Z-Numbers: How They Describe Student Confidence and How They Can Explain (and Improve) Laplacian and Schroedinger Eigenmap Dimension Reduction in Data Analysis

Authors

Vladik Kreinovich, The University of Texas at El PasoFollow
Olga Kosheleva, The University of Texas at El PasoFollow
Michael Zakharevich, SeeCure Systems, Inc.Follow

Publication Date

12-2017

Comments

Technical Report: UTEP-CS-17-92

Abstract

Experts have different degrees of confidence in their statements. To describe these different degrees of confidence, Lotfi A. Zadeh proposed the notion of a Z-number: a fuzzy set (or other type of uncertainty) supplemented by a degree of confidence in the statement corresponding to fuzzy sets. In this chapter, we show that Z-numbers provide a natural formalization of the competence-vs-confidence dichotomy, which is especially important for educating low-income students. We also show that Z-numbers provide a natural theoretical explanation for several empirically heuristic techniques of dimension reduction in data analysis, such as Laplacian and Schroedinger eigenmaps, and, moreover, show how these methods can be further improved.