Data-informed fuzzy measures for fuzzy integration of intervals and fuzzy numbers

The fuzzy integral (FI) with respect to a fuzzy measure (FM) is a powerful means of aggregating information. The most popular FIs are the Choquet and Sugeno, and most research focuses on these two variants. The arena of the FM is much more populated, including numerically derived FMs such as the Sugeno λ-measure and decomposable measure, expert-defined FMs, and data-informed FMs. The drawback of numerically derived and expert-defined FMs is that one must know something about the relative values of the input sources. However, there are many problems where this information is unavailable, such as crowdsourcing. This paper focuses on data-informed FMs, or those FMs that are computed by an algorithm that analyzes some property of the input data itself, gleaning the importance of each input source by the data they provide. The original instantiation of a data-informed FM is the agreement FM, which assigns high confidence to combinations of sources that numerically agree with one another. This paper extends upon our previous work in datainformed FMs by proposing the uniqueness measure and additive measure of agreement for interval-valued evidence. We then extend data-informed FMs to fuzzy number (FN)-valued inputs. We demonstrate the proposed FMs by aggregating interval and FN evidence with the Choquet and Sugeno FIs for both synthetic and real-world data

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

Extension of the fuzzy integral for general fuzzy set-valued information

The fuzzy integral (FI) is an extremely flexible aggregation operator. It is used in numerous applications, such as image processing, multicriteria decision making, skeletal age-at-death estimation, and multisource (e.g., feature, algorithm, sensor, and confidence) fusion. To date, a few works have appeared on the topic of generalizing Sugeno's original real-valued integrand and fuzzy measure (FM) for the case of higher order uncertain information (both integrand and measure). For the most part, these extensions are motivated by, and are consistent with, Zadeh's extension principle (EP). Namely, existing extensions focus on fuzzy number (FN), i.e., convex and normal fuzzy set- (FS) valued integrands. Herein, we put forth a new definition, called the generalized FI (gFI), and efficient algorithm for calculation for FS-valued integrands. In addition, we compare the gFI, numerically and theoretically, with our non-EP-based FI extension called the nondirect FI (NDFI). Examples are investigated in the areas of skeletal age-at-death estimation in forensic anthropology and multisource fusion. These applications help demonstrate the need and benefit of the proposed work. In particular, we show there is not one supreme technique. Instead, multiple extensions are of benefit in different contexts and applications

Reshare an Operational Ontology Framework for Research Modeling, Combining and Sharing

Scientists always face difficulties dealing with disjointed information. There is a need for a standardized and robust way to represent and exchange knowledge. Ontology has been widely used for this purpose. However, since research involves semantics and operations, we need to conceptualize both of them. In this thesis, we propose ReShare to provide a solution for this problem. Maximizing utilization while preserving the semantics is one of the main challenges when the heterogeneous knowledge is combined. Therefore, operational annotations were designed to allow generic object modeling, binding and representation. Furthermore, a test bed is developed and preliminary results are presented to show the usefulness and robustness of our approach. Moreover, two aggregation techniques for fusing ontology matchers are investigated as an initial work for building an algorithm which converts descriptive ontologies into operational ones

Novel similarity measure for interval-valued data based on overlapping ratio

In computing the similarity of intervals, current similarity measures such as the commonly used Jaccard and Dice measures are at times not sensitive to changes in the width of intervals, producing equal similarities for substantially different pairs of intervals. To address this, we propose a new similarity measure that uses a bi-directional approach to determine interval similarity. For each direction, the overlapping ratio of the given interval in a pair with the other interval is used as a measure of uni-directional similarity. We show that the proposed measure satisfies all common properties of a similarity measure, while also being invariant in respect to multiplication of the interval endpoints and exhibiting linear growth in respect to linearly increasing overlap. Further, we compare the behavior of the proposed measure with the highly popular Jaccard and Dice similarity measures, highlighting that the proposed approach is more sensitive to changes in interval widths. Finally, we show that the proposed similarity is bounded by the Jaccard and the Dice similarity, thus providing a reliable alternative

Fusion of an Ensemble of Augmented Image Detectors for Robust Object Detection

A significant challenge in object detection is accurate identification of an object's position in image space, whereas one algorithm with one set of parameters is usually not enough, and the fusion of multiple algorithms and/or parameters can lead to more robust results. Herein, a new computational intelligence fusion approach based on the dynamic analysis of agreement among object detection outputs is proposed. Furthermore, we propose an online versus just in training image augmentation strategy. Experiments comparing the results both with and without fusion are presented. We demonstrate that the augmented and fused combination results are the best, with respect to higher accuracy rates and reduction of outlier influences. The approach is demonstrated in the context of cone, pedestrian and box detection for Advanced Driver Assistance Systems (ADAS) applications.Comment: 21 pages, 12 figures, journal paper, MDPI Sensors, 201

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

A Similarity Measure Based on Bidirectional Subsethood for Intervals

With a growing number of areas leveraging interval-valued data—including in the context of modelling human uncertainty (e.g., in Cyber Security), the capacity to accurately and systematically compare intervals for reasoning and computation is increasingly important. In practice, well established set-theoretic similarity measures such as the Jaccard and Sørensen-Dice measures are commonly used, while axiomatically a wide breadth of possible measures have been theoretically explored. This paper identifies, articulates, and addresses an inherent and so far not discussed limitation of popular measures—their tendency to be subject to aliasing—where they return the same similarity value for very different sets of intervals. The latter risks counter-intuitive results and poor automated reasoning in real-world applications dependent on systematically comparing interval-valued system variables or states. Given this, we introduce new axioms establishing desirable properties for robust similarity measures, followed by putting forward a novel set-theoretic similarity measure based on the concept of bidirectional subsethood which satisfies both the traditional and new axioms. The proposed measure is designed to be sensitive to the variation in the size of intervals, thus avoiding aliasing. The paper provides a detailed theoretical exploration of the new proposed measure, and systematically demonstrates its behaviour using an extensive set of synthetic and real-world data. Specifically, the measure is shown to return robust outputs that follow intuition—essential for real world applications. For example, we show that it is bounded above and below by the Jaccard and Sørensen-Dice similarity measures (when the minimum t-norm is used). Finally, we show that a dissimilarity or distance measure, which satisfies the properties of a metric, can easily be derived from the proposed similarity measure

From interval-valued data to general type-2 fuzzy sets

In this paper, a new approach is presented to model interval-based data using fuzzy sets (FSs). Specifically, we show how both crisp and uncertain intervals (where there is uncertainty about the endpoints of intervals) collected from individual or multiple survey participants over single or repeated surveys can be modeled using type-1, interval type-2, or general type-2 FSs based on zSlices. The proposed approach is designed to minimize any loss of information when transferring the interval-based data into FS models, and to avoid, as much as possible, assumptions about the distribution of the data. Furthermore, our approach does not rely on data preprocessing or outlier removal, which can lead to the elimination of important information. Different types of uncertainty contained within the data, namely intra- and inter-source uncertainty, are identified and modeled using the different degrees of freedom of type-2 FSs, thus providing a clear representation and separation of these individual types of uncertainty present in the data. We provide full details of the proposed approach, as well as a series of detailed examples based on both real-world and synthetic data. We perform comparisons with analogue techniques to derive FSs from intervals, namely the interval approach and the enhanced interval approach, and highlight the practical applicability of the proposed approach