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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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