Ranking of QSAR Models to Predict Minimal Inhibitory Concentrations Toward Mycobacterium tuberculosis for a Set of Fluoroquinolones

<p>CP-ANN technique was used to build 54 different QSAR models. The models were built for three sets (assays) of fluo-roquinolones considering their antituberculosis activity and using different technical parameters (dimension of network and number of learning epochs). The models served as a reliable basis for ranking by a new powerful method based on sum of ranking differences (SRD). With the applied SRD procedure we can find the optimal ones. The best model can be selected easily for the first assay. Two models can be recommended for the second assay, and no recommended mo-del was found for the assay3.</p

Pareto front of the sparse NMF.

The α = 0.4 and β = 0.4 parameters are selected to incorporate a fair number of members to the Pareto front, and also have most of the objectives incorporated into a component.</p

Relationship between objectives by PCA.

The biplot depicts both scores and loadings, however, the scores are scaled differently in Fig 8. The axes split the objectives; the vertical axis distinguishes between the effects of the objectives that either impair or benefit the solutions, in this case, universities. In the fourth quarter, the objectives are quantity indicators, while in other quarters, the proportional indicators can be identified. Six clusters can be observed: three gender proportional indicators, the proportional variables of collaboration and open-access indicators, the proportional variables of scientific indicators, the quantity indicators, and the two drawback clusters. There is an outlier objective (S8). When a university improves their indicators in the drawback clusters, their performance decrease, moving the farther from the Pareto front.</p

Pareto front representation.

Pareto fronts can be convex (A), concave (B), or the mixture of both. Additionally, the set of solutions can have multiple Pareto fronts, which can be identified by ignoring previously determined non-dominated solutions and repeating the search for Pareto fronts. Consider an onion; peel off the first layer to see the second, which will then become the outermost layer. The Pareto front can also be clustered.[34, 35].</p

NMF-based Pareto front with scaled data as an input.

Each indicator in the components is positively correlating with each other. Therefore there are no reverse objectives, only perpendicular ones. For this reason, the Pareto front is determined with the help of maximizing the objective functions. The NMF is applied to the scaled data before the Pareto front ranks the institutions. This method clusters the institutions into Eastern and Western cultures concerning education. The Chinese and Taiwanese universities are placed into a cluster on the left-hand side, while the European, American, and the remaining academies can be found on the right-hand side. South Korean and Iranian universities connect the two dominant scientific circles.</p

Pareto Front of the two objectives with the highest entropy.

The Pareto fronts only consider those non-dominated solutions that are optimal from the viewpoints of the ‘total number of open-access publications’ (O2) and the ‘number of green access publications’ (O6). Coincidentally, the chosen objectives are one of the most correlating pairs. The other 44 objectives have no input to the ranking. Should another two dominant objectives taken into consideration, different non-dominated solutions would be obtained.</p

CWTS Leiden Ranking 2020 database variables.

Non-negative matrix factorization (NMF) efficiently reduces high dimensionality for many-objective ranking problems. In multi-objective optimization, as long as only three or four conflicting viewpoints are present, an optimal solution can be determined by finding the Pareto front. When the number of the objectives increases, the multi-objective problem evolves into a many-objective optimization task, where the Pareto front becomes oversaturated. The key idea is that NMF aggregates the objectives so that the Pareto front can be applied, while the Sum of Ranking Differences (SRD) method selects the objectives that have a detrimental effect on the aggregation, and validates the findings. The applicability of the method is illustrated by the ranking of 1176 universities based on 46 variables of the CWTS Leiden Ranking 2020 database. The performance of NMF is compared to principal component analysis (PCA) and sparse non-negative matrix factorization-based solutions. The results illustrate that PCA incorporates negatively correlated objectives into the same principal component. On the contrary, NMF only allows non-negative correlations, which enable the proper use of the Pareto front. With the combination of NMF and SRD, a non-biased ranking of the universities based on 46 criteria is established, where Harvard, Rockefeller and Stanford Universities are determined as the first three. To evaluate the ranking capabilities of the methods, measures based on Relative Entropy (RE) and Hypervolume (HV) are proposed. The results confirm that the sparse NMF method provides the most informative ranking. The results highlight that academic excellence can be improved by decreasing the proportion of unknown open-access publications and short distance collaborations. The proportion of gender indicators barely correlate with scientific impact. More authors, long-distance collaborations, publications that have more scientific impact and citations on average highly influence the university ranking in a positive direction.</div

The interpretation of the principal components.

Non-negative matrix factorization (NMF) efficiently reduces high dimensionality for many-objective ranking problems. In multi-objective optimization, as long as only three or four conflicting viewpoints are present, an optimal solution can be determined by finding the Pareto front. When the number of the objectives increases, the multi-objective problem evolves into a many-objective optimization task, where the Pareto front becomes oversaturated. The key idea is that NMF aggregates the objectives so that the Pareto front can be applied, while the Sum of Ranking Differences (SRD) method selects the objectives that have a detrimental effect on the aggregation, and validates the findings. The applicability of the method is illustrated by the ranking of 1176 universities based on 46 variables of the CWTS Leiden Ranking 2020 database. The performance of NMF is compared to principal component analysis (PCA) and sparse non-negative matrix factorization-based solutions. The results illustrate that PCA incorporates negatively correlated objectives into the same principal component. On the contrary, NMF only allows non-negative correlations, which enable the proper use of the Pareto front. With the combination of NMF and SRD, a non-biased ranking of the universities based on 46 criteria is established, where Harvard, Rockefeller and Stanford Universities are determined as the first three. To evaluate the ranking capabilities of the methods, measures based on Relative Entropy (RE) and Hypervolume (HV) are proposed. The results confirm that the sparse NMF method provides the most informative ranking. The results highlight that academic excellence can be improved by decreasing the proportion of unknown open-access publications and short distance collaborations. The proportion of gender indicators barely correlate with scientific impact. More authors, long-distance collaborations, publications that have more scientific impact and citations on average highly influence the university ranking in a positive direction.</div

Literature of Matrix factorization and MOO.

The figure shows the cooccurrences of keywords in articles from 497 publications queried using Scopus. The search expressions/keywords were matrix factorization and ranking, multi-objective optimization, decision support and multi-objective decision-making. Unique keywords, denoted by the nodes, are connected if they cooccur at least four times. Only publications before the end of December, 2021 are included.</p

Coefficients of the ranking by NMF with ranked data as an input.

The non-negativity constraint forces the objectives into the positive quarter of the Cartesian coordinate system. By providing ranked input to NMF, an insight was gained into the relationship between the objectives. The ones close to the axes are almost identical to the SRD result of best/worst objectives, but the randomly located ones are in between. Furthermore, clusters of indicators appear as well. Cluster A) describes the alignment of proportional collaboration variables (C9–10).</p