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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.

Here, the approach of using partially ordered sets is described by applying it to a battery of tests on sediments of the Lake Ontario. In our analysis we found: (1) the dimension analysis of partially ordered sets suggests that there is a considerable redundancy with respect to ranking. The partial ranking of the sediment sites can be visualized within a two-dimensional grid. (2) Information, obtained from the structure of the Hasse diagram: For example six classes of sediment sites have high priority, each class exhibits a different pattern of results. (3) The sensitivity analysis identifies one test as most important, namely the test for Fecal Coliforms/ . This means that the ranking of samples is heavily influenced by the results of this specific test.

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).

A ranking model is a relationship between a set of dependent attributes, experimentally investigated, and a set of independent attributes, i.e. model variables. As in regression and classification models the variable selection is one of the main step to find predictive models. In the present work, the Genetic Algorithm (GA-VSS) approach is proposed as the variable selection method to search for the best ranking models within a wide set of predictor variables. The ranking based on the selected subsets of variables is compared with the experimental ranking and evaluated both in partial and total ranking by a set of similarity indices and the Spearman’s rank index, respectively. A case study application is presented on a partial order ranking model developed for 12 congeneric phenylureas selected as similarly acting mixture components and analysed according to their toxicity on .

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.