Piecewise-linear approximations of uncertain functions

Abstract

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). © 2011 Springer-Verlag.