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Using Parameter Elimination to Solve Discrete Linear Chebyshev Approximation Problems

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  • Nikolai Krivulin

    (Faculty of Mthematics and Mechanics, St. Petersburg State University, Universitetskaya Emb. 7/9, 199034 St. Petersburg, Russia)

Abstract

We consider discrete linear Chebyshev approximation problems in which the unknown parameters of linear function are fitted by minimizing the least maximum absolute deviation of errors. Such problems find application in the solution of overdetermined systems of linear equations that appear in many practical contexts. The least maximum absolute deviation estimator is used in regression analysis in statistics when the distribution of errors has bounded support. To derive a direct solution of the problem, we propose an algebraic approach based on a parameter elimination technique. As a key component of the approach, an elimination lemma is proved to handle the problem by reducing it to a problem with one parameter eliminated, together with a box constraint imposed on this parameter. We demonstrate the application of the lemma to the direct solution of linear regression problems with one and two parameters. We develop a procedure to solve multidimensional approximation (multiple linear regression) problems in a finite number of steps. The procedure follows a method that comprises two phases: backward elimination and forward substitution of parameters. We describe the main components of the procedure and estimate its computational complexity. We implement symbolic computations in MATLAB to obtain exact solutions for two numerical examples.

Suggested Citation

  • Nikolai Krivulin, 2020. "Using Parameter Elimination to Solve Discrete Linear Chebyshev Approximation Problems," Mathematics, MDPI, vol. 8(12), pages 1-16, December.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2210-:d:461436
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    References listed on IDEAS

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    1. Castillo, Enrique & Minguez, Roberto & Castillo, Carmen & Cofino, Antonio S., 2008. "Dealing with the multiplicity of solutions of the l1 and l[infinity] regression models," European Journal of Operational Research, Elsevier, vol. 188(2), pages 460-484, July.
    2. Stefan R. Jaschke, 1997. "Arbitrage bounds for the term structure of interest rates," Finance and Stochastics, Springer, vol. 2(1), pages 29-40.
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    Cited by:

    1. Nikolai Krivulin, 2021. "Algebraic Solution of Tropical Polynomial Optimization Problems," Mathematics, MDPI, vol. 9(19), pages 1-18, October.
    2. Nikolai Krivulin, 2023. "Algebraic Solution of Tropical Best Approximation Problems," Mathematics, MDPI, vol. 11(18), pages 1-17, September.

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