Arthur Ramer

Uncertainty in the Dempster-Shafer theory: a critical re-examination

Measures of two types of uncertainty that coexist in the Dempster-Shafer theory are overivewed. A measure of one type of uncertainty, which expresses nonspecificity of evidential claims, is well justified on both intuitive and mathermatical grounds. Proposed measures of the other types of uncertainty, which attempt to capture conflicts among evidential claims, are shown to have some deficiencies. To alleviate these deficiencies, a new measure is proposed. This measure, which is called a measure of discord, is not only satisfactory on intuitive grounds, but has alos desirable mathematical properties. A measure of total uncertainty, which is defined as the sum of nonspecificity and discord, is also discussed. The paper focuses on conceptual issues. Mathematical properties of the measure of idscord are only stated ; their proofs are given in a companion paper

Conditional possibility measures

Possibility theory is the formalization of the methods of reasoning about uncertainty and information, derived from the principles of fuzzy sets and systems. Derived possibilistic assignments can be constructed within this theory. This paper defines the notion of conditional possibility—an assignment subject to conditioning by other possibilistic random variables. The proposed definition can be based on one of the two principles: proper interaction with marginal distributions, or minimization of possibilistic information distance between the original and derived distributions. Both approaches independently lead to the same definition, thus strongly suggesting its wider applicability.

CONCEPTS OF FUZZY INFORMATION MEASURES ON CONTINUOUS DOMAINS

Measures of information based on fuzzy sets (possibility distributions) had been defined only for finite domains of discourse. This paper presents a method of defining such information functions on a continuous universe of discourse—a domain which is a measurable space of measure 1. The method is based on the concept of “rearangement” of a function, used in lieu of sorting discrete possibility values. For technical reasons, it is preferred to express information value as information distance to the most “uninformed” (constant possibility 1) distribution. The final form of the information for possibility distribution f is The paper then discusses related information distances and approximations using discrete information functions.

Measures of discord in the Dempster-Shafer theory

Abstract This paper is a companion of a previous paper, in which a new measure of uncertainty, called a measure of discord, was introduced. While the previous paper focuses on intuitive justification of this new measure and identification of deficiencies of some previously considered measures, this paper is oriented toward a mathematical consideration of the measure of discord. Also discussed in the paper is a total measure of uncertainty in the Dempster-Shafer theory, defined as the sum of discord and nonspecificity.

Structure of possibilistic information metrics and distances : properties

Abstract Systematic analysis of identities and inequalities satisfied by possibilistic information distances is conducted. The analysis is based on their representations as discrete sums and on certain inequalities for rearrangements of sequences. These identities and inequalities express several properties that are usually deemed characteristic of information distances and measures. In the companion paper those properties are used to obtain several axiomatic characterizations of possibility distances. The basic distance is g(p,q) defined for the distributions p = (Pi,[tdot],pn) and q = (qi,[tdot],qn) such that Pi≤qi. =i=1,[tdot],n They serve to define a metric G(p,q)=g<p,p ∨ q)+g(q,p [tdot] q) and a distanceH(p,q) = g(p∧q,p)+g(p ∧ q,q). All these distances are, in turn based on the U-uncertainty information function.

A note on defining conditional probability

The notion of conditional probability is usually introduced on the basis of frequencies of events belonging to a specified subset (of the set all admissible events) events. Given P(xi) = Pi, i = 1,.. ., n, not identically 0 on the set of events {xi}, = 1, .. ., m, m < n, let us suppose that we wish to define an associated conditional distribution Q(xi) = qi, i = 1,..., m, on that subset. It is then defined as

Predictive analysis of linewidth parameters of microwave garnets

Abstract The linearity of microwave parameters—resonance linewidth ΔH and effective linewidth ΔHeff—is demonstrated and its use in the CAD/CAM of new microwave garnets proposed. Such an approach would combine a numerical database of microwave data and several computational programs. This paper complements the discussion of the magnetic properties of microwave materials and their CAD/CAM in R. Puflea-Ramer et al., Electr. Power Syst. Res., 24 (1992) 141–148.

Possibilistic information metrics and distances : characterizations of structure

Abstract Axiomatic characterizations of possibilistic distances and measures are studied. The basic distance g(p,q), defined for p≦q, is characterized using axioms of translation invariance, monotonic sum, metric and additivity with respect to cartesian products. To extend this definition to arbitrary pairs p, q one of the latter properties must be relaxed. Retaining the additive property gives H(p,q), while retaining the metric property leads to a class of metric distances, of which G(p,q) is maximal. A new metric K(p,q) = max(g(p, p ∨ q),g(q,p ∨ q)) is introduced. It is in a certain sense a minimal metric extension of g(p,q).

Axioms of uncertainty measures: dependence and independence

Abstract Axiom systems for uncertainty functions in possibility and evidence theories are studied from the point of view of the logical relationships. First, it is proven that monotonicity and subadditivity properties are equivalent in the theory of possibility, however the latter is a strictly stronger property in the theory of evidence. Then the role of linearity and branching property in possibility theory is analyzed. The paper extends the research initiated by Higashi and Klir [5], Higashi [4], Klir and Mariano [6], Ramer and Lander [10] and Ramer [8,9].

CONFLUENT POSSIBILITY DISTRIBUTIONS AND THEIR REPRESENTATIONS

Abstract Possibilistic distributions admit both measures of uncertainty and (metric) distances defining their information closeness. For general pairs of distributions these measures and metrics were first introduced in the form of integral expressions. Particularly important are pairs of distributions p and q which have consonant ordering—for any two events x and y in the domain of discourse p(x)l p(y) if and only if q(x) l q(y). We call such distributions confluent and study their information distances. This paper presents discrete sum form of uncertainty measures of arbitrary distributions, and uses it to obtain similar representations of metrics on the space of confluent distributions. Using these representations, a number of properties like additivity. monotonicity and a form of distributivity are proven. Finally, a branching property is introduced, which will serve (in a separate paper) to characterize axiomatically possibilistic information distances.