Least-squares estimation for linear models with certain ranges

As a signal processing method, the least-squares method plays a crucial role in parameter estimation, and great progress has been made in recent decades. However, errors may occur when the parameters to be estimated have some actual physical meaning, e.g., if the human-body temperature is estimated to be 70 °C by a general least-squares method. In this study, we consider solving a particular problem, named ranged least-squares estimation (RLSE), where the parameters are restricted to certain meaningful ranges. By using a theoretical analysis, we prove that the solution of the RLSE problem is unique and can be obtained in finite number of steps when the system matrix has a full column rank. Two programmable algorithms are proposed: a basic algorithm and another with improved efficiency. We also present a numerical experiment of an actual RLSE problem for hydrological parameter estimation, which validates the proposed method.