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Piecewise-linear approximations of uncertain functions

Abam, M. A ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1007/978-3-642-22300-6_1
  3. Abstract:
  4. We study the problem of approximating a function F:ℝ → ℝ by a piecewise-linear function F̄ when the values of F at {x 1,...,xn} are given by a discrete probability distribution. Thus, for each xi we are given a discrete set y i,1,..., yi,mi of possible function values with associated probabilities pi,j such that Pr[F(xi) = yi,j] = pi,j. We define the error of F̄ as error(F, F̄) = maxi=1n E[|Fxi) - F̄(xi)|]. Let m = ∑i=1nmi be the total number of potential values over all F(xi). We obtain the following two results: (i) an O(m) algorithm that, given F and a maximum error ε, computes a function F̄ with the minimum number of links such that error(F, F̄) ≤ ε; (ii) an O(n4/3+δ + mlogn) algorithm that, given F, an integer value 1 ≤ k ≤ n and any δ > 0, computes a function F̄ of at most k links that minimizes error(F, F̄)
  5. Keywords:
  6. Discrete probability distribution ; Discrete sets ; Function values ; Integer values ; Maximum error ; Piecewise linear approximations ; Piecewise linear functions ; Potential values ; Uncertain functions ; Algorithms ; Data structures ; Probability distributions ; Piecewise linear techniques
  7. Source: 12th International Symposium on Algorithms and Data Structures, WADS 2011, New York, NY, 15 August 2011 through 17 August 2011 ; Volume 6844 LNCS , 2011 , Pages 1-12 ; 03029743 (ISSN) ; 9783642222993 (ISBN)
  8. URL: http://link.springer.com/chapter/10.1007%2F978-3-642-22300-6_1