Transverse Least Squares Gives Comparable Results To Those of Ordinary Least Squares

  I did some data analysis to compare transverse least square with ordinary least squares and got comparable results. But we start by taking a closer look at some formulas for the transverse least square for the z-values ξ and η.

It turns out that the expected values ⟨ξ⟩ and ⟨η⟩ both equal 0, ⟨ξ²⟩ and ⟨η²⟩ both equal 1 and the slope, σ,  of the fitted line in the ξ,η-plane is ±1. In the x,y-plane the following formulas are useful.

The data analysis did fits of 25 lines with 20 data points per lines having small random normal errors in the values of the coordinates. Each of the two fit methods gave comparable results with the uncertainties in the mean values of the slope, s, and y-intercept, y₀, indicated by the standard deviation, sd, greater than the deviations from the original values as these two samples show.

In the first example ordinary least squares gave the better fit while the opposite was true in the second example. The two methods give independent fits of the lines.

The positions of the data points along the lines ranged from approximately 0 to 2 units from the y-intercept of the original line.