9O
From: AAAI Technical Report SS-99-01. Compilation copyright © 1999, AAAI (www.aaai.org). All rights reserved.
Rule Generation by Means of Lattice Theory
R.Brtiggemann 1, S.Pudenz 1, 2H.-G.Bartel
l) Institute of Freshwater
EcologyandInlandFisheries, Dep.of Ecohydrology
RudowerChaussee6a
D- 12484Berlin
brg@igb-berlin.de
2) Institute of Chemistry
Department
of AnalyticalandEnvironmental
Chemistry
Humboldt
UniversityBerlin
HessischeStr. 1-2
D-10II5Berlin
hans-georg=bartel
@rz.hu-berlin,de
Abstract
Arather small data matrixof sevenchemicalsand17 different ecotoxicological
endpoints is examined
by lattice theory.
Especially the FormalConceptAnalysis, developedsince
1980by the schoolof Willeis applied.Theheart of this mathematicalmethodis the concept,consistingof a pair of sets,
whichare related to eachother (GaloisConnection).Theone
set is a subsetof objects,the othera subsetof properties.The
conceptsare partially ordereddue a subset relation among
thosesets. Fromthe subsetrelation implicationsare derived.
Here-after giving somereadingexamples-implicationsare
discussed. For examplethe followingquestion can be answered:
Is there a chemical
that has a highecotoxicological
effect for
exampleon bothTanytarsusdiss. (insect) and Ranacatesbeiana (amphibium)?
Ourchemicalset has no one, whichhas
these properties in common.
Otherexamplesare discussed.
Introduction
ity of visualizing rather complexrelationships in high-dimensionalspaces. Somefirst steps to apply lattice theory in
environmental sciences are already published, see e.g.
Briiggemann,Voigt and Steinberg (1997).
Data
Sevenchemicalsand 17 test species of different trophic levels are analyzed (see table 1). The species are commonly
used in the frame of environmentalprotection studies. From
Table1: Test speciesof different trophic levels taken fromDevillers, Thioulouse and Karcher (1993).
Species
Identifier
Type
Daphniamagna(LC50)
A
Crustacea
Tanytarsus
diss.
B
Insect
Orconectesimmunis
C
Crustacea
In our paper we concentrate on the purely qualitative point
of view, i.e. we wouldlike to generate ,,if.-.then.- rules".
Ecotoxicologicaltest data designedto evaluate the load status of the aquatic environmentare of specific interest. The
premises of such rules should be structural properties of the
chemicals. The conclusions should be statements about the
order of amountof toxicities. A lack of numerical accuracy
may be compensated for by the potency to derive many
rules.
Ranacatesbeiana
D
Amphibium
Oncorynchusmykiss
E
Fish
Lepomis macrochirus
F
Fish
Gambusia
affinis
G
Fish
Ictalurus punctatus
H
Fish
Carassiusduratus
I
Fish
Thetool to generatesuch rules is derivedfromlattice theory,
which is a rather new discipline of Discrete Mathematics
(since around 1930) (Birkhoff 1984, Daveyand Priestley
1990). In particular the variant of FormalConceptAnalysis,
a sub discipline of lattice theory dating from around1980,
turns out to be useful in deriving such rules in an automatic,
i.e. algorithmic way(Daveyand Priestley 1990, Ganter and
Wille 1996). One of the main advantages of this methodis
its connectionto graphtheory and throughthis the possibil-
Pimephalespromelas
J
Fish
Arbactapunctulata(early embryo)
K
Echinoderm
Arbactapunctulata(spermcell)
L
Echinoderm
Photobacteriumphosphoreum
M
Bacterium
Arbactapunctulata
(early embryo,DANN)
N
Echinoderm
Daphniamagna(EC50)
O
Crustacea
Dahniapulex
P
Crustacea
Ceriodaphnia
reticulata
Q
Crustacea
Copyright © 1999, AmericanAssociation for Artifical Intelligence (www.aaai.org).All rights reserved.
91
a systematic point of view, moredifferent species per group
wouldbe favored; howeverthe data availability for such an
ambitious aim is rather poor.Devillers, Thioulouse and
Karcher (1993) selected mainly the chemicals by their different physiological mechanisms.There are oxy- or halogenatedcompounds
(see table 2). The full data matrix can
examinedby looking into the original paper. Wecall the
chemicalsthe objects of the analysis and the set of objects
G. The toxicological data are called the attributes of each
object. Theattribute set is called M.
Table2: Sevenchemicalsandtheir identifier.
Chemical
H H OH
I I I
H3C~C~C~C~CH
3
I I I
OH H CH
3
H
I
H3C--C--CHE--OH
Identifier
2-Methyl-propanol
Y
.4-Pentadione
8
OH
2
1.89 - 3.00
3
3.09- 4.02
4
4.24 - 5.06
5
5.22 - 6.52
.....
hi7
)
Lattice theory
Anorder relation has to be
¯ reflexive
¯ antisymmetric and
¯ transitive.
2-Chloroethanol
E
For example the comparison of two numbers follows these
three axioms. Sets equippedwith an order relation are called
partially orderedsets (posets).
For any two objects one may look for upper and lower
bounds. Simplified spokenif such boundsexist and there is
an unique upper and lower boundresp. then the mathematical structureof a lattice arises.
CI
Cl " "’~’~"Cl
0.68- 1.83
For introductory literature the reader is referred to Birkhoff
(1984), Daveyand Priestley (1990). Consider a set of
jects G, then binary relations amongthe elements of this set
maybe considered. For examplethe order relation:
Hexachloroethane
CI
1
corresponding to the 17 different species/end points. The
quantities ni are integer numbers,as defined by table 3. A
simplifying notation can be introduced: <value in terms of
ni><identifier for the species>.
Cl
CI
attribute-interval
n(x) = (n1, nz
CI@C--CI-I2--OH
H H
I I
C1--CDCmOH
I I
H H
°Interval N
Instead of the raw measureddata Pi, each object x is now
ecotoxicologically characterized by a 17-tuple
2.2.2-Trichloroethanol
H
I
H3C--CmCnC--CH3
II I II
O H O
Table3: Classificationof the attribute values.
-Methyl-2.4-pentadiol
CH
3
CI
In order of robustnessan attribute-wise classification is performed. Eachattribute is divided into five intervals as follows(table 3):
Pentachlorophenol
n
Here we are interested in lattices of specific constructions,
namely lattices of Formal Concept Analysis (FCA)(Ganter
and Wille 1996).
The basic notion of FCAis a triple (G, M, 1) called formal
concept. G is the set of objects under consideration and M
the set of their (original or suitable transformed)attributes.
Anelement glm (g ~ G, m ~ M) of the binary relation I c_
G x M(glm - (g, m) ~ is read: "Object g h asattr ibute m".
Anotherform for representing a context (G, M,/) is a cross
table. Its rowsare labeled by the objects g ~ G and its columnsby the attributes m¢ M. A"X" at the position of cross-
92
ing row g with columnm meansthat glm holds.
A formal concept corresponding to a given context (G, M,/)
is defined as follows: Let X’ be the set of attributes shared
by all objects of the subset X c_ G, X’ := {m~MI glm: all g
X}, andY’ the object set sharing all attributes of the subset
YcM, Y’ := {g~Glglm: all m E Y}. Then a formal concept
can be defined as the pair (X, Y), if X’ = Y and Y’ = Xhold
(,,Galois connection"). Thesubset X is called the extent and
the subset Ythe intent of the concept(X, Y).
Withinthe set of the concepts B(G,M,/) of the context (G,
M,/) a partially ordered relation denoted by < (subconceptsuperconceptrelation) can be defined: the concept(X, Y)
the subconcept of the superconcept (U, V), if the extent
contains all objects of the extent X and the intent Y has all
attributes of the intent V, (E Y) < (U, IO:¢:~Xc_U(c:~VcY).
The
partially ordered set (B(G,M,/), <) correspondsto the mathematical structure of a completelattice, the concept lattice
of the context (G, M,/). It reflects the generalizedhierarchy
and conceptual structure correspondingto (G, M,/).
Conceptlattices are effectively visualized by labeled line diagrams(HASSE
diagrams). Little circles or dots represent
the concepts and the ascending paths of line segmentsrepresent the neighborhoodrelation -<. (X,Y) -< (U,V) holds
(the concept(U, V) is upper neighbor of the concept(X,
if (X, Y) < (U, V) and if there does not exist any other concept (T,S) whichis subconceptto (U, V) and superconcept
(X,Y), that means (X, Y)<(T,S)<(U, V)is false. An
equivalent description of the conceptual knowledgegiven
by the concept lattice or the line diagram to a context
(G, M,/) is the description by attribute implications:
context ¢:v conceptlattice
¢:~ attribute implications
(data)
(conceptualhierarchy)
(logicalstructure)
Animplication between attributes, PR~ CO, PR, COc_ M,
holds in (G, M,/), if all objects sharing all attributes of PR
also share all attributes of CO.For each finite context a reducedand complete implication basis can be found.
Results
To construct an extendedcontext consisting of the ecotoxicological informationfromn(x) and the structure, the structure is codedas follows: -CI, -OH, =Oor aromatic (planar
4n+2system). Thus in the context table a chemical becomes
a ,,X" if it has the correspondingstructural element:
Chemical
-OH
X
X
X
--C1
E
X
11
X
X
X
X
aromatic
=O
The HASSEdiagram of (B(G,M,I), <)the context is
shownin figure 1. By such a diagram the information summarized in a data matrix can be easily deduced. Starting
from an object and following the lines upwardsthe information of the objects can be obtained. For examplechemical13:
Three other points are reachable: 2C, 1A and -OH. This
meanschemical~ has the toxicity 2C, 1Aand the structural
element -OH.
Figure1: HASSE
diagramof the lattice.
[]
Following the lines downwardsand starting for example
from properties one finds chemicals with a given spectrum
of properties: Theproperty 2M(i.e. a specific ecotoxicological reaction with mediumextent) is fulfilled for all chemicals but 13. However the structural elements cannot
characterize this exceptionalcase.
Somemore skillness is evident if several properties or
chemicalsare to be examined.For example: Is there a chemical that has a high ecotoxicological effect for exampleon
both Tanytarsus diss. (insect) and Rana catesbeiana (amphibium), i.e. 4B and 5D ? Our chemical set has no one,
which has these properties in common.Relaxing the demandfor the toxicity to the amphibiumto only 3Dthen the
chemical~ is found. Againthe structural code is by far to
crude to characterize this specifity.
Findingrules:
Here, the implications are selected, for which the premises
are concernedsolely with chemicalstructural elements:
X
8
X
X
C1 ~ 2A, 2C, 2M
=O ~ 3A, 2C, 5D, 3E, 4L, 2M
aromatics ~ all with exception 4B
93
The third rule however has not a firm basis because its
premise is only fulfilled by one compound,whichis the only
one in the wholeset of substances.
information from them.
The problem becomesmore evident by another rule:
Birkhoff, G. 1984. Lattice theory. Providence, RhodeIsland: AmericanMathematical Society, Vol 25.
References
aromatics ~ -CI
This is true within the actual chosenset but can clearly not
generalized. Therules are extracted froma finite set of objects and attributes, therefore the generationof rules should
better be seen as an automatic hypotheses-generation.
Discussion
A data matrix is converted into a HASSE
diagramof concepts
by several steps (Classification, extending by structural
codes, forming simple valued context). Fromthe HASSE
diagramand the subset-superset relation implications can be
derived. Among
the manyimplications (the total of 63 rules
is the outcomeof that simple data matrix), whichrelate the
attributes, those are of primaryinterest, whichrelate structural informationwith ecotoxicological end points.
There are somedistinct drawbacks:
Clearly the substancebasis is still by far too weakto establish real rules; howeverthe methodallows in principle to do
this, as was shownwith a few examples.
The real measureddata are converted into discrete values
without any statistical guidance. Thusthere is no safety
against slight changesin the data transformation. This however could be done and is not on the focus of this paper.
The HASSEdiagram may consist of a complex system of
lines and therefore difficult to read. However,
there are techniques to circumvent such unreadable diagrams.
One mayargue that the type of ecotoxicological end point
was not discussed so far. However
here someresults are given in Bartel (1999), which are moreor less methodological
and whichare therefore suppressed here.
Whatare the advantages?
Clearly if a graphicalpresentation is possible, it is favoured
against boring tables. As pointed out there can be performed
manyinteresting studies, which are as more non trivial as
morenot only single relations like glm are examinedbut relations like G’I? or ?IM’with G’ _c G or M’_c M.
Finally the subset-superset relation allows to deduceimplications. Examplesare discussed. A crucial point is, whether
a premise is emptyor not. If the premise is not emptythen
one gets some kind of probability. As more often the
premiseis fulfilled as moreweightit has as a real rule. However it should be clarified that eventhe machineryof lattice
theory cannot ,,generate" newinsights. Theyare all already
includedin the data set,- the problemis only, to distillate the
Davey,B. A. and Priestley, H. A. 1990. Introduction to Lattices and Order. Cambridge:CambridgeUniversity Press.
Ganter, B. and Wille, R. eds. 1996. FormaleBegriffsanalyse
MathematischeGrundlagen. Berlin: Springer-Verlag.
Briiggemann,R., Voigt, K. and Steinberg, C. 1997b. Application of Formal Concept Analysis to Evaluate Environmental Databases. Chemosphere35(3): 479-486.
Devillers, J., Thioulouse, J. and Karcher, W. 1993. Chemometrical Evaluation of Multispecies-MultichemicalData by
Means of Graphical Techniques Combinedwith Multivariate Analyses. Ecotoxicol. Environ. Saf. 26: 333-345.
Bartel, H.-G. (1999). Formal ConceptAnalysis in Ecotoxicology. In Proceedings of L Workshopon Order Theoretical
Tools in EnvironmentalSciences. Germany,Berlin. To appear as IGBReport.