Applying d-XChoquet integrals in classification problems

Several generalizations of the Choquet integral have been applied in the Fuzzy Reasoning Method (FRM) of Fuzzy Rule-Based Classification Systems (FRBCS's) to improve its performance. Additionally, to achieve that goal, researchers have searched for new ways to provide more flexibility to those generalizations, by restricting the requirements of the functions being used in their constructions and relaxing the monotonicity of the integral. This is the case of CT-integrals, CC-integrals, CF-integrals, CF1F2-integrals and dCF-integrals, which obtained good performance in classification algorithms, more specifically, in the fuzzy association rule-based classification method for high-dimensional problems (FARC-HD). Thereafter, with the introduction of Choquet integrals based on restricted dissimilarity functions (RDFs) in place of the standard difference, a new generalization was made possible: the d-XChoquet (d-XC) integrals, which are ordered directional increasing functions and, depending on the adopted RDF, may also be a pre-aggregation function. Those integrals were applied in multi-criteria decision making problems and also in a motor-imagery brain computer interface framework. In the present paper, we introduce a new FRM based on the d-XC integral family, analyzing its performance by applying it to 33 different datasets from the literature.Supported by Navarra de Servicios y Tecnologías, S.A. (NASERTIC), CNPq (301618/2019-4, 305805/2021-5), FAPERGS (19/2551-0001660-3), the Spanish Ministry of Science and Technology (TIN2016-77356-P, PID2019- 108392GB I00 (MCIN/AEI/10.13039/501100011033)

Enabling Explainable Fusion in Deep Learning with Fuzzy Integral Neural Networks

Information fusion is an essential part of numerous engineering systems and biological functions, e.g., human cognition. Fusion occurs at many levels, ranging from the low-level combination of signals to the high-level aggregation of heterogeneous decision-making processes. While the last decade has witnessed an explosion of research in deep learning, fusion in neural networks has not observed the same revolution. Specifically, most neural fusion approaches are ad hoc, are not understood, are distributed versus localized, and/or explainability is low (if present at all). Herein, we prove that the fuzzy Choquet integral (ChI), a powerful nonlinear aggregation function, can be represented as a multi-layer network, referred to hereafter as ChIMP. We also put forth an improved ChIMP (iChIMP) that leads to a stochastic gradient descent-based optimization in light of the exponential number of ChI inequality constraints. An additional benefit of ChIMP/iChIMP is that it enables eXplainable AI (XAI). Synthetic validation experiments are provided and iChIMP is applied to the fusion of a set of heterogeneous architecture deep models in remote sensing. We show an improvement in model accuracy and our previously established XAI indices shed light on the quality of our data, model, and its decisions.Comment: IEEE Transactions on Fuzzy System

Pre-aggregation functions: construction and an application

In this work we introduce the notion of preaggregation function. Such a function satisfies the same boundary conditions as an aggregation function, but, instead of requiring monotonicity, only monotonicity along some fixed direction (directional monotonicity) is required. We present some examples of such functions. We propose three different methods to build pre-aggregation functions. We experimentally show that in fuzzy rule-based classification systems, when we use one of these methods, namely, the one based on the use of the Choquet integral replacing the product by other aggregation functions, if we consider the minimum or the Hamacher product t-norms for such construction, we improve the results obtained when applying the fuzzy reasoning methods obtained using two classical averaging operators like the maximum and the Choquet integral.This work was supported in part by the Spanish Ministry of Science and Technology under projects TIN2008-06681-C06-01, TIN2010- 15055, TIN2013-40765-P, TIN2011-29520

Transformation-Based Fuzzy Rule Interpolation With Mahalanobis Distance Measures Supported by Choquet Integral

Fuzzy rule interpolation (FRI) strongly supports approximate inference when a new observation matches no rules, through selecting and subsequently interpolating appropriate rules close to the observation from the given (sparse) rule base. Traditional ways of implementing the critical rule selection process are typically based on the exploitation of Euclidean distances between the observation and rules. It is conceptually straightforward for implementation but applying this distance metric may systematically lead to inferior results because it fails to reflect the variations of the relevance or significance levels amongst different domain features. To address this important issue, a novel transformation-based FRI approach is presented, on the basis of utilising the Mahalanobis distance metric. The new FRI method works by transforming a given sparse rule base into a coordinates system where the distance between instances of the same category becomes closer while that between different categories becomes further apart. In so doing, when an observation is present that matches no rules, the most relevant neighbouring rules to implement the required interpolation are more likely to be selected. Following this, the scale and move factors within the classical transformation-based FRI procedure are also modified by Choquet integral. Systematic experimental investigation over a range of classification problems demonstrates that the proposed approach remarkably outperforms the existing state-of-the-art FRI methods in both accuracy and efficiency

Neuro-inspired edge feature fusion using Choquet integrals

It is known that the human visual system performs a hierarchical information process in which early vision cues (or primitives) are fused in the visual cortex to compose complex shapes and descriptors. While different aspects of the process have been extensively studied, such as lens adaptation or feature detection, some other aspects, such as feature fusion, have been mostly left aside. In this work, we elaborate on the fusion of early vision primitives using generalizations of the Choquet integral, and novel aggregation operators that have been extensively studied in recent years. We propose to use generalizations of the Choquet integral to sensibly fuse elementary edge cues, in an attempt to model the behaviour of neurons in the early visual cortex. Our proposal leads to a fully-framed edge detection algorithm whose performance is put to the test in state-of-the-art edge detection datasets.The authors gratefully acknowledge the financial support of the Spanish Ministry of Science and Technology (project PID2019-108392GB-I00 (AEI/10.13039/501100011033), the Research Services of Universidad Pública de Navarra, CNPq (307781/2016-0, 301618/2019-4), FAPERGS (19/2551-0001660) and PNPD/CAPES (464880/2019-00)