Estimating correlation under interval and fuzzy uncertainty: Case of hierarchical estimation

In many situations, we are interested in finding the correlation ρ between different quantities x and y based on the values xi and yi of these quantities measured in different situations i. The correlation is easy to compute when we know the exact sample values xi and yi. In practice, the sample values come from measurements or from expert estimates; in both cases, the values are not exact. Sometimes, we know the probabilities of different values of measurement errors, but in many cases, we only know the upper bounds Δxi and Δyi on the corresponding measurement errors. In such situations, after we get the measurement results x̃i and ỹi, the only information that we have about the actual (unknown) values xi and yi is that they belong to the corresponding intervals [x̃i - Δxi, x̃i + Δxi] and [ỹi - Δyi, ỹi + Δyi]. For expert estimates, we get different intervals corresponding to different degrees of certainty - i.e., fuzzy sets. Different values of xi and yi lead, in general, to different values of the correlation ρ. It is therefore desirable to find the range [ρ_, ρ̅] of possible values of the correlation when xi and yi 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 ρ̅ when ρ̅ >; 0 and for computing ρ_ when ρ_ <; 0.