Least Squares Fitting of Data to Pseudolinear Relationships

A general method is presented for applying the principle of least squares to determine the constants $C$⁠, $D$⁠, $E$⁠, and $k$ that optimize the fit of experimental data to a relationship between a dependent variable $y$ and an independent variable $x$ expressible either in the form $h(y) = Cf(x) + Dg(x) + E$ or in the form $h(y) = Cf(k; x) + Dg(k; x) + E$⁠, where $h$⁠, $f$⁠, and $g$ are arbitrary functions of the indicated arguments. Analysis of the first form is essentially identical with analysis of a bilinear form, and an analytic solution can be obtained; analysis of the second form involves an iterative process in which the value of $k$ is assumed, tried, and successively revised until the principle of least squares is satisfied to a specified accuracy. The resulting formulation makes the broad utility of the principle of least squares more apparent than is often the case, stresses particularly the importance of weighting to the correct application of the principle and can be easily specialized to cover a wide variety of commonly occurring relationships such as straight lines, quadratic polynomials, exponentials, power laws, Gaussian curves, the Lorentz line shape, and sinusoidal functions.