=Paper=
{{Paper
|id=Vol-1197/paper12
|storemode=property
|title=Visual Analytics in FCA-based Clustering
|pdfUrl=https://ceur-ws.org/Vol-1197/paper12.pdf
|volume=Vol-1197
|dblpUrl=https://dblp.org/rec/conf/aist/Kashnitsky14
}}
==Visual Analytics in FCA-based Clustering==
https://ceur-ws.org/Vol-1197/paper12.pdf
Visual Analytics in FCA-based Clustering
Yury Kashnitsky
Higher School of Economics, Moscow, Russia ykashnitsky@hse.ru
Abstract. Visual analytics is a subdomain of data analysis which com-
bines both human and machine analytical abilities and is applied mostly
in decision-making and data mining tasks. Triclustering, based on Formal
Concept Analysis (FCA), was developed to detect groups of objects with
similar properties under similar conditions. It is used in Social Network
Analysis (SNA) and is a basis for certain types of recommender systems.
The problem of triclustering algorithms is that they do not always pro-
duce meaningful clusters. This article describes a specific triclustering
algorithm and a prototype of a visual analytics platform for working
with obtained clusters. This tool is designed as a testing frameworkis
and is intended to help an analyst to grasp the results of triclustering
and recommender algorithms, and to make decisions on meaningfulness
of certain triclusters and recommendations.
Keywords: visual analytics, formal concept analysis, triclustering, social
network analysis.
1 Introduction
Classical Formal Concept Analysis (FCA) deals with data which describe a
relationship between a set of objects and a set of attributes and provides methods
to derive a concept hierarchy or formal ontology in them [1]. FCA is a powerful
tool for revealing dependencies in data and is commonly applied to data mining
(in particular, text mining), machine learning, knowledge management, semantic
webs, software development, and biology.
As a natural extension of FCA, Triadic Concept Analysis (TCA) manages
triadic data in a form of objects, their attributes, and conditions under which
these objects have certain attributes [2]. A common example is a social network
analysis with a context including users (objects), events they take part in (at-
tributes) and interests (which might be regarded as conditions under which a
user participates in a certain event).
As the task of finding all concepts or triconcepts is computationally chal-
lenging, certain relaxations of these terms have been introduced: biclusters [3]
and triclusters [4]. Here we address triclusters, i.e. combinations of sets of ob-
jects, their attributes, and conditions where not every object must have each
attribute. Triclustering provides an output in the form of object clusters with
similar attributes under similar conditions. Therefore, it is applied to mining
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users with common interests, applicants with similar competences or books la-
belled by close tags [5], [6]. Triclustering is also a basis for a certain type of
recommender systems [7], [8].
Visual analytics is an increasingly popular branch of Computer Science which
combines both human and computer qualities to solve a range of problems that
might lay beyond the power of man or machine separately. Actually, it is a sub-
domain of data analysis focusing on decision-making through data preprocessing,
data mining and interactive user interfaces. For instance, Siemens PLM software
allows developers to collect, process, visualize report data in the 3D engineering
environment, and make real-time decisions in the process of developing new ve-
hicles. The same method is used in situational and decision-making centres, in
nuclear power energetics, and in crime investigations.
In this paper, we explore these topics and describe a framework which uses
visual analytics to solve some problems in FCA.
2 Visual analytics
2.1 Definition and specificity
Generalizing and selecting crucial aspects of various definitions of visual an-
alytics [9], [10], here we propose the following one:
Visual analytics is a subdomain of data analysis focusing on analytical rea-
soning on the basis of interactive user interfaces in process of data mining, data
preprocessing, knowledge representation, discovering dependencies, and decision-
making.
Let us further consider core peculiarities of visual analytics and the tasks it
is designed to solve: [11]
1. Visual analytics usually deals with complicated problems with big amounts
of data requiring both human and machine resources.
2. The final goal of visual analysis is to enable users to obtain deep insight
in problems to be solved which might include processing of large amounts
of data from various sources. For this purpose visual analytics combines
both human and technological resources. On one hand, data mining and
statistics are the driving force of any automatic data analysis. On the other
hand, human brain’s aptitude for information perception and discovering
dependencies in data complies to machine techniques and thus turns visual
analytics into a promising sphere for further development.
3. In its development, visual analytics fosters in its turn the development of data
mining, data representation and visualization, and analytical reporting.
4. Visual analytics also deals with human cognition, information perception,
Computer Science, interactive and graphical design.
5. Visual analytics combines methods of information visualization and graphi-
cal data representation where visualization fosters human perception by the
following means:
(a) Enlarging data resources capacities makes user memorize less
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(b) Reducing search, such as by representing a large amount of data in small
space
(c) Enhancing recognition of patterns, such as when information is organized
in space by its time relationships
(d) Supporting easy relationship inference
(e) Monitoring large amounts of potential events
(f) Providing techniques for dynamic data monitoring
2.2 Siemems
Siemens uses visual analytics techniques in its product lifecycle management
(PLM) software enabling developers to collect, process, visualize report data in
the 3D engineering environment, and make real-time decisions in the process of
developing new vehicles. 1
Fig. 1. One of development stages with Siemens PLM Software
The crucial point is that this system allows real-time visual interaction. This
speeds up the processes of testing production for meeting given criteria, and
eliminating product quality problems.
2.3 Supernova modelling
A highly powerful implementation of visual analytics paradigm was fulfilled
by astrophysicists in Terascale Supernova Initiative (TSI) project. 2 The goal of
1
http://www.plm.automation.siemens.com
2
science.energy.gov/∼/media/ascr/ascac/pdf/meetings/mar03/Mezzacappa.pdf
71
the project is to give scientists from various fields access to powerful computation
resources in order to produce knowledge in the sphere of fundamental science. In
particular, the question of supernova birth was studied which encompassed 3D
turbulence, gravitation and magnetic field modelling. The scale of the investiga-
tion was impressive - the modelling resulted in terabytes of data. The analysis
of such amount of data lays beyond human power but combining human and
machine capabilities allowed to make some inferences from all the bulk of data.
3 Formal Concept Analysis and OA-biclustering
3.1 Main definitions
A formal context in FCA is a triple K = (G, M, I) where G is a set of
objects, M is a set of attributes, and the binary relation I ⊆ G × M shows
which object possesses which attribute. gIm denotes that object g has attribute
m. For subsets of objects and attributes A ⊆ G and B ⊆ M Galois operators
are defined as follows:
A0 = {m ∈ M | gIm ∀g ∈ A},
B 0 = {g ∈ G | gIm ∀m ∈ B}.
A pair (A, B) such that A ⊂ G, B ⊂ M, A0 = B and B 0 = A, is called a formal
concept of a context K. The sets A and B are closed and called the extent and
the intent of a formal concept (A, B) respectively. For the set of objects A the
set of their common attributes A0 describes the similarity of objects of the set
A and the closed set A00 is a cluster of similar objects (with the set of common
attributes A0 ).
The number of formal concepts of a context K = (G, M, I) can be quite large
(2min{|G|,|M |} in the worst case), and the problem of computing this number
is #P-complete [12]. There exist some ways to reduce the number of formal
concepts, for instance, choosing concepts by stability, index or extent size [13].
An alternative way is to make a relaxation of the definition of a formal
concept. One of them is an OA-bicluster [3].
If (g, m) ∈ I , then (m0 , g 0 ) is called an object-attribute bicluster with the density
T
|I (m0 × g 0 )|
ρ(m0 , g 0 ) = .
(|m0 ||g 0 |)
Bicluster density represents the percent of object-attribute pairs from the initial
context in a certain bicluster.
Here are the main properties of OA-biclusters:
1. For any bicluster (A, B) ⊆ 2G × 2M it is true that 0 ≤ ρ(A, B) ≤ 1,
2. An OA-bicluster (m0 , g 0 ) is a formal concept if ρ = 1,
3. If (m0 , g 0 ) is a bicluster, then (g 00 , g 0 ) ≤ (m0 , m00 ).
A bicluster (A, B) is called dense if its density is greater than a predefined
minimum threshold, i.e. ρ((A, B)) ≥ ρmin . The above mentioned properties show
that OA-biclusters differ from formal concepts since unit density is not required.
Below follows an illustrative example for triconcepts and triclusters.
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4 Triadic FCA and OAC-triclustering
As a solution for three-way data in FCA, Triadic Concept Analysis (TCA)
was introduced [2].
A triadic context K = (G, M, B, I) consists of sets G (objects), M (at-
tributes), B (conditions), and ternary relation I ⊆ G × M × B. An incidence
(g, m, b) ∈ I shows that the object g has the attribute m under condition b.
We denote a triadic context by (X1 , X2 , X3 , I). A triadic context K = (X1 , X2 , X3 , I)
gives rise to the following dyadic contexts:
K (1) = (X1 , X2 × X3 , I (1) ),
K (2) = (X2 , X3 × X1 , I (2) ),
K (3) = (X3 , X1 × X2 , I (3) ),
where gI (1) (m, b) ⇔ mI (1) (g, b) ⇔ bI (1) (g, m) ⇔ (g, m, b) ∈ I.
The derivation operators (or prime operators) induced by K (i) are denoted by
(.) . For each induced dyadic context we have two kinds of derivation operators.
(i)
That is, for {i, j, k} = {1, 2, 3} with j < k and for Z ⊆ Xi and W ⊆ Xj × Xk ,
the (i)-derivation operators are defined by:
Z → Z (i) = {(xj , xk ) ∈ Xj × Xk | xi , xj , xk are related by I for all xi ∈ Z},
W → W (i) = {xi ∈ Xi | xi , xj , xk are related by I for all (xj , xk ) ∈ W }
A triadic concept of a triadic context K = (G, M, B, I) is a triple (A1 , A2 , A3 )
of A1 ⊆ X1 , A2 ⊆ X2 , A3 ⊆ X3 such that for every {i, j, k} = {1, 2, 3} with
(i)
j < k we have Ai = (Aj × Ak ).
A1 , A2 and A3 are called the extent, the intent and the modus of (A1 , A2 , A3 ).
A set T = ((m, b)0 , (g, b)0 , (g, m)0 ) for a triple (g, m, b) ∈ I is called an OAC-
tricluster (or object-attribute-condition tricluster or just tricluster) based on
prime operators. Here
(g, m)0 = {b | (g, m, b) ∈ I},
(g, b)0 = {m | (g, m, b) ∈ I},
(m, b)0 = {g | (g, m, b) ∈ I}.
The density of a tricluster (A, B, C) of a triadic context K = (G, M, B, I) is
given by the fraction
T of all triples of I in the tricluster, that is
ρ(A, B, C) = |I |A||B||C|
A×B×C|
.
The tricluster T = (A, B, C) is called dense if its density is greater than a
predefined minimum threshold, i.e. ρ(T ) ≥ ρmin . Just similarly to biclusters,
triclusters have the following properties:
1. For every triconcept (A, B, C) of a triadic context K = (G, M, B, I) with
nonempty sets A, B and C we have ρ(A, B, C) = 1,
2. For every tricluster (A, B, C) of a triadic context K = (G, M, B, I) with
nonempty sets A, B and C we have 0 ≤ ρ(A, B, C) ≤ 1.
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4.1 Example
Let us consider a sample context K = (U, I, S, Y ), where U = {Ed, Leo,
Max} is a set of users, I = {soccer, hockey} — their interests, S = {soccer.com,
nhl.com, fifa.com, hockeycanada.ca} — sites they have added to bookmarks,
Y ⊆ U × I × S is a ternary relation between U, I, S which can be expressed by
Table 1:
i1 i2 s1 s2 s3 s4
s1 s2 s3 s4
u1 x x u1 x x x x
i1 x x
u2 x x u2 x x x
i2 x x
u3 x x u3 x x x x
Table 1. Sample context. Designations: u1 - Ed, u2 - Leo, u3 - Max, i1 - soccer, i2 -
hockey, s1 - soccer.com, s2 - nhl.com, s3 - fifa.com, s4 - hockeycanada.ca.
Here, generally, we have |U ||I||S| = 24 triples to analyze. But actually, this
number is reduced to 11, as there are lots of void triples present.
Actually, users Ed, Leo and Max share the same interests and almost the
same sites (all the difference is that Leo has not bookmarked hockeycanada.ca).
The idea of clustering here is presented by a tricluster T = ({u1 , u2 , u3 }, {i1 , i2 },
{s1 , s2 , s3 , s4 }) with density ρ = 11/24 ∼
= 0.46.
It is just one pattern to analyze instead of 11 in case of triples.
5 Implemented algorithms
The algorithms, described below, were implemented in Python 2.7.3 on a 2-
processor machine (Core i3-370M, 2.4 HGz) with 3.87 GB RAM. One can find
a description of testing procedure for these algorithms in [14] and [15].
5.1 OAC-prime triclustering algorithm
The hard core of the algorithm is quite simple: for all incidences (g, m, b) ∈ I
for a triadic context K = (G, M, B, I) we build a tricluster T = ((m, b)0 , (g, b)0 , (g, m)0 ).
If a tricluster is unique and its density exceeds a predefined minimum thresh-
old then it is added to an array of triclusters. A pseudocode of algorithm for
OAC-triclustering based on prime operators is presented below.
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Algorithm 1 OAC-triclustering based on prime operators
Input: K = (G, M, B) - tricontext,
ρmin - density threshold
Output: T dic = {X1 , X2 , X3 } — a tricluster dictionary. X1 ⊆ G, X2 ⊆ M, X3 ⊆
B
for (g, m, b) ∈ I do
T = ((m, b)0 , (g, b)0 , (g, m)0 )
HashKey = hash(T )
if HashKey ∈ / T dic.keys() and ρ(T ) ≥ ρmin then
T dic[hashKey] = T
end if
end for
5.2 Recommender algorithm based on triclustering
Algorithm 2 Recommender algorithm
Input: K = (U, T, R, I) - tricontext, T r - a set of triclusters
Output: T agrec , Resrec - sets of recommended tags and re-
sources
for u ∈ U do
for i = 1,...,|Tr| do
|Ru ∩R | |Tu ∩T |
simu (T ri ) = 12 ( |Ru ∪RTT rri | + |Tu ∪TTT rri | )
i i
T rbest = argmax(simu (T ri ))
T ag rec [i] = TT rbest \ Tu
Resrec [i] = RT rbest \ Ru
end for
end for
The recommender algorithm applied to sets of a tricontext is analogous to
the one described in [7]. It takes as an input a context of three sets (objects,
attributes, conditions), and the set of triclusters obtained as a result of the OAC-
prime triclustering algorithm. For each user among all triclusters the one most
similar to triples with this user is selected. The similarity of triclusters and triples
is defined by function simu (T ri ). The algorithm returns sets T agrec , Resrec - tag
and resource recommendations for all users.
6 The challenge and visual tricluster analysis framework
The challenge of the problem of triclustering (as of clustering on the whole) is
to output meaningful, well-interpreted clusters. Actually, the term "meaningful"
is not formally defined and is used by people to express their own subjective
opinion on how well the task of clustering is solved, i.e. how similar the objects
75
in same clusters are, how distant - in different ones, how it corresponds to real
world problems etc. Therefore, here an expert opinion might be useful, and
a prototype of a visual analytics framework, described below, provides visual
feedback for expert, and gives him ability to explore clusters in details.
Fig. 2. Highlighting a largest tricluster for a user-tag pair (u6, t4)
In figure 2, we can see a map of triclusters produced by algorithm 1 for a
context of 20 users, 20 tags, and 200 resources. The map is projected on the User-
Tag plane. The more a certain user-tag pair is presented in triclusters the darker
the corresponding square. A user-tag pair (u6, t12), for instance, is included in
73 triclusters (a dark red square) while (u5, t9) - just in 1 (a red square), and
no triclusters have a pair (u9, t10) (a grey one).
All triclusters including a certain user-tag pair can be listed by clicking on the
"Triclusters" menu label. Similarly, triconcepts can be listed. One can also high-
light the biggest tricluster with a certain user-tag pair or output all triclusters of
the initial context ordered by density. Moreover, through the "Recommend at-
tributes" context menu option an analyst can depict the results of recommender
algorithm for a certain user (in this case, to show recommended tags).
The tool is intended to help an analyst to grasp the results of triclustering
and recommender algorithms, and to make decisions on meaningfulness of certain
triclusters and recommendations. The map helps the expert to quickly detect the
concentrated regions (dark squares) and visualize dense triclusters including the
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Fig. 3. Recommended tags for several users
corresponding triples. Further, it helps to make the decision whether the selected
dense tricluster is meaningful or not, i.e. if it really combines similar users, tags,
and resources.
7 Further work
There are several important issues to be regarded:
1. Limited human contribution: human contribution to triclustering in this vi-
sual analytics approach is limited and might only reach some hundreds of de-
cisions on certain triclusters (less plausible, a thousand). Therefore, machine
learning approach might help to learn the algorithm to classify meaningful
clusters. The distance metric on triclusters should be carefully chosen.
2. Scalability: the issue of scalability is quite challenging in the described tech-
nique, and is to be solved. In current state, the application can support only
contexts with one long dimension, for instance, a context of 20 users, 20
tags, and 400000 resources which can be projected onto a user-tag plane.
One possible way to address the scalability issue is to perform preliminary
clustering of objects, attributes, and conditions separately, and then choose
representatives from each cluster.
3. Extending the idea of a human-machine approach to other problems in FCA
or data mining, such as exploring implications and association rules in order
to find meaningful ones.
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8 Conclusion
Visual analytics, as one of the flourishing domains of data analysis, can be
useful in mining objects with similar attributes under similar conditions in a
context of social network data. A special algorithm was developed for uniting
such objects, attributes, and conditions in triclusters. The program framework
under development is intended to graphically display the results of this algorithm
and to empower an analyst to decide on the meaningfulness of clusters and tags
or resources recommendations for objects.
Acknowledgements The author would like to thank his colleagues from Higher
School of Economics Sergei Kuznetsov and Dmitry Ignatov for their well-timed
advice and support during this work. He also expresses gratitude to Stanislav
Klimenko from Moscow Institute of Physics and Technology for consulting in
visual analytics.
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79
Визуальная аналитика в задаче
трикластеризации, основанной на анализе
формальных понятий
Юрий Кашницкий
НИУ ВШЭ, Москва, Россия
ykashnitsky@hse.ru
Аннотация Трикластеризация — это способ обнаружения объектов
со схожими свойствами в контексте из трех множеств сущностей. На-
пример, в задаче анализа данных социальных сетей, такими множе-
ствами могут быть пользователи, их интересы и события, в которых
они принимают участие. Трикластеризация здесь может помочь най-
ти группы пользователей с похожими интересами и, делать им реко-
мендации событий на основе этих интересов. В статье описывается
конкретный алгоритм трикластеризации и прототип программной
платформы для визуального анализа полученных трикластеров.
Ключевые слова: визуальная аналитика, анализ формальных по-
нятий, трикластеризация, анализ социальных сетей.
80