Navigating and Exploring RDF Data using Formal Concept Analysis

Navigating and Exploring RDF Data using Formal
Concept Analysis
Mehwish Alam Amedeo Napoli
LORIA (CNRS – Inria Nancy Grand Est – Université de Lorraine), BP 239, Vandoeuvre les
Nancy, F-54506, France,
{firstname.lastname}@loria.fr
September 11th 2015
Linked Data for Knowledge Discovery - ECML/PKDD - Porto, Protugal.
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Motivation
A Scenario
A patient takes some drug which is a cardiovascular agent and which
causes some side effect i.e., allergic reaction.
The domain expert (a medical doctor) should look for drugs which are
cardiovascular agents but do not cause any side effect or allergic
reactions (for this patient).
The domain expert chooses some datasets, resources and knowledge
sources to be searched, e.g. Drugbank (drug with their categories),
SIDER (for side effect), Mesh, MedDRA. . .
The objective of this research work is to provide a platform supporting
the querying of such resources.
Requirements:
Drug
Information
(DrugBank)
Side Effect
Information
(SIDER)
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Motivation
Problem Statement
3 / 31

Motivation
RDF Graph for Drugbank
Drug
DB01193 P10635 Fatty acids
acebutolol Cytochrome P450
targets
rdf:type
rdfs:label
hasPathway
rdfs:subClassOf
4 / 31

Motivation
RDF Triples as Entity and Description
tid Subject Predicate Object Provenance
t1 s1 p1 o11 dataset1
t4 s1 rdf:type Drug dataset2
t5 o11 rdf:type C1 dataset3
t6 C1 rdfs:subClassOf C3 dataset3
t7 o26 rdf:type D6 dataset4
t8 D6 rdfs:subClassOf D8 dataset4
...
...
...
...
...
Table : Sample RDF triple store from several resources.
Entities S Descriptions Ds
s1 {(p1:{C1, C5, C7}), (p2:{D6})}
s2 {(p1:{C1, C13, C11}), (p2:{D7})}
s3 {(p1:{C5}), (p2:{D7})}
s4 {(p1:{C13, C11, C7}), (p2:{D1})}
s5 {(p1:{C1, C7}), (p2:{D6})}
Table : RDF Triples as entities S and descriptions Ds.
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FCA: formal context and binary attributes
A formal context is a triple pG, M, Iq where G is a set of objects, M a set of
attributes, and I a binary relation such as pg, mq P I means that “object g
owns attribute m”.
A Galois connection characterizes formal concepts:
A1
“ tm P M | @g P A Ď G : pg, mq P Iu
B1
“ tg P G | @m P B Ď M : pg, mq P Iu
pA, Bq is a formal concept with extent A “ B1
and intent B “ A1
, e.g.
ptg3, g4, g5u, tm2, m3uq.
Concepts are pairs of maximal sets of objects with corre-
sponding maximal sets of attributes.
m1 m2 m3
g1 ˆ ˆ
g2 ˆ ˆ
g3 ˆ ˆ
g4 ˆ ˆ
g5 ˆ ˆ ˆ

FCA: concept lattice
pA1, B1q ď pA2, B2q ô A1 Ď A2 pô B2 Ď B1q
ptg1, g5u, tm1, m3uq ď ptg1, g2, g5u, tm1uq
The concept lattice is based on two dual partial orderings:
Larger extents and smaller intents are in the “higher levels” of the
concept lattice.
Smaller extents and larger intents are in the “lower levels” of the
concept lattice.
Reduced labeling:
intents are “inherited” from top to bottom (top-down),
and extents are “inherited” from bottom to top (bottom-up).

FCA and Pattern Structures
Conceptual scaling
The formal context is the basic data type of Formal Concept Analysis.
However data are often given in form of a many-valued context.
Many-valued contexts are translated to one-valued context via
conceptual scaling.
But this is not automatic and some arbitrary choices have to be made.
Examples of scalings:
Nominal: K “ pN, N, “q
Ordinal: K “ pN, N, ďq
Interordinal: K “ pN, N, ď Y ěq
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A numerical example
G / M m1 m2 m3
g1 1 3 4
g2 2 2 3
g3 4 1 1
g4 3 2 1
Nominal Scaling:
G / M m1=1 m1=2 m1=4 m2=1 m2=2 m2=3 m3=1 m3=3 m3=4
g1 x x x
g2 x x x
g3 x x x
g4 x x x
Interordinal Scaling:
G / M m1.lt.1 m1.gt.1 m1.lt.2 m1.gt.2 m1.lt.3 m1.gt.3 m1.lt.4 m1.gt.4 m2.lt.1 m2.gt.1
g1 x x x x x x
g2 x x x x x x
g3 x x x x x x x
g4 x x x x x x

Computing similarity between descriptions
Intersection considered as a similarity operator:
X behaves like a similarity operator: tm1, m2u X tm1, m3u “ tm1u
m1 m2 m3
g1 ˆ ˆ
g2 ˆ ˆ
g3 ˆ ˆ
g4 ˆ ˆ
g5 ˆ ˆ ˆ
X induces a partial ordering relation Ď as follows:
S1 X S2 “ S1 ðñ S1 Ď S2
tm1u X tm1, m2u “ tm1u ðñ tm1u Ď tm1, m2u
X has the properties of a meet [ in a semi lattice, i.e. a
commutative, associative and idempotent operation:
c [ d “ c ðñ c Ď d

The deﬁnition of a Pattern Structure
A pattern structure pG, pD, [q, δq is composed of:
G a set of objects,
pD, [q a semi-lattice of descriptions or patterns,
δ a mapping such as δpgq P D describes object g.
The Galois connection for pG, pD, [q, δq is deﬁned as:
The maximal description representing the similarity of a set of objects:
A˝
“ [gPAδpgq for A Ď G
The maximal set of objects sharing a given description:
d˝
“ tg P G|d Ď δpgqu for d P pD, [q

Standard FCA as a Pattern Structure pG, pD, [q, δq
Considering a standard formal context pG, M, Iq:
G is the set of objects,
pD, [q corresponds to ℘pMq where M is the set of attributes.
δpgq corresponds to the description of g in terms of attributes.
The Galois connection:
m1 m2 m3
g1 ˆ ˆ
g2 ˆ ˆ
g3 ˆ ˆ
g4 ˆ ˆ
g5 ˆ ˆ ˆ
A˝ “ [gPAδpgq for A Ď G
tg1, g2u
1
“ g
1
1 X g
1
2 “ tm1, m2u X tm1, m3u “ tm1u
d˝ “ tg P G|d Ď δpgqu for d P pD, [q
tm1u
1
“ tgi P G|tm1u Ď g
1
i u “ tg1, g2, g5u

From FCA to Pattern Structures
A formal context pG, M, Iq is based on a
set of objects G, a set of attributes M,
and a binary relation I Ď G ˆ M.
Two derivation operators are defined as
follows, @A Ď G, B Ď M:
A1
“ tm P M|@g P A, pg, mq P Iu
B1
“ tg P G|@m P B, pg, mq P Iu
A formal concept pA, Bq verifies A1 “ B
and A “ B1.
Formal concepts are partially ordered
w.r.t. inclusion of extents (or dually of
intents):
pA1, B1q ď pA2, B2q iff A1 Ď A2
A pattern structure pG, pD, [q, δq is
based on a set of objects G, a meet
semi-lattice of object descriptions
pD, [q, and a mapping δ : G ÝÑ D
which associates a description to each
object.
Two derivation operators are defined as
follows, @A Ď G, d P pD, [q:
A˝
“ [gPAδpgq
d˝
“ tg P G|d Ď δpgq
A formal concept pA, dq verifies A˝ “ d
and A “ d˝
Pattern concepts are partially ordered
w.r.t. inclusion of extents (or dually
inclusion of intents):
pA1, d1q ď pA2, d2q iff A1 Ď A2

Interval Pattern Structure
Let D be a set of intervals with integer bounds (for simplicity),
let [ be a meet operator deﬁned on D as the convex hull of intervals:
ra1, b1s [ ra2, b2s “ rminpa1, a2q, maxpb1, b2qs
r4, 5s [ r5, 5s “ r4, 5s
ra1, b1s Ď ra2, b2s ðñ ra2, b2s Ď ra1, b1s
r4, 5s Ď r5, 5s ðñ r5, 5s Ď r4, 5s

Interval Pattern Structure
m1 m2 m3
g1 5 7 6
g2 6 8 4
g3 4 8 5
g4 4 9 8
g5 5 8 5
tg1, g2u˝
“ [gPtg1,g2uδpgq
“ x5, 7, 6y [ x6, 8, 4y
“ xr5, 6s, r7, 8s, r4, 6sy
xr5, 6s, r7, 8s, r4, 6sy˝
“ tg P G|xr5, 6s, r7, 8s, r4, 6sy Ď δpgqu
“ tg1, g2, g5u
ptg1, g2, g5u, xr5, 6s, r7, 8s, r4, 6syq is a pattern concept

FCA and Pattern Structures
Interval pattern concept lattice
ptg1, g2, g5u, xr5, 6s, r7, 8s, r4, 6syq is a pattern concept
Highest concepts: largest extents and smallest intents (but the largest
intervals),
Lowest concepts: smallest extents and largest intents (but the smallest
intervals),
Problem: eﬃcient pattern mining.
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Interval pattern concept lattice

RDF Pattern Structures
Back to RDF Triple Descriptions
tid Subject Predicate Object Provenance
t4 s1 rdf:type Drug dataset2
t5 o11 rdf:type C1 dataset3
t6 C1 rdfs:subClassOf C3 dataset3
t7 o26 rdf:type D6 dataset4
t8 D6 rdfs:subClassOf D8 dataset4
...
...
...
...
...
Table : Sample RDF triple store from several resources.
Entities S Descriptions Ds
s1 {(p1:{C1, C5, C7}), (p2:{D6})}
s2 {(p1:{C1, C13, C11}), (p2:{D7})}
s3 {(p1:{C5}), (p2:{D7})}
s4 {(p1:{C13, C11, C7}), (p2:{D1})}
s5 {(p1:{C1, C7}), (p2:{D6})}
Table : RDF Triples as entities S and descriptions Ds.
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RDF Schema
C4
C3
C2
C1
C6
C5
C10
C9
C8
C7
C12
C11 C13
Figure : RDF Schema for p1.
D5
D4
D3
D2
D1
D9
D8
D6 D7
Figure : RDF Schema for p2.
19 / 31

Similarity Operation between Two Classes
Deﬁnition (Least Common Subsumer)
Given a partially ordered set pS, ďq, a least common subsumer E of two
classes C and D (lcs(C,D) for short) in a partially ordered set is a class
such that C ď E and D ď E and E is least i.e., if there is a class E1 such
that C ď E1 and D ď E1 then E ď E1.
C4
C3
C2
C1
C6
C5
C10
C9
C8
C7
C12
C11 C13
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Similarity Operation between Two Set of classes
δps1q “ xp1 : tC1, C5, C7uy
C4
C3
C2
C1
C6
C5
C10
C9
C8
C7
C12
C11 C13
21 / 31

C4
C3
C2
C1
C6
C5
C10
C9
C8
C7
C12
C11 C13
C1 , C3 and C4 are comparable i.e., C4 ě C3 ě C1 then the most speciﬁc
element is considered i.e., C1.
22 / 31

C4
C3
C2
C1
C6
C5
C10
C9
C8
C7
C12
C11 C13
δps1q [ δps2q “ xp1 : tC1, C9uy
22 / 31

δps1q “ xp1 : tC1, C5, C7uyxp2 : tD6uy
δps2q “ xp1 : tC1, C13, C11uyxp2 : tD7uy
C4
C3
C2
C1
C6
C5
C10
C9
C8
C7
C12
C11 C13
D5
D4
D3
D2
D1
D9
D8
D6 D7
Similarity between s1 and s2:
δps1q [ δps2q “ xp1 : tC1, C9uy, xp2 : tD8uy.
23 / 31

Building a Pattern Concept Lattice
S Ds
s1 {(p1:{C1, C5, C7}), (p2:{D6})}
s2 {(p1:{C1, C13, C11}), (p2:{D7})}
s3 {(p1:{C5}), (p2:{D7})}
s4 {(p1:{C13, C11, C7}), (p2:{D1})}
s5 {(p1:{C1, C7}), (p2:{D6})}
ts1, s2ul
“
ę
sPts1,s2u
δpsq
“ δps1q [ δps2q
“ xpp1 : tC1, C9uqpp2 : tD8uqy
xpp1 : tC1, C9uqpp2 : tD8uqyl
“ ts P S|xpp1 : tC1, C9uqpp2 : tD8uqy Ď δpsqu
“ ts1, s2, s5u
Pattern Concept
xts1, s2, s5u, pp1 : tC1, C9uqpp2 : tD8uqy is a pattern concept.
24 / 31

Pattern Concept Lattice
K#8
K#4 K#9
K#5 K#6 K#2 K#10 K#11
K#13
K#7 K#3 K#1 K#12
K#0
Figure : Pattern Concept lattice
K#ID Extent Intent
K#1 {s1} (p1:{C7, C1, C5}), (p2:{D6})
K#2 {s1, s2, s5} (p1:{C1, C9}), (p2:{D8})
K#3 {s2} (p1:{C1, C11, C13}), (p2:{D7})
K#4 {s1, s2, s3, s5} (p1:{C3}), (p2:{D8})
K#5 {s1, s3} (p1:{C5}), (p2:{D8})
K#6 {s2, s3} (p1:{C3}), (p2:{D7})
K#7 {s3} (p1:{C5}), (p2:{D7})
K#8 {s1, s2, s3, s4, s5} (p1:{C4}), (p2:{D5})
K#9 {s1, s2, s4, s5} (p1:{C9}), (p2:{D5})
K#10 {s1, s4, s5} (p1:{C7}), (p2:{D5})
K#11 {s2, s4} (p1:{C11, C13}), (p2:{D5})
K#12 {s4} (p1:{C7, C11, C13}), (p2:{D1})
K#13 {s1, s5} (p1:{C1, C7}), (p2:{D6})
Table : Details of Pattern Concept lattice
25 / 31

Pattern Concept Lattice
K#ID Extent Intent
K#2 {s1, s2, s5} (p1:{C1, C9}), (p2:{D8})
K#4 {s1, s2, s3, s5} (p1:{C3}), (p2:{D8})
C “ pp1 : tC3uq, pp2 : tD8uq
D “ pp1 : tC1, C9uq, pp2 : tD8uq
C [ D “ pp1 : tC3uq, pp2 : tD8uq [
pp1 : tC1, C9uq, pp2 : tD8uq
C [ D “ pp1 : tlcspC3, C1q, lcspC3, C9quq,
pp2 : tlcspD8, D8quq
C [ D “ pp1 : tC3, C4uq, pp2 : tD8uq
C [ D “ pp1 : tC3uq, pp2 : tD8uq “ C
C [ D “ C ô C Ď D
@c1 P C, Dd1 P D, d1 ď c1
26 / 31

Navigating the Pattern Concept Lattice
Navigating Concept Lattice
Search for Cardiovascular Agents causing Allergic Conditions.
C3 “ Cardiac Disorders, D8 “ Cardiovascular Agents, C9 “ Allergic
Conditions
K#8
K#4 K#9
K#5 K#6 K#2 K#10 K#11
K#13
K#7 K#3 K#1 K#12
K#0
K#ID Extent Intent
K#1 {s1} (p1:{C7, C1, C5}), (p2:{D6})
K#2 {s1, s2, s5} (p1:{C1, C9}), (p2:{D8})
K#3 {s2} (p1:{C1, C11, C13}), (p2:{D7})
K#4 {s1, s2, s3, s5} (p1:{C3}), (p2:{D8})
K#5 {s1, s3} (p1:{C5}), (p2:{D8})
K#6 {s2, s3} (p1:{C3}), (p2:{D7})
K#7 {s3} (p1:{C5}), (p2:{D7})
K#8 {s1, s2, s3, s4, s5} (p1:{C4}), (p2:{D5})
K#9 {s1, s2, s4, s5} (p1:{C9}), (p2:{D5})
K#10 {s1, s4, s5} (p1:{C7}), (p2:{D5})
K#11 {s2, s4} (p1:{C11, C13}), (p2:{D5})
K#12 {s4} (p1:{C7, C11, C13}), (p2:{D1})
K#13 {s1, s5} (p1:{C1, C7}), (p2:{D6})
27 / 31

Search for Cardiovascular Agents causing Allergic Conditions.
Conditions, C1 “ Acute Coronary Syndrome.
K#8
K#4 K#9
K#5 K#6 K#2 K#10 K#11
K#13
K#7 K#3 K#1 K#12
K#0
K#ID Extent Intent
K#1 {s1} (p1:{C7, C1, C5}), (p2:{D6})
K#2 {s1, s2, s5} (p1:{C1, C9}), (p2:{D8})
K#3 {s2} (p1:{C1, C11, C13}), (p2:{D7})
K#4 {s1, s2, s3, s5} (p1:{C3}), (p2:{D8})
K#5 {s1, s3} (p1:{C5}), (p2:{D8})
K#6 {s2, s3} (p1:{C3}), (p2:{D7})
K#7 {s3} (p1:{C5}), (p2:{D7})
K#8 {s1, s2, s3, s4, s5} (p1:{C4}), (p2:{D5})
K#9 {s1, s2, s4, s5} (p1:{C9}), (p2:{D5})
K#10 {s1, s4, s5} (p1:{C7}), (p2:{D5})
K#11 {s2, s4} (p1:{C11, C13}), (p2:{D5})
K#12 {s4} (p1:{C7, C11, C13}), (p2:{D1})
K#13 {s1, s5} (p1:{C1, C7}), (p2:{D6})
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Search for Cardiovascular Agents not causing Allergic Conditions.
Conditions, C1 “ Acute Coronary Syndrome.
K#8
K#4 K#9
K#5 K#6 K#2 K#10 K#11
K#13
K#7 K#3 K#1 K#12
K#0
K#ID Extent Intent
K#1 {s1} (p1:{C7, C1, C5}), (p2:{D6})
K#2 {s1, s2, s5} (p1:{C1, C9}), (p2:{D8})
K#3 {s2} (p1:{C1, C11, C13}), (p2:{D7})
K#4 {s1, s2, s3, s5} (p1:{C3}), (p2:{D8})
K#5 {s1, s3} (p1:{C5}), (p2:{D8})
K#6 {s2, s3} (p1:{C3}), (p2:{D7})
K#7 {s3} (p1:{C5}), (p2:{D7})
K#8 {s1, s2, s3, s4, s5} (p1:{C4}), (p2:{D5})
K#9 {s1, s2, s4, s5} (p1:{C9}), (p2:{D5})
K#10 {s1, s4, s5} (p1:{C7}), (p2:{D5})
K#11 {s2, s4} (p1:{C11, C13}), (p2:{D5})
K#12 {s4} (p1:{C7, C11, C13}), (p2:{D1})
K#13 {s1, s5} (p1:{C1, C7}), (p2:{D6})
27 / 31

Experiments and Conclusion
Experimentation
Coded in C++.
Real datasets Biomedical Data and DBLP.
Subsets of drug data were considered, i.e., Cardiovascular Agents and
Central Nervous System.
The selected datasets:
Drugbank (RDF triples)
Sider (RDF triples)
MedDRA (RDF Schema - BioPortal)
MeSH (RDF Schema - Bio2RDF)
28 / 31

Experimentation
Datasets No. of Triples No. of Subjects No. of Objects Runtime
Cardiovascular Agents 31098 145 927 0-22 sec
Central Nervous System 22680 105 1050 0-25 sec
Table : Statistics of two datasets and index lattice.
(a) Index Size for Drugbank Dataset
Index Size Reduction
Hiding non-interesting
parts.
Removing general classes
from RDF Schema.
Support.
Stability.
29 / 31

Conclusion and Future Work
Obtained index provide the following benefits:
Classification of RDF triples with respect to RDF Schema.
Simultaneous access over RDF triples as well as RDF schema.
One platform for navigating and querying several resources.
Heterogeneous data i.e., when the reference schema is not present such
as proteins can be easily dealt with.
Results are visualized using RV-Xplorer1
.
Future Work:
Provide a better formalization of the framework.
Provide more realistic usecases.
Define a new similarity measure to deal with graph-based structures.
1
A new tool developed by our team for allowing lattice interaction, (Alam et. al.
International Conference on Concept Lattice and their Applications, 2015.)
30 / 31

Navigating and Exploring RDF Data using Formal Concept Analysis

Navigating and Exploring RDF Data using Formal Concept Analysis

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