Rainer Brüggemann
Lars Carlsen
Partial Order
in Environmental Sciences and Chemistry
Rainer Brüggemann
Lars Carlsen
(Editors)
Partial Order
in Environmental Sciences
and Chemistry
With 140 Figures and 50 Tables
DR. RAINER BRÜGGEMANN
Leibniz Institute
of Freshwater Ecology and Inland Fisheries
Dept. Ecohydrology
Müggelseedamm 310
12587 Berlin-Friedrichshagen
Germany
E-mail:
[email protected]
PROF. DR. LARS CARLSEN
Awareness-Center
Hyldeholm 4
4000 Roskilde-Veddelev
Denmark
E-mail:
[email protected]
Library of Congress Control Number: 2006924685
ISBN-10
ISBN-13
3-540-33968-X Springer Berlin Heidelberg New York
978-3-540-33968-7 Springer Berlin Heidelberg New York
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Preface
When you edit a book, the editors should ask themselves, why are we doing this and whom are we doing this for? To whom could this book be
valuable as a source of information and possibly inspiration and of course
are there other books with similar topics on the market? Indeed the mathematical structure 'partial order' is explained in many mathematical textbooks, which require different degrees of mathematical skills to comprehend. Thus, as far as we can tell, all these books are dedicated directly
towards mathematician working in the area of Discrete Mathematics and
Theoretical Informatics. Although partial order is very well known in
quantum mechanics, especially within the context of Young-diagrams, literature stressing the application aspect of partial order seems to be not
available. However, an increasing number of publications in scientific
journals have in recent years appeared, applying partial order to various
fields of chemistry and environmental sciences. A recent summary can be
found in a special issue of the journal Match - Commun.Math.Comput.
Chem. 2000, edited by Klein and Brickmann. However, we believe that
this journal possibly is too specific and as such it may not reach scientists
actually applying partial order in various fields of research. Hence, we
dared to initiate the editing of this book in order to address a broader audience and we were happy to convincing distinguished scientists working
with different aspects of partial order theory to contribute to this book. We
are indeed indebted to all of them.
What is a partial order? A general explanation can be found just in the
first chapters of this book and according to the different application aspects, correspondingly adopted definitions can be found in many other
chapters; however, it might be useful briefly to explain the concept here by
a simple example. Thus, if a chemical is toxic and is bioaccumulating then
obviously the chemical may exert an environmental risk. If there are two
other chemicals, one exhibiting a lower toxicity but a higher bioaccumulation potential and another with a much higher toxicity but a lower bioaccumulation potential, we may have a problem to assess their individual environmental risks. This kind of problems can be analyzed with partial
order. The only mathematical operation needed is the comparison, i.e. is a
larger or smaller than b. Hence, partial order in its various application aspects is the science of comparisons! Comparisons of chemical properties,
comparisons of environmental systems, and even comparisons of strategies
or management options are all topic that advantageously may be analyzed
using partial order theory. Our objective with this book is to demonstrate
how to use partial order in the field of pure chemistry, in substance prop-
vi
Preface
erty estimations, and in environmental sciences. Some chapters will show
how partial order can be applied in field monitoring studies, in deriving
decisions and in judging the quality of databases in the context of environmental systems and chemistry. The charming aspect of partial order is
just that by comparison we learn something about the objects, which are to
be compared!
Most of the readers will probably be trained within differential calculus,
with linear algebra, or with statistics. All the mathematical operations
needed in these disciplines are by far more complex than that single one
needed in partial order. The point is that operating without numbers may
appear somewhat strange. The book aims to reduce this uncomfortable
strange feeling.
Thus, we hope that this book will broaden the circle of scientists, which
find partial order as a useful tool for their work. The theoretical and practical aspects of partial order are discussed in, e.g., the INDO-US-workshop
on Mathematical Chemistry, a series of scientific symposia initialized by
Basak and Sinha, 1998, and in specific workshops about partial order in
chemistry and environmental systems. We urge scientist, newcomers as
well as established partial order users to contribute to these workshops,
contacts can be found by our E-Mail-addresses ([email protected] or
[email protected] (Brueggemann) or [email protected]
(Carlsen)).
April 2006
Rainer Brüggemann and Lars Carlsen
Preface
vii
Acknowledgement
This book could not have been reality without the enthusiasm of all our
contributing authors. We are truly grateful and thank each of them cordially. We thank Alexandra Sakowsky for her help and her patience in rewriting texts in the correct layout, Dagmar Schwamm, Grit Siegert, Barbara Kobisch and Dr. Torsten Strube for helping us. Last not least we
thank the Leibniz-Institute of Freshwater Ecology and Inland Fisheries for
supporting this work.
We thank the publishing house 'Springer' for its patience.
Contents
Preface by R. Brüggemann and L. Carlsen
v
1 Chemistry and Partial Order
Partial Ordering of Properties: The Young Diagram Lattice
and Related Chemical Systems
SHERIF EL-BASIL
3
Hasse Diagrams and their Relation to Molecular Periodicity
RAY HEFFERLIN
27
Directed Reaction Graphs as Posets
D. J. KLEIN AND T. IVANCIUC
35
2 Environmental Chemistry and Systems
Introduction to partial order theory exemplified by the
Evaluation of Sampling Sites
RAINER BRÜGGEMANN AND LARS CARLSEN
61
Comparative Evaluation and Analysis of Water Sediment Data
STEFAN PUDENZ
111
Prioritizing PBT Substances
LARS CARLSEN, JOHN D. WALKER
153
3 Quantitative Structure Activity Relationships
Interpolation Schemes in QSAR
LARS CARLSEN
163
New QSAR Modelling Approach Based on Ranking Models
by Genetic Algorithms – Variable Subset Selection (GA-VSS)
MANUELA PAVAN, VIVIANA CONSONNI, PAOLA GRAMATICA
ROBERTO TODESCHINI
AND
181
4 Decision support
Aspects of Decision Support in Water Management: Data
based evaluation compared with expectations
UTE SIMON, RAINER BRÜGGEMANN, STEFAN PUDENZ, HORST
BEHRENDT
221
x
Contents
A Comparison of Partial Order Technique with Three Methods
of Multi-Criteria Analysis for Ranking of Chemical Substance
RAINER BRÜGGEMANN, LARS CARLSEN, DORTE B. LERCHE
PETER B. SØRENSEN
237
AND
5 Field, Monitoring and Information
Developing decision support based on field data and partial
order theory
PETER B. SØRENSEN, DORTE B. LERCHE AND MARIANNE THOMSEN
259
Evaluation of Biomonitoring Data
DIETER HELM
285
Exploring Patterns of Habitat Diversity Across Landscapes
Using Partial Ordering
WAYNE L. MYERS, G. P. PATIL AND YUN CAI
309
Information Systems and Databases
KRISTINA VOIGT, RAINER BRÜGGEMANN
327
6 Rules and Complexity
Contexts, Concepts, Implications and Hypotheses
ADALBERT KERBER
355
Partial Orders and Complexity: The Young Diagram Lattice
WILLIAM SEITZ
367
7 Historical remarks
Hasse Diagrams and Software Development
EFRAIM HALFON
385
8 Introductory References
393
Index
399
List of Contributors
BEHRENDT, H.
Leibniz-Institute of Freshwater Ecology and Inland Fisheries
Müggelseedamm 310, D-12587 Berlin, Germany
e-mail: [email protected]
BRÜGGEMANN, R.
Leibniz-Institute of Freshwater Ecology and Inland Fisheries
Müggelseedamm 310, D-12587 Berlin, Germany
e-mail: [email protected] or [email protected]
CAI, Y.
Department of Statistics, The Pennsylvania State University
Univ. Park, PA 16802, USA
e-mail: [email protected]
CARLSEN, L.
Awareness Center
Veddelev, Hyldeholm 4, 4000 Roskilde, Denmark
e-mail: [email protected]
CONSONNI, V.
Milano Chemometrics and QSAR Research Group
Dept. of Environmental Sciences, University of Milano-Bicocca
P.za della Szienza, I-20126 Milano, Italy
e-mail: [email protected]
EL-BASIL, S.
Faculty of Pharmacy, University of Cairo
Kasr Al-Aini st. Cairo 11562, Egypt
e-mail: [email protected]
GRAMATICA, P.
QSAR and Environmental Chemistry Research Unit
Dept. of Structural and Functional Biology, University of Insubria
via Dunant 3, I-21100 Varese, Italy
e-mail: [email protected]
xii
List of Contributors
HALFON, E.
Burlington, Ontario, 4481 Concord Place, Canada L7L1J5
e-mail: [email protected]
HEFFERLIN, R.
Southern Adventist University, Collegedale, Tennessee 37315, USA
e-mail: [email protected]
HEININGER, P.
Federal Institute of Hydrology (BfG), Dept. Qualitative Hydrology
P.O. Box 200253, D-56002 Koblenz, Germany
e-mail: [email protected]
HELM, D.
Robert Koch-Institute, Seestr. 10, D-13353 Berlin, Germany
e-mail: [email protected]
IVANCIUC, T.
Texas A&M University, Galveston, Texas, USA
e-mail: [email protected]
KERBER, A.
Department of Mathematics, University of Bayreuth, Germany
e-mail: [email protected]
KLEIN, D. J.
Texas A&M University, Galveston, Texas, USA
e-mail: [email protected]
LERCHE, D. B.
The National Environmental Research Institute, Department of Policy
Analysis, Frederiksborgvej 399, DK-4000 Roskilde, Denmark
e-mail: [email protected]
MYERS, W. L.
124 Land & Water Research Bildg, The Pennsylvania State University,
Univ. Park, PA 16802, USA
e-mail: [email protected]
List of Contributors
xiii
PATIL, G. P.
Department of Statistics, The Pennsylvania State University, Univ. Park,
PA 16802, USA
e-mail: [email protected]
PAVAN, M.
Milano Chemometrics and QSAR Research Group, Dept. of Environmental Sciences, University of Milano-Bicocca, P.za della Szienza,
I-20126 Milano, Italy.
e-mail: [email protected] (recently: [email protected])
PUDENZ, S.
Criterion-Evaluation & Information Management
Mariannenstr. 33, D-10999 Berlin, Germany
e-mail: [email protected]
SEITZ, W.
Department of Marine Sciences, University at Galveston, Texas 77539,
P.O. Box 1675, USA
e-mail: [email protected]
SIMON, U.
Leibniz-Institute of Freshwater Ecology and Inland Fisheries
Müggelseedamm 310, D-12587 Berlin, Germany
e-mail: [email protected]
SØRENSEN, P. B.
Department of Policy Analysis, National Environmental Research Institute, Vejlsoevej 25, DK-8600 Silkeborg, Denmark
e-mail: [email protected]
THOMSEN, M.
The National Environmental Research Institute, Department of Policy
Analysis, Frederiksborgvej 399, DK-4000 Roskilde, Denmark
e-mail: [email protected]
TODESCHINI, R.
Milano Chemometrics and QSAR Research Group, Dept. of Environmental Sciences, University of Milano-Bicocca, P.za della Szienza,
I-20126 Milano, Italy
e-mail: [email protected]
xiv
List of Contributors
VOIGT, K.
GSF-Research Centre for Environment and Health,
Institute for Biomathematics and Biometry,
Ingolstädter Landstr. 1, D-85758 Oberschleissheim, Germany
e-mail: [email protected]
WALKER, J. D.
TSCA Interagency Testing Committee (ITC), Office of Pollution Prevention and Toxics (7401), Washington, D.C. 20460, USA
e-mail: [email protected]
1 Chemistry and Partial Order
In this section the fundamentals of partial orders are introduced in three
chapters, which are rather different, albeit they point to the same item: partial order in chemistry. The reader will learn basic concepts and a manifold
how to derive a partial order from chemical concepts.
In the first chapter, by El-Basil, the main terms and concepts of partial
order are explained. It shows that there are many different ways to apply
the axioms of partial order. Especially the important theorem of Muirhead
and its generalization are broadly discussed. The reader may learn how to
develop Young diagrams and how to extract useful results form the partially ordered set of Young diagrams. The examples are mainly following
the chemistry of aromatics. Hence, the reader will become familiar with
the broad topic of Kekulé structures and counting them.
The detection of the periodic system of chemical elements was a break
through in the theoretical understanding of chemistry. Hefferlin discusses
periodicities of chemical elements and small molecules. He shows how
general the concept of posets is. Why not explore the properties of small
molecules by means of a Hasse diagram? Hefferlin shows by the example
of Phosphorus oxides how this may be done.
The first two chapters are devoted to a static presentation of chemical
concepts. However, chemistry is the science of reactions and interactions.
In the third chapter Klein and Ivanciuc show, how partial order can be applied within the context of substitution patterns. The authors demonstrate
for example that partial order relations and an order based on environmental toxicities match very well and how a parameter free approach to
QSAR can be found (see also topic 3). Methodologically the reader will
learn how chemical structures and partially ordered sets can be related and
how interpolation schemes are working. Finally, the important idea to extend the field of chemical property estimations by the concept or quantitative super-structure activity relationships is discussed.
Partial Ordering of Properties: The Young
Diagram Lattice and Related Chemical Systems
Sherif El-Basil
Faculty of Pharmacy, University of Cairo, Kasr Al-Aini st. Cairo 11562,
Egypt
e-mail: [email protected]
Abstract
The basic definitions related to the general topic of ordering are reviewed
and exemplified including: partial ordering, posets, Hasse diagrams, majorization of structures and comparable / incomparable structures.
Young Diagram lattice (of Ruch) and the ordering scheme of tree graphs
(of Gutman and Randiü) are described and it is shown, how the two
schemes coincide with each other, i.e. generate identical orders.
The role of Young diagrams in the ordering of chemical structures is
explained by their relation to alkane hydrocarbons and unbranched catacondensed benzenoid systems.
The Basic Terms: Examples of Posets, The Hasse
Diagram
The concept of a partial order appears to be very useful in environmental
science when evaluation and comparative study of properties are required.
The object to be studied form an object set and the partially order set (Ł
poset) depends on the ” , (greater than- or equal to-) relation (Luther et al.
(2000). We now introduce some of the popular definitions in an intuitive
approach, which avoids the “dryness” of mathematical rigor.
4
El-Basil, S.
Partially ordered set (poset)
It may be helpful to consider the following graph and analyze some parts
of it: (cf. Fig. 1)
8
12
4
6
2
3
1
Fig. 1. A labelled graph, which corresponds to a relation on a set of numbers
Obviously, the above graph describes some sort of a relation, R, on the
components of the set of integers:
S = {1, 2, 3, 4, 6, 8, 12}
(1)
We consider S as ground set (object set), whose elements are labelled
vertices of a graph. The relation among the vertices, graphically displayed
by lines (called "edges") depends on the questions one has. For example:
One observes that numbers, which divide others are connected, those that
do not divide each other are not. One, then, says that the above graph
represents some sort of ordering relation expressed as.
{(a,b) | ˨a divides b} on S = {1, 2, 3, 4, 6, 8, 12}
(2)
The relations among integers are described as follows:
a) Because every element of S is related to itself, i. e., (a, a)  R ; R
is said to be reflexive.
b) While, e.g., 2 divides 4, 4 does not divide 2 and so on. Such a relation is said to be anti-symmetric.
Partial Ordering of Properties: The Young Diagram Lattice
5
c) The last property may be exemplified on the subset {2, 4, 8}: 2 divides 4; 4 divides 8 hence 2 divides 8, which is true for other components, i. e.: if (a, b)  R and (b, c)  R then (a, b, c)  R.
The above property is called the transitive character of R. A poset may
then be defined as a relation R, on a set S if R is reflexive, anti-symmetric
and transitive.
The graph, which describes a particular poset, is called a Hasse diagram
after the 20th century German mathematician Helmut Hasse (1898-1979)
(Rosen 1991). See also chapter by Halfon p. 385.
A word on Hasse diagrams:
Actually the object shown in Fig. 1 is just a graph (not a diagram!): perhaps the word diagram is associated to it from the way it is used to be
drawn. In fact all self-evident edges are now removed such as all loops,
which describe the reflexive relation and also which result from the transitive character, e.g., edges (2, 8), (3, 12) and (1, all other vertices) are removed. Also arrows that indicate relative positions of components are no
longer indicated, yet the “old name”: diagram, (instead of graph) remained.
The Hasse diagram can be drawn in different ways maintaining the main
information, the order relations. Such Hasse diagrams are isomorphic to
each other.
Majorization of Structures: Relative Importance
Sometimes in (partial) ordering problems one may be interested in the relative importance of the components of a set. This situation reminds us with
the relation A ” B i.e., “A is a descendent of B” or that: “B majorizes A”.
A popular example is the partial ordering {(A, B) | A Ž B} on the power
set S = {a, b, c} where A Ž B means that A is a subset of B. Whenever this
relation exists one says that B majorizes A. The power set S contains
23 = 8 elements, viz., {a}, {b}, {a, b}, {a, c}, {b, c}, {a, b, c} and ‡,
where ‡ is the empty set.
For this particular case the Hasse diagram is simply a cube, labelled as
shown in Fig. 2.
6
El-Basil, S.
{a, b, c}
{b, c}
{a, c}
{a, b}
{c}
{a}
{b}
‡
Fig. 2. The Hasse diagram of S = {a, b, c}. Each subset is attached to its direct offspring, so that the descendant (less important components) lies in lower levels
One observes that {a, b}, {a, c} and {b, c} are subsets of {a, b, c} and
therefore of lower relative importance and analogously for the singlecomponent subsets {a}, {b}, {c}. The above example represents one of the
simplest cases of relative importance ordering problems, which finds
chemical applications (section ‘Relative importance of Kekulé Structures
of Benzenoid Hydrocarbons: Chain ordering’).
Comparable and incomparable elements: Chain and Anti-chain
The elements a and b of a poset (S,<) are called comparable if either a ” b
or b ” a. When a and b are elements of S such that neither a ” b nor b ” a, a
and b are called incomparable. For example the subsets {a,c}, {b,c} and
{a, b} are incomparable with each other: (they are not directly connected
(= adjacent) to each other, cf. Fig. 2). On the other hand, because {a, b, c}
majorizes {a, c}, e.g., they are comparable components of S.
Partial ordering may, then, be viewed as first weakening (Ł relaxation)
of the usual total ordering which is required for every pair of elements,
a,b S, that it must be a ” b or b ” a or a = b. Of course the standard total
ordering is that of “greater than or equal to” on the set of real members. In
Fig. 2, the subset of vertices, labelled {{a, b, c}, {a, c}, {c}, ‡} is called a
chain because every two elements of this subset are comparable. On the