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 xi related to y and then estimate y from the results $\tilde x_i$ of these measurements as $f(\tilde x_1 ,...,\tilde x_n )$ by using a known relation f. Interval computations are used "to find the range of f(x1,...,xn) when xi are known to belong to intervals $x_i = [\tilde x_i - \Delta _i ,\tilde x_i + \Delta _i ]$," where Δi 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 xi; usually, the errors xi - $\tilde x_i$ are independent Gaussian random variables with 0 average and known standard deviations σi. For such situations, we can formulate a similar probabilistic problem: "given σi, compute the standard deviation of f(x1,...,xn) ." 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 xi 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.