Ranking probability

Lerche, D., Sorensen, P.B. and Briiggemann, R. (2003) Improved estimation of the ranking probabilities in partial orders using random linear extensions by approximation of the mutual ranking probability. J. Chem. Inf. Comput. Sci., 43, 1471-1480. 

A further use of linear extension is the probability scheme (ranking probabilities) that they provide. Probability plots are depicted for the three sites 1, 17 and 91 (Fig. 21a) and for site 5 (Fig. 21b), respectively. (See also the contributions, chapters by Voigt and Bruggemann, p. 327 Bruggemann et al., p. 237 Carlsen, p. 163. 

The analysis by linear extensions is very attractive as it helps to derive a linear ranking, without any subjective preferences. The data lead to a poset, the poset may be analyzed with respect to its structure, this is a combinatorial problem, and finally a ranking probability can be derived. Crucially in this procedure is that very different attribute profiles may lead to the same Hasse diagram and thus to the same set of linear extensions and therefore finally to the same probability characteristics Thus, the attribute profiles a) (0,0), (1,0), (0,1), (1,1) and b) (0,0), (1,0), (0,5), (4,7) lead to identical Hasse diagrams. 

Reidel Publishing Company, Dordrecht, pp 3-41 Lerche D, Briiggemann R, Sorensen PB, Carlsen L, Nielsen OJ (2002) A Comparison of Partial Order Technique with three Methods of Multicriteria Analysis for Ranking of Chemical Substances. J Chem Inf Comp Sc 42 1086-1098 Lerche D, Sorensen PB (2003) Evaluation of the ranking probabilities for partial orders based on random linear extensions. Chemosphere 53 981-992... 

Lerche D, Sorensen PB, Briiggemann R (2003) Improved Estimation of the Ranking Probabilities in Partial Orders Using Random Linear Extensions by Approximation of the Mutual Probability. J Chem Inf Comp Sc 53 1471-1480 Muirhead RF (1900) Inequalities relating to some Algebraic Means. Proc Edinburgh Math Soc 19 36-45... 

Fig. lb. Calculation of the Ranking frequencies (left table) and ordinal ranking probabilities (right table) and of averaged ranks... 

For the substance x3 the ordinal ranking probability for rank no. 5 is equal to 0 and for rank no. 4 it is 0.375. Note that one can apply symmetry considerations (i.e. in mathematical terms, analyze the automorphism group of a Hasse diagram (see Schroder 2003)) For example x3 "sees" the... 

Using all linear extensions the probability distribution of ranks for each substance can be found. Table 3 gives the ordinal probability for the individual substances to occupy a certain rank. The ranking probabilities are spread out in the interval of possible ranking positions. Clearly the probabilities of isolated objects, like malathion (MAL) (compare Fig. 2) will be spread equally out over the whole interval providing an averaged rank of 6.5. 

Table 5. The distributions of ranking probabilities when using a Monte Carlo simulation on the weights for a the Utility Function and b PROMETHEE...

Lerche D, Sorensen PB (2003) Evaluation of the Ranking Probabilities for Partial Orders based on Random Linear Extensions. Chemosphere 53 981-992... 

A discussion of the derivation of the failure probability from the numbers of failures (particularly for relatively small numbers) will be found in Refs. 21 and 22. Thi.s probability is not generally equal to the ratio of the number failed to the total number but to the "median rank" probability. A widely accepted close approximation to this is... 

Rank Probability That the Defect Will Be Detected... 

D. Lerche and P. B S0rensen, Evaluation of the ranking probabilities for partial orders based on random linear extensions, Chemosphere 53 (2003) 981-992. 

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