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Book ChapterDOI

A Clustering Model for Uncertain Preferences Based on Belief Functions

03 Sep 2018-pp 111-125
TL;DR: The model for uncertain preferences is based on the theory of belief functions with an appropriate dissimilarity measure when performing the clustering steps and has an equivalent quality with traditional preference representations for certain cases while it has better quality confronting imperfect cases.
Abstract: Community detection is a popular topic in network science field. In social network analysis, preference is often applied as an attribute for individuals’ representation. In some cases, uncertain and imprecise preferences may appear. Moreover, conflicting preferences can arise from multiple sources. From a model for imperfect preferences we proposed earlier, we study the clustering quality in case of perfect preferences as well as imperfect ones based on weak orders (orders that are complete, reflexive and transitive). The model for uncertain preferences is based on the theory of belief functions with an appropriate dissimilarity measure when performing the clustering steps. To evaluate the quality of clustering results, we used Adjusted Rand Index (ARI) and silhouette score on synthetic data as well as on Sushi preference data set collected from real world. The results show that our model has an equivalent quality with traditional preference representations for certain cases while it has better quality confronting imperfect cases.

Summary (3 min read)

1 Introduction

  • Community detection is a very popular topic in network science field, and has received a great deal of attention.
  • It is a key task for identifying groups (i.e. clusters) of objects that share common properties and/or interact with each other.
  • In [17], the authors introduced a new community detection algorithm based on preference network.
  • To form meaningful groups of agents according to their preferences, a clustering algorithm need to capture the preference data structure and to cope with imperfect information.
  • In previous works [19, 11], a qualitative and expressive preference modeling strategy based on the theory of belief functions to model imperfect preferences was proposed.

2.1 Preference Order

  • A binary relation satisfies any of the following properties: reflexive, irreflexive, symmetric, antisymmetric, asymmetric, complete, strongly complete, transitive, negatively transitive, semitransitive, and Ferrers relation [13].
  • The detailed definitions of the properties are not in the scope of this article.
  • Inspired by a four-valued logic introduced in [1, 11, 4], the authors introduce four relations between alternatives.

2.2 Dissimilarity between orders

  • To measure the dissimilarity between two preferences represented by total orders, metrics such as Euclidean distance and Kendall distance are often adopted.
  • Given two preference orders O1 and O2 on the same alternatives, the authors give some basic concepts on such metrics.
  • The rank function r(O, a) denotes the position of the alternative a according to the order O.
  • Intuitively, both partial orders agree that ai and aj are tied.

3.2 Preference model on belief functions

  • The objective is to cluster the experts represented by their preferences under uncertainty.
  • The authors consider the model from [19] to represent this uncertain preference by the theory of belief functions.
  • This procedure consists in two steps: 1. Initialization of mass functions 2. Clustering on quasi orders represented by mass functions.

4 Contribution: agent clustering based on their preferences

  • The authors explain how the agents are represented and clustered from two sources of preferences.
  • The first block concerns the representation of agents and modeling of mass functions from two preference sources S1, S2 (sections 4.1 and 4.2).
  • The second block concerns the measure of dissimilarity between agents (section 4.3).
  • The third block concerns clustering algorithm, the authors use EkNNclus algorithm in their work (section 4.4).

4.3 Dissimilarity between different agents

  • The dissimilarity measure is based on Jousselme distance [8] for mass functions.
  • Given two mass functions modeling preference relations between alternatives i and j from agents u1 and u2 expressing preference orders O1, O2.
  • To simplify the expression, the authors use BF model to refer to their model and the corresponding dissimilarity function.

4.4 Unsupervised classifier–Ek-NN [3]

  • For dissimilarity spaces in which only pairwise distances are given (such as Kendall distance), the centroid of several agents is a metric k-center problem and is proved to be NP-hard.
  • Therefore, the authors avoid using clustering methods requiring the calculation of centroid, such as k-means.
  • The authors applied Ek-NNclus method [3] as classifier.

5 Experiments

  • The authors still wonder its quality for clustering on certain preferences.
  • Thus the clustering quality of their model can be divided into two aspects: on certain preferences and on uncertain preferences.

5.1 Evaluation criteria

  • With the similar aforementioned reasons, it’s NP-hard to calculate centroids.
  • Thus, the authors choose two evaluation criteria that do not require a cluster centroid calculation: Adjusted Rand Index (ARI) [7] for data with ground truth and silhouette coefficient [15] for any dataset.
  • The authors tested different metrics on synthetic certain and uncertain preferences.
  • The authors also compared different metrics on a real world certain preferences from SUSHI data set [9].
  • In the following parts, the authors introduce the method of generating synthetic preferences and compare the clustering quality of different metrics.

5.2 Certain preferences

  • On synthetic data Certain preferences are those who are from non-conflicting sources.
  • To study the clustering quality, the authors firstly generate preferences with different ranges to their centroids.
  • The authors test on various K and choose the one that returns the largest ARI and average silhouette coefficient4 as their result.
  • On real data SUSHI preference dataset [9] is collected from a survey on Japanese consumer preferences over different sushis.
  • 6 Kendall distance and BF model have similar quality.

5.3 Uncertain preferences

  • The authors suppose a case that two preferences are given with different representations: ranking and score.
  • Indifference relations are introduced, causing conflicts between two preference sources.
  • Or of 10 alternatives a1 to a10, the scores are generated by the following rules: – For least preferred two alternatives (2 alternatives at the end of the Or, i.e. ranking no. 9 and 10), the authors give score 1. – For alternatives sorted at the positions 7 and 8, they give score 2. –.
  • As indifference relations exist in Os, the authors apply Fagin distance for Os.
  • The results illustrated by these figures show the advantage of BF model over Euclidean distance and Kendall distance when dealing with two sources.

6 Conclusion and perspectives

  • The authors investigate the problem of clustering individuals according to their preferences, when dealing with multiple and conflicting sources (two in their case study).
  • To cope with this issue, the authors apply the theory of belief functions (BF model) to express and interpret the contradictions and conflicts from different sources as uncertainty and ignorance.
  • To highlight the relevance of the proposed solution, the authors perform experiments on synthetic and real data to compare their method with other preference models, and found the advantage in the expressiveness of the uncertainty and the incomparability of the preference orders.
  • In certain cases, BF model has equivalent clustering-quality with Kendall distance and outperforms Euclidean distance.

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A clustering model for uncertain preferences based on
belief functions
Yiru Zhang, Tassadit Bouadi, Arnaud Martin
To cite this version:
Yiru Zhang, Tassadit Bouadi, Arnaud Martin. A clustering model for uncertain preferences based
on belief functions. DaWaK: Data Warehousing and Knowledge Discovery, Sep 2018, Regensburg,
Germany. �10.1007/978-3-319-98539-8_9�. �hal-01880391�

A clustering model for uncertain preferences
based on belief functions
Yiru Zhang, Tassadit Bouadi, and Arnaud Martin
Univ Rennes 1, CNRS, IRISA
firstname.lastname@irisa.fr
http://www-druid.irisa.fr/
Abstract. Community detection is a popular topic in network science
field. In social network analysis, preference is often applied as an attribute
for individuals’ representation. In some cases, uncertain and imprecise
preferences may appear in some cases. Moreover, conflicting preferences
can arise from multiple sources. From a model for imperfect preferences
we proposed earlier, we study the clustering quality in case of perfect
preferences as well as imperfect ones based on weak orders (orders that
are complete, reflexive and transitive). The model for uncertain pref-
erences is based on the theory of belief functions with an appropriate
dissimilarity measure when performing the clustering steps. To evaluate
the quality of clustering results, we used Adjusted Rand Index (ARI) and
silhouette score on synthetic data as well as on Sushi preference data set
collected from real world. The results show that our model has an equiv-
alent quality with traditional preference representations for certain cases
while it has better quality confronting imperfect cases.
Keywords: Clustering for orders; Imperfect preference modeling; The-
ory of belief functions
1 Introduction
Community detection is a very popular topic in network science field, and has
received a great deal of attention. It is a key task for identifying groups (i.e.
clusters) of objects that share common properties and/or interact with each
other. Many algorithms have been developed for efficient community detection.
Depending on the information source used to perform the clustering task, these
algorithms can be classified into tow main categories: graph structure based
techniques [12], and node attribute based techniques [18]. The first one consid-
ers the relationships and the connections between the objects (e.g. friendships
or professional relationships between social network agents
1
, proteins interac-
tions, etc.), while the second one analyses the similarity between objects based
1
In decision making theory, different terms may refer to the same concepts. To avoid
ambiguity, we unify the terminology concerning preferences. In this article, “agents”
is used for individuals expressing their preferences, “alternatives” for items which
are compared in preferences.

2 Y. Zhang et al.
on their attribute and feature information (e.g. gender, location, personal in-
terests, etc.). More recently, some works [14] discuss hybrid techniques using
both node attributes and network topology for community detection. In this pa-
per, we are particularly interested in attribute based techniques in the context
of social networks. More precisely, we consider the case of interest-based so-
cial networks (e.g. Pinterest, Flickr, etc.) where agents sharing similar interests,
opinions or viewpoints on some topics belong to the same community. In many
real life applications, preferences (i.e. a preference describes how an agent orders
any two alternatives) are considered to be very useful to efficiently express and
model agent’s interests, needs or wishes. The aim of this work is to propose a
novel community detection method based on clustering agents according to their
preferences.
Few work has been done on clustering agents based on their preferences.
Kamishima et al. in [10] proposed the k’o-means clustering method, an adapta-
tion of the k-means method, adjusted to support preference orders. In [17], the
authors introduced a new community detection algorithm based on preference
network. The communities are constructed according to the node preferences
(i.e. each node gives information about its preferred nodes in order to be in the
same group).
However, preferences are not always expressed firmly or consistently, some-
times a preference may be uncertain or imprecise facing an unknown situation,
or conflicting when dealing with multiple sources. To the best of our knowledge,
none of the work mentioned above has investigated preference-based clustering
methods when preferences are imperfect (i.e. uncertain, imprecise or conflicting).
To form meaningful groups of agents according to their preferences, a cluster-
ing algorithm need to capture the preference data structure and to cope with
imperfect information.
In previous works [19, 11], a qualitative and expressive preference modeling
strategy based on the theory of belief functions to model imperfect preferences
was proposed. In this paper, starting from this model for agent’s preference mod-
eling, we develop a preference-based clustering approach in the space of theory of
belief functions. We discuss the clustering quality of our method by considering
the Adjusted Rand Index (ARI) and silhouette coefficient as evaluation criteria.
To highlight the relevance of the proposed solution, we perform experiments on
synthetic and real data to compare our method with different preference model-
ings, reference in the field, and found the advantage in the expressiveness of the
uncertainty and the conflict of the preferences.
Outline of the paper is as follows. In section 2, we give background infor-
mation related to preference orders, similarity measures over orders, and theory
of belief functions. We then explain in section 3 our previous preference model
based on theory of belief functions and in section 4 our clustering approach in
detail. Experiments and their analysis are given in section 5. We finish with the
conclusion and perspectives in section 6.

A clustering model for uncertain preferences based on belief functions 3
2 Basic notions
2.1 Preference Order
Definition 1 (Binary Relation) Let A = {a
1
, a
2
, . . . , a
|A|
} be a finite set of
alternatives, a binary relation O on the set A is a subset of the Cartesian
product A × A, that is, a set of ordered pairs (a
i
, a
j
) such that a
i
and a
j
are in
A : O A × A [13].
A binary relation satisfies any of the following properties: reflexive, irreflexive,
symmetric, antisymmetric, asymmetric, complete, strongly complete, transitive,
negatively transitive, semitransitive, and Ferrers relation [13]. The detailed def-
initions of the properties are not in the scope of this article.
Based on definition 1, we denote the binary relation “prefer” by . The
relation
2
a
i
a
j
means a
i
is at least as good as a
j
”. Inspired by a four-valued
logic introduced in [1, 11, 4], we introduce four relations between alternatives.
Given the alternative set A and a preference order defined on A, we have
a
i
, a
j
A, the four possible relations defined by:
Strict preference denoted by P : a
i
a
j
(a
i
is strictly preferred to a
j
) a
i
a
j
¬(a
j
a
i
)
Inverse strict preference denoted by ¬P : a
j
a
i
(a
i
is inversely strictly preferred to a
j
) ¬(a
j
a
i
) a
j
a
i
Indifference denoted by I: a
i
a
j
(a
i
is indifferent, or equally preferred, to a
j
) a
i
a
j
a
j
a
i
Incomparability denoted by J: a
i
a
j
(a
i
is incomparable to a
j
) ¬(a
i
a
j
) ¬(a
j
a
i
)
A preferences structure hP, I, Ji on multiple alternatives can therefore be
presented by a binary relation [13].
Definition 2 (Preference Structure) A preference structure is a collection of
binary relations defined on the set A such that for each pair a
i
, a
j
in A:
at least one relation is satisfied
if one relation is satisfied, another one cannot be satisfied.
The model for uncertain preferences detailed in [19] is compatible with quasi-
orders while our dissimilarity measure is suitable for weak orders, which is a
subset of quasi-orders, defined as [13]:
Definition 3 (Weak Order) Let O be a binary relation (O = P I) on the set
A, O being a characteristic relation of hP, Ii, the following three definitions are
equivalent:
1. O is a weak order.
2. O is reflexive, strongly complete and transitive.
3.
I is transitive
P is transitive
P I is reflexive and complete.
2
As a
i
a
j
is equivalent to a
j
a
i
, to avoid repetitive comparisons between two
alternatives, we assume i > j in this article.

4 Y. Zhang et al.
2.2 Dissimilarity between orders
To measure the dissimilarity between two preferences represented by total orders,
metrics such as Euclidean distance and Kendall distance are often adopted.
Given two preference orders O
1
and O
2
on the same alternatives, we give some
basic concepts on such metrics.
Euclidean distance The rank function r(O, a) denotes the position of the alter-
native a according to the order O. For example, for the order
O = a
1
a
3
a
2
, r(O, a
1
) = 1 and r(O, a
2
) = 3. Thus, for two orders O
1
and O
2
on the same alternative set A, Euclidean distance (l
2
norm) between
two orders is defined by:
d
l
2
(O
1
, O
2
) =
s
X
aA
(r(O
1
, a) r(O
2
, a))
2
(1)
Kendall’s τ distance and Fagin distance Kendall τ distance measures the
dissimilarity with “penalty”. Fagin proposed a more general metric in [6] adapt-
ing for orders with indifference based on Kendall distance, we name it Fagin
distance in this article. In Fagin distance, for alternatives a
i
, a
j
, the penalty be-
tween two orders O
1
and O
2
on a
i
and a
j
, denoted as
¯
K
(p)
i,j
(O
1
, O
2
), is defined
as follows:
Case 1: a
i
and a
j
are in both O
1
and O
2
. If a
i
and a
j
are ordered in the same
way (such as a
i
a
j
in both O
1
, O
2
),
¯
K
(p)
i,j
(O
1
, O
2
) = 0, this corresponds to
“no penalty” for a
i
and a
j
. If a
i
, a
j
are ordered reversely (such as a
i
a
j
in
O
1
while a
i
a
j
in O
2
), the penalty of a
i
, a
j
¯
K
(p)
i,j
(O
1
, O
2
) = 1.
Case 2: a
i
and a
j
are tied in both O
1
and O
2
,
¯
K
(p)
i,j
(O
1
, O
2
) = 0. Intuitively,
both partial orders agree that a
i
and a
j
are tied.
Case 3: a
i
and a
j
are indifference in one of the partial order (say O
1
) and of
different rank in the other order (therefore O
2
), we give a penalty parameter
¯
K
(p)
i,j
(O
1
, O
2
) = p
3
.
Based on these cases, the Kendall distance with penalty parameter p (i.e. Fagin
distance) is defined as follow:
K
(p)
(O
1
, O
2
) =
X
i,j[1,|A|]
¯
K
(p)
i,j
(O
1
, O
2
) (2)
2.3 Belief functions
The theory of belief functions (also referred to as Dempster-Shafer or Evidence
Theory) was firstly introduced by Dempster [2] then developed by Shafer [16]
3
In our work, we take p = 0.5

Citations
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Journal ArticleDOI
TL;DR: A novel distance named Unequal Singleton Pair (USP) distance is proposed, able to discriminate specific singletons from others when measuring the dissimilarity, and shows that USP distance effectively improves the quality of decision results.

11 citations

Journal ArticleDOI
TL;DR: In this article , the authors focus on the problem of aggregating evidence corpus to obtain a representative one, and they show through an impossibility theorem that in this case, there is a fundamental contradiction between the use of conjunctive combination rules on the one hand, and use of distances on the other hand.
Abstract: The theory of belief functions (TBFs) is now a widespread framework to deal and reason with uncertain and imprecise information, in particular to solve information fusion and clustering problems. Combination functions (rules) and distances are essential tools common to both the clustering and information fusion problems in the context of TBF, which have generated considerable literature. Distances and combination between evidence corpus of TBF are indeed often used within various clustering and classification algorithms, however, their interplay and connections have seldom been investigated, which is the topic of this article. More precisely, we focus on the problem of aggregating evidence corpus to obtain a representative one, and we show through an impossibility theorem that in this case, there is a fundamental contradiction between the use of conjunctive combination rules on the one hand, and the use of distances on the other hand. Rather than adding new methodologies, such results are instrumental in guiding the user among the many methodologies that already exist. To illustrate the interest of our results, we discuss different cases where they are at play.

2 citations

Journal ArticleDOI
TL;DR: In this article , a new preference clustering algorithm is proposed that incorporates the key features of preferences, usually represented by order vectors, and it takes ideas from Social Choice Theory, Decision Making Theory and Cluster Analysis as sources of inspiration.
Abstract: Preferences and their classification are essential in many decision making processes. However, grouping preferences is not an easy matter because their very nature. In this paper a new preference clustering algorithm is proposed that incorporates the key features of preferences, usually represented by order vectors, and it takes ideas from Social Choice Theory, Decision Making Theory and Cluster Analysis as sources of inspiration. Additionally, a study of the main properties of our proposal is included as well as several internal validation measurements. Finally and in order to improve understanding of the proposed approach, assorted experiments on real data are included.
Journal ArticleDOI
TL;DR: In this article , the rank aggregation problem under the generalized Kendall-tau distance is studied, which contains Kemeny aggregation as a special case and introduces a social choice property (GXCC) that encloses existing variations of the Condorcet criterion as special cases.
Abstract: Rank Aggregation has ubiquitous applications in operations research, artificial intelligence, computational social choice, and various other fields. Interest in this problem has increased due in part to the need to consolidate lists of rankings and scores output by different decision-making processes and algorithms. Although most attention has focused on the variant of this problem induced by the Kemeny-Snell distance, other robust rank aggregation problems have been proposed. This work delves into the rank aggregation problem under the generalized Kendall-tau distance —a parameterizable-penalty distance measure for comparing rankings with ties— which contains Kemeny aggregation as a special case. First, it derives exact and heuristic solution methods. Second, it introduces a social choice property (GXCC) that encloses existing variations of the Condorcet criterion as special cases, thereby expanding this seminal social choice concept beyond Kemeny aggregation for the first time. GXCC offers both computational and theoretical advantages. In particular, GXCC may help to divide the original problem into smaller subproblems, while still ensuring that solving them independently yields the optimal solution to the original problem. Experiments on two benchmark datasets conducted herein show that the featured exact and heuristic solution methods can benefit from GXCC. Finally, this work derives new theoretical insights into the effects of the generalized Kendall-tau distance penalty parameter on the optimal ranking and on the proposed social choice property.
Journal ArticleDOI
TL;DR: In this paper , the authors focus on the problem of aggregating evidence corpus to obtain a representative one, and they show through an impossibility theorem that in this case, there is a fundamental contradiction between the use of conjunctive combination rules on the one hand, and use of distances on the other hand.
Abstract: The theory of belief functions (TBFs) is now a widespread framework to deal and reason with uncertain and imprecise information, in particular to solve information fusion and clustering problems. Combination functions (rules) and distances are essential tools common to both the clustering and information fusion problems in the context of TBF, which have generated considerable literature. Distances and combination between evidence corpus of TBF are indeed often used within various clustering and classification algorithms, however, their interplay and connections have seldom been investigated, which is the topic of this article. More precisely, we focus on the problem of aggregating evidence corpus to obtain a representative one, and we show through an impossibility theorem that in this case, there is a fundamental contradiction between the use of conjunctive combination rules on the one hand, and the use of distances on the other hand. Rather than adding new methodologies, such results are instrumental in guiding the user among the many methodologies that already exist. To illustrate the interest of our results, we discuss different cases where they are at play.
References
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"A Clustering Model for Uncertain Pr..." refers methods in this paper

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    [...]

  • ...The theory of belief functions (also referred to as Dempster-Shafer or Evidence Theory) was firstly introduced by Dempster [2] then developed by Shafer [16] 3 In our work, we take p = 0.5 A clustering model for uncertain preferences based on belief functions 5 as a general model of uncertainties....

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"A Clustering Model for Uncertain Pr..." refers methods in this paper

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5,437 citations


"A Clustering Model for Uncertain Pr..." refers methods in this paper

  • ...The theory of belief functions (also referred to as Dempster-Shafer or Evidence Theory) was firstly introduced by Dempster [2] then developed by Shafer [16]...

    [...]

  • ...The theory of belief functions (also referred to as Dempster-Shafer or Evidence Theory) was firstly introduced by Dempster [2] then developed by Shafer [16] 3 In our work, we take p = 0.5 A clustering model for uncertain preferences based on belief functions 5 as a general model of uncertainties....

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4,637 citations

Frequently Asked Questions (2)
Q1. What have the authors contributed in "A clustering model for uncertain preferences based on belief functions" ?

From a model for imperfect preferences the authors proposed earlier, they study the clustering quality in case of perfect preferences as well as imperfect ones based on weak orders ( orders that are complete, reflexive and transitive ). 

In the future, the authors will work on an ameliorated BF model 14 Y. Moreover, a more general dissimilarity measure method for incomplete orders ( i. e. quasiorders ) is also in the scope of their future work. In fact, the combination of preferences from multiple sources is a social choice problem, and different combination rules can be applied, corresponding to different complexity.