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%.

[1]  Rainer Brüggemann,et al.  A Comparison of Partial Order Technique with Three Methods of Multi-Criteria Analysis for Ranking of Chemical Substances , 2002, J. Chem. Inf. Comput. Sci..

[2]  Peter B. Sørensen,et al.  Improving the Predicting Power of Partial Order Based QSARs through Linear Extensions , 2002, J. Chem. Inf. Comput. Sci..

[3]  R. Graham.  Linear Extensions of Partial Orders and the FKG Inequality , 1982 .

[4]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[5]  Takayuki Hibi,et al.  A recurrence for linear extensions , 1989 .

[6]  Trine S. Jensen,et al.  Ranking contaminated sites using a partial ordering method , 2003, Environmental toxicology and chemistry.

[7]  Peter Winkler.  Average height in a partially ordered set , 1982, Discret. Math..

[8]  F Balk,et al.  Selecting chemical substances for the UN-ECE POP protocol. , 2002, Chemosphere.

[9]  P. Sørensen,et al.  Evaluation of the ranking probabilities for partial orders based on random linear extensions. , 2003, Chemosphere.

[10]  Rainer Brüggemann,et al.  Applying the Concept of Partially Ordered Sets on the Ranking of Near-Shore Sediments by a Battery of Tests , 2001, J. Chem. Inf. Comput. Sci..

[11]  M. D. Atkinson.  The Complexity of Orders , 1989 .

[12]  W. Trotter,et al.  Combinatorics and Partially Ordered Sets: Dimension Theory , 1992 .

[13]  B. Bollobás.  Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Combinatorial Probability , 1986 .

[14]  Ole John Nielsen,et al.  Comparison of the combined monitoring-based and modelling-based priority setting scheme with partial order theory and random linear extensions for ranking of chemical substances. , 2002, Chemosphere.