Get Computing Statistics under Interval and Fuzzy Uncertainty: PDF

By Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang

ISBN-10: 3642249043

ISBN-13: 9783642249044

In many useful events, we're attracted to facts characterizing a inhabitants of items: e.g. within the suggest peak of individuals from a definite area.

Most algorithms for estimating such statistics suppose that the pattern values are specified. In perform, pattern values come from measurements, and measurements are by no means totally actual. occasionally, we all know the precise chance distribution of the dimension inaccuracy, yet usually, we in simple terms understand the higher sure in this inaccuracy. as a consequence, now we have period uncertainty: e.g. if the measured price is 1.0, and inaccuracy is bounded through 0.1, then the particular (unknown) worth of the amount should be anyplace among 1.0 - 0.1 = 0.9 and 1.0 + 0.1 = 1.1. In different circumstances, the values are professional estimates, and we simply have fuzzy information regarding the estimation inaccuracy.

This booklet exhibits find out how to compute facts lower than such period and fuzzy uncertainty. The ensuing equipment are utilized to machine technology (optimal scheduling of other processors), to info know-how (maintaining privacy), to machine engineering (design of computing device chips), and to info processing in geosciences, radar imaging, and structural mechanics.

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Extra resources for Computing Statistics under Interval and Fuzzy Uncertainty: Applications to Computer Science and Engineering

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A normal distribution is uniquely determined by its mean and variance. When we add several independent variables, their means and variances add up. For each uniform distribution Δxi on the interval [−Δx , Δx ] of width 1 2Δx , the probability density is equal to ρ(x) = , so the mean is 0 and 2Δx the variance is Δx V = −Δx x2 ·ρ(x) dx = 1 · 2Δx Δx −Δx x2 dx = 1 1 3 · ·x 2Δx 3 Δx −Δx = 1 2 ·Δ . 1) 3 x 5 Uniform Distributions Approach to Interval Uncertainty 27 Thus, for the sum Δy of n such variables, the mean E is 0, and the variance √ 2 is equal √ to (n/3) · Δx .

Y (N ) are therefore a sample from the Cauchy distribution with the desired parameter Δ. Based on this sample, we can estimate the value Δ. + 2 δy (k) Δ 1+ 1 + ... + 1+ δy (N ) Δ = 2 N . 10) 2 The left-hand side of this equation is an increasing function that is equal to 0 (hence smaller than N/2) for Δ = 0 and larger than N/2 for Δ = max δy (k) ; therefore the solution to this equation can be found by applying a bisection method to the interval 0, max δy (k) . 11) where ri is uniformly distributed on the interval [0, 1].

Com 20 4 Fuzzy Computations Reduced to Interval Computations μS (xi ) – which values xi should we then classify as possible ones and which as impossible? Under uncertainty, a reasonable idea is to select a threshold α ∈ (0, 1]. , for which μS (xi ) < α – are classified as impossible. The resulting set of possible elements def xi (α) = {xi : μS (xi ) ≥ α} is called the α-cut of the membership function μS (xi ). The choice of a threshold α depends on the practical problem. For example, if we are looking for a potentially very valuable mineral deposit, then it makes sense to continue prospecting even when our degree of confidence is not very high.

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Computing Statistics under Interval and Fuzzy Uncertainty: Applications to Computer Science and Engineering by Hung T. Nguyen, Vladik Kreinovich, Berlin Wu, Gang Xiang

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