GM(1,1;λ) with Constrained Linear Least Squares

The only parameters of the original GM(1,1) that are generally estimated by the ordinary least squares method are the development coefficient a and the grey input b. However, the weight of the background value, denoted as lambda, cannot be obtained simultaneously by such a method. This study, therefore, proposes two simple transformation formulations such that the unknown parameters a, b and lambda can be simultaneously estimated by the least squares method. Therefore, such a grey model is termed the GM(1,1;lambda). On the other hand, because the permission zone of the development coefficient is bounded, the parameter estimation of the GM(1,1) could be regarded as a bound-constrained least squares problem. Since constrained linear least squares problems generally can be solved by an iterative approach, this study applies the Matlab function lsqlin to solve such constrained problems. Numerical results show that the proposed GM(1,1;lambda) performs better than the GM(1,1) in terms of its model fitting accuracy and its forecasting precision.