Interval-valued and fuzzy-valued random variables: from computing sample variances to computing sample covariances

Abstract

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