=Paper=
{{Paper
|id=Vol-42/paper-4
|storemode=property
|title=Structuration of phenotypes/genotypes through Galois lattices and implications
|pdfUrl=https://ceur-ws.org/Vol-42/paper2_duquenne.pdf
|volume=Vol-42
}}
==Structuration of phenotypes/genotypes through Galois lattices and implications==
https://ceur-ws.org/Vol-42/paper2_duquenne.pdf
Structuration of Phenotypes / Genotypes through
Galois Lattices and Implications
Vincent Duquenne 1,4, Caroline Chabert 1, Améziane Cherfouh 1,
Jean-Maurice Delabar 2, Anne-Lise Doyen 1, and Douglas Pickering 3
1 FRE 2134, Institut de Transgénose, 3B rue de la Férollerie, F-45071 Orléans Cedex 02
2 UMR 8602, Lab. de Biochimie Génétique, 156 rue de Vaugirard, F-75730 Paris Cedex 15
3 Department of Mathematics, Brandon University, Manitoba, Canada R7A 6A9
4 To whom correspondence should be addressed at FRE 2134
v.duquenne@wanadoo.fr
Abstract. The Galois Lattice of a binary relation formalizes it as a concept
system, dually ordered in "extension" / "intension". All implications between
conjunctions of properties holding in it are summarized by a (recursive)
canonical basis -all basis having the same cardinality (see [MR #87k:08009]).
We report here how these tools structure phenotypes / genotypes in behavior
genetics. On a generic viewpoint, both situations comprise two binary data sets
that are paired through either a column or a row matching, which raises specific
questions. If the data are small, as compared with data bases in bioinformatics,
this illustrates how these abstract tools can unfold and better interpretations.
1 Introduction
The outstanding developments of data and knowledge bases in bioinformatics raise
the questions of "knowledge extracting" and putting data bases in "canonical forms" -
in order to speed up access to information-, and of extracting classifications and rules
providing some explanation on the biological topics under study.
On the other hand, it is now well established (see [1,12,15]) that the Galois Lattice
of a binary relation formalizes any kind of duality -here between a set of objects
(subjects, patients...) O and a set of properties (attributes...) P- and can be used as a
general model for structuring it as a concept system that is dually ordered in
"extension" / "intension": the lattice elements -sometimes called "concepts"- are
simply the ordered pairs (X,Y)∈OxP, where X (extension) is the maximal group of
objects "having" Y, and reciprocally for Y (intension) which is the maximal subset of
properties shared by X. The lattice elements are just ordered along their extensions.
This model was also used to enrich techniques and models that are implicitly based
on trees (classifications, inheritance of properties...), and -through the equivalence
between (finite) lattices / closure operator- could even be used as a general model for
formal languages (see [5]) and for evolving data bases (see [2,3,24], and [14,16,17]
for the algorithms). Lattices are more general and hopefully flexible than trees.
In a previous work (see [18, MR #87k:08009]) it has been shown that all the
implications between conjunctions of properties holding in such binary data can be
21
V. Duquenne & al. ICCS’01 Int’l. Workshop on Concept Lattices−based KDD
summarized by a (recursive) canonical basis -and that all basis do have the same
cardinality-, which can be expressed within the lattice through the existence of a
minimal sub-structure -it's meet-core (see [10, MR #92g:06011])- made of meet-
irreducible and essential meet-reducible elements on which the meet operation is
restricted, out of which the whole lattice can be reconstructed -that generalizes G.
Birkhoff's correspondence between partial orders and distributive lattices. Several
papers have revisited these canonical form theorems (see for instance [15 §2]), and
have either shown that the canonical implication basis could be helpful for contracting
some classical results by D. Maier on data bases (see [6,22,23]), or have used them in
statistics (see [9,13,20]) and in the analysis of symptoms in psychiatry (see [11,21]).
In this paper, we will take a more concrete approach and report on two current
collaborations where these latticial and implicational tools have been used to structure
the phenotypes and genotypes of subjects in behavior genetics (see [4,8]). Although
the underlying data are in both cases quite small, as compared with the development
of large data bases, it is worthwhile to illustrate how these abstract tools can unfold
and better interpretations. On a technical viewpoint, both situations comprise two data
sets that are paired through either a column or a row matching, respectively, which
extends the usual situation with one binary table, and will raise new specific questions.
2 Implications for two groups of subjects: laterality questionnaire
For assessing handedness, several questionnaires are used on which multivariate
analyses have been performed, showing some lack of agreement yet. Hence, the main
goal of this report (see [8] for more interpretations) is to show how lattice analysis can
help in understanding the associations among items, by comparing the results of
left / right-handed writers for a questionnaire reporting which hand they use in life.
Right-handed writers. Basic data: a S61xA11 0/1-matrix where (s,a)=1 when the
right-handed writer s uses exclusively his / her right hand for action a. Many subject's
profiles are equal, and many right-handers use their right-hands for nearly all the
actions. The profiles are ordered by (reverse) set inclusion -the closer they are to the
bottom the more they are consistent right-handers- which is completed by set
intersection that generates a Galois lattice. An element represents a maximal sub-table
S'xA' filled with ones, for the subjects S' that share the actions A', S' is the extension of
A', and dually A' is the intension of S'. The lattice is minimally labeled: an element's
extension S' is restored by listing the subjects which are below it, and dually for A'.
Hence, the lattice is directly encoding the extension / intension duality, and gives an
exact representation of which actions the subjects are sharing, and how this sharing is
structured. The observed Galois lattice (simplified in Fig.1, see below) is small (190
el. / the potential 2**11=2048 combinations ), which reflects a strong structure of the
right-handers / association of these actions. The actions are ordered by their
extensions, which can be read as implications of which the premises are single
actions. Hence, T < r can be read T → r : T:Throw implies r:racket. Similarly,
B:Broom implies h:hammer, and C:Cards implies h:hammer and r:racket. Read from
top to bottom, the lattice starts with (s:scissors, b:tooth-brush, m:matches, S:Shovel)
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V. Duquenne & al. ICCS’01 Int’l. Workshop on Concept Lattices−based KDD
Fig. 1. Galois lattice and implications (right-handers)
Fig. 2. Gluing decomposition in intervals (right-handers)
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V. Duquenne & al. ICCS’01 Int’l. Workshop on Concept Lattices−based KDD
which are independent (generating a 2**4 interval), then r:racket and T:Throw are
independent with some of these, while c:cap associates only with m:matches. All these
actions associate more together when they are in conjunction with h:hammer. All the
implications that hold can be summarized by a canonical basis (see Fig.1).
Those implications of which the premises are conjunctions reflect discrepancy to
independence. They are weighted by the extension of their premise+conclusion's
union. Most imply h:hammer or b:tooth-brush. Fig.1 have been constructed by
"blocking together" all the actions not below h:hammer, and then by introducing the
remaining actions, to simplify the drawing by erasing some lines ("boxed" lattices in
[15]). What is left by the action of implications is a kind of gluing of Boolean intervals
which expresses (local) independence, which is extended by regular decreasing of the
extensions' cardinalities. The lattice is decomposable by un-gluing (see [15], and
Fig.2), which expresses exchangeability properties between actions that fall in the
same intervals: they behave the same way for the two lattice operations which reflect
the sharing of actions, and the intersections of the subject's groups.
Left-handed writers. Basic data: a S23xA11 0/1-matrix, with the same set of
actions. The profiles are nearly all distinct, except for one group. Here, h:hammer and
C:Cards are equivalent, and both implied by T:Throw (see Fig.3). Up to these
implications, all the eleven actions but three are independent. On the other hand,
scissors, Shovel, and Broom don't associate much with all the other actions, due to the
fact that the remaining implications comprise exactly one of these in their premises.
Among the complex implications of the basis, most of them are satisfied by more than
half the subjects, and several groups express equivalencies (#2-6 in Fig.3).
Out of the implications, C ↔ h and T → Ch and the fact that Broom, Shovel and
scissors don't associate with the other actions, the structure of the intervals reveals that
the conjunctions of actions are mostly (locally) independent, that is also comforted by
a regular decreasing of extensions. The left-handedness lattice is again decomposable
in two intervals (see Fig.4), a decomposition also governed by hammer (Cards) that
concerns a majority of subjects (18/23). The lower interval is generated downwards by
Throw, as compared with Broom and Cards, for the group of right-handed subjects.
Comparison right / left-handed writers. The profiles of the left-handers are more
diverse than for the right-handed subjects, which are somehow more stereotyped. In
both populations C:Cards implies h:hammer, while T:Throw implies another action
involving energy (r:racket / h:hammer), and it is interesting that both un-gluing are
commended by h:hammer. The structure of left-handedness can be summarized by
independence of eleven actions (up to the equivalence C:Cards ↔ h:hammer, which
are both implied by T:Throw), together with non-association with the three remaining
actions: Broom, Shovel and scissors, which are thus sufficient to be quasi-consistent
left-handers. The previous graphics give a clear picture on the hierarchies of actions
for these populations taken apart, but it would be desirable to be in a position to
characterize what is specifically true for the right-handers being not true for the left-
handers, and symmetrically.
To this end, we designed the following new scheme: First, we construct the basis
BRL of implications which are holding for the two populations -joined by union into a
S84xA11 matrix-, and which represents a consensus of what can be inferred from
them. Then, we construct the list of all implications which have to be added to the
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V. Duquenne & al. ICCS’01 Int’l. Workshop on Concept Lattices−based KDD
Fig. 3. Galois lattice and implications (left-handers)
Fig. 4. Gluing decomposition in intervals (left-handers)
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V. Duquenne & al. ICCS’01 Int’l. Workshop on Concept Lattices−based KDD
14 rmBc ---> b 14 TBhc ---> C 62 C ---> h 16 mBS ---> hb
16 rmtB ---> b 15 rBhc ---> C 25 TB ---> h
25 mtSh ---> b 38 Tmhc ---> C 28 Sc ---> h 27 Bb ---> m
29 TmSh ---> b 40 rmhc ---> C 29 tS ---> h
14 TmtBh ---> b 18 TtShc ---> C 44 Tc ---> h 57 Ts ---> r
27 rmsSh ---> b 19 rtShc ---> C 50 Tt ---> h 58 sh ---> r
12 TmBChc ---> b 33 Tthbc ---> C 56 Tb ---> h
34 rmsChc ---> b 34 rthbc ---> C 14 rBS ---> h 40 sc ---> rh
23 rsB ---> h 47 st ---> rh
33 rSb ---> h 21 msB ---> rh
51 rsb ---> h
Basis BRL (84 subjects)
10 mtBh ---> b
23 mthc ---> b
09 mBhc ---> b
T:Throw 18 h ---> C
39 rt ---> h
r:racket 14 T ---> Ch
33 mt ---> h
m:matches 10 tB ---> m
21 B ---> h
s:scissors
33 tb ---> h
t:thread 09 BCh ---> mt
33 rc ---> h
B:Broom 09 Bc ---> mt
27 tc ---> h
S:Shovel 07 rB ---> mtb
30 bc ---> h
C:Cards 09 sb ---> rtCh
51 T ---> r
h:hammer 08 ms ---> rtChb
44 Ch ---> r
b:t.brush 10 mB ---> t
c:cap 12 SCh ---> tb
38 Trth ---> s
Actions 12 Sb ---> tCh
32 Trhc ---> s
09 rs ---> tChb
50 rs ---> T
10 rS ---> tChb
20 rBh ---> T
11 mS ---> tChb
43 rb ---> Th
32 rmth ---> Ts Basis BL-RL
26 rthc ---> Ts (23subjects)
Basis BR-RL
(61 subjects)
r h b C m
hr bh
c o n s e n s u s r+ l
C t m
s T r h b
Ch bt mt
sT hT
r ig h t-h a n d e r s
Cht bm t
C h rt bCht
b C h rt le ft-h a n d e r s
Fig. 5. The implications that are common / specific to right / left-handers
26
V. Duquenne & al. ICCS’01 Int’l. Workshop on Concept Lattices−based KDD
basis BRL in order to generate a list being equivalent to the basis BR of the right-
handers, which can be summarized by a new specific basis that will be denoted by
BR-RL. Symmetrically, for left-handers, the specific basis BL-RL is defined.
Comparing and commenting on the consensus basis and these two specific basis
should reveal what is common to both and what is specific to each of these
populations, which addresses directly the cognitive structure of left / right-handedness.
For the results concerning the consensus by union of right / left-handed subjects,
the 35 implications of the common basis BRL have only seven distinct conclusions,
which can be organized in two groups along an axis, from the actions that require
more energy, r:racket, h:hammer and b:tooth-brush (and hr and bh, see Fig.5 top), up
to those involving dexterity, such as C:Cards and m:matches.
Specific properties of the right(/ left)-handed subjects: The 19 implications of the
specific basis BR-RL are also structured in two groups. The first twelve ones refine
the previous group of the consensus concerning energy, and imply either r:racket,
h:hammer or b:tooth-brush, while the second group implies new conclusions:
T:Throw and s:scissors. Hence, most of the implications for the right-handers imply
an action which requires more energy than dexterity, to the exception of s:scissors
which plays a peculiar role. As it is the case for the consensus by union, the partial
order of conclusions is simple, being of order dimension five and length two.
Specific properties of the left(/ right)-handed subjects: As opposed to the right-
handers, all the conclusions of the specific basis BL-RL involve more dexterity, with
C:Cards and m:matches, as in the consensus, but also the new conclusion t:thread, in
conjunction with other actions, up to generate a complex partial order of conclusions
of dimension three, and length five (see Fig.5 bottom).
Hence, to temporarily conclude with this study, as the items of the laterality
questionnaire do not have the same categorical impact for left / right-handers, it can be
stressed that they do not generate the same structures, which confirms and extends
previous data from literature. The behavior of right-handers seems globally more
stereotyped, with a minority of subjects generating a rich set of dependencies between
conjunctions of actions. The behavior of left-handers appears less stereotyped and
governed more by independence and avoidance among actions. While the implications
holding in both populations are clearly scaled on an axis energy / dexterity, it is
significant that right-handers refine the conclusions governed by energy, while left-
handers those involving dexterity. Now, when some questionnaires pretend to evaluate
laterality along a continuum, this analysis seems to question such a strong assumption.
This calls for cooperation to investigate other populations of subjects, to be in a
position to refine and to assess the cognitive structure of handedness.
3 Implications between two sets of variables: partial trisomies 21
A main question in Behavioral Genetics concerns the assessment of relationships
between genotypic / phenotypic variables. We now report how these questions can be
revisited through lattice analysis in the case of Down syndrome (see [4] for more
interpretations). Its phenotype and genotype involve morphologic and anatomic
abnormalities with more or less severe mental retardation, and a partial or complete
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V. Duquenne & al. ICCS’01 Int’l. Workshop on Concept Lattices−based KDD
triplication of chromosome 21. This study will extend a previous work (see [7]) on the
"Molecular mapping of twenty-four features of Down syndrome on chromosome 21"
that showed that a candidate region for Down syndrome ("DCR") is the band q22.2.
Matching model. After an analysis of the genotypes (11 cytogenetic bands) and
phenotypes (24 features) of a population of ten patients suffering from partial trisomy,
which can be coded into a binary table S10x(B11+F24), the matching was defined
there by a set theoretic procedure: each feature f ∈ F24 was ascribed to the subset of
bands -say fFB ⊆ B11- that are shared by all patients "having f", which can be read as
an implication f → fFB (and not a correlation as sometimes said). The molecular
mapping onto minimal regions is a representation of these implications in terms of
their locations on chromosome 21.
Our discussion of the original model starts with a basic question: which are the
respective interests and genetic significance of the implications "features → bands"
and "bands → features"? Both kinds are equally informative -and cannot be ignored-
as they address the two sides of genetic causation of observed phenotypes.
The latter point out to the common features of a group of patients which are sharing
a set of bands, which expresses one -out of several possible- sufficient condition for
having these common features -sufficient, since other patients, while not sharing these
bands, may also have these features, due to "some other causes".
Conversely, an implication "feature → bands": f → fFB expresses a -unique, in
contrast- maximal necessary genotypic condition for having this phenotype -necessary,
since all patients having it do share these bands in common without exception, while
other patients may also share these bands without having this phenotype due to "some
variability in the expression of responsible genes". This variability could also be
ascribed by assigning a penetrance to different genetic configurations. Even mental
impairment, the only constant finding, varies in its expression and severity. Hence, the
viewpoint taken in the original papers (see [7,19]) stresses the "feature → bands"
implications as formalizing a set theoretic matching, hence assuming a
genotypic / phenotypic asymmetry and focusing only on the unique maximal necessary
genotypic condition for having each single features f ∈ F24.
In this report, we reinvestigate the model, in order to evaluate how far these
implications "feature → bands": (f → fFB for all f∈F) are from defining equivalencies
f ↔ fFB -in which case all the patients sharing fFB do have f as well, so that fFB is of
maximal penetrance-, and this question can be either raised locally, or globally.
Locally: such an equivalence is of greater confidence in view of further gene
identification since it expresses a necessary and sufficient condition for having feature
f, hence providing a genotypic characterization of it without intervention of "other
causes" and "expression variability": all possible minimal sufficient conditions are
then confused with the maximal necessary one in a unique region. Globally: if all
these equivalencies held, that would assess that the set of bands is sufficient to
characterize the polymorphism of the phenotypic diversity of trisomy 21 in this
population. It turns out that only five of the 24 features bear such an equivalence,
which partly explains the original genotype / phenotype asymmetry assumption that
can be found in literature (see [7,19]), and justifies to extend the model by
relaxing / extending the assumptions to refine the evaluation.
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V. Duquenne & al. ICCS’01 Int’l. Workshop on Concept Lattices−based KDD
Extended model. This evaluation requires some steps: we re-defined the matching
model by introducing appropriate denotations to study the redundancy of these
implications f → fFB regarding the phenotypic structure of the population -whenever
they appear to be the consequence of some implications between features or differ
greatly in their extensions in the population, being therefore far from equivalencies.
This more generally leads to consider implications / equivalencies between
conjunctions of bands as well as features, and therefore to evaluate the complexity of
the genotypic / phenotypic structures by using the Galois Lattices associated to the
binary tables S10xB11 / S10xF24. Extending the model up to conjunctions involves a
risk of "combinatorial explosion" which is tamed by the Galois lattice construction
that focus only on pertinent (i.e. maximal shared by some group of patients)
conjunctions and has two major benefits. First, this treats the syndromes as they are:
conjunctions of features shared by some group of patients, and this increases the
chance of detecting genotypic characterizations through equivalencies, since feature
conjunctions obviously have smaller extensions than their single features.
It should also be noted that lattice analysis confirms and makes more precise a
previous conclusion that DCR is structured around band q22.2 (see [7]), but unfolds it
along three independent directions q22.1(S54] (up to S54), q22.1[SOD1) (from
SOD1), and a chain of nested intervals of q22.3 (see Fig.6 top).
The genotype / phenotype lattices are labeled with basis of implications going from
feature conjunctions into band conjunctions, and conversely, which completely
characterizes the matching of the geno / phenotypic structures in terms of necessary /
sufficient conditions for having the phenotypes. It is shown that half of the band
conjunctions which characterize the genotypic structure bear equivalencies -hence of
maximal penetrance-, which define a core structure "already assessed" (see Fig.6).
Hence, the cytogenetic bands B11 actually provide a coherent core structure, even
if they do not completely characterize the phenotypes yet -which would require more
patients or a finer description of their genotypes. This can be interpreted as an
encouraging result that also gives some strength to the matching model's extension,
which both treats the syndromes as conjunctions of features and takes in charge the
potential polymorphism of trisomy 21. Notice that the extended model departs only
from the original one by: (1) assuming a more symmetric view on the
genotypes / phenotypes and necessary / sufficient genotypic conditions, (2) weighting
the implication premises / conclusions for evaluating their discrepancy to equivalence,
and (3) extending the matching procedure to the conjunctions of features.
The resulting combinatorial complexity is minimized by the Galois lattices which
focus only on pertinent conjunctions, while this extension is essential to reveal the
phenotypes / genotypes for which there is actually a genotypic characterization of the
phenotypes -or not, suggesting some further refinements-, for the cytogenetic bands
under study. This should lead to raise local questions for bettering the genetic
description of features, and to search for some minimal sets of questions on
conjunctions -not bearing equivalencies- of middle size extensions -i.e. reasonably
high in the lattices-, which should be investigated through considering new smaller
chromosomic regions, and to extend the study by constructing larger data bases.
29
V. Duquenne & al. ICCS’01 Int’l. Workshop on Concept Lattices−based KDD
ABCDEFGHIJK
01110000000 BCD ---> 6/9
#ML 11110000000 ABCD ---> 5/9
00000100000 7,5,7/8 e,h,p -->> F ---> 8/9
01110100000 4/5 hlp -->> BCDF -->> lp 5/6
00001100000 5/5 elp <->> EF <->> elp 5/5
#FG 01111100000 3/4 elpt -->> BCDEF ---> elp 4/5
#IG 00000110000 7/7 b <->> FG <->> b 7/7
00000111000 4,5/6 bh,blp -->> FGH ---> b 6/7
11110111000 4/4 ablp <->> ABCDFGH <->> ablp 4/4
00001111000 EFGH <->> bcelp 4/4
#TY 11111111000 ABCDEFGH <--> abcelp 3/3
00000111100 FGHI ---> b 5/7
11110111100 ABCDFGHI ---> ablp 3/4
00001111100 3/3 bceklpqr <->> EFGHI <->> bceklpqr 3/3
#LI 11111111100 ABCDEFGHI <->> abceklpqrt 2/2
00000111110 2,3/4 bcehistvw,bdlp -->> FGHIJ ---> b 4/7
11110111110 ABCDFGHIJ <->> abdhlp 2/2
00001111110 EFGHIJ <->> bcdeklpqrvw 2/2
#AL 11111111110 ABCDEFGHIJ <->> abcdehiklopqrstuvw 1/1
#AB 00000111111 2/3 bcevwx -->> FGHIJK ---> b 3/7
#SC 11110111111 ABCDFGHIJK ---> abdhlp 1/2
#DL 00001111111 EFGHIJK <->> bcdejklmnpqrvwx 1/1
ABCDEFGHIJK
. . . . . K:q22.3(qter]
. . . . .J:q22.3(CD18]
. . . . I:q22.3(CRYA1]
. . . .H:q22.3(S42]
. . . G:q22.3(MX1]
. . .F:q22.2
. . E:q22.1[SOD1)
. .D:q22.1(S54]
. C:q21
.B:q11.2
A:[p,11.1]
Fig. 6. Top: the band lattice is labeled with all implications "bands → features", and
non-redundant implications "features → bands". Bottom: the band lattice is listed as
an extended molecular mapping with the subject bands and implications.
30
V. Duquenne & al. ICCS’01 Int’l. Workshop on Concept Lattices−based KDD
4 Conclusion
On a generic level, these two studies show that Galois lattices and implication basis
are most useful for exploring the matching of two binary relations (0/1 data tables),
which are paired either "by the columns" or "by the rows". The former situation leads
to the fundamental problem of comparing two Galois lattices -or closure
operators / implication basis- by taking a "differential" viewpoint. Dually, the latter
case leads to search what can be inferred, in the same population, from a set of
variables to another one, with an extra bonus for the characterization of equivalencies
that are indeed approaching "causality". On an interpretation level, both studies call
for exploring larger populations for refining the provisory conclusions.
This raises two kinds of questions, which are addressed either to the
psychologist / biologist in charge of the data and its interpretations, or to the computer
scientist who would like to extend these approaches to much larger data bases in the
successful topic of KDD (Knowledge Discovery in Data bases), or to both.
First, the underlying structures are here simpler and therefore more natural than
what is commonly used in these disciplines: extra methodological assumptions cost
nothing, for sure, but ... plausibility. To the computer scientist a first remark: our
experience is that scanning thousands of lattice elements / implications cost nowadays
no more than one second with reasonable programs and pc-computers. Hence, the
current question is perhaps more to do something about it, to promote and unfold
semantic interpretations, than to cut this time by a quarter: it is a question of priority.
Another wisdom message will be that whenever one adds a lot more subjects (rows,
information), all observed lattices become Boolean -without implications!- that
stresses an urge for a real, funded, sensitive approximation theory of these discrete
structures. Facing this apparent contradiction between practical / theoretical needs, all
the announced developments for pruning, navigating ... or putting more flexibility
through this combinatorial complexity will be welcome: the remaining questions will
of course be their canonicness and significance, their efficiency and usefulness.
Acknowledgments
This research was supported by the Action Bioinformatique 2000, Centre National de
la Recherche Scientifique - Centre Auvergne Limousin, Conseil de la Région Centre
and the Fondation Jérome Lejeune. The 2nd and 5th authors received a grant from the
Fondation pour la Recherche Médicale. The development of the program GLAD was
aided by CNRS, University R. Descartes, and Maison des Sciences de l'Homme-Paris.
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