=Paper=
{{Paper
|id=Vol-2668/paper15
|storemode=property
|title=Practical Comparison of FCA Extensions to Model Indeterminate Value of Ternary Data
|pdfUrl=https://ceur-ws.org/Vol-2668/paper15.pdf
|volume=Vol-2668
|authors=Priscilla Keip,Sébastien Ferré,Alain Gutierrez,Marianne Huchard,Pierre Silvie,Pierre Martin
|dblpUrl=https://dblp.org/rec/conf/cla/KeipFGHS020
}}
==Practical Comparison of FCA Extensions to Model Indeterminate Value of Ternary Data==
https://ceur-ws.org/Vol-2668/paper15.pdf
Practical Comparison of FCA Extensions to
Model Indeterminate Value of Ternary Data ?
Priscilla Keip1 , Sébastien Ferré2 , Alain Gutierrez, Marianne Huchard3 ,
Pierre Silvie1,4 , and Pierre Martin1
1
CIRAD, UPR AIDA, F-34398 Montpellier, France
AIDA, Univ Montpellier, CIRAD, Montpellier, France
{priscilla.keip,pierre.silvie,pierre.martin}@cirad.fr
2
Univ Rennes, CNRS, IRISA, 35042 Rennes, France
ferre@irisa.fr
3
LIRMM, Université de Montpellier, CNRS, Montpellier, France
marianne.huchard@lirmm.fr
4
IRD, UMR IPME, F-34394 Montpellier, France
Abstract. The Knomana knowledge base brings together knowledge
from the scientific literature on the use of plants with pesticidal or an-
tibiotic effects on animals, plants, and human beings to propose pro-
tection solutions using local plants. In this literature, the elements of
the 3-tuple (protected organism, protecting plant, pest) are named us-
ing the binomial nomenclature consisting of the genus name followed by
the species name. In some instances, authors use the abbreviation “sp.”
in the singular or “spp.” in the plural, as species name, to indicate the
indeterminate status of the species for a guaranteed genus. To suggest
protection solutions, the indeterminacy of the species has to be hypoth-
esized based on assigning the sp./spp. to the other species in the same
genus and conversely. This paper discusses the classification of ternary
data containing some indeterminate values generated by three extensions
of Formal Concept Analysis.
Keywords: Graph-FCA · Relational Concept Analysis · Triadic Con-
cept Analysis
1 Introduction
According to the Committee on World Food Security, governments have to im-
prove small food producers’ access to knowledge on the use of biodiversity for
crop protection to enhance food security and nutrition [7]. With the aim of
proposing protection solutions based on the use of local plants, the Knomana
Knowledge Base (KKB) brings together knowledge from the scientific literature
on the use of plants with pesticidal or antibiotic effects on animals, plants, and
?
This work was supported by the French National Research Agency under the Invest-
ments for the Future Program, referred as ANR-16-CONV-0004.
Copyright c 2020 for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0). Published in Francisco
J. Valverde-Albacete, Martin Trnecka (Eds.): Proceedings of the 15th International
Conference on Concept Lattices and Their Applications, CLA 2020, pp. 197–208,
2020.
198 Priscilla Keip et al.
human beings [13]. The methodology used to suggest protection solutions is
based on Formal Concept Analysis [9].
In KKB, a plant use is characterized using 70 descriptors, for example, the
part of plant used to prepare the bio-product, the method of preparation and
its efficacy. In early January 2020, KKB contained 42,400 plant use descriptions
from 630 documents, dated 1957 to 2019. These descriptions include a total
of 1,782 plants to protect 64 (crop, animal and human) systems against 451
bio-aggressors (pests, diseases, bacteria, viruses, etc.). This 3-tuple (protected
organism, protecting plant, pest5 ) is the core of the protection solution as its
efficacy is linked to the interactions between these three elements.
In the literature, each element of the 3-tuple is named using the binomial
nomenclature, which consists of the genus name followed by the species name.
This designation is a widely accepted convention for the formal scientific naming
of the biological organisms, and avoids the ambiguity associated to vernacular
names that are not shared by all communities. For the name of some species
within the 3-tuples, authors use the abbreviation “sp.” in the singular or “spp.”
in the plural. The abbreviation sp. indicates that the genus name of the organ-
ism is known by the authors but its species name is not; the abbreviation spp.
indicates that at least two organisms of the same genus are considered without
their names being mentioned.
To classify 3-tuples using Formal Concept Analysis (FCA), the presence of
the sp. or spp. abbreviation in a formal context corresponds to the absence of
the name of the species, and thus to indeterminate value of data. This indeter-
minacy has to be hypothesized to suggest protection solutions. This paper deals
with ternary relationships with FCA and discusses how to account for the in-
determinate specification of species. Three extensions of FCA, Triadic Concept
Analysis (TCA) [11], Relational Concept Analysis (RCA) [5], and Graph-FCA
(GFCA) [3], were qualitatively and quantitatively evaluated on two reduced
KKB datasets.
Section 2 presents a short excerpt of KKB to show the two issues addressed in
this paper. Sections 3, 4 and 5 present solutions using, respectively, TCA, RCA,
and GFCA. Section 6 compares the different solutions and Section 7 concludes
the paper.
2 Indeterminate Information and Ternary Relationships
in the Knomana Knowledge Base
For this work, two datasets were extracted from KKB. They describe known
protection solutions against species of Spodoptera, including the pest Spodoptera
frugiperda that is currently spreading very rapidly in Africa and is devastat-
ing maize, tomato and cabbage crops. The first dataset is composed of 12 3-
tuples describing relations between six organisms, nine plants, and three pests
(S. littoralis, S. litura, and Spodoptera spp.). This dataset enabled our qualita-
tive evaluation that consisted in comparing the classifications provided by the
5
Taken in this paper in the broadest sense to designate bio-aggressors.
FCA for indeterminate data in ternary relationships 199
S. spp+any
S. litu+spp S. litto+spp
S. litu S. spp S. litto
Fig. 1. Aggregation hierarchy used to propagate the spp value in the first dataset.
FCA extensions to highlight similarities and differences among them. The second
dataset was used for the quantitative evaluation of scalability. It is composed of
34 3-tuples describing relations between six organisms, 30 plants and four pests,
Spodoptera spp., S. frugiperda, S. littoralis, and S. litura. This dataset includes
the first one composed of three pests.
In both these datasets, the pests all belong to the genus Spodoptera. One is
called Spodoptera spp., showing it refers to at least two species of the Spodoptera
genus. It is not mentioned if the species are S. littoralis, S. litura, and S.
frugiperda or others. To deal with this indeterminate value, we defined the three
levels of aggregation presented in Fig. 1. The known species and the Spodoptera
spp. indeterminate species are located at the bottom of the hierarchy. The hypo-
thetical species, resulting from the aggregation of the ones located at the bottom,
are located above: S. litu+spp (resp. S. litto+spp) gathers information on S.litu
(resp. S. litto) and S.spp. That is, it encompasses known information on S. litu
(resp. S. litto) or at least two Spodoptera species, possibly including S.litu (resp.
S. litto). At the top of the hierarchy, S. spp+any gathers information on S. litto,
S. litu, and S. spp. The use of this hierarchy differs depending on the FCA
extension concerned and is described in the following sections.
Another characteristic of KKB is a central ternary relationship connecting
the three main components: protected organisms, protecting plants and pests.
FCA [4] and RCA consider binary relationships that require application of con-
ceptual model transformations as proposed in the database domain [8]. The two
other FCA extensions studied, TCA and Graph-FCA, are directly applicable to
this kind of data: TCA because it was introduced for this purpose, Graph-FCA
because it allows the definition of n-ary relationships of any dimension.
Both datasets and all files, corresponding to each evaluation of the three next
sections, are available online [12].
3 Triadic Concept Analysis (TCA)
Triadic concept analysis (TCA) was introduced by Lehmann and Wille to deal
with situations where “an object g has the attribute m under the condition b”
[11]. By extension, it can be considered to deal with any ternary relationship.
In our case, the ternary relationship can be embedded in such a scheme. Pro-
200 Priscilla Keip et al.
tected organisms, protecting plants, and pests can be used respectively as ob-
jects, attributes and conditions. A triadic context is a 4-tuple K = (G, M, B, Y )
where G is the set of objects, M the set of attributes, B the set of condi-
tions, and Y ⊆ G × M × B associates an object and an attribute under a
condition. For example, Table 1 (the three tables in the top row) is a triadic
context in which the objects are protected organisms, the attributes are pro-
tecting plants, and the conditions are pests. It associates object A.escu with
attribute A.indi under condition S.litto, meaning that A.indi treats A.escu when
attacked by S.litto. Let us denote this relationship IT , and IT[P rotected,P lant]
the mapping of IT on the first two components. Triplets inferred from this ini-
tial knowledge with the addition of S.litto+spp, S.litu+spp and S.spp+any, to
deal with indeterminate value, are presented in the three tables in the bottom
row in Table 1: ∀(po, pl, pe) ∈ IT[P rotected,P lant] × {S.litto, S.litu}, (po, pl, pe +
spp); ∀(po, pl, pe) ∈ IT[P rotected,P lant] × {S.spp}, (po, pl, S.litto + spp) and
(po, pl, S.litu + spp); ∀(po, pl) ∈ IT[P rotected,P lant] , (po, pl, S.spp + any). It is easy
to generalize this addition to species other than S.litto and S.litu.
A triadic concept (A1, A2, A3) of (G, M, B, Y ) satisfies A1 ⊆ G, A2 ⊆ M
and A3 ⊆ B as well as: A1 × A2 × A3 ⊆ Y ((A1, A2, A3) is a rectangular
parallelepiped full of ×); and X1 × X2 × X3 ⊆ Y , A1 ⊆ X1, A2 ⊆ X2
and A3 ⊆ X3 implies that (A1, A2, A3) = (X1, X2, X3) (formal concept max-
imality). A1, A2 and A3 are respectively called extent, intent and modus of
the triadic concept (A1, A2, A3). Table 1 highlights in red the triadic concept
TC5 = ({A.escu, B.ole}, {A.indi, C.papa}, {S.litto, S.litto + spp, S.spp + any}).
Selected triadic concepts for the triadic context of Table 1 are presented in Ta-
ble 2 and were built using FCA Tools Bundle6 . They reveal several facts on the
Spodoptera excerpt. We chose them because they are non-trivial and provide
valuable information on data. TC2 indicates that no plant protects all organ-
isms against Spodoptera. TC5 groups A.escu and B.ole, which are protected
against S.litto by A.indi and C.papa. This shows that A.indi and C.papa can
replace one another to control S.litto. As shown by TC6, A. indi controls some
Spodoptera species on the three plants A.escu, B.ole and Z.mays.
4 Relational Concept Analysis (RCA)
Relational Concept Analysis (RCA) [5] was developed to view a dataset (called
a relational context family, or RCF) as a set of objects of several categories
(e.g. protecting plants, protected organisms and pests), connected through sev-
eral relationships (e.g. an organism is protected by a plant and a plant treats
a pest). Three object-attribute contexts (Plants and Protected, with nominal
scaling, and Pests with hierarchical scaling) describe the three object cate-
gories, while two object-object contexts (protectedBy and treats) encode the
relationships. These relationships contain the same information as that listed
in Table 1. RCA acts as an iterative process with several steps. Relations are
considered through the addition of relational attributes based on quantifiers
6
https://fca-tools-bundle.com/
FCA for indeterminate data in ternary relationships 201
Table 1. Triadic context, with highlighted concept TC5 (in red)
D. dume
D. dume
D. dume
W. pros
W. pros
W. pros
C. papa
C. papa
C. papa
V. parv
V. parv
V. parv
V. cane
V. cane
V. cane
C. opul
C. opul
C. opul
V. fusc
V. fusc
V. fusc
A. indi
A. indi
A. indi
C. spp
C. spp
C. spp
S. litu S. spp S. litto
Z. mays Z. mays × × × Z. mays
A. escu A. escu A. escu × ×
B. ole B. ole B. ole × ×
G. hirs G. hirs G. hirs ×
S. lyco S. lyco S. lyco × × ×
R. com × R. com R. com
W. prost
D. dume
D. dume
D. dume
W. pros
W. pros
C. papa
C. papa
C. papa
V. parv
V. parv
V. cane
V. cane
V. parv
V. cane
C. opul
C. opul
C. opul
V. fusc
V. fusc
A. indi
A. indi
V. fusc
A. indi
C. spp
C. spp
C. spp
S.litu+spp S.spp+any S.litto+spp
Z. mays × × × Z. mays × × × Z. mays × × ×
A. escu A. escu × × A. escu × ×
B. oler B. oler × × B. oler × ×
G. hirs G. hirsu × G. hirs ×
S. lyco S. lyco × × × S. lyco × × ×
R. com × R. com × R. com
Table 2. Three triadic concepts among the 9 concepts built using Table 1
TC2 Protected A.escu B.ole G.hirsu R.com S.lyco Z.mays
Plants
Pests S.litto S.litto+spp S.litu S.litu+spp S.spp S.spp-any
TC5 Protected A.escu B.ole
Plants A.indi C.papa
Pests S.litto S.litto+spp S.spp-any
TC6 Protected A.escu B.ole Z.mays
Plants A.indi
Pests S.litto+spp S.spp-any
inspired by description logic operators [1]. The detailed process is described
in [5] and for this paper, we used the RCAExplore tool [2] available online7 .
Figure 2 shows an excerpt of the result. The concept number in the lattices
is respectively 12 (protected organisms), 14 (plants), and seven (pests). Con-
cept CP lant13 (Fig. 2 (middle)) groups plants (A. indi, C. papa) that treat
at least one pest in CP est1 (S.litto) (relational attribute ∃treats(CP est1)).
Concept CP rot8 (Fig. 2 (left)) groups organisms (A.escu, B.ole) that are
protected by at least one plant in CP lant2, CP lant1 and CP lant13
(relational attributes ∃protectedBy(CP lant2), ∃protectedBy(CP lant1) and
∃protectedBy(CP lant13), the latter being inherited). This concept chain, which
crosses the three lattices, corresponds to Triadic concept TC5. It would also be
found with quantifier ∃⊇. Similarly, the Concept chain CP rot9, CP lant1 and
CP est6 corresponds to Triadic concept TC6. Triadic concept TC2 does not
appear with quantifier ∃, because all protected organisms have at least one pro-
tecting plant against one pest: situation (CP o, CP l, CP e) such that ∀po ∈ CP o,
∃pl ∈ CP l, (po, pl) ∈ protectedBy, and ∃pe ∈ CP e, (pl, pe) ∈ treats, that could
be summarized as pattern ∃protectedBy∃treats. TC2 would appear with quan-
tifier ∃⊇.
RCA enables various encodings of datasets. For example, two alternatives can
be proposed for the above encoding named RCAC (for RCA with Chain). They
7
http://dataqual.engees.unistra.fr/logiciels/rcaExplore
202 Priscilla Keip et al.
CProt7
∃ protectedBy(CPlant10)
CProt11 CPlant10
CProt10
∃ protectedBy(CPlant13) ∃ treats(CPest6)
∃ protectedBy(CPlant12)
∃ protectedBy(CPlant14)
CProt6 CProt3 CPlant14 CPlant12
CProt9
R.com G.hirsu ∃ treats(CPest4) ∃ treats(CPest5)
∃ protectedBy(CPlant11)
∃ protectedBy(CPlant9) ∃ protectedBy(CPlant1) ∃ protectedBy(CPlant3)
R.com G.hirsu CPest6
CPlant9
S.spp+any
CProt4 CPlant13 CPlant11 W.pro
CProt5
∃ treats(CPest1)
S.lyco ∃ treats(CPest3) ∃ treats(CPest2)
Z.mays CProt8 ∃ protectedBy(CPlant6) CPest4 CPest5
∃ protectedBy(CPlant8) ∃ protectedBy(CPlant2) ∃ protectedBy(CPlant5) W.pro
∃ protectedBy(CPlant7) S.litto+spp S.litu+spp
∃ protectedBy(CPlant4)
Z.mays
S.lyco
CPlant3 CPlant4 CPlant5 CPlant6 CPlant2 CPlant1 CPlant8 CPlant7
D.dume V.cane V.fusca V.parvi C.papa A.indi C.opu C.spp
CPest1 CPest3 CPest2
CProt1 CProt2
A.escuD.dume
B.ole V.cane V.fusca V.parvi C.papa A.indi C.opu C.spp
S.litto S.spp S.litu
A.escu B.ole S.litto S.spp S.litu
CPlant0
CProt0 ∃ treats(CPest0) CPest0
Fig. 2. RCAC Concept lattices: protected org. (left), plants (middle), pests (right)
∃ protectedBy(CPlant0)
are illustrated with quantifier ∃. The conceptual models corresponding to these
variants are shown on the left in Figure 4. A first alternative (called RCAR for
RCA with Reification) consists of reifying the ternary relation that is thus added
as a fourth object-attribute context to the RCF, whose objects are the 3-tuples
of the relation. This results in 4 lattices. Figure 3 shows an excerpt of the central
lattice (24 concepts) where the objects are the 3-tuples. The other lattices are
quite flat, or with a hierarchy (the case of pests). Concept CP ion22 groups the
3-tuples corresponding to Triadic Concept TC5, and the relational attribute
∃manages(CP est1) (S.litto). Their respective plants and protected organisms
are separated into sub-concepts of CP ion22. Triadic concept TC6 corresponds
to CP ion19, and the relational attributes indicate that S.litto+spp is managed
(∃manages(CP est4)) and the plant used is A.indi (∃uses(CP lant1)). A second
alternative (called RCAG for RCA with Graph view) consists of an RCF com-
posed of two categories that describe protected organisms and pests, respectively,
and one relationship per protecting plant. This results in two lattices, with nine
protected organism concepts (Fig. 4 (middle)), and seven pest concepts (Fig. 4
(right)). Triadic concept TC5 (resp. TC6) appears as CP rot8 (resp. CP rot9).
The shared relational attributes describe protecting plants as roles pointing to
controlled pests.
5 Graph-FCA (GFCA)
Graph-FCA (or GFCA) [3] was introduced as an extension of classical FCA
to knowledge graphs, or to be more precise, to directed multi-hyper-graphs.
A graph context is a 3-tuple K = (O, A, I) like in FCA except that objects
play the role of nodes in the graph, and elements of the incidence relation I
are directed hyper-edges connecting objects and labeled with attributes. Unary
attributes connect a single node: they act as node labels and correspond to
CPion0
∃ manages(CPest6)
∃ protects(CProt7)
∃ uses(CPlant10)
FCA for indeterminate data in ternary relationships 203
CPion23
∃ manages(CPest4)
CPion21 CPion19 CPion22
∃ manages(CPest5) ∃ uses(CPlant1) ∃ manages(CPest1)
CPion10
CPion20 C
∃ manages(CPest2) CPion14 CPion17 CPion16 CPion18
CPion15 ∃ prote
∃ protects(CProt6) ∃ manages(CPest3)
∃ protects(CProt1) ∃ protects(CProt2) ∃ uses(CPlant2) ∃ protects(CProt4) ∃ uses
∃ uses(CPlant9) ∃ protects(CProt5)
G.hirsu_D
R.com_W.pro_S.litu
CPion13 CPion12 CPion8 CPion7 CPion
CPion11 CPion2 CPion4 CPion3 CPion5
∃ uses(CPlant8) ∃ uses(CPlant7) ∃ uses(CPlant5) ∃ uses(CPlant4) ∃ uses(CP
Z.mays_A.indi_S.spp A.escu_A.indi_S.litto B.ole_A.indi_S.litto A.escu_C.papa_S.litto B.ole_C.papa_S.litto
Z.mays_C.opu_S.spp Z.mays_C.spp_S.spp S.lyco_V.fusca_S.litto S.lyco_V.cane_S.litto S.lyco_V.parv
CPion1
∃ manages(CPest0)
Fig. 3. Pests (RCAR)
CProt7
∃ protects(CProt0)
∃ uses(CPlant0)
CProt6
R.com CProt9
∃ W.pro(CPest5) ∃ A.indi(CPest4)
∃ W.pro(CPest2) ∃ A.indi(CPest6)
∃ W.pro(CPest6)
R.com
CProt4
S.lyco
∃ V.cane(CPest4)
CProt3 CProt8 ∃ V.cane(CPest1)
G.hirsu ∃ A.indi(CPest1) ∃ V.cane(CPest6)
∃ D.dume(CPest4) ∃ C.papa(CPest4) ∃ V.fusca(CPest4)
∃ D.dume(CPest1) ∃ C.papa(CPest1) ∃ V.fusca(CPest1)
∃ D.dume(CPest6) ∃ C.papa(CPest6) ∃ V.fusca(CPest6)
G.hirsu ∃ V.parvi(CPest4)
∃ V.parvi(CPest1)
∃ V.parvi(CPest6)
S.lyco
CPest6
CProt5 S.spp+any
Z.mays
∃ A.indi(CPest5)
∃ A.indi(CPest3)
∃ C.spp(CPest5) CPest4 CPest5
CProt1 CProt2 ∃ C.spp(CPest3) S.litto+spp S.litu+spp
A.escu B.ole ∃ C.spp(CPest4)
A.escu B.ole ∃ C.spp(CPest6)
∃ C.opu(CPest5)
∃ C.opu(CPest3) CPest1 CPest3 CPest2
∃ C.opu(CPest4)
∃ C.opu(CPest6) S.litto S.spp S.litu
Z.mays S.litto S.spp S.litu
CProt0 CPest0
Fig. 4. Data models (left), RCAG lattices: Protected
∃ A.indi(CPest2) org. (middle), pests (right)
∃ A.indi(CPest0)
∃ C.papa(CPest5)
∃ C.papa(CPest3)
∃ C.papa(CPest2)
∃ C.papa(CPest0)
∃ D.dume(CPest5)
∃ D.dume(CPest3)
classical FCA attributes. Binary attributes connect one node to another, and ∃ D.dume(CPest2)
∃ D.dume(CPest0)
therefore represent relationships or links between objects. N-ary attributes can ∃ V.cane(CPest5)
∃ V.cane(CPest3)
encode more complex relationships involving more than two participants (e.g., ∃ V.cane(CPest2)
∃ V.cane(CPest0)
∃ V.fusca(CPest5)
“plant X protects organism Y against a pest Z”). The intents of graph concepts ∃ V.fusca(CPest3)
∃ V.fusca(CPest2)
are like conjunctive queries, i.e. graph patterns with a distinguished node. GFCA ∃ V.fusca(CPest0)
∃ V.parvi(CPest5)
also defines n-ary graph concepts, but here we only consider unary concepts. ∃ V.parvi(CPest3)
∃ V.parvi(CPest2)
∃ V.parvi(CPest0)
Choosing a Graph-FCA representation of the 3-tuples (Protected,Plant,Pest) ∃ C.spp(CPest2)
∃ C.spp(CPest0)
amounts to choosing what the objects and the attributes are. An immediate ∃ C.spp(CPest1)
∃ C.opu(CPest2)
∃ C.opu(CPest0)
representation (GFCA-Ternary) would be to introduce a ternary attribute for ∃ C.opu(CPest1)
∃ W.pro(CPest3)
the ternary relation between protected organisms, protecting plants, and pests: ∃ W.pro(CPest0)
∃ W.pro(CPest4)
e.g., protection(A.escu, A.indi, S.litto). Unary attributes are used with nominal ∃ W.pro(CPest1)
scaling on organisms and plants, and with hierarchical scaling on pests: e.g.,
A.indi (A.indi), S .spp any(S.litto). As a result, we obtain 36 concepts (excluding
bottom concepts): 13 concerning protected organisms, 17 concerning plants, and
204 Priscilla Keip et al.
Fig. 5. Concept lattices (excerpt) for GFCA-Ternary (edges as relational attributes)
6 concerning pests. They are grouped in three connected graph patterns: pattern
Q1 is specific to the 3-tuple (R.com,W.pro,S.litu), and corresponds to triadic
concept TC1; pattern Q3 contains all the other 3-tuples because they are all
interconnected; the concepts in pattern Q2 generalize the concepts of the two
other patterns.
Figure 5 shows an excerpt of GFCA output8 . For instance, concept Q2g
characterizes the four organisms (out of six) that are protected by plants (Q2d)
that protect against S.litu+spp (Q2c). It contains two organisms, A.escu and
B.ole, that are not known to be protected against S.litu+spp by any of those
plants (they are not in concept Q2a, a sub-concept of Q2g). The concerned
plants, i.e. C.opu, C.spp, A.indi, W.pro, are therefore candidates for protection
of the two organisms against S.litu+spp. The triadic concepts can be found in
the GFCA output. TC2 corresponds to concept Q2e, which characterizes the
set of protecting plants, showing that no plant protects all organisms because
its intent contains no specific plant. TC5 corresponds to concept Q3u, which
states that A.indi and C.papa have in common to protect both A.escu (Concept
Q3a) and B.ole (Concept Q3b) against S.litto (Concept Q3n). TC6 corresponds
to concept Q3w, which is the most general concept of the organisms that are
specifically protected by A.indi. Its extent contains A.escu, B.ole, and Z.mays;
and its intent states that the protection is against S.litto+spp.
A more compact representation of the 3-tuples (GFCA-Binary) would
be to use plants as binary relations from organisms to pests: e.g.
A.indi (A.escu, S.litto). There are several possible justifications for this repre-
8
Source code and user manual at https://bitbucket.org/sebferre/graph-fca/
FCA for indeterminate data in ternary relationships 205
Fig. 6. Concept lattices and graph patterns for GFCA-Binary
sentation. First, protecting plants is the focus of the study, and we prefer con-
cept intents that contain explicit references to plants: using them as attributes
is one way to enforce this. Second, the plants concerned can be considered as
agents intervening between organisms and pests. This representation resembles
that of RDF triples where the middle element is used as an edge label [6]. We
applied the same scaling as above for organisms and pests, and simple hierarchi-
cal scaling for plants with a binary edge plant(X, Z) for every 3-tuple (X, Y, Z).
Figure 6 shows the complete GFCA output, here with a graphical representa-
206 Priscilla Keip et al.
Table 3. Classifications obtained by FCA extensions (NA=Not applicable)
TCA RCAC RCAR RCAG GFCA-Ter. GFCA-Binary
#concepts 2nd dataset 12 59 99 18 63 22
2nd dataset content protection 50 (10-NA-9)
(12-38-9) (12-41-10) (12-NA-10)
(organism-plant-pest) (8-32-9) + 30 relations
#concepts 1st dataset 9 33 50 16 36 18
tion of intents: solid arrows represent the edges of the graph patterns that define
concept intents. We obtained 18 concepts grouped in six patterns: 12 concern-
ing organisms, and 6 concerning pests. Protecting plants are displayed as edge
labels. The exclusive use of unary and binary attributes makes the output much
easier to read. Despite the simplification, triadic concepts can still be found in
the GFCA-Binary output, e.g.: TC2 as the edge from Concept Q3b to Q3a;
TC5 as the edge from Q6g to Q6a; TC6 as the edge from Q5b to Q5a.
6 Results of the Whole Use Case and Discussion
This section presents the results of the complete Spodoptera dataset, which is
almost three times larger than the first one (an increase from 12 to 34 3-tuples).
We then discuss the different methods.
Results of the complete Spodoptera dataset. The second dataset contains 34 3-
tuples using six organisms, 30 plants, and four pests, while the first one has
12 3-tuples, using respectively, six, nine and three species. Table 3 presents the
number of concepts per classification for each method. For all the classifications,
more concepts are built by methods using the second dataset. The increase is al-
most 50% for TCA and RCAC and twice as high for RCAR and GFCA-Ternary.
Despite the notable difference in the number of plants (30 vs 9), the classifi-
cations of RCAG and GFCA-Binary contain two and four additional concepts,
respectively. This slight difference results from considering plants as relations be-
tween organisms and pests, which increases the number of relational attributes.
Regarding the other methods, the additional concepts originate from the mod-
eling of the 22 new 3-tuples, which implement the protection of Z.mays against
the additional pest S.frugi using 22 new plants.
Readability and ease of use. TCA and GFCA both present their results as a
single package. TCA lists the triadic concepts, and GFCA connects them in
a single graph. RCA also connects the concepts but a switch between lattices
is required to access the overall classification. When a classification is used as
a support for exploration, classifying the objects per category, as provided by
RCAG, is a good solution. For instance, RCAC and RCAG show that A.escu and
B.ole are equivalent (CProt8) and share more with Z.mays than with the other
species (CProt9). GFCA-Ternary is more difficult to read than GFCA-Binary
because it has more concepts and uses ellipse nodes (not shown in this paper)
in addition to concept nodes to represent ternary relationships.
FCA for indeterminate data in ternary relationships 207
Modeling symmetry. TCA, RCAR, and GFCA-Ternary modeling considers the
three object categories symmetrically. Thus, 3-tuples sharing characteristics at
a given position are recognizable at a glance. RCAG, RCAC, and GFCA-Binary
modeling does not consider the three object categories symmetrically. For in-
stance, RCAC classifies protecting plants in relation to targeted pests and pro-
tected organisms. Reverse and transitive relationships can be included in the for-
mal contexts to achieve symmetry, but the cost is increased complexity. GFCA-
Binary and RCAG handle plants as relation labels instead of objects, implying
the absence of plant concepts in the lattice. Two other representations are also
possible, where protected organisms or pests are used as relation labels.
Visibility of concept hierarchies and relational patterns. Triadic lattices are dif-
ficult to fully visualize and understand, even with the projection paradigm used
by FCA Tools Bundle [10]. RCA clearly depicts the concept hierarchies, one
for each object type, but not the relational patterns. GFCA has three output
modes: only showing hierarchies such as RCA, only showing relational patterns,
or showing the two combined. The latter is the richest representation but it is
more difficult to grasp.
Extensibility and limits. All the methods allow the user to return to the orig-
inal 3-tuples, provided a nominal scaling was used for RCA and GFCA. Like
classical FCA, all the implementations of TCA, RCA (with ∃, ∃∀ and ∃⊇), and
GFCA are sensitive to noisy data, in that missing 3-tuples increase the number
of concepts. For RCA, noisy data can be processed with percentage quantifiers.
TCA is restricted to ternary relations: objects, attributes and conditions cannot
have their own description. These limitations reduce the modeling options. RCA
and GFCA methods can apply different models depending on the expert’s needs.
Using ∃ quantifier, RCA is less precise than the other FCA extensions, but this
allows experts to hypothesize and suggest new protection solutions. As RCA
and GFCA are relatively flexible, adding information on objects of the 3-tuples
is possible, such as taxonomy for the species. GFCA is less flexible than RCA, as
its patterns are exclusively existential, and the relations are two-way. But GFCA
patterns can contain cycles while RCA patterns are (possibly infinite) trees, like
class expressions in description logics [1].
7 Conclusion
The Knomana Knowledge Base (KKB) is a typical example of a dataset in the
environmental domain, characterized by a lack of information concerning the
name of some species and a main ternary relationship. In this paper, we address
these two questions using a KKB excerpt. We introduce an aggregation hierarchy
on species names and abbreviation spp. We define several representations of the
ternary relationship to analyze the KKB excerpt with three FCA extensions.
The discussion highlights the complementarity of the different approaches.
In a future study, we plan to deepen the qualitative analysis on pests, starting
with Spodoptera species and to define an analytical process that takes advantage
208 Priscilla Keip et al.
of the three FCA extensions. We also plan to explore variants of the represen-
tations proposed in this paper. Our long-term objective is to provide domain
experts with efficient tools to analyze their dataset, not only to extract exist-
ing knowledge in a deductive way, but also to assist them in setting up new
hypotheses, and suggesting new experiments for bio-pesticides.
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