Error Estimation for Indirect Measurements: Interval Computation Problem is (Slightly) Harder than a Similar Probabilistic Computational Problem

Abstract

One of main applications of interval computations is estimating errors of indirect measurements. A quantity y is measured indirectly if we measure some quantities x_i related to y and then estimate y from the results x̄, of these measurements as f(x̄₁,..., x̄_n) by using a known relation f. Interval computations are used “to find the range of f(x₁,...,x_n) when x_i are known to belong to intervals x_i = [x̄_i — Δ_i,x̄_i + Δ_i],” where Δ_i are guaranteed accuracies of direct measurements. It is known that the corresponding problem is intractable (NP-hard) even for polynomial functions f.

In some real-life situations, we know the probabilities of different value of x_i; usually, the errors x_i — x̄_i are independent Gaussian random variables with 0 average and known standard deviations σ_i. For such situations, we can formulate a similar probabilistic problem: “ given σ_i, compute the standard deviation of f(x₁,..., x_n)”. It is reasonably easy to show that this problem is feasible for polynomial functions f. So, for polynomial f, this probabilistic computation problem is easier than the interval computation problem.

It is not too much easier: Indeed, polynomials can be described as functions obtained from x_i by applying addition, subtraction, and multiplication. A natural expansion is to add division, thus getting rational functions. We prove that for rational functions, the probabilistic computational problem (described above) is NP-hard.

The results of this chapter appear in Kosheleva et al. [186].