Partial Order in Environmental Sciences and Chemistry

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 Randić) 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.

Hasse diagrams are applied to molecules and radiation phenomena. Then the relation of these diagrams to periodicity in atoms is noted. The possibility is raised that Hasse diagrams can also be related to the growing body of evidence that periodicity exists in molecules with two, three, and four atoms; in binary inorganic molecules; and in some organic molecules.

Reaction diagrams are considered especially for the circumstance of progressive substitution (or addition) on a fixed molecular skeleton, and it is noted that these naturally form Hasse diagrams for a partially ordered set (or poset) of the substituted structures. The possibility that different properties are similarly ordered is a further natural consideration, and is here illustrated for several different properties for (methyl & chloro) substituted benzenes.

This posetic approach thence provides a novel approach to structure/ property and structure/bioactivity correlations, with focus in some sense beyond simple molecular structure, in that this approach attends to how a structure fits into a systematic (reaction) network of structures. Different manners for fitting and prediction of properties are noted, with illustration of an especially simple “poset-average” scheme. Some numerical evidence indicates that such approaches are quite reasonable. It is emphasized that such directed reaction graphs admitting posetic treatment are widespread.

The first part of this chapter gives a detailed introduction to partial order ranking and Hasse Diagram Technique (HDT). Thus, the construction of Hasse diagrams is elucidated as is the different concepts associated with the diagrams. The analysis of Hasse diagrams is disclosed including structural analysis, dimension analysis and sensitivity analysis. Further the concept of linear extensions is introduced including ranking probability and averaged rank. The evaluation of sampling sites is, in the second part of the chapter, used as an illustrative example of the advantageous use of partial order ranking and Hasse Diagram Technique.

When a ranking of some objects (chemicals, geographical sites, river sections etc.) by a multicriteria analysis is of concern, it is often difficult to find a common scale among the criteria and therefore even the simple sorting process is performed by applying additional constraints, just to get a ranking index. However, such additional constraints, often arising from normative considerations are controversial. The theory of partially ordered sets and its graphical representation (Hasse diagrams) does not need such additional information just to sort the objects.

With respect to sediment pollution responses of ecotoxicological tests may differ from those of biochemical test systems and moreover both tests are indicating effects instead of simply measuring of chemical concentrations. Because most test results of sediment investigations are commonly given as inhibition values and sediment pollution by chemicals is measured by their concentrations a comparative evaluation of sediments by means of both test results and chemicals at the same time has to consider different scales. Both data transformations on a common scale (standardization) and aggregations lead to loss of information and hamper the interpretation of results. In order to avoid merging of data and to circumvent often-crucial data transformations, partial ordering is used for evaluation of sediment samples from German rivers. The aim here is to compare the evaluation of river sections by different parameter groups, namely biochemical and ecotoxicological tests, as well as concentrations of organic pollutants, heavy metals etc. Fuzzy cluster analysis as a pre-processing step is additionally used to understand the pollution pattern that is given by each test result. It is shown that for most of the river sections, test systems among each other and also compared to chemical concentrations yield different quality pattern and therefore lead to different Hasse diagrams. Sole exception is a bayou where the sediment is undisturbed by shipping traffic and sewage. Moreover, as a consequence of varying pollution pattern during the sampling period (over several years), only for a few river sections it is possible to derive distinct temporal changes: Except for the nematode sediment contact test, where all parameters are significantly correlated, this holds for both ecotoxicological and biochemical tests, and for chemical concentrations. Furthermore, for one river section it could be observed that chemical concentrations indicate a decline of contamination, whereas ecotoxicological parameters point to an increased toxicity. With respect to the development of a classification system for river sediments it is recommended to take care in the selection of parameters and to base it at least at two parameter groups.

The interplay between partial order ranking and Quantitative Structure Activity Relationships (QSARs) constitute a strong decision support tool. By means of partial order ranking it is possible to prioritize and select chemicals for decision-making among a group of substances based on simultaneous evaluation of data related to different endpoints. In the absence of experimental data, QSARs are used to provide estimates. In the present chapter, the identification of chemicals with Persistence and Bioconcentration (PB) potential is used to illustrate the interplay between partial order ranking and QSARs. The endpoints biodegradation and bioconcentration were obtained using the BioWin and BCFWin modules from http://www.epa.gov/oppt/exposure/docs/episuitedl.htm. Partial order theory was used to rank chemicals for PB potential based on QSAR estimates. The proposed approach is suggested as a decision support tool to facilitate pollution prevention activities by regulated and regulatory communities.

The interplay between Quantitative Structure-Activity Relationships (QSARs) and partial order ranking appears as an advantageous method to assess and prioritize chemical substances, e.g., due to their potential environmental hazard taking several parameters simultaneously into account. Especially the application of so-called ‘noise-deficient’ descriptors is emphasized in order to eliminate the natural fluctuation of experimental as well as simple QSAR derived data. Further partial order ranking appears as an attractive alternative to conventional QSAR methods that typically rely on the application of stochastic methods. The latter use of partial order ranking may be applicable both to direct QSARs as well to solving inverse QSAR problems. The present chapter summarizes the various types of interplay between of partial order ranking and QSAR modelling.

Partial and total order ranking strategies, which from a mathematical point of view are based on elementary methods of Discrete Mathematics, appear as an attractive and simple tool to perform data analysis. Moreover order ranking strategies seem to be a very useful tool not only to perform data exploration but also to develop order-ranking models, being a possible alternative to conventional QSAR methods. In fact, when data material is characterised by uncertainties, order methods can be used as alternative to statistical methods such as multiple linear regression (MLR), since they do not require specific functional relationship between the independent variables and the dependent variables (responses).

In the cities of Berlin and Potsdam nine water management strategies (scenarios) were evaluated with respect to their ecological effects to the system of surface water. Scenarios were generated by combining different water management measures such as wastewater and storm water treatment. Indicators were qualitatively modelled as well as quantitatively evaluated by experts’ knowledge. For decision support Hasse Diagram Technique (HDT) was used. The scenario modular structure increases the transparency of the evaluation process and brought up the question whether time and work consuming calculation of data by mathematical models is needed or experts’ knowledge is sufficient for evaluation. To clarify this question, the results of two evaluation examples were compared: (a) data based and (b) experts expectations. Beyond the concept of antagonistic indicators the similarity-profile is introduces as a new tool to compare HDT evaluation results. Our study revealed that in the present investigation evaluation by expert knowledge is not satisfactory. The shift in the type of indicators from state to pressure and the effect of up scaling from local to regional may be the reason.

An alternative to time-consuming risk assessments of chemical substances could be more reliable and advanced priority setting methods. Hasse Diagram Technique (HDT) and/or Multi-Criteria Analysis (MCA) provide an elaboration of the simple scoring methods. The present chapter evaluates HDT relative to two MCA techniques. The main methodological step in the comparison is the use of probability concepts based on mathematical tools such as linear extensions of partially ordered sets and Monte Carlo simulations. A data set consisting of 12 High Production Volume Chemicals (HPVCs) is used for illustration.

It is a paradigm in this investigation to claim that the need of external input (often subjective weightings of criteria) should be minimized and that the transparency should be maximized in any multicriteria prioritisation. This study illustrates that the Hasse Diagram Technique (HDT) needs least external input, is most transparent and is therefore the least subjective of the techniques studied. However, HDT has some weaknesses if there are criteria, which exclude each other. In such cases weighting is needed. Multi-Criteria Analysis (i.e. Utility function approach and PROMETHEE as examples) can deal with such mutual exclusions because their formalisms to quantify preferences allow participation e.g. weighting of criteria. Consequently MCA include more subjectivity and loose transparency. The recommendation, which arises from this study, is that a first step in decision-making is to run HDT and as a second step possibly to run one of the MCA algorithms.

The construction of posets or Hasse diagrams is a profitable means for the evaluation of biomonitoring data. In contrast to other statistical approaches the Hasse Diagram Technique enables the consideration of multiple attributes at the same time and will result in at least partially ordered data sets. Moreover, the calculation of averaged ranks allows the construction of a total order for a given data set. For the evaluation of biomonitoring data, as obtained for the German Environmental Specimen Bank, the Hasse diagram technique was applied to achieve partially or totally ordered data. The following scheme was applied: i) Careful rounding of the original data to increase the number of comparabilities; ii) splitting of the data in smaller sub-sets; iii) construction of the posets for each sub-set; iv) construction of the total order for each sub-set (by means of averaged ranks) and, v) synopsis of the sub-sets.

Potential habitat suitability was assessed for species groupings of vertebrate fauna in the State of Pennsylvania, USA as part of a nationally coordinated GAP Analysis Program to find gaps in provision for conservation of important habitats. Diversity values were compiled spatially at a resolution of one square kilometre from species models developed at 30-meter resolution. Diversity patterns differ in varying degrees among species groups for mammals, birds, amphibians, snakes/lizards, turtles, and fishes. Comparing the patterns for partial ordering on watershed extents using statistical indices of ranking can facilitate determination of inter-group commonality and contrast. This helps to designate watersheds as having importance from multi-group and particular group perspectives. Partial ordering on the basis of rank-range runs is particularly informative when combined with levels of counter-indication corresponding to levels in a Hasse diagram. This serves to segregate sets having combinatorial clarity of condition relative to conservation from settings where disparate conditions may offer opportunities for targeted restoration. Disparity of conditions on multiple bio-indicators may arise from habitat heterogeneity as well as differential degradation. Broadening the spectrum of indicators will usually increase the apparent complexity of the conservation context.

The main objective of the European Commission’s White Paper on a future chemicals strategy (EEC 2001) is to facilitate the risk assessment of chemicals leading to, where necessary, risk reduction. Important roles play the chemical and environmental databases, which can be regarded as an information turnover. In this paper the emphasis lies on the evaluation of 12 numerical databases available on the free Internet, which focus on environmental fate and ecotoxicity as well as on high production volume chemicals. Hence we analyse a 12×27 data-matrix in the first place. Two multi-criteria evaluation and decision support instruments are applied: The Hasse Diagram Technique (HDT), a method derived from discrete mathematics, and the Method of Evaluation by Order Theory (METEOR). The original data-matrix of 12 databases (objects) and 27 parameters (attributes) will be subject to several logical aggregation steps. The aim of the aggregation procedure that can be performed by applying iteratively the Hasse Diagram Technique (HDT) is to get a unique prioritisation scheme. Significant data gaps even on the chosen well-known high production volume chemicals as well as on ecotoxicity and environmental fate parameters are identified by the chosen methods and weighting procedures. This indicates an alarming signal concerning the new existing chemicals policy of the EEC.

We give a brief introduction to the notion of concept, a mathematical model of conceptual thinking. It serves very well in the organization of interviews, tests and evaluations, since it allows a systematic way of drawing conclusions and establishing hypotheses. Thus, it can be considered as an efficient tool for decision support, for example, in environmental risk management. In fact, it models a certain way of doing research by gathering examples and trying pattern recognition.

A partial order of longstanding interest to mathematicians and chemists, the Young Diagram Lattice (YDL) is discussed in the context of complexity. Ruch’s (1975) identification of this partially ordered set with that appropriate to a general partial ordering for mixing is discussed. A mathematical quantity associated with each member of the set (the cardinality of maximal anti-chains for that member) is argued to provide a quantitative measure for complexity for members of the set. The measure has the desirable feature that low complexity is associated with both highly ordered and very random systems, while systems that have intermediate “structure” have larger complexity. Several quantitative examples based on the YDL are briefly discussed including statistical mechanics, diffusion, and biopolymeric complexity. Finally, a metaphor for complexity suggested by the YDL associates high complexity with posetic incomparability. Examples from sociology, ecology, and politics are discussed.

This chapter describes the evolution of the use of Hasse diagrams in the environmental field and the development of the related software. Two Italian scientists, Marcello Reggiani and Roberto Marchetti, used Hasse diagrams to study the problem of model order estimation. Halfon extended the use of Hasse diagrams to ecological modelling and later to environmental chemistry. The HASSE software was initially developed in FORTRAN to be run on mainframe computers. Later Halfon in Canada and Brüggemann in Germany reprogrammed it for use on personal computers and code was added to display the diagrams interactively. Nowadays, scientific groups in Denmark, Germany, Italy and Sweden are actively developing new applications and developing new theoretical concepts.