TY - JOUR
T1 - A Polynomial Fitting Problem
T2 - The Orthogonal Distances Method
AU - Cantera-Cantera, Luis Alberto
AU - Vargas-Jarillo, Cristóbal
AU - Palomino-Reséndiz, Sergio Isaí
AU - Lozano-Hernández, Yair
AU - Montelongo-Vázquez, Carlos Manuel
N1 - Publisher Copyright:
© 2022 by the authors.
PY - 2022/12
Y1 - 2022/12
N2 - 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.
AB - 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.
KW - least squares
KW - orthogonal distances
KW - parameter estimation
KW - polynomial fitting
KW - total least squares
UR - http://www.scopus.com/inward/record.url?scp=85143602326&partnerID=8YFLogxK
U2 - 10.3390/math10234596
DO - 10.3390/math10234596
M3 - Artículo
AN - SCOPUS:85143602326
SN - 2227-7390
VL - 10
JO - Mathematics
JF - Mathematics
IS - 23
M1 - 4596
ER -