Applying the Concept of Partially Ordered Sets on the Ranking of Near-Shore Sediments by a Battery of Tests

Abstract

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.

Measurement of fiscal rules: Introducing the application of partially ordered set (POSET) theory

“…Notwithstanding these differences in the point estimates of the effects of fiscal rules, the high correlation of the two indices deserves some discussion. As outlined in Section 3.2.1, 23 1990-2005 1985-2012 1985-2012 1990-2005 1985-2012 1985-2012 there is always one weighting scheme that implies equivalence between the composite and the POSET index (Brüggemann et al, 2001). Coincidentally, in the present sample it happens to be the case that the IMF index by Schaechter et al (2012), using equal weights for the different properties, is quite close to this particular weighting scheme.…”

“…It has been employed, e.g., in order to rank chemicals according to environmental hazards (Halfon and Reggiani, 1986), for the ranking of near-shore sediments (Brüggemann et al, 2001), to evaluate the quality of air pollutant monitoring systems (Voigt et al, 2004) or to explore patterns of habitat diversity across US landscapes (Myers et al, 2006). In the next section we outline the theoretical concepts underlying the POSET approach.…”

“…for our empirical analysis, the averaged ranks, ranging from potentially 1 to N , will be normalized to take values between zero and one. Relating the POSET to the index function approach, the latter is a very special case of the former in the sense that each ranking implied by a particular weighting scheme in an index function corresponds to exactly one particular linear extension of the POSET (Brüggemann et al, 2001). …”

Summarizing data using partially ordered set theory: An application to fiscal frameworks in 97 countries

“…The latter distinguishes itself from the former by the fact that at least one possible linear extension contained in ε(G) corresponds to any ranking obtained by a weighting scheme in an index function (Brüggemann et al, 2001). As shown in Brüggemann et al (2004), the computation of the approximate rank probabilities of more than n = 25 objects exceeds today's computer capacities.…”

METEOR: a step‐by‐step procedure to explore effects of indicator aggregation in multi criteria decision aiding – application to water management in Berlin, Germany

“…Alternatively, there are evaluation approaches purely based on partial order theory, e. g., the formal concept analysis (FCA) [10] or the Hasse diagram technique (HDT) [11]. Options are sorted on the basis of a simple f comparison where each indicator induces a linear order of the options.…”