Improved estimation of the ranking probabilities in partial orders using random linear extensions by approximation of the mutual ranking probability

The application of partial order theory and Hasse diagram technique in environmental science is getting increasing attention. One of the latest developments in the field of Hasse diagram technique is the use of random linear extensions to estimate ranking probabilities. In the original algorithm for estimating the ranking probability it is assumed that the order between two incomparable pair of objects can be chosen randomly. However, if the total set of linear extensions is considered there is a specific probability that one object will be larger than another, which can be far from 50%. In this study it is investigated if an approximation of the mutual ranking probability can improve the algorithm. Applying an approximation of the mutual ranking probability the estimation of the ranking probabilities are significantly improved. Using a test set of 39 partial orders with randomly chosen values the relative mean root square difference (MRSD) decrease in average from 7.9% to 2.2% and a maximum relative improvement of 90% can be found. In the most successful case the relative MRSD goes as low as 0.77%.     

Applying the concept of partially ordered sets on the ranking of near-shore sediments by a battery of tests

When a ranking of some objects (chemicals, geographical sites, river sections, etc.) by a multicriteria analysis is of concern, then 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 controversially discussed. 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, developed by Dutka et al. In our analysis we found the following: (1) The dimension analysis of partially ordered sets suggests that, at least in the case of the 55 analyzed samples and the evaluation by the scores, developed by Dutka et al., there is a considerable redundancy with respect to ranking. The visualization of the sediment sites can be performed 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, and each class exhibits a different pattern of results. (3) Loss of information, when an aggregation of test results is used in order to guarantee complete comparability among all objects. A relation between information drawn from the graphic and the uncertainty of ranking after using an aggregation is given. (4) The sensitivity analysis identifies one test as most important, namely the test for Fecal Coliforms/Escherichia coli. This means that the ranking of samples is heavily influenced by the results of this specific test.     

Basic principles of Hasse diagram technique in chemistry

Principles of partial order applied to ranking are explained. The Hasse diagram technique (HDT) is the application of partial order theory based on a data matrix. In this paper, HDT is introduced in a stepwise procedure, and some elementary theorems are exemplified. The focus is to show how the multivariate character of a data matrix is realized by HDT and in which cases one should apply other mathematical or statistical methods. Many simple examples illustrate the basic theoretical ideas. Finally, it is shown that HDT is a useful alternative for the evaluation of antifouling agents, which was originally performed by amoeba diagrams.     

Data mining by total ranking methods: a case study on optimisation of the "pulp and bleaching" process in the paper industry

Total order ranking methods are multicriteria decision making techniques used for the ranking of various alternatives on the basis of more than one criterion. The criteria, which are the standards by which the elements of the system are judged are not always in agreement, they can be conflicting, motivating the need to find an overall optimum that can deviate from the optima of one or more of the single criteria. Total order ranking methods are based on an aggregation of the criteria in a scalar function, i.e. an order or ranking index, which allow to sort elements according to its numerical value. Several evaluation methods which define a ranking parameter generating a total order ranking have been proposed in the literature. Four total order ranking methods are here described: Desirability functions, Utility functions, Dominance functions and Absolute Reference method. These methods have been compared to each other by applying them to a decision making problem in the paper industry. Various bleaching processes have been analysed and compared on the basis of multiple criteria, the aim being to find out best bleaching process among the ones proposed in the last years as alternative to chlorine bleaching process which is of high environmental impact due to the potential for chlorinated dioxin production.     

New operations for informative combination of two partial order relations with illustrations on pollution data

We discuss various ways in which to construct and process partial order relations or partially ordered sets (posets) in the context of ranking objects on the basis of multiple criteria. Oftentimes, it is undesirable or even impossible to devise a weighting scheme to compute a final score on the basis of the criteria. An alternative is then to restrict oneself to the information contained in the partial ordering of all objects implied by the criteria. We will consider some ways in which one can exploit partial order relations to determine a ranking of a collection of objects. More exactly, we will examine how to combine information coming from two sources, both for the case in which the sources are considered to be equally important, as well as for the case in which one source of information should take priority. We illustrate the concepts on pollution data coming from 59 regions in Baden-Württemberg.     

A chemometric approach based on a novel similarity/diversity measure for the characterisation and selection of electronic nose sensors

Electronic nose sensor signals provide a digital fingerprint of the product in analysis, which can be subsequently investigated by means of chemometrics. In this paper, the fingerprint characterisation of electronic nose data has been studied by means of a novel chemometric approach based on the partial ordering technique and the Hasse matrix. This matrix can be associated to each data sequence and the similarity between two sequences can be evaluated with the definition of a distance between the corresponding Hasse matrices. Since all the signals achieved along time are intrinsically ordered, the data provided by electronic nose can be also considered as sequential data and consequently characterized by means of the proposed approach. The similarity/diversity measure has been here applied in order to characterize the class discrimination capability of each electronic nose sensor: extra virgin olive oil samples of different geographical origin have been considered and Hasse distances have been used to select the sensors which appear more able to discriminate the olive oil origins. The distance based on the Hasse matrix has showed some useful properties and proved to be able to link each electronic nose time profile to a meaningful mathematical term (the Hasse matrix), which can be consequently studied by multivariate analysis.     

A comparison of partial order technique with three methods of multi-criteria analysis for ranking of chemical substances

An alternative to the often cumbersome and time-consuming risk assessments of chemical substances could be more reliable and advanced priority setting methods. An elaboration of the simple scoring methods is provided by Hasse Diagram Technique (HDT) and/or Multi-Criteria Analysis (MCA). The present study provides an in depth evaluation of HDT relative to three MCA techniques. The new and 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. The study illustrates that the Hasse diagram technique (HDT) needs least external input, is most transparent and is least subjective. However, HDT has some weaknesses if there are criteria which exclude each other. Then weighting is needed. Multi-Criteria Analysis (i.e. Utility Function approach, PROMETHEE and concordance analysis) 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 the first step in decision making is to run HDT and as the second step possibly is to run one of the MCA algorithms.     

Hasse diagram technique as a tool for water quality assessment

The management of the quality large water catchments is a complex problem which requires intelligent data analysis on various levels – analytical, spatial, and temporal. Recently, a successful approach is developed combining advanced multivariate data treatment approaches like self-organizing maps of Kohonen (SOM) and Hasse diagram technique (HDT). In the first step of the environmetric analysis the monitoring data were subject to pre-processing using SOMs to reduce the number of objects and/or water quality parameters. In the next step HDT for partial ranking (both in spatial and temporal aspect) was applied according to the pre-selected set of the water quality parameters. The use of the water quality norms issued by the Bulgarian environmental authorities revealed important details in assessing the Maritsa River water quality. Thus, the relations between different water quality patterns and sampling stations could be used by water management authorities during the period of observation.     

Comparative evaluation of chemical and environmental online and CD-ROM databases

In a constantly expanding world of chemical and environmental information sources, the need for their evaluation gains more and more importance. This paper presents a comparative evaluation of datasources of online databases and databases on CD-ROM (called CD-ROMs in this paper) in the field of environmental chemicals. The approach is based on research results gained in the years 1996/1997. The authors are aware that changes in the database industry may lead to different results. Before the actual evaluation process can be carried out, two major procedures are necessary, namely, the selection of sets of datasources and the definition of evaluation criteria. In order to perform the difficult task of an evaluation based on several criteria, a general order relation has to be introduced. Methods of partially ordered set theory are applied, and the results are visualized by the technique of Hasse diagrams. On the basis of these evaluation results, the datasources are grouped and then evaluated. It will be shown that there are groups of datasources with quite specific property profiles, and only two groups turn out to be relatively better than the others.     

Order theoretical tools for the evaluation of complex regional pollution patterns

The mathematical and statistical evaluation of environmental data gains an increasing importance in environmental chemistry as the data sets become more complex. It is inarguable that different mathematical and statistical methods should be applied in order to compare results and to enhance the possible interpretation of the data. Very often several aspects have to be considered simultaneously, for example, several chemicals entailing a data matrix with objects (rows) and variables (columns). In this paper a data set is given concerning the pollution of 58 regions in the state of Baden-Württemberg, Germany, which are polluted with metals lead, cadmium, zinc, and with sulfur. For pragmatic reasons the evaluation is performed with the dichotomized data matrix. First this dichotomized 58 x 13 data matrix is evaluated by the Hasse diagram technique, a multicriteria evaluation method which has its scientific origin in Discrete Mathematics. Then the Partially Ordered Scalogram Analysis with Coordinates (POSAC) method is applied. It reduces the data matrix in plotting it in a two-dimensional space. A small given percentage of information is lost in this method. Important priority objects, like maximal and minimal objects (high and low polluted regions), can easily be detected by Hasse diagram technique and POSAC. Two variables attained exceptional importance by the data analysis shown here: TLS, Sulfur found in Tree Layer, is difficult to interpret and needs further investigations, whereas LRPB, Lead in Lumbricus Rubellus, seems to be a satisfying result because the earthworm is commonly discussed in the ecotoxicological literature as a specific and highly sensitive bioindicator.     

A hitchhiker's guide to poset ranking

When ranking objects (like chemicals, geographical sites, river sections, etc.) by multicriteria analysis, it is in most cases controversial and difficult to find a common scale among the criteria of concern. Therefore, ideally, one should not resort to such artificial additional constraints. The theory of partially ordered sets (or posets for short) provides a solid formal framework for the ranking of objects without assigning a common scale and/or weights to the criteria, and therefore constitutes a valuable alternative to traditional approaches. In this paper, we aim to give a comprehensive literature review on the topic. First we formalize the problem of ranking objects according to some predefined criteria. In this theoretical framework, we focus on several algorithms and illustrate them on a toy example. To conclude, a more realistic real-world application shows the power of some of the algorithms considered in this paper.     

Total ranking models by the genetic algorithm variable subset selection (GA–VSS) approach for environmental priority settings

Total order ranking (TOR) strategies, which are mathematically based on elementary methods of discrete mathematics, seem to be attractive and simple tools for performing 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, a possible alternative to conventional quantitative structure–activity relationship (QSAR) methods. In fact, when data material is characterised by uncertainties, order methods can be used as alternative to statistical methods such as multilinear regression (MLR), because they do not require specific functional relationships between the independent and 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 attributes, which are calculated attributes. As in regression and classification models, the variable selection model is one of the main steps in finding predictive models. In this work the genetic algorithm–variable subset selection (GA–VSS) approach is proposed as the variable selection method for searching for the best ranking models within a wide set of variables. The models based on the selected subsets of variables are compared with the experimental ranking and evaluated by the Spearmans rank index. A case study application is presented on a TOR model developed for polychlorinated biphenyl (PCB) compounds, which have been analysed according to some of their physicochemical properties which play an important role in their environmental impact.     

Monte Carlo phase diagram for diblock copolymer melts

A partial phase diagram is constructed for diblock copolymer melts using lattice-based Monte Carlo simulations. This is done by locating the order-disorder transition (ODT) with the aid of a recently proposed order parameter and identifying the ordered phase over a wide range of copolymer compositions (0.2相似文献