Non-negative least-squares variance and covariance component estimation using the positive-valued function for errors-in-variables models

Although (co)variance component estimation has been widely applied in the errors-in-variables (EIV) model, the occurrence of negative variance components is still a major issue in the estimated variance components. This problem may be due to the following unfavorable factors: 1) unreasonable selection of initial variance values; 2) low redundancy in the EIV functional model; 3) improper design in the EIV stochastic model, and 4) other data quality problems. Many attempts have been made to prevent the appearance of negative variance components. In this contribution, a novel and efficient non-negative least-squares variance component estimation using the PVF (PVF-NLS-VCE) is introduced, which can simultaneously estimate the unknown (co)variance components in the EIV stochastic model and the parameters in the EIV functional model. Its principle is to implicitly impose a non-negative constraint by replacing the variance component with the positive-valued function (PVF) whose range is the set of positive real numbers. Two numerical examples using real and simulated data are presented. The numerical results of linear regression are identical to those obtained based on least-squares variance component estimation (LS-VCE) with positive initial values of variance components. The numerical results of two-dimensional affine transformation are shown to prevent negative variance components and precede those obtained by LS-VCE with a negative initial value of variance component. Both numerical examples verify the effectiveness of the PVF-NLS-VCE method whether the initial values of variance components are positive or negative. The proposed PVF-NLS-VCE is a simple, convenient and flexible method to achieve the non-negative estimates of variance components, which can reduce sensitivity to initial value selection and automatically guarantee a non-negative definite covariance matrix.