Interval-Valued and Fuzzy-Valued Random Variables: From Computing Sample Variances to Computing Sample Covariances

Authors

Jan B. Beck
Vladik Kreinovich, The University of Texas at El PasoFollow
Berlin Wu

Publication Date

4-2004

Comments

UTEP-CS-04-03a.

Published in: M. Lopez, M. A. Gil, P. Grzegorzewski, O. Hrynewicz, and J. Lawry (eds.), Soft Methodology and Random Information Systems, Springer-Verlag, 2004, pp. 85-92.

Abstract

Due to measurement uncertainty, often, instead of the actual values xi of the measured quantities, we only know the intervals [xi]=[Xi-Di,Xi+Di], where Xi is the measured value and Di is the upper bound on the measurement error (provided, e.g., by the manufacturer of the measuring instrument). In such situations, instead of the exact value of the sample statistics such as covariance C(x,y), we can only have an interval [C](x,y) of possible values of this statistic. It is known that in general, computing such an interval [C](x,y) for C(x,y) is an NP-hard problem. In this paper, we describe an algorithm that computes this range [C](x,y) for the case when the measurements are accurate enough -- so that the intervals corresponding to different measurements do not intersect much.