A Polynomial Fitting Problem: The Orthogonal Distances Method

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.