Vector valued information measures and integration with respect to fuzzy vector capacities

[EN] Integration with respect to vector-valued fuzzy measures is used to define and study information measuring tools. Motivated by some current developments in Information Science, we apply the integration of scalar functions with respect to vector-valued fuzzy measures, also called vector capacities. Bartle-Dunford-Schwartz integration (for the additive case) and Choquet type integration (for the non-additive case) are considered, showing that these formalisms can be used to define and develop vector-valued impact measures. Examples related to existing bibliometric tools as well as to new measuring indices are given.The authors would like to thank both Prof. Dr. Olvido Delgado and the referee for their valuable comments and suggestions which helped to prepare the manuscript. The first author gratefully acknowledges the support of the Ministerio de Economia, Industria y Competitividad (Spain) under project MTM2016-77054-C2-1-P.Sánchez Pérez, EA.; Szwedek, R. (2019). Vector valued information measures and integration with respect to fuzzy vector capacities. Fuzzy Sets and Systems. 355:1-25. https://doi.org/10.1016/j.fss.2018.05.004S12535

Choquet type L-1-spaces of a vector capacity

[EN] Given a set function Lambda with values in a Banach space X, we construct an integration theory for scalar functions with respect to Lambda by using duality on Xand Choquet scalar integrals. Our construction extends the classical Bartle-Dunford-Schwartz integration for vector measures. Since just the minimal necessary conditions on Lambda are required, several L-1-spaces of integrable functions associated to Lambda appear in such a way that the integration map can be defined in them. We study the properties of these spaces and how they are related. We show that the behavior of the L-1-spaces and the integration map can be improved in the case when Xis an order continuous Banach lattice, providing new tools for (non-linear) operator theory and information sciences. (C) 2017 Elsevier B.V. All rights reserved.The first and second authors gratefully acknowledge the support of the Ministerio de Economia y Competitividad under projects MTM2015-65888-C4-1-P and MTM2016-77054-C2-1-P, respectively. The first author also acknowledges the support of the Junta de Andalucia (project FQM-7276), Spain.Delgado Garrido, O.; Sánchez Pérez, EA. (2017). Choquet type L-1-spaces of a vector capacity. Fuzzy Sets and Systems. 327:98-122. https://doi.org/10.1016/j.fss.2017.05.014S9812232

Using Feature Weighting as a Tool for Clustering Applications

The weighted variant of k-Means (Wk-Means), which assigns values to features based on their relevance, is a well-known approach to address the shortcoming of k-Means with data containing noisy and irrelevant features. This research aims first to explore how feature weighting can be used for feature selection, second to investigate the performance of Minkowski weighted k- Means (MWk-Means), and its intelligent variant, on datasets defined in different p-norms, and third to address the problem of missing values with a weighted variant of k-Means. A partial distance approach is used to address the problem of missing values for weighted variant of k- Means. Anomalous clustering has been successfully used to detect natural clusters and initialize centroids in k-means type algorithms. Similarly, extensive work has been carried out on using feature weights to rescale features under Minkowski Lp metrics for p ≥ 1 . In this thesis, aspects from both of these approaches enable feature weights to be detected based on natural clusters present in the training data, but the clusters are not limited to spherical shape. Two methods, mean-FSFW and max-FSFW, are developed as further extensions of intelligent Minkowski Weighted k-Means(iMWk-Means), where feature weights are used as indices for feature selection with no requirement for user-specified parameters. The proposed feature selection methods are able to significantly reduce the number of noisy features. These methods are further extended to mean-FSFWextPD and max-FSFWextPD to address missing values and are found to be better alternatives than existing imputation methods. The effect of feature weighting on clustering of dataset defined in varying p-norms is further explored in the thesis. An algorithm that translates a dataset into different p-norms has been proposed. The capability of MWk-Means to read true shapes of clusters defined in different p- norms is explored. To address the problem of missing feature values in weighted variant of k-Means, different missing-value imputation methods are tested. The MWk-Means and its intelligent variant are further extended to incorporate the partial distance approach, specifically to address the problem of missing values. All these methods are tested in both synthetic and real-world datasets against three models of noise - noisy feature added, feature blurring and cluster-wise feature blurring - where applicable. These noises are generated from Gaussian and uniform distribution with three different strength of noise, i.e., no noise, half noise and full noise Overall, results demonstrate that feature weighting can improve feature selection. The partial- distance approach, with feature weights, is effective at ignoring missing values, and cluster retrieval in various p-norm spaces is effective