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A Characterization Theorem for the Best L 1 Piecewise Monotonic Data Approximation Problem

In: Contributions in Mathematics and Engineering

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  • Ioannis C. Demetriou

    (National and Kapodistrian University of Athens, Division of Mathematics and Informatics, Department of Economics)

Abstract

Let a sequence of n univariate data that include random errors be given. We consider the problem of calculating a best L 1 approximation to the data subject to the condition that the first differences of the approximated values have at most k − 1 sign changes, where k is a prescribed integer. The choice of the positions of sign changes by considering all possible combinations of positions can be of magnitude n k−1, so that it is not practicable to test each one separately. We provide a theorem that decomposes the problem into the best L 1 monotonic approximation (case k = 1) problems to disjoint sets of adjacent data. The decomposition allows the development of a dynamic programming procedure that provides a highly efficient calculation of the solution.

Suggested Citation

  • Ioannis C. Demetriou, 2016. "A Characterization Theorem for the Best L 1 Piecewise Monotonic Data Approximation Problem," Springer Books, in: Panos M. Pardalos & Themistocles M. Rassias (ed.), Contributions in Mathematics and Engineering, pages 117-126, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-31317-7_7
    DOI: 10.1007/978-3-319-31317-7_7
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