Population Variance under Interval Uncertainty: A New Algorithm

Population Variance under Interval Uncertainty: A New Algorithm In statistical analysis of measurement results, it is often beneficial to compute the range V of the population variance $$v=\frac{1}{2}\cdot\sum^{n}_{i=1}(x_{i}-E)^{2}\,(where E=\frac{1}{n}\sum^{n}_{i=1}x_{i})$$ when we only know the intervals $$[{\tilde x}_{i}-\Delta_{i},{\tilde x}_{i}+\Delta_{i}]$$ of possible values of the xi. In general, this problem is NP-hard; a polynomialtime algorithm is known for the case when the measurements are sufficiently accurate, i.e., when $$|{\tilde x}_{i}-{\tilde x}_{j}|\geq\frac{\Delta_{i}}{n}+\frac{\Delta_{j}}{n}$$ for all $$i\neq j.$$ In this paper, we show that we can efficiently compute V under a weaker (and more general) condition $$|{\tilde x}_{i}-{\tilde x}_{j}|\geq\frac{|\Delta_{i}-\Delta_{j}|}{n}$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

Population Variance under Interval Uncertainty: A New Algorithm

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Publisher
Springer Netherlands
Copyright
Copyright © 2006 by Springer Science + Business Media, Inc.
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/s11155-006-9001-x
Publisher site
See Article on Publisher Site

Abstract

In statistical analysis of measurement results, it is often beneficial to compute the range V of the population variance $$v=\frac{1}{2}\cdot\sum^{n}_{i=1}(x_{i}-E)^{2}\,(where E=\frac{1}{n}\sum^{n}_{i=1}x_{i})$$ when we only know the intervals $$[{\tilde x}_{i}-\Delta_{i},{\tilde x}_{i}+\Delta_{i}]$$ of possible values of the xi. In general, this problem is NP-hard; a polynomialtime algorithm is known for the case when the measurements are sufficiently accurate, i.e., when $$|{\tilde x}_{i}-{\tilde x}_{j}|\geq\frac{\Delta_{i}}{n}+\frac{\Delta_{j}}{n}$$ for all $$i\neq j.$$ In this paper, we show that we can efficiently compute V under a weaker (and more general) condition $$|{\tilde x}_{i}-{\tilde x}_{j}|\geq\frac{|\Delta_{i}-\Delta_{j}|}{n}$$ .

Journal

Reliable ComputingSpringer Journals

Published: Aug 1, 2006

References

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