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**Contents:**

Additional file 6 shows the XML files of the projects described. These projects on each chemical dataset mentioned above see Table 1 were computed. Posteriorly, the GA-MLR procedure was used to build several models for 3, 5 and 7 variables for each operator-type.

The best model for each dimension on each dataset was retained Additional file 7.

A pool with the non-fuzzy 3D-MDs and other pool with the fuzzy 3D-MDs included in the best models built on each dataset were created. From these pools, non-fuzzy and fuzzy models for 7 variables were built on each dataset, and the models with the best bootstrapping value were selected as the best ones Additional file 8. In this sense, an analysis by means of a boxplot graphic box-and-whisker graphic was firstly performed, in order to examine the shape of the distributions of the results achieved.

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Then, a Wilcoxon signed-rank test [ 89 ] was carried out to know whether the predictive abilities achieved by the fuzzy models and the predictive abilities achieved by the non-fuzzy models differ. The SPSS software was used to perform the first analysis mentioned above, while the Keel [ 90 ] software was employed to perform the other one. So, it is evidenced that fuzzy QuBiLS-MIDAS MDs calculated both for a low and a high synergism contribute to codify useful chemical information, and that their performances depend on the molecular structures under study.

Therefore, both types of fuzzy 3D-MDs should be jointly used with the purpose of creating models with better predictive ability. In this sense, Fig.

The fuzzy MDs determined for a low and a high synergism among atomic contributions were used, while the non-fuzzy MDs used are those computed from the norm-, mean- and statistic-based operators. Thus, it can be stated that MDs with better modeling ability can be calculated using the Choquet integral-based operator, if compared with the MDs computed from the traditional non-fuzzy operators.

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Moreover, Fig. Additionally, Fig. In general, it can be seen that, albeit the superadditivity is exclusive for the ACHE and THR datasets, the amount of superadditivity on each dataset is moderate. This behavior can be due to the fact that the datasets used are comprised of congeneric compounds. That is, since the compounds are structurally similar, then MDs computed of additive way, or considering a low synergism among atomic contributions, may be those in achieving better correlations into a QSAR model.

This assumption is supported in the average amount of low synergism obtained.

As it can be seen, the behavior for a low synergism is from moderate to high, except in the BZR dataset. Thus, at least preliminarily, it can be stated that high amounts of superadditivity will contribute to compute better MDs in non-congeneric datasets than in congeneric datasets. Additional file 9 shows the descriptive statistics calculated. On one hand, it can be firstly seen that there are not outlier predictive abilities. In this sense, if the performances achieved by the models built on each dataset are examined Additional file 8 , it can be appreciated that for the ACHE dataset, the fuzzy model built presents the best progress of all, for a Only in the GPB dataset, the improvement of the fuzzy model is insignificant 0.

The Choquet integral is rather different from the other norm-, mean- and statistic-based non-fuzzy operators used to date. It performs a reordering step to fuse according to the magnitude of the criteria and, in addition, it considers the interrelation among criteria by using a fuzzy measure. In this way, fuzzy descriptors can be derived from traditional or recent descriptors; e. It was demonstrated that with the Choquet integral-based descriptors, models with better predictive power can be built, if compared to the models built with the descriptors computed from the other non-fuzzy operators.

These outcomes were statistically corroborated using the Wilcoxon signed-rank test. All in all, it can be concluded that the use of the Choquet integral as a fuzzy aggregation operator constitutes a prominent way to extract useful structural information of the molecules and, in this way, enhance the modeling capacity of several existing molecular descriptors in ADME-Tox and pharmacological endpoints. Tan C Generalized intuitionistic fuzzy geometric aggregation operator and its application to multi-criteria group decision making. Soft Comput 15 5 — Comput Electron Agric — Fuzzy Set Syst — Calvo T et al Aggregation operators: properties, classes and construction methods.

Physica-Verlag, Heidelberg, pp 3— In: Aggregation functions: a guide for practitioners. Springer, Berlin, pp 39— Yager RR On ordered weighted averaging aggregation operators in multicriteria decisionmaking.

Fuzzy Set Syst 59 2 — Inf Sci 6 — Sugeno M Theory of fuzzy integrals and its applications. Tokyo Institute of Technology, Tokyo. Burkill JC The Lebesgue integral, vol Cambridge University Press, Cambridge, p Physica-Verlag, New York. Springer, New York, pp — Choquet G Theory of capacities.

Marichal JL An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria. Marichal J-L Aggregation of interacting criteria by means of the discrete Choquet integral. Physica-Verlag, Heidelberg, pp — Springer, Berlin, pp 56— Fuzzy Set Syst Supplement C — Int J Fuzzy Syst 20 3 — Barrenechea E et al Using the Choquet integral in the fuzzy reasoning method of fuzzy rule-based classification systems.

Axioms 2 2 In: Zadeh LA ed Nonlinear integrals and their applications in data mining. World Scientific, Singapore, pp — J Bus Res — Demirel T et al Choquet integral-based hesitant fuzzy decision-making to prevent soil erosion. Geoderma — Liu B et al An interval-valued 2-tuple linguistic group decision-making model based on the Choquet integral operator. Int J Inf Sci 49 2 — Bajorath J Molecular similarity concepts for informatics applications.

J Mol Struct — Curr Comput Aided Drug Des 13 4 — Todeschini R, Consonni V Molecular descriptors for chemoinformatics. Methods and principles in medicinal chemistry, 2nd edn.

Hongxia Wang, Email: moc. Methods The aim of this paper is to study the upper bound and the lower bound of the Choquet integral for log-convex functions. Ivan Kojadinovic. Here we specially mention that Mesiar et al. We shall provide an easy example below.

Wiley-VCH, Weinheim. Chem Rev 96 3 — Barigye SJ et al Trends in information theory-based chemical structure codification. Mol Divers 18 3 — J Chem Inf Comput Sci 35 3 — Balaban AT Local atomic and global molecular graph-theoretical descriptors. Theory and QSPR application. J Comput Aided Mol Des 26 11 — Generalization of global indices defined from local vertex invariants.

Curr Comput Aided Drug Des 9 2 — J Comput Chem 35 18 — J Math Chem 53 9 — J Cheminform 9 1 Mol Inf. Marrero-Ponce Y et al Optimum search strategies or novel 3D molecular descriptors: is there a stalemate? Curr Bioinform 10 5 — The functions contained in Kappalab can for instance be used in the framework of multicriteria decision making or cooperative game theory.

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Non-Additive Measure and Integral Manipulation Functions S4 tool box for capacity or non-additive measure, fuzzy measure and integral manipulation in a finite setting. API documentation. Denneberg: Non-additive Measure and Integral.

Non-Additive Measure and Integral is the first systematic approach to the subject. Much of the additive theory (convergence theorems, Lebesgue spaces, representation theorems) is generalized, at least for submodular measures which are characterized by having a subadditive integral. Key words and phrases: non-additive measure, aggregation function, Choquet integral, Sugeno integral, triangular conorm, triangular norm, pseudo-additive.

Non-Additive Measure and Integral is the first systematic approach to the subject. Much of the additive theory convergence theorems, Lebesgue spaces, representation theorems is generalized, at least for submodular Denneberg, Dieter. Denneberg states comonotonic additivity of the Choquet integral in Proposition 5.

Conditioning updating non-additive measures - ebsco. A simple proof for the convexity of the Choquet integral. Representation of the Choquet integral with the a-additive M6bius.

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