Estimating Correlation under Interval Uncertainty

Authors

Ali Jalal-Kamali, The University of Texas at El PasoFollow
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

11-2011

Comments

Technical Report: UTEP-CS-11-57

Abstract

In many engineering situations, we are interested in finding the correlation ρ between different quantities x and y based on the values x_i and y_i of these quantities measured in different situations i. Measurements are never absolutely accurate; it is therefore necessary to take this inaccuracy into account when estimating the correlation ρ. Sometimes, we know the probabilities of different values of measurement errors, but in many cases, we only know the upper bounds Δ_xi and Δ_yi on the corresponding measurement errors. In such situations, after we get the measurement results X_i and Y_i, the only information that we have about the actual (unknown) values x_i and y_i is that they belong to the corresponding intervals [X_i - Δ_xi, X_i + Δ_xi] and [Y_i - Δ_yi, Y_i + Δ_yi]. Different values from these intervals lead, in general, to different values of the correlation ρ. It is therefore desirable to find the range [ρ^-, ρ⁺] of possible values of the correlation when x_i and y_i take values from the corresponding intervals. In general, the problem of computing this range is NP-hard. In this paper, we provide a feasible (= polynomial-time) algorithm for computing at least one of the endpoints of this interval: for computing ρ⁺ when ρ⁺ > 0 and for computing ρ^- when ρ^- < 0.