The Knomana knowledge base brings together knowledge from the scientific literature on the use of plants with pesticidal or antibiotic effects on animals, plants, and human beings to propose protection solutions using local plants. In this literature, the elements of the 3-tuple (protected organism, protecting plant, pest) are named using 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 hypothesized 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.
According to the Committee on World Food Security, governments have to
improve small food producers’ access to knowledge on the use of biodiversity for
crop protection to enhance food security and nutrition [
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 organism 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
indeterminacy has to be hypothesized to suggest protection solutions. This paper deals
with ternary relationships with FCA and discusses how to account for the
indeterminate specification of species. Three extensions of FCA, Triadic Concept
Analysis (TCA) [
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
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 devastating maize, tomato and cabbage crops. The first dataset is composed of 12 3tuples describing relations between six organisms, nine plants, and three pests (S. littoralis, S. litura, and Spodoptera spp.). This dataset enabled our qualitative evaluation that consisted in comparing the classifications provided by the 5 Taken in this paper in the broadest sense to designate bio-aggressors. 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 hypothetical 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 [
Both datasets and all files, corresponding to each evaluation of the three next
sections, are available online [
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”
[
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 maximality). 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 Table 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 organisms 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) [
S. litu Z. mays A. escu B. ole G. hirs S. lyco R. com ×
WV V V A C C C DS. spp
Z. mays A. escu B. ole G. hirs S. lyco R. com .srop .cean .fscu .ravp ii.nd .spp .lopu .aapp .eudm WV V V A C C C DS. litto × × × ZA.. mesacyus
B. ole G. hirs S. lyco R. com sro en scu ravp iind spp lopu aapp edum .pW.caV .fV .V .A .C .C .C .D × × × × × × × × S.litu+spp Z. mays A. escu B. oler G. hirs S. lyco R. com × .srop .cean .fscu .ravp ii.nd .spp .lopu .aapp .eudm WV V V A C C C DS.spp+any × × × ZA.. mesacyus
B. oler
G. hirsu
SR.. lcyocmo × × × ×
sro en scu ravp iind spp lopu aapp edum tsro en scu ravp iind spp lopu aapp edum
.pW.caV .fV .V .A .C .C .C .DS.litto+spp .pW.caV .fV .V .A .C .C .C .D
××× × × ×× ZBA... moeslaecyrus ××× × × ××
× SRG...lchyoicrmos × × × ×
inspired by description logic operators [
RCA enables various encodings of datasets. For example, two alternatives can be proposed for the above encoding named RCAC (for RCA with Chain). They
CProt7 ∃ protectedBy(CPlant10)
∃ protectedBy(CPlant12) ∃ protectedBy(CPlant14) CProt6
R.com ∃ protectedBy(CPlant9)
R.com
∃ protectedBy(CPlant11) ∃ protectedBy(CPlant3) ∃ protectedBy(CPlant1) G.hirsu
CPlant10 ∃ treats(CPest6)
CPlant14 CPlant12 ∃ treats(CPest4) ∃ treats(CPest5)
CPest6
CPlant11 CWP.lapSnr.ost9pp+any ∃ treats(CPest3) ∃ treats(CPest2)
CPWe.sptr4o
S.litto+spp CPlant8 CPlant7 C.opu C.sCpPpest1 CPest3 C.opu C.sSpp.litto S.spp
S.litto S.spp
CPest5 S.litu+spp
CPest2 S.litu S.litu CProt5
Z.mays CProt8 ∃ protectedBy(CPlant8) ∃ protectedBy(CPlant2) ∃ protectedBy(CPlant7)
Z.mays
CPlant3 CPlant4 CA.Persoctu1DD..ddCuuBPmm.oroeelet2 VV..ccaannee A.escu B.ole
CProt4 CPlant13
S.lyc∃otreats(CPest1) ∃ protectedBy(CPlant6) ∃ protectedBy(CPlant5) ∃ protectedBy(CPlant4)
CPlaSn.lty5co CPlant6 V.fusca V.parvi V.fusca V.parvi
CPlant2 C.papa C.papa
CPlant1 A.indi
A.indi
CPlant0
CProt0 ∃ treats(CPest0) CPest0
Fig. 2. RCA C∃proCtecotendBcye(CpPltantl0a)ttices: protected org. (left), plants (middle), pests (right) 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 composed 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 (or GFCA) [
CPion23 ∃ uses(CFPlaCnt1A0) for indeterminate data in ternary relationships classical FCA attributes. Binary attribute∃∃sDD..ddcuummoeen((CCnPPeesstt23c)) t one node to another, and e ∃ D.dume(CPest0) therefore represent relationships or links b∃eVt.cwanee(CePnest5o)bjects. N-ary attributes can ∃ V.cane(CPest3) encode more complex relationships involvin∃Vg.canme(CoPerste2) than two participants (e.g., ∃ V.cane(CPest0) “plant X protects organism Y against a pe∃∃sVVt..ffuussZccaa((CCPPeessttT35))he intents of graph concepts ”). ∃ V.fusca(CPest2) are like conjunctive queries, i.e. graph patte∃rVn.fusscaw(CPitesht0) a distinguished node. GFCA ∃ V.parvi(CPest5) also defines n-ary graph concepts, but here∃ Vw.paervi(oCPneslt3y) consider unary concepts. ∃ V.parvi(CPest2) ∃ V.parvi(CPest0) Choosing a Graph-FCA representation o∃fC.tsphp(eCPe3st-2t)uples (Protected,Plant,Pest) ∃ C.spp(CPest0) amounts to choosing what the objects and∃C.stpph(CePesat1)ttributes are. An immediate ∃ C.opu(CPest2) representation (GFCA-Ternary) would be ∃∃tCCo..ooppuui((CCPPteersstto01)) duce a ternary attribute for n ∃ W.pro(CPest3) the ternary relation between protected org a∃Wn.pirso(mCPess,t0)protecting plants, and pests: ∃ W.pro(CPest4) e.g., protection(A.escu, A.indi, S.litto). Un a∃Wry.pro(aCPtetstr1)ibutes are used with nominal 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 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
repre8 Source code and user manual at https://bitbucket.org/sebferre/graph-fca/
sentation. First, protecting plants is the focus of the study, and we prefer
concept 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 [
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 3tuples 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 almost 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 classifications of RCAG and GFCA-Binary contain two and four additional concepts, respectively. This slight difference results from considering plants as relations between organisms and pests, which increases the number of relational attributes. Regarding the other methods, the additional concepts originate from the modeling 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.
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
instance, RCAC classifies protecting plants in relation to targeted pests and
protected organisms. Reverse and transitive relationships can be included in the
formal contexts to achieve symmetry, but the cost is increased complexity.
GFCABinary 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
difficult to fully visualize and understand, even with the projection paradigm used
by FCA Tools Bundle [
Extensibility and limits. All the methods allow the user to return to the
original 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 [
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 of the three FCA extensions. We also plan to explore variants of the representations proposed in this paper. Our long-term objective is to provide domain experts with efficient tools to analyze their dataset, not only to extract existing knowledge in a deductive way, but also to assist them in setting up new hypotheses, and suggesting new experiments for bio-pesticides.