Abstract
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 λ, cannot be obtained simultaneously by such a method. This study, therefore, proposes two simple transformation formulations such that the unknown parameters a, b and λ can be simultaneously estimated by the least squares method. Therefore, such a grey model is termed the GM(1,1;λ). 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;λ) performs better than the GM(1,1) in terms of its model fitting accuracy and its forecasting precision.
Highlights
Uncertain systems with small samples and poor information exist commonly in the real world
The grey model is the kernel of grey prediction, where the latter is one of the most important and widely used fields in the grey system theory
With the proposed simple transformation, the development coefficient a, the grey input b, and the weight of background value λ can be simultaneously obtained from the ordinary least squares method (19) and the transformation formulations given in (20) and (21)
Summary
Uncertain systems with small samples and poor information exist commonly in the real world. The response of the classic GM(1,1) is essentially an exponential model with two internal parameters: the development coefficient a and the grey input b Speaking, these internal parameters are estimated by the least squares method with the background values being part of the observed data. The internal parameters of a classic GM(1,1), i.e., the development coefficient a and the grey input b, are generally estimated by the ordinary least squares method. Since both a and b are bounded [16], the parameter estimation could be regarded as a bound-constrained least squares problem. It should be noted that the GM(1,1) with a constant parameter λ, usually specified to 0.5, is the so-called classic GM(1,1), while the GM(1,1) with the adjustable parameter λ is denoted as GM(1,1;λ) for differentiation
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