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A Polynomial Fitting Problem: The Orthogonal Distances Method

Author

Listed:
  • Luis Alberto Cantera-Cantera

    (Automation and Control Engineering Department, ESIME-IPN, Unidad Profesional Adolfo López Mateos, Zacatenco, Av. Luis Enrique Erro s/n, Mexico City C.P. 07738, Mexico)

  • Cristóbal Vargas-Jarillo

    (Automatic Control Department, CINVESTAV-IPN, Av. Instituto Politecnico Nacional 2508, Col. San Pedro Zacatenco, Mexico City C.P. 07360, Mexico)

  • Sergio Isaí Palomino-Reséndiz

    (Automation and Control Engineering Department, ESIME-IPN, Unidad Profesional Adolfo López Mateos, Zacatenco, Av. Luis Enrique Erro s/n, Mexico City C.P. 07738, Mexico)

  • Yair Lozano-Hernández

    (Unidad Profesional Interdisciplinaria de Ingeniería Campus Hidalgo, Instituto Politécnico Nacional, San Agustín Tlaxiaca C.P. 42162, Mexico)

  • Carlos Manuel Montelongo-Vázquez

    (Automation and Control Engineering Department, ESIME-IPN, Unidad Profesional Adolfo López Mateos, Zacatenco, Av. Luis Enrique Erro s/n, Mexico City C.P. 07738, Mexico)

Abstract

The classical curve-fitting problem to relate two variables, x and y , deals with polynomials. Generally, this problem is solved by the least squares method (LS), where the minimization function considers the vertical errors from the data points to the fitting curve. Another curve-fitting method is total least squares (TLS), which takes into account errors in both x and y variables. A further method is the orthogonal distances method (OD), which minimizes the sum of the squares of orthogonal distances from the data points to the fitting curve. In this work, we develop the OD method for the polynomial fitting of degree n and compare the TLS and OD methods. The results show that TLS and OD methods are not equivalent in general; however, both methods get the same estimates when a polynomial of degree 1 without an independent coefficient is considered. As examples, we consider the calibration curve-fitting problem of a R-type thermocouple by polynomials of degrees 1 to 4, with and without an independent coefficient, using the LS, TLS and OD methods.

Suggested Citation

  • Luis Alberto Cantera-Cantera & Cristóbal Vargas-Jarillo & Sergio Isaí Palomino-Reséndiz & Yair Lozano-Hernández & Carlos Manuel Montelongo-Vázquez, 2022. "A Polynomial Fitting Problem: The Orthogonal Distances Method," Mathematics, MDPI, vol. 10(23), pages 1-17, December.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:23:p:4596-:d:993009
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    References listed on IDEAS

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    2. Carl Eckart & Gale Young, 1936. "The approximation of one matrix by another of lower rank," Psychometrika, Springer;The Psychometric Society, vol. 1(3), pages 211-218, September.
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