Least-squares fit of a linear combination of functions

Abstract

We propose that given a data-set $S=\{(x_i,y_i)/i=1,2,{\dots}n\}$ and real-valued functions $\{f_\alpha(x)/\alpha=1,2,{\dots}m\},$ the least-squares fit vector $A=\{a_\alpha\}$ for $y=\sum_\alpha a_{\alpha}f_\alpha(x)$ is $A = (F^TF)^{-1}F^TY$ where $[F_{i\alpha}]=[f_\alpha(x_i)].$ We test this formalism by deriving the algebraic expressions of the regression coefficients in $y = ax + b$ and in $y = ax^2 + bx + c.$ As a practical application, we successfully arrive at the coefficients in the semi-empirical mass formula of nuclear physics. The formalism is {\it generic} - it has the potential of being applicable to any {\it type} of $\{x_i\}$ as long as there exist appropriate $\{f_\alpha\}.$ The method can be exploited with a CAS or an object-oriented language and is excellently suitable for parallel-processing.