Journal of Computational and Applied Mathematics - Special issue: Scientific computing, computer arithmetic, and validated numerics (SCAN 2004)
Information Sciences: an International Journal
Towards adding probabilities and correlations to interval computations
International Journal of Approximate Reasoning
Computing best-possible bounds for the distribution of a sum of several variables is NP-hard
International Journal of Approximate Reasoning
Hi-index | 0.00 |
When we have only interval ranges [\underline {x},\bar {x}_i] of sample values x1,…,xn, what is the interval [\underline {V},\bar {V}] of possible values for the variance V of these values? We show that the problem of computing the upper bound \bar {V} is NP-hard. We provide a feasible (quadratic time) algorithm for computing the exact lower bound \underline {V} on the variance of interval data. We also provide feasible algorithms that computes \bar {V} under reasonable easily verifiable conditions, in particular, in case interval uncertainty is introduced to maintain privacy in a statistical database. We also extend the main formulas of interval arithmetic for different arithmetic operations x1 op x2 to the case when,\break for each input xi, in addition to the interval {\bf x}_i=[\underline {x}_i,\bar {x}_i] of possible values, we also know its mean Ei (or an interval Ei of possible values of the mean), and we want to find the corresponding bounds for y=x1 op x2 and its mean. In this case, we are interested not only in the bounds for y, but also in the bounds for the mean of y. We formulate and solve the corresponding optimization problems, and describe remaining open problems.