Computational Complexity and Feasibility of Data Processing and Interval Computations pp 331-346 | Cite as

# 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(x̄_{1},..., x̄_{n}) by using a known relation f. Interval computations are used “to find the range of f(x_{1},...,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(x_{1},..., 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].

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