Computing Population Variance and Entropy under Interval Uncertainty: Linear-Time Algorithms

Abstract

In statistical analysis of measurement results it is often necessary to compute the range $[\underline V ,\,\overline V]$ of the population variance $V = \frac{1}{n}\, \cdot \,\sum\limits_{i = 1}^n (x_i \, - \,E)^2 \,\left({\rm where}\,E = \frac{1}{n}\, \cdot \,\sum\limits_{i = 1}^n {x_i }\,\right)$ when we only know the intervals $[\tilde x_i - \Delta _i,\,\tilde x_i \, + \,\Delta _i]$ of possible values of the x _i . While $\underline {V}$ can be computed efficiently, the problem of computing $\overline {V}$ is, in general, NP-hard. In our previous paper “Population Variance under Interval Uncertainty: A New Algorithm” (Reliable Computing 12 (4) (2006), pp. 273–280) we showed that in a practically important case we can use constraints techniques to compute $\overline {V}$ in time O(n · log(n)). In this paper we provide new algorithms that compute $\underline {V}$ (in all cases) and $\overline {V}$ (for the above case) in linear time O(n).

Similar linear-time algorithms are described for computing the range of the entropy $S = - \sum\limits_{i = 1}^n {p_i\,\cdot\,{\rm log} (p_i )}$ when we only know the intervals ${\bf P}_i \, = \,[p_{-i},\,\bar p_i]$ of possible values of probabilities p _i .

In general, a statistical characteristic ƒ can be more complex so that even computing ƒ can take much longer than linear time. For such ƒ, the question is how to compute the range $[\underline y,\,\overline y]$ in as few calls to ƒ as possible. We show that for convex symmetric functions ƒ, we can compute $\bar {y}$ in n calls to ƒ.