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

Computing Population Variance and Entropy under Interval Uncertainty: Linear-Time Algorithms 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). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

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

, Volume 13 (6) – Feb 5, 2008
22 pages

/lp/springer_journal/computing-population-variance-and-entropy-under-interval-uncertainty-JEGNksTMp0
Publisher
Springer Journals
Subject
Mathematics; Numeric Computing; Mathematical Modeling and Industrial Mathematics; Approximations and Expansions; Computational Mathematics and Numerical Analysis
ISSN
1385-3139
eISSN
1573-1340
D.O.I.
10.1007/s11155-007-9045-6
Publisher site
See Article on Publisher Site

### 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).

### Journal

Reliable ComputingSpringer Journals

Published: Feb 5, 2008

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