On The Effects of Equal Weights on the Derived Parameters From Least Squares Solution

Abstract

Equal weighting is a general strategy in the least squares solutions to reflect the equal contribution of observations that were obtained, for example, by identical measurement systems or similar measurement procedures or algorithms. This type of weighting can be imposed either implicitly or explicitly. Implicit weighting takes the form of an identity weight matrix while explicit weighting is imposed by a weight matrix of equal and known variance value of the observations. Through theoretical and numerical demonstrations, this paper shows that equal weights do not affect the estimated parameters and the residuals in the least squares solution. Moreover, for a relatively large set of observations, the estimated variance component converges to the variance of the original observations in the case of the implicit weighting; and it converges to a value that is very close to one in the case of explicit weighting. In addition, the posterior variance-covariance or dispersion matrices in the implicit and explicit cases are very close to each other after the adjustment. In this study, Monte Carlo simulation was used to generate numerical values of random noise from a normal distribution. This random noise was added to the coordinates of a straight-line for practical evaluation of the proposed arguments.