On-line algorithms for computing mean and variance of interval data, and their use in intelligent systems

Authors:
Vladik Kreinovich;Hung T. Nguyen;Berlin Wu
Affiliations:
Computer Science Department, University of Texas, El Paso, TX 79968, USA;Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA;Department of Mathematical Sciences, National Chengchi University, Taipei, Taiwan
Venue:
Information Sciences: an International Journal
Year:
2007

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Abstract

When we have only interval ranges [x@?"i,x"i@?] of sample values x"1,...,x"n, what is the interval [V@?,V@?] of possible values for the variance V of these values? There are quadratic time algorithms for computing the exact lower bound V on the variance of interval data, and for computing V@? under reasonable easily verifiable conditions. The problem is that in real life, we often make additional measurements. In traditional statistics, if we have a new measurement result, we can modify the value of variance in constant time. In contrast, previously known algorithms for processing interval data required that, once a new data point is added, we start from the very beginning. In this paper, we describe new algorithms for statistical processing of interval data, algorithms in which adding a data point requires only O(n) computational steps.