Karel De Loof

A Hitchhikers Guide to Poset Ranking

When ranking objects (like chemicals, geographical sites, river sections, etc.) by multicriteria analysis, it is in most cases controversial and difficult to find a common scale among the criteria of concern. Therefore, ideally, one should not resort to such artificial additional constraints. The theory of partially ordered sets (or posets for short) provides a solid formal framework for the ranking of objects without assigning a common scale and/or weights to the criteria, and therefore constitutes a valuable alternative to traditional approaches. In this paper, we aim to give a comprehensive literature review on the topic. First we formalize the problem of ranking objects according to some predefined criteria. In this theoretical framework, we focus on several algorithms and illustrate them on a toy example. To conclude, a more realistic real-world application shows the power of some of the algorithms considered in this paper.

Filtering and ranking techniques for automated selection of high-quality 16S rRNA gene sequences

StrainInfo has augmented its type strain and species/subspecies passports with a recommendation for a high-quality 16S rRNA gene sequence available from the public sequence databases. These recommendations are generated by an automated pipeline that collects all candidate 16S rRNA gene sequences for a prokaryotic type strain, filters out low-quality sequences and retains a high-quality sequence from the remaining pool. Due to thorough automation, recommendations can be renewed daily using the latest updates of the public sequence databases and the latest species descriptions. We discuss the quality criteria constructed to filter and rank available 16S rRNA gene sequences, and show how a partially ordered set (poset) ranking algorithm can be applied to solve the multi-criteria ranking problem of selecting the best candidate sequence. The proof of concept of the recommender system is validated by comparing the results of automated selection with an expert selection made in the All-Species Living Tree Project. Based on these validation results, the pipeline may reliably be applied for non-type strains and developed further for the automated selection of housekeeping genes.

Product Triplets in Winning Probability Relations

It is known that the winning probability relation of a dice model, which amounts to the pairwise comparison of a set of independent random variables that are uniformly distributed on finite integer multisets, is dice transitive. The condition of dice transitivity, also called the 3-cycle condition, is, however, not sufficient for an arbitrary rational-valued reciprocal relation to be the winning probability relation of a dice model. An additional necessary condition, called the 4-cycle condition, is introduced in this contribution. Moreover, we reveal a remarkable relationship between the 3-cycle condition and the number of so-called product triplets of a reciprocal relation. Finally, we experimentally count product triplets for several families of winning probability relations.

The Omnipresence of Cycle-Transitivity in the Comparison of Random Variables

In this paper, the transitivity properties of reciprocal relations, also called probabilistic relations, are investigated within the framework of cycle-transitivity. Interesting types of transitivity are highlighted and shown to be realizable in applications. For example, given a collection of random variables (X k )k ∈ I, pairwisely coupled by means of a same copula C ∈ {T M , T P , T L }, the transitivity of the reciprocal relation Q defined by \(Q (X_i,X_j) = {\rm Prob}\{X_i X_j\} + 1/2 {\rm\ Prob}\{X_i=X_j\}\) can be characterized within the cycle- transitivity framework. Similarly, given a poset (P, ≤ ) with P = {x 1, ..., x n }, the transitivity of the mutual rank probability relation Q P , where Q P (X i ,X j ) denotes the probability that x i precedes x j in a random linear extension of P, is characterized as a type of cycle-transitivity for which no realization had been found so far.