Partial order ranking

Partial order ranking (POR) is based on elementary methods of discrete mathematics (e.g., Hasse diagrams) — if A < B and B < C, then A < C in the ranking procedures. POR does not assume linearity or any assumptions about distribution properties such as normality. The disadvantage is that often a preprocessing of data is needed to avoid the effects of stochastic noise. Combining POR with PCA may improve its usefulness. POR can only be applied for interpolation. 

Examples of applications of partial-order ranking methods and other theoretical aspects are reported in [Walczak and Massart, 1999 Klein and Bytautas, 2000 Carlsen, Sorensen et al,... 

DART (Decision Analysis by Ranking Techniques) is a free available sofiware implementing both partial-order ranking and several total ranking methods [DART- Milano Chemometrics, 2007]. 

Carlsen, L. (2004) Giving molecules an identity On the interplay between QSARs and partial order ranking. Molecules, 9, 1010-1018. 

Carlsen, L., Sorensen, P.B. and Thomsen, M. (2001) Partial order ranking-based QSARs estimation of solubilities and octanol—water partitioning. Chemosphere, 43, 295—302. 

Sorensen, P.B., Briiggemann, R., Carlsen, L., Mogensen, B.B., Kreuger, J. and Pudenz, S. (2003) Analysis of monitoring data ofpesticide residues in surface waters using partial order ranking theory. Environ. Toxicol. Chem., 22, 661-670. 

Carlsen L, Sorensen PB, Thomsen M (2001) Partial Order Ranking-Based QSAR s Estimation of Solubilities and Octanol-Water Partitioning. Chemosphere 43 295-302... 

Carlsen L., Sorensen PB, Thomsen M, Bruggemann R (2002) QSAR s Based on Partial Order Ranking. SAR and QSAR in Environ Res 13 153-165 Carlsen L (2004) Giving Molecules an Identity. On the Interplay between QSARs and Partial Order Ranking. Molecules 9 1010-1018... 

The questions put here for sediments could be applied to other environmental evaluation problems just as well, for instance to soil or groundwater pollution. However, a common difficulty with these evaluations is that many of the methods mask and aggregate the data, and therefore both valuable information and transparency are lost. An alternative is partial order ranking that avoids the merging of data and thus preserves important elements of the evaluation. Here we will show that partial order ranking has useful qualities in data analysis and it can be applied for preprocessing in the development of classification systems. 

Heininger P, Tippmann P (1995) Determination of Enzymatic Activities for the Characterization of Sediments. Toxicol Environm Chem 52 25-33 Luther B, Bruggemann R, Pudenz S (2000) An Approach to Combine Cluster Analysis with Order Theoretical Tools in Problems of Environmental Pollution. MATCH - Comm in Math Chem 42 119-143 Pudenz S (2004) ProRank - Software for Partial Order Ranking. MATCH -Comm in Math Chem 54(3) 611-622... 

Partial order ranking of the substances was made using the 50 BCF and BDP estimates (cf. Table 1) and applying the WHASSE software (Fig. 1). 

Fig. 1. Hasse Diagram displaying the partial order ranking of the substances studied using the BCF and BDP estimates as descriptors...

Partial order techniques may also be applied directly as QSAR method as illustrated by the use of the QSAR descriptors as input to the ranking (Carlsen et al. 2001 Carlsen et al. 2002). On the other hand partial order ranking based on QSAR descriptors may also be applied as inverse QSARs, i.e. to disclose specific characteristics of new substances to be synthesized, e.g., as substitutes for environmentally harmful counterparts (Bruggemann et al. 2001b) or simply to give a given chemical compound an identity by comparing specific characteristics to those of other, possibly less harmful substances (Carlsen 2004). 

Obviously the successful interplay between QSARs and partial order ranking depends on the single techniques. Thus, in the following QSARs and partial order ranking, including linear extensions will be shortly presented. 

Subsequently, these QSAR generated endpoints may be applied for a partial order ranking of the substances using two or more of the endpoints as descriptors for the ranking exercise. 

In partial order ranking - in contrast to standard multidimensional statistical analysis - neither assumptions about linearity nor any assumptions about distribution properties are made. In this way the partial order ranking can be considered as a non-parametric method. Thus, there is no preference among the descriptors. However, due to the simple mathematics outlined above, it is obvious that the method a priori is rather sensitive to noise, since even minor fluctuations in the descriptor values may lead to non-comparability or reversed ordering. An approach how to handle loss of information by using an ordinal in stead of a matrix can also be found in the chapter by Pavan et al., see p. 181). 

In partial order ranking - in contrast to standard multidimensional statistical analysis - neither assumptions about linearity nor any assumptions about distribution properties are made. Partial order ranking may be considered as a parameter-free method. Thus, there is no preference among the... 

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