Complementary Lipschitz continuity results for the distribution of intersections or unions of independent random sets in finite discrete spaces

We prove that intersections and unions of independent random sets in finite spaces achieve a form of Lipschitz continuity. More precisely, given the distribution of a random set $\Xi$, the function mapping any random set distribution to the distribution of its intersection (under independence assumption) with $\Xi$ is Lipschitz continuous with unit Lipschitz constant if the space of random set distributions is endowed with a metric defined as the $L_k$ norm distance between inclusion functionals also known as commonalities. Moreover, the function mapping any random set distribution to the distribution of its union (under independence assumption) with $\Xi$ is Lipschitz continuous with unit Lipschitz constant if the space of random set distributions is endowed with a metric defined as the $L_k$ norm distance between hitting functionals also known as plausibilities. Using the epistemic random set interpretation of belief functions, we also discuss the ability of these distances to yield conflict measures. All the proofs in this paper are derived in the framework of Dempster-Shafer belief functions. Let alone the discussion on conflict measures, it is straightforward to transcribe the proofs into the general (non necessarily epistemic) random set terminology

Combining case based reasoning with neural networks

This paper presents a neural network based technique for mapping problem situations to problem solutions for Case-Based Reasoning (CBR) applications. Both neural networks and CBR are instance-based learning techniques, although neural nets work with numerical data and CBR systems work with symbolic data. This paper discusses how the application scope of both paradigms could be enhanced by the use of hybrid concepts. To make the use of neural networks possible, the problem's situation and solution features are transformed into continuous features, using techniques similar to CBR's definition of similarity metrics. Radial Basis Function (RBF) neural nets are used to create a multivariable, continuous input-output mapping. As the mapping is continuous, this technique also provides generalisation between cases, replacing the domain specific solution adaptation techniques required by conventional CBR. This continuous representation also allows, as in fuzzy logic, an associated membership measure to be output with each symbolic feature, aiding the prioritisation of various possible solutions. A further advantage is that, as the RBF neurons are only active in a limited area of the input space, the solution can be accompanied by local estimates of accuracy, based on the sufficiency of the cases present in that area as well as the results measured during testing. We describe how the application of this technique could be of benefit to the real world problem of sales advisory systems, among others

Combining case based reasoning with neural networks

This paper presents a neural network based technique for mapping problem situations to problem solutions for Case-Based Reasoning (CBR) applications. Both neural networks and CBR are instance-based learning techniques, although neural nets work with numerical data and CBR systems work with symbolic data. This paper discusses how the application scope of both paradigms could be enhanced by the use of hybrid concepts. To make the use of neural networks possible, the problem's situation and solution features are transformed into continuous features, using techniques similar to CBR's definition of similarity metrics. Radial Basis Function (RBF) neural nets are used to create a multivariable, continuous input-output mapping. As the mapping is continuous, this technique also provides generalisation between cases, replacing the domain specific solution adaptation techniques required by conventional CBR. This continuous representation also allows, as in fuzzy logic, an associated membership measure to be output with each symbolic feature, aiding the prioritisation of various possible solutions. A further advantage is that, as the RBF neurons are only active in a limited area of the input space, the solution can be accompanied by local estimates of accuracy, based on the sufficiency of the cases present in that area as well as the results measured during testing. We describe how the application of this technique could be of benefit to the real world problem of sales advisory systems, among others

Mining and visualizing uncertain data objects and named data networking traffics by fuzzy self-organizing map

Uncertainty is widely spread in real-world data. Uncertain data-in computer science-is typically found in the area of sensor networks where the sensors sense the environment with certain error. Mining and visualizing uncertain data is one of the new challenges that face uncertain databases. This paper presents a new intelligent hybrid algorithm that applies fuzzy set theory into the context of the Self-Organizing Map to mine and visualize uncertain objects. The algorithm is tested in some benchmark problems and the uncertain traffics in Named Data Networking (NDN). Experimental results indicate that the proposed algorithm is precise and effective in terms of the applied performance criteria.Peer ReviewedPostprint (published version

Implication functions in interval-valued fuzzy set theory

Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a particular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory

Designing an expert knowledge-based Systemic Importance Index for financial institutions

Defining whether a financial institution is systemically important (or not) is challenging due to (i) the inevitability of combining complex importance criteria such as institutions’ size, connectedness and substitutability; (ii) the ambiguity of what an appropriate threshold for those criteria may be; and (iii) the involvement of expert knowledge as a key input for combining those criteria. The proposed method, a Fuzzy Logic Inference System, uses four key systemic importance indicators that capture institutions’ size, connectedness and substitutability, and a convenient deconstruction of expert knowledge to obtain a Systemic Importance Index. This method allows for combining dissimilar concepts in a non-linear, consistent and intuitive manner, whilst considering them as continuous –non binary- functions. Results reveal that the method imitates the way experts them-selves think about the decision process regarding what a systemically important financial institution is within the financial system under analysis. The Index is a comprehensive relative assessment of each financial institution’s systemic importance. It may serve financial authorities as a quantitative tool for focusing their attention and resources where the severity resulting from an institution failing or near-failing is estimated to be the greatest. It may also serve for enhanced policy-making (e.g. prudential regulation, oversight and supervision) and decision-making (e.g. resolving, restructuring or providing emergency liquidity).Systemic Importance, Systemic Risk, Fuzzy Logic, Approximate Reasoning, Too-connected-to-fail, Too-big-to-fail. Classification JEL: D85, C63, E58, G28.