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

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

Insights and Characterization of l1-norm Based Sparsity Learning of a Lexicographically Encoded Capacity Vector for the Choquet Integral

This thesis aims to simultaneously minimize function error and model complexity for data fusion via the Choquet integral (CI). The CI is a generator function, i.e., it is parametric and yields a wealth of aggregation operators based on the specifics of the underlying fuzzy measure. It is often the case that we desire to learn a fusion from data and the goal is to have the smallest possible sum of squared error between the trained model and a set of labels. However, we also desire to learn as “simple’’ of solutions as possible. Herein, L1-norm regularization of a lexicographically encoded capacity vector relative to the CI is explored. The impact of regularization is explored in terms of what capacities and aggregation operators it induces under different common and extreme scenarios. Synthetic experiments are provided in order to illustrate the propositions and concepts put forth

Genetic Algorithm Optimization for Determining Fuzzy Measures from Fuzzy Data

Fuzzy measures and fuzzy integrals have been successfully used in many real applications. How to determine fuzzy measures is a very difficult problem in these applications. Though there have existed some methodologies for solving this problem, such as genetic algorithms, gradient descent algorithms, neural networks, and particle swarm algorithm, it is hard to say which one is more appropriate and more feasible. Each method has its advantages. Most of the existed works can only deal with the data consisting of classic numbers which may arise limitations in practical applications. It is not reasonable to assume that all data are real data before we elicit them from practical data. Sometimes, fuzzy data may exist, such as in pharmacological, financial and sociological applications. Thus, we make an attempt to determine a more generalized type of general fuzzy measures from fuzzy data by means of genetic algorithms and Choquet integrals. In this paper, we make the first effort to define the σ-λ rules. Furthermore we define and characterize the Choquet integrals of interval-valued functions and fuzzy-number-valued functions based on σ-λ rules. In addition, we design a special genetic algorithm to determine a type of general fuzzy measures from fuzzy data

Feature and Decision Level Fusion Using Multiple Kernel Learning and Fuzzy Integrals

The work collected in this dissertation addresses the problem of data fusion. In other words, this is the problem of making decisions (also known as the problem of classification in the machine learning and statistics communities) when data from multiple sources are available, or when decisions/confidence levels from a panel of decision-makers are accessible. This problem has become increasingly important in recent years, especially with the ever-increasing popularity of autonomous systems outfitted with suites of sensors and the dawn of the ``age of big data.\u27\u27 While data fusion is a very broad topic, the work in this dissertation considers two very specific techniques: feature-level fusion and decision-level fusion. In general, the fusion methods proposed throughout this dissertation rely on kernel methods and fuzzy integrals. Both are very powerful tools, however, they also come with challenges, some of which are summarized below. I address these challenges in this dissertation. Kernel methods for classification is a well-studied area in which data are implicitly mapped from a lower-dimensional space to a higher-dimensional space to improve classification accuracy. However, for most kernel methods, one must still choose a kernel to use for the problem. Since there is, in general, no way of knowing which kernel is the best, multiple kernel learning (MKL) is a technique used to learn the aggregation of a set of valid kernels into a single (ideally) superior kernel. The aggregation can be done using weighted sums of the pre-computed kernels, but determining the summation weights is not a trivial task. Furthermore, MKL does not work well with large datasets because of limited storage space and prediction speed. These challenges are tackled by the introduction of many new algorithms in the following chapters. I also address MKL\u27s storage and speed drawbacks, allowing MKL-based techniques to be applied to big data efficiently. Some algorithms in this work are based on the Choquet fuzzy integral, a powerful nonlinear aggregation operator parameterized by the fuzzy measure (FM). These decision-level fusion algorithms learn a fuzzy measure by minimizing a sum of squared error (SSE) criterion based on a set of training data. The flexibility of the Choquet integral comes with a cost, however---given a set of N decision makers, the size of the FM the algorithm must learn is 2N. This means that the training data must be diverse enough to include 2N independent observations, though this is rarely encountered in practice. I address this in the following chapters via many different regularization functions, a popular technique in machine learning and statistics used to prevent overfitting and increase model generalization. Finally, it is worth noting that the aggregation behavior of the Choquet integral is not intuitive. I tackle this by proposing a quantitative visualization strategy allowing the FM and Choquet integral behavior to be shown simultaneously

EXPLAINABLE FEATURE- AND DECISION-LEVEL FUSION

Information fusion is the process of aggregating knowledge from multiple data sources to produce more consistent, accurate, and useful information than any one individual source can provide. In general, there are three primary sources of data/information: humans, algorithms, and sensors. Typically, objective data---e.g., measurements---arise from sensors. Using these data sources, applications such as computer vision and remote sensing have long been applying fusion at different levels (signal, feature, decision, etc.). Furthermore, the daily advancement in engineering technologies like smart cars, which operate in complex and dynamic environments using multiple sensors, are raising both the demand for and complexity of fusion. There is a great need to discover new theories to combine and analyze heterogeneous data arising from one or more sources. The work collected in this dissertation addresses the problem of feature- and decision-level fusion. Specifically, this work focuses on fuzzy choquet integral (ChI)-based data fusion methods. Most mathematical approaches for data fusion have focused on combining inputs relative to the assumption of independence between them. However, often there are rich interactions (e.g., correlations) between inputs that should be exploited. The ChI is a powerful aggregation tool that is capable modeling these interactions. Consider the fusion of m sources, where there are 2m unique subsets (interactions); the ChI is capable of learning the worth of each of these possible source subsets. However, the complexity of fuzzy integral-based methods grows quickly, as the number of trainable parameters for the fusion of m sources scales as 2m. Hence, we require a large amount of training data to avoid the problem of over-fitting. This work addresses the over-fitting problem of ChI-based data fusion with novel regularization strategies. These regularization strategies alleviate the issue of over-fitting while training with limited data and also enable the user to consciously push the learned methods to take a predefined, or perhaps known, structure. Also, the existing methods for training the ChI for decision- and feature-level data fusion involve quadratic programming (QP). The QP-based learning approach for learning ChI-based data fusion solutions has a high space complexity. This has limited the practical application of ChI-based data fusion methods to six or fewer input sources. To address the space complexity issue, this work introduces an online training algorithm for learning ChI. The online method is an iterative gradient descent approach that processes one observation at a time, enabling the applicability of ChI-based data fusion on higher dimensional data sets. In many real-world data fusion applications, it is imperative to have an explanation or interpretation. This may include providing information on what was learned, what is the worth of individual sources, why a decision was reached, what evidence process(es) were used, and what confidence does the system have on its decision. However, most existing machine learning solutions for data fusion are black boxes, e.g., deep learning. In this work, we designed methods and metrics that help with answering these questions of interpretation, and we also developed visualization methods that help users better understand the machine learning solution and its behavior for different instances of data

Multiple Instance Choquet Integral for multiresolution sensor fusion

Imagine you are traveling to Columbia, MO for the first time. On your flight to Columbia, the woman sitting next to you recommended a bakery by a large park with a big yellow umbrella outside. After you land, you need directions to the hotel from the airport. Suppose you are driving a rental car, you will need to park your car at a parking lot or a parking structure. After a good night's sleep in the hotel, you may decide to go for a run in the morning on the closest trail and stop by that recommended bakery under a big yellow umbrella. It would be helpful in the course of completing all these tasks to accurately distinguish the proper car route and walking trail, find a parking lot, and pinpoint the yellow umbrella. Satellite imagery and other geo-tagged data such as Open Street Maps provide effective information for this goal. Open Street Maps can provide road information and suggest bakery within a five-mile radius. The yellow umbrella is a distinctive color and, perhaps, is made of a distinctive material that can be identified from a hyperspectral camera. Open Street Maps polygons are tagged with information such as "parking lot" and "sidewalk." All these information can and should be fused to help identify and offer better guidance on the tasks you are completing. Supervised learning methods generally require precise labels for each training data point. It is hard (and probably at an extra cost) to manually go through and label each pixel in the training imagery. GPS coordinates cannot always be fully trusted as a GPS device may only be accurate to the level of several pixels. In many cases, it is practically infeasible to obtain accurate pixel-level training labels to perform fusion for all the imagery and maps available. Besides, the training data may come in a variety of data types, such as imagery or as a 3D point cloud. The imagery may have different resolutions, scales and, even, coordinate systems. Previous fusion methods are generally only limited to data mapped to the same pixel grid, with accurate labels. Furthermore, most fusion methods are restricted to only two sources, even if certain methods, such as pan-sharpening, can deal with different geo-spatial types or data of different resolution. It is, therefore, necessary and important, to come up with a way to perform fusion on multiple sources of imagery and map data, possibly with different resolutions and of different geo-spatial types with consideration of uncertain labels. I propose a Multiple Instance Choquet Integral framework for multi-resolution multisensor fusion with uncertain training labels. The Multiple Instance Choquet Integral (MICI) framework addresses uncertain training labels and performs both classification and regression. Three classifier fusion models, i.e. the noisy-or, min-max, and generalized-mean models, are derived under MICI. The Multi-Resolution Multiple Instance Choquet Integral (MR-MICI) framework is built upon the MICI framework and further addresses multiresolution in the fusion sources in addition to the uncertainty in training labels. For both MICI and MR-MICI, a monotonic normalized fuzzy measure is learned to be used with the Choquet integral to perform two-class classifier fusion given bag-level training labels. An optimization scheme based on the evolutionary algorithm is used to optimize the models proposed. For regression problems where the desired prediction is real-valued, the primary instance assumption is adopted. The algorithms are applied to target detection, regression and scene understanding applications. Experiments are conducted on the fusion of remote sensing data (hyperspectral and LiDAR) over the campus of University of Southern Mississippi - Gulfpark. Clothpanel sub-pixel and super-pixel targets were placed on campus with varying levels of occlusion and the proposed algorithms can successfully detect the targets in the scene. A semi-supervised approach is developed to automatically generate training labels based on data from Google Maps, Google Earth and Open Street Map. Based on such training labels with uncertainty, the proposed algorithms can also identify materials on campus for scene understanding, such as road, buildings, sidewalks, etc. In addition, the algorithms are used for weed detection and real-valued crop yield prediction experiments based on remote sensing data that can provide information for agricultural applications.Includes biblographical reference

Explainable contextual data driven fusion

Numerous applications require the intelligent combining of disparate sensor data streams to create a more complete and enhanced observation in support of underlying tasks like classification, regression, or decision making. This presentation is focused on two underappreciated and often overlooked parts of information fusion, explainability and context. Due to the rapidly increasing deployment and complexity of machine learning solutions, it is critical that the humans who deploy these algorithms can understand why and how a given algorithm works, as well as be able to determine when an algorithm is suitable for use in a particular instance of the problem. The first half of this paper outlines a new similarity measure for capacities and integrals. This measure is used to compare machine learned fusion solutions and explain what a single fusion solution learned. The second half of the paper is focused on contextual fusion with respect to incomplete (limited knowledge) models and metadata for unmanned aerial vehicles (UAVs). Example UAV metadata includes platform (e.g., GPS, IMU, etc.) and environmental (e.g., weather, solar position, etc.) data. Incomplete models herein are a result of limitations of machine learning related to under-sampling of training data. To address these challenges, a new contextually adaptive online Choquet integral is outlined

On Feature Selection and Rule Extraction for High Dimensional Data: A Case of Diffuse Large B-Cell Lymphomas Microarrays Classification

Neurofuzzy methods capable of selecting a handful of useful features are very useful in analysis of high dimensional datasets. A neurofuzzy classification scheme that can create proper linguistic features and simultaneously select informative features for a high dimensional dataset is presented and applied to the diffuse large B-cell lymphomas (DLBCL) microarray classification problem. The classification scheme is the combination of embedded linguistic feature creation and tuning algorithm, feature selection, and rule-based classification in one neural network framework. The adjustable linguistic features are embedded in the network structure via fuzzy membership functions. The network performs the classification task on the high dimensional DLBCL microarray dataset either by the direct calculation or by the rule-based approach. The 10-fold cross validation is applied to ensure the validity of the results. Very good results from both direct calculation and logical rules are achieved. The results show that the network can select a small set of informative features in this high dimensional dataset. By a comparison to other previously proposed methods, our method yields better classification performance

Fraud detection in the banking sector : a multi-agent approach

Fraud is an increasing phenomenon as shown in many surveys carried out by leading international consulting companies in the last years. Despite the evolution of electronic payments and hacking techniques there is still a strong human component in fraud schemes. Conflict of interest in particular is the main contributing factor to the success of internal fraud. In such cases anomaly detection tools are not always the best instruments, since the fraud schemes are based on faking documents in a context dominated by lack of controls, and the perpetrators are those ones who should control possible irregularities. In the banking sector audit team experts can count only on their experience, whistle blowing and the reports sent by their inspectors. The Fraud Interactive Decision Expert System (FIDES), which is the core of this research, is a multi-agent system built to support auditors in evaluating suspicious behaviours and to speed up the evaluation process in order to detect or prevent fraud schemes. The system combines Think-map, Delphi method and Attack trees and it has been built around audit team experts and their needs. The output of FIDES is an attack tree, a tree-based diagram to ”systematically categorize the different ways in which a system can be attacked”. Once the attack tree is built, auditors can choose the path they perceive as more suitable and decide whether or not to start the investigation. The system is meant for use in the future to retrieve old cases in order to match them with new ones and find similarities. The retrieving features of the system will be useful to simplify the risk management phase, since similar countermeasures adopted for past cases might be useful for present ones. Even though FIDES has been built with the banking sector in mind, it can be applied in all those organisations, like insurance companies or public organizations, where anti-fraud activity is based on a central anti-fraud unit and a reporting system