On the minimization of correlated residuals

Abstract

SUMMARY A variety of methods for solving overdetermined linear systems of equations are considered. The Gaussian method of least squares assumes a noise-free matrix of coefficients. The maximum-likelihood method assumes the same amount of noise in each column of the matrix and the resulting vector. Both methods represent special cases of the general problem of deriving unbiased solutions in the presence of noise. Here, the minimization of correlated residuals is introduced. A set of boundary conditions is required to adjust the statistical properties of the solution. This method minimizes the second term of the autocovariance function of the residuals and is independent of the noise. The operation of the method is outlined analytically using an univariate scatter problem and applied numerically to synthetic data of a linear regression problem. To demonstrate the applicability of the method in practical situations, geomagnetic response functions are derived from observational data using the Z : H method.