuzzy and/or Possibilistic Partitions with the Rand Index

Quéré, Romain; Capitaine, Hoël Le; Fraisseix, N.; Frélicot, Carl

doi:10.1109/icdm.2010.130

Cited by 8 publications

(19 citation statements)

References 14 publications

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“…For fuzzy partitions, Anderson et al., Campello, Brouwer, Hullermeier and Rifqi, Quere et al., and Hullermeier et al . proposed different extensions according to various directions for evaluating

R I

to identify the similarity between two clusterings.…”

Section: Previous Methodsmentioning

confidence: 99%

“…For reinforcing the results of Borgelt and Kruse and Campello, Quere et al . used the cosine‐correlation idea of Brouwer to generalize coincidence matrices by means of t ‐norms and normalization.…”

Section: Previous Methodsmentioning

confidence: 99%

“…Borgelt and Kruse extended indices for similarities between clusterings by defining a coincidence matrix

ϕ_{M^{false(r false)}}^{T}

M^{false(r false)}

based on a t ‐norm (triangular norm) T as

ϕ_{M^{(r)}}^{T} = {[ϕ {_{M^{false(r false)}}^{T}}_{, i j}]}_{n \times n}

with

0true ϕ {_{M^{false(r false)}}^{T}}_{, i j} = \sum_{h = 1}^{k_{r}} T true(m_{h i}^{false(r false)}, m_{h j}^{false(r false)} true)

. However, Quere et al . pointed out that this extension may produce undesirable results.…”

Section: Previous Methodsmentioning

confidence: 99%

“…This is because, whatever the t ‐norm, diagonal terms

ϕ_{M^{false(r false)}, i i}^{T}

, representing the degree with which each

x_{i}

is as in the same class as itself, are no more equal to 1 so that the normalization is necessary. Quere et al . used the cosine‐correlation idea of Brouwer to normalize coincidence matrices as follows.…”

Section: Previous Methodsmentioning

confidence: 99%

“…In recent years, to widen the use of similarity between two clusterings, many researchers had devoted considerable effort to generalizing the RI into fuzzy partitions. In set theory, a binary relation between the object data set

O = {o_{1}, o_{2}, \dots, o_{n}}

and the crisp partition

P = {S_{1}, S_{2}, \dots, S_{k}}

, where

⋃_{h = 1}^{k} S_{h} = P

and

S_{h} \cap S_{h^{'}} = φ

for all

h \neq h^{'}

, is defined as

m false(o_{i}, S_{h} false) = \{\begin{matrix} 1 & if o_{i} \in S_{h} \\ 0 & if o_{i} \notin S_{h} \end{matrix}

for all

i = 1, 2, \dots, n

and

h = 1, 2, \dots, k .

To generalize the binary relation, Zadeh first extended

m (o_{i}, S_{h})

to represent the strength of belonging for the object

o_{i}

S_{h}

as a fuzzy membership with

0 \leq m (o_{i}, S_{h}) \leq 1

and

0true \sum_{h = 1}^{k} m (o_{i}, S_{h}) = 1

.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Fuzzy Generalization and Comparisons for the Rand Index

Yang

Yeh

2017

Int. J. Intell. Syst.

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To generalize the Rand index (RI) from crisp partitions to fuzzy partitions, we first propose a graph method in which color edges in the graph for crisp partitions are used to determine the relation matrix between objects such that the matrix trace can be employed to calculate the RI. This approach is then introduced into fuzzy partitions to generalize the RI to the fuzzy RI (FRI). Compared with previous fuzzy generalizations, the most unique aspect of our method has the following important characteristics that for any two partition matrices M(1) and M(2), the result with M(1)=M(2) is the necessary and sufficient condition for the result that the FRI is equal to 1. This important characteristic renders our fuzzy generalization of the RI is not only able to determine the similarities between fuzzy partitions and crisp reference partitions, but also to identify the similarity between fuzzy partitions and fuzzy reference partitions. The method can even be used to explore and compare the similarities between various data sets and the same fuzzy reference partition. Finally, we use synthetic data and real data to give more demonstrations, and further perform comparisons of our method with those existing fuzzy extensions of the RI.

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R I

to identify the similarity between two clusterings.…”

Section: Previous Methodsmentioning

confidence: 99%

Section: Previous Methodsmentioning

confidence: 99%

“…Borgelt and Kruse extended indices for similarities between clusterings by defining a coincidence matrix

ϕ_{M^{false(r false)}}^{T}

M^{false(r false)}

based on a t ‐norm (triangular norm) T as

ϕ_{M^{(r)}}^{T} = {[ϕ {_{M^{false(r false)}}^{T}}_{, i j}]}_{n \times n}

with

0true ϕ {_{M^{false(r false)}}^{T}}_{, i j} = \sum_{h = 1}^{k_{r}} T true(m_{h i}^{false(r false)}, m_{h j}^{false(r false)} true)

. However, Quere et al . pointed out that this extension may produce undesirable results.…”

Section: Previous Methodsmentioning

confidence: 99%

“…This is because, whatever the t ‐norm, diagonal terms

ϕ_{M^{false(r false)}, i i}^{T}

, representing the degree with which each

x_{i}

Section: Previous Methodsmentioning

confidence: 99%

O = {o_{1}, o_{2}, \dots, o_{n}}

and the crisp partition

P = {S_{1}, S_{2}, \dots, S_{k}}

, where

⋃_{h = 1}^{k} S_{h} = P

and

S_{h} \cap S_{h^{'}} = φ

for all

h \neq h^{'}

, is defined as

m false(o_{i}, S_{h} false) = \{\begin{matrix} 1 & if o_{i} \in S_{h} \\ 0 & if o_{i} \notin S_{h} \end{matrix}

for all

i = 1, 2, \dots, n

and

h = 1, 2, \dots, k .

To generalize the binary relation, Zadeh first extended

m (o_{i}, S_{h})

to represent the strength of belonging for the object

o_{i}

S_{h}

as a fuzzy membership with

0 \leq m (o_{i}, S_{h}) \leq 1

and

0true \sum_{h = 1}^{k} m (o_{i}, S_{h}) = 1

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Fuzzy Generalization and Comparisons for the Rand Index

Yang

Yeh

2017

Int. J. Intell. Syst.

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A New Index Based on Sparsity Measures for Comparing Fuzzy Partitions

Quéré

Frélicot

2012

Lecture Notes in Computer Science

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This article adresses the problem of assessing how close two strict and/or fuzzy partitions are. A new index based on a measurement of the sparsity of the contingency matrix crossing the partitions is proposed that satisfies the required properties formulated within the paper and presents a low complexity. It is compared to well-known existing indices of the literature, such as the Rand and the Jaccard indices, the transfert distance and some of their recent fuzzy counterparts.

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Generalization of clustering agreements and distances for overlapping clusters and network communities

Rabbany

Zaı̈ane

2015

Data Min Knowl Disc

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A measure of distance between two clusterings has important applications, including clustering validation and ensemble clustering. Generally, such distance measure provides navigation through the space of possible clusterings. Mostly used in cluster validation, a normalized clustering distance, a.k.a. agreement measure, compares a given clustering result against the ground-truth clustering. The two widely-used clustering agreement measures are adjusted rand index and normalized mutual information. In this paper, we present a generalized clustering distance from which these two measures can be derived. We then use this generalization to construct new measures specific for comparing (dis)agreement of clusterings in networks, a.k.a. communities. Further, we discuss the difficulty of extending the current, contingency based, formulations to overlapping cases, and present an alternative algebraic formulation for these (dis)agreement measures. Unlike the original measures, the new co-membership based formulation is easily extendable for different cases, including overlapping clusters and clusters of inter-related data. These two extensions are, in particular, important in the context of finding communities in complex networks.

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On Normalizing Fuzzy Coincidence Matrices to Compare Fuzzy and/or Possibilistic Partitions with the Rand Index

Cited by 8 publications

References 14 publications

Fuzzy Generalization and Comparisons for the Rand Index

Fuzzy Generalization and Comparisons for the Rand Index

A New Index Based on Sparsity Measures for Comparing Fuzzy Partitions

Generalization of clustering agreements and distances for overlapping clusters and network communities

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