This report summarizes a variety of the most useful and commonly applied methods for obtaining Dempster-Shafer structures, and their mathematical kin probability boxes, from empirical information or theoretical knowledge. The report includes a review of the aggregation methods for handling agreement and conflict when multiple such objects are obtained from different sources.

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Constructing Probability Boxes and Dempster-Shafer Structures

The Karnik-Mendel (KM) algorithms are iterative procedures widely used in fuzzy logic theory. They are known to converge monotonically and superexponentially fast; however, several (usually two to six) iterations are still needed before convergence occurs. Methods to reduce their computational cost are proposed in this paper. Extensive simulations show that, on average, the enhanced KM algorithms can save about two iterations, which corresponds to more than a 39% reduction in computation time. An additional (at least) 23% computational cost can be saved if no sorting of the inputs is needed.

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Enhanced Karnik--Mendel Algorithms

This chapter is an overview of past and present works dealing with fuzzy intervals and their operations. A fuzzy interval is a fuzzy set in the real line whose level-cuts are intervals. Particular cases include usual real numbers and intervals. Usual operations on the real line canonically extend to operations between fuzzy quantities, thus extending the usual interval (or error) analysis to membership functions. What is obtained is a counterpart of random variable calculus, but where, contrary to the latter case, there is no compensation between variables. Many results pertaining to basic properties of fuzzy interval analysis are summed up in the chapter. Computational methods are presented, exact or approximate ones, based on parametric representations, or level-cut approximations. The generalized fuzzy variable calculus involving interactive variables is also discussed with emphasis on triangular-norm based fuzzy additions. Dual ‘optimistic’ operations on fuzzy intervals, i.e., with maximal error compensation are also presented; its interest lies in providing tools for solving fuzzy interval equations. This chapter also contains a reasoned survey of methods for comparing and ranking fuzzy intervals. The chapter includes some historical background, as well as pointers to applications in mathematics and engineering.

Fuzzy Interval Analysis

Intrusion detection has gain a broad attention and become a fertile field for several researches, and still being the subject of widespread interest by researchers. The intrusion detection community still confronts difficult problems even after many years of research. Reducing the large number of false alerts during the process of detecting unknown attack patterns remains unresolved problem. However, several research results recently have shown that there are potential solutions to this problem. Anomaly detection is a key issue of intrusion detection in which perturbations of normal behavior indicates a presence of intended or unintended induced attacks, faults, defects and others. This paper presents an overview of research directions for applying supervised and unsupervised methods for managing the problem of anomaly detection. The references cited will cover the major theoretical issues, guiding the researcher in interesting research directions.

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Machine Learning Techniques for Anomaly Detection: An Overview

We study concept lattices with hedges. The principal aim is to control, in a parametrical way, the size of a concept lattice. The paper presents theoretical insight, comments, and examples. We show that a concept lattice with hedges is indeed a complete lattice which is isomorphic to an ordinary concept lattice. We describe the isomorphism and its inverse. These mappings serve as translation procedures. As a consequence, we obtain a theorem characterizing the structure of concept lattices with hedges which generalizes the so-called main theorem of concept lattices. Furthermore, the isomorphism and its inverse enable us to compute a concept lattice with hedges using algorithms for ordinary concept lattices. Further insight is provided in case one uses hedges only for attributes. We demonstrate by experiments that the size reduction using hedges as a parameter is smooth

http://phoebe.inf.upol.cz/~havrlanl/articles/BeVy_Rsfclh.pdf

Reducing the Size of Fuzzy Concept Lattices by Hedges

IEEE International Conference on Fuzzy Systems

Most practical applications of statistical methods are based on the implicit assumption that if an event has a very small probability, then it cannot occur. For example, the probability that a kettle placed on a cold stove would start boiling by itself is not 0, it is positive, but it is so small, that physicists conclude that such an event is simply impossible. This assumption is difficult to formalize in traditional probability theory, because this theory only describes measures on sets and does not allow us to divide functions into 'random' and non-random ones. This distinction was made possible by the idea of algorithmic randomness, introduce by Kolmogorov and his student Martin- Loef in the 1960s. We show that this idea can also be used for inverse problems. In particular, we prove that for every probability measure, the corresponding set of random functions is compact, and, therefore, the corresponding restricted inverse problem is well-defined. The resulting techniques turns out to be interestingly related with the qualitative esthetic measure introduced by G. Birkhoff as order/complexity.© (1998) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

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Kolmogorov complexity, statistical regularization of inverse problems, and Birkhoff's formalization of beauty

Fast implementations of fuzzy arithmetic operations using fast Fourier transform (FFT)

In engineering applications of fuzzy logic, the main goal is not to simulate the way the experts really think, but to come up with a good engineering solution that would (ideally) be better than the expert's control. In such applications, it makes perfect sense to restrict ourselves to simplified approximate expressions for membership functions. If we need to perform arithmetic operations with the resulting fuzzy numbers, then we can use simple and fast algorithms that are known for operations with simple membership functions. In other applications, especially the ones that are related to humanities, simulating experts is one of the main goals. In such applications, we must use membership functions that capture every nuance of the expert's opinion; these functions an therefore complicated, and fuzzy arithmetic operations with the corresponding fuzzy numbers become a computational problem. In this paper, we design a new algorithm for performing such operations. This algorithm uses fast Fourier transforms (FFTs) to reduce computation time from O(n/sup 2/) to O(nlog(n)) (when n is the number of points x at which we know the membership functions /spl mu/(x)). To compute FFT even faster, we propose to use special hardware.

If several physical theories are consistent with the same experimental data, which theory should we choose? Physicists often choose the simplest theory; this principle (explicitly formulated by Occam) is one of the basic principles of physical reasoning. However, until recently, this principle was mainly a heuristic because it uses the informal notion of simplicity.

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Misha Koshelev

Papers

IEEE International Conference on Fuzzy Systems

Kolmogorov complexity, statistical regularization of inverse problems, and Birkhoff's formalization of beauty

Fast implementations of fuzzy arithmetic operations using fast Fourier transform (FFT)

Fast implementations of fuzzy arithmetic operations using fast Fourier transform (FFT)

We Must Choose the Simplest Physical Theory: Levin-Li-Vitanyi Theorem and its Potential Physical Applications