Average Heights in Partially Ordered Sets

作者：Bruggemann Rainer^*; Annoni Paola

来源：MATCH-Communications in Mathematical and in Computer Chemistry, 2014, 71(1): 117-142.

Ranking of chemicals is an important step in risk assessments. The hazard of chemicals exerted on human or on environments is not directly measurable, instead a set of indicators serve as proxy for this purpose. This so-called multi-indicator system is of high scientific value because it contains specific knowledge about chemicals. Often a ranking is obtained by aggregating this detailed information to get a scalar, the ranking index. Such an aggregation may be considered as subjective, at least up to certain extent. An alternative to aggregation is possible by applying simple elements of partial order theory. Instead of unique, linear orders, partial order theory can provide the so-called average heights from the concept of linear extensions. The calculation of average heights is of high interest because, once estimated, a weak order can be derived. Once the weak order is found, (tied) ranks can be determined. This is why average heights are often called average ranks. The calculation of the needed average heights is most often computationally intractable and approximations are needed. The quality of the approximations cannot be checked because of the computational restrictions in calculating exactly all needed average heights. A possible way out of this conundrum is the definition of models of partially ordered sets - model-posets - which are simple enough to derive analytical formulas for the average heights and, at the same time, provide enough structural diversity to allow for generalizations. Once model-posets are set-up, the quality of approximations can be checked and the role of different parts of the graph, representing the partially ordered set, can be studied by sensitivity analysis. We start the analysis with an example taken from environmental chemistry, where persistent organic pollutants are ranked according to three indicators, namely persistence, bioaccumulation and toxicities.
It is found that i) an approximation model, based on a local partial order concept, provides results in a satisfying coincidence with the exact values and ii) that the structure parameters can be ranked with respect to their impact on the average height value.

• 出版日期2014