Modelling fraud detection by attack trees and Choquet integral

Modelling an attack tree is basically a matter of associating a logical ÒndÓand a logical ÒrÓ but in most of real world applications related to fraud management the Ònd/orÓlogic is not adequate to effectively represent the relationship between a parent node and its children, most of all when information about attributes is associated to the nodes and the main problem to solve is how to promulgate attribute values up the tree through recursive aggregation operations occurring at the Ònd/orÓnodes. OWA-based aggregations have been introduced to generalize ÒndÓand ÒrÓoperators starting from the observation that in between the extremes Òor allÓ(and) and Òor anyÓ(or), terms (quantifiers) like ÒeveralÓ ÒostÓ ÒewÓ ÒomeÓ etc. can be introduced to represent the different weights associated to the nodes in the aggregation. The aggregation process taking place at an OWA node depends on the ordered position of the child nodes but it doesnÕ take care of the possible interactions between the nodes. In this paper, we propose to overcome this drawback introducing the Choquet integral whose distinguished feature is to be able to take into account the interaction between nodes. At first, the attack tree is valuated recursively through a bottom-up algorithm whose complexity is linear versus the number of nodes and exponential for every node. Then, the algorithm is extended assuming that the attribute values in the leaves are unimodal LR fuzzy numbers and the calculation of Choquet integral is carried out using the alpha-cuts.Fraud detection; attack tree; ordered weighted averaging (OWA) operator; Choquet integral; fuzzy numbers.

A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid

The main advances regarding the use of the Choquet and Sugeno integrals in multi-criteria decision aid over the last decade are reviewed. They concern mainly a bipolar extension of both the Choquet integral and the Sugeno integral, interesting particular submodels, new learning techniques, a better interpretation of the models and a better use of the Choquet integral in multi-criteria decision aid. Parallel to these theoretical works, the Choquet integral has been applied to many new fields, and several softwares and libraries dedicated to this model have been developed.Choquet integral, Sugeno integral, capacity, bipolarity, preferences

Aggregation on bipolar scales

The paper addresses the problem of extending aggregation operators typically defined on $[0,1]$ to the symmetric interval $[-1,1]$, where the ``0'' value plays a particular role (neutral value). We distinguish the cases where aggregation operators are associative or not. In the former case, the ``0'' value may play the role of neutral or absorbant element, leading to pseudo-addition and pseudo-multiplication. We address also in this category the special case of minimum and maximum defined on some finite ordinal scale. In the latter case, we find that a general class of extended operators can be defined using an interpolation approach, supposing the value of the aggregation to be known for ternary vectors.bipolar scale; bi-capacity; aggregation

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

Fuzzy integral for rule aggregation in fuzzy inference systems

The fuzzy inference system (FIS) has been tuned and re-vamped many times over and applied to numerous domains. New and improved techniques have been presented for fuzzification, implication, rule composition and defuzzification, leaving one key component relatively underrepresented, rule aggregation. Current FIS aggregation operators are relatively simple and have remained more-or-less unchanged over the years. For many problems, these simple aggregation operators produce intuitive, useful and meaningful results. However, there exists a wide class of problems for which quality aggregation requires non- additivity and exploitation of interactions between rules. Herein, we show how the fuzzy integral, a parametric non-linear aggregation operator, can be used to fill this gap. Specifically, recent advancements in extensions of the fuzzy integral to \unrestricted" fuzzy sets, i.e., subnormal and non- convex, makes this now possible. We explore the role of two extensions, the gFI and the NDFI, discuss when and where to apply these aggregations, and present efficient algorithms to approximate their solutions

Fuzzy measures and integrals in MCDA

This chapter aims at a unified presentation of various methods of MCDA based onfuzzy measures (capacity) and fuzzy integrals, essentially the Choquet andSugeno integral. A first section sets the position of the problem ofmulticriteria decision making, and describes the various possible scales ofmeasurement (difference, ratio, and ordinal). Then a whole section is devotedto each case in detail: after introducing necessary concepts, the methodologyis described, and the problem of the practical identification of fuzzy measuresis given. The important concept of interaction between criteria, central inthis chapter, is explained in details. It is shown how it leads to k-additivefuzzy measures. The case of bipolar scales leads to thegeneral model based on bi-capacities, encompassing usual models based oncapacities. A general definition of interaction for bipolar scales isintroduced. The case of ordinal scales leads to the use of Sugeno integral, andits symmetrized version when one considers symmetric ordinal scales. Apractical methodology for the identification of fuzzy measures in this contextis given. Lastly, we give a short description of some practical applications.Choquet integral; fuzzy measure; interaction; bi-capacities

Efficient Data Driven Multi Source Fusion

Data/information fusion is an integral component of many existing and emerging applications; e.g., remote sensing, smart cars, Internet of Things (IoT), and Big Data, to name a few. While fusion aims to achieve better results than what any one individual input can provide, often the challenge is to determine the underlying mathematics for aggregation suitable for an application. In this dissertation, I focus on the following three aspects of aggregation: (i) efficient data-driven learning and optimization, (ii) extensions and new aggregation methods, and (iii) feature and decision level fusion for machine learning with applications to signal and image processing. The Choquet integral (ChI), a powerful nonlinear aggregation operator, is a parametric way (with respect to the fuzzy measure (FM)) to generate a wealth of aggregation operators. The FM has 2N variables and N(2N − 1) constraints for N inputs. As a result, learning the ChI parameters from data quickly becomes impractical for most applications. Herein, I propose a scalable learning procedure (which is linear with respect to training sample size) for the ChI that identifies and optimizes only data-supported variables. As such, the computational complexity of the learning algorithm is proportional to the complexity of the solver used. This method also includes an imputation framework to obtain scalar values for data-unsupported (aka missing) variables and a compression algorithm (lossy or losselss) of the learned variables. I also propose a genetic algorithm (GA) to optimize the ChI for non-convex, multi-modal, and/or analytical objective functions. This algorithm introduces two operators that automatically preserve the constraints; therefore there is no need to explicitly enforce the constraints as is required by traditional GA algorithms. In addition, this algorithm provides an efficient representation of the search space with the minimal set of vertices. Furthermore, I study different strategies for extending the fuzzy integral for missing data and I propose a GOAL programming framework to aggregate inputs from heterogeneous sources for the ChI learning. Last, my work in remote sensing involves visual clustering based band group selection and Lp-norm multiple kernel learning based feature level fusion in hyperspectral image processing to enhance pixel level classification