Linear Fit: Best Choice for the Reference Point

  In the last blog the reference point for the line was assumed to be the average of the data points. This can be deduced by treating the components of the reference as parameters to be determined as part of the variation. In assigning r₀ for the reference point the zero subscript was chosen to indicate that it was the value for the function r(ρ) when ρ was set equal to zero. To avoid the possible confusion resulting from it being mistaken for a component of the vector r a better choice for it would be r_ref. We can start with the result that was found for the estimate of ρ and the reduced expression for δ.

The second line up from the bottom line gives a result involving r_ref and the average with some additional factors involving P-bar which is a function of e and the unit vectors for the axes. This expression has to be satisfied for all choices for the unit vectors and all possible P-bars since e is still a variable. This requires that the first factor in the expression be set equal to zero and as a result we get r_ref =<r>.

  This example illustrates how one can reduce the variation problem by solving for independent variables separately. This leaves the problem of the constraint and simultaneous variation of parameters till last but the expressions involved have been simplified. Each constraint adds a Lagrange multiplier.