1. The Eigen-Coordinates Method: Reduction of Non-linear Fitting Problems

This chapter provides essential information about the fitting procedure. Everybody should know the conventional linear least squares method (LLSM), which was proposed by K. F. Gauss in the 18th century and later modified by other mathematicians. This method is very efficient when dealing with linear fitting coefficients. On the contrary, nonlinear fitting represents a serious problem without a general solution at present. This statement is related to the fact that the search of the global minimum in the space of the fitting parameters is an unsolved problem. The introduced eigen-coordinates method is based on the following result. It can be shown that many differential equations for functions having initially nonlinear fitting parameters possess a new set of linear parameters. Therefore, in this case, one can calculate a basic linear relationship and reframe the solution of the fitting problem in the application of the LLSM again. In this chapter, the reader can see how to fit the combination of exponential, power-law, and other functions. Some exercises help to evaluate this innovation and use it in various research problems. Besides, the chapter explains the procedure of optimal linear smoothing (POLS) that helps to smooth data.