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

Due to measurement uncertainty, often, instead of the actual values xi of the measured quantities, we only know the intervals xi = [x̃i − ∆i, x̃i + ∆i], where x̃i is the measured value and ∆i 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 covariance Cx,y, 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 Cx,y 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.