Mark J. Wierman

Information entropy, rough entropy and knowledge granulation in incomplete information systems

Rough set theory is a relatively new mathematical tool for use in computer applications in circumstances that are characterized by vagueness and uncertainty. Rough set theory uses a table called an information system, and knowledge is defined as classifications of an information system. In this paper, we introduce the concepts of information entropy, rough entropy, knowledge granulation and granularity measure in incomplete information systems, their important properties are given, and the relationships among these concepts are established. The relationship between the information entropy E(A) and the knowledge granulation GK(A) of knowledge A can be expressed as E(A)+GK(A) = 1, the relationship between the granularity measure G(A) and the rough entropy E r(A) of knowledge A can be expressed as G(A)+E r(A) = log2|U|. The conclusions in Liang and Shi (2004) are special instances in this paper. Furthermore, two inequalities − log2 GK(A) ≤ G(A) and E r(A) ≤ log2(|U|(1 − E(A))) about the measures GK, G, E and E r are obtained. These results will be very helpful for understanding the essence of uncertainty measurement, the significance of an attribute, constructing the heuristic function in a heuristic reduct algorithm and measuring the quality of a decision rule in incomplete information systems.

MEASURING UNCERTAINTY IN ROUGH SET THEORY

A well justified measure of uncertainty for rough set theory is presented along with an axiomatic derivation. The connection between this measure and classical measures of uncertainty is provided.

Consensus and dissention: A measure of ordinal dispersion

A new measure of dispersion is introduced as a representation of consensus (agreement) and dissention (disagreement). Building on the generally accepted Shannon entropy, this measure utilizes a probability distribution and the distance between categories to produce a value spanning the unit interval. The measure is applied to the Likert scale (or any ordinal scale) to determine degrees of consensus or agreement. Using this measure, data on ordinal scales can be given a value of dispersion that is both logically and theoretically sound.

Empirical study of defuzzification

The most important application of fuzzy logic is designing controllers. Fuzzy logic controllers (FLC) are much easier to design than non-linear controllers of similar capabilities. The rules that a designer needs to create are often based on their current experience and knowledge. Conventional FLCs use Center of Gravity or Mean of Maxima defuzzification methods, though other methods have been studied. This paper compares the efficiency of many different models of the defuzzification process. The goal is to examine the accuracy of the output data and the amount of processing time required. A simple controller that backs a truck up to a gate is used in the study. All of the variables are granulated with trapezoidal fuzzy numbers. Some of the defuzzification methods examined are Fast Center of Gravity, Mean of Maxima, True Center of Gravity and various new methods that have shown promise in application.

Central values of fuzzy numbers—defuzzification

Abstract Axioms for defining a central value of a fuzzy set are developed. Various proposed central values are examined. Implications for defuzzification are discussed.

Consensus and dissention: a new measure of agreement

This paper introduces a new measure of ordinal scales, specifically that of the Likert scale. An overview of the Likert scale is given and a set of requirements is presented that should be satisfied to assure a proper meaning of consensus within the confines of the Likert scale. Illustrations compare the consensus measure with entropy and other statistical measures. The consensus measure also serves as an effective measure of dispersion.

Measuring conflict in evidence theory

The Shannon entropy measures conflict in probabilistic evidence. The evidence theory also presents information that is inherently conflicting. Uncertainty measures for conflict in the evidence theory include aggregate uncertainty, dissonance, confusion, discord, granularity and strife. This paper compares and critiques the properties of these measures.

Consensus and dissention: theory and properties

The Likert scale is a popular tool for assessing peoples attitude about various questions. This papers introduces two new measures, dissension and consensus, that provide a better statistic for assessing the amount of overall agreement in a population when data is extracted from the population using a Likert scale. The measures are applicable to any discrete random variable.

Using Consensus to Measure Weighted Targeted Agreement

The information-theoretic measures of consensus, dissent and agreement are used to address the problem of the assignment of weights in recognition of expert opinions, and interval weights to reflect categorical weights. All measures are bounded in the 0 to 1 interval. Dissent is also interpreted as an indicator of dispersion. Thus, the values selected by a panel of experts is calculated for each targeted category and the category with the highest resulting value is the one chosen to represent the overall expert judgment. Further, the distances between threat levels can be calculated and the dispersion for the distribution may also be calculated. This is different from the standard statistical measures of variance for categorical values are based on an ordinal scale of ordered categories and the standard deviation requires the presence of an interval or ratio scale. Illustrations are shown to describe the functionality of the measures.

Extending set functions and relations

Abstract Zadehs extension principle is applicable to functions defined upon elements of a set. A new extension principle is developed for functions defined upon power sets. Another extension is proposed for relations between sets, such as subsethood. A formal relation connects the two new extension principles. The new extension principles are not a consequences of Zadehs original.