A Systematic Study of the Best Fitting Line Problem
AbstractThe classical approach for tackling the problem of drawing the 'best fitting line' through a plot of experimental points (here called a scenario) is the least square process applied to the errors along the vertical axis. However, more elaborate processes exist or may be found. In this report, we present a comprehensive study on the subject. Five possible processes are identified: two of them (respectively called VE, HE) measure errors along one axis, and the remaining three (respectively called YE, PE, and AE) take into consideration errors along both axes. Since the axes and their corresponding errors may have different physical dimensions, a procedure is proposed to compensate for this difference so that all processes could express their answers in the same consistent dimensions. As usual, to avoid mutual cancellation, errors are squared or taken in their absolute value. The two cases are separately studied.In the case of squared errors, the five processes are tested in many scenarios of experimental points, both analytically (using the software Mathematica) and numerically (with programs written on Python platform employing the Nelder-Mead optimization method). The investigation showed the possible existence of multiple solutions. Different answers originating from different starting points in Nelder?Mead correspond to solutions revealed by analytic search with Mathematica. For each scenario of experimental points, it was found that the best lines of the five processes intercept at a common point. Furthermore, the point of intercept happens to coincide with the 'center of mass' of the scenario. This fact is described by stating the existence of an empirical 'Meeting Point Law'. The case of absolute errors is only treated numerically, with Nelder?Mead minimizer. As expected, the absolute error option shows greater robustness against outliers than the square error option, for all processes. The Meeting Point Law is not valid in this case.By taking the value of minimized error as a criterion, the five processes are compared for efficiency. On average, processes PE and AE, that consider errors along both axes, resulted in the smallest minimized error and may be considered the best processes. Processes that rely on errors along a single axis (VE, HE) stay at the second place. In all cases, YE is the process that results in the largest minimized errors
