The effectiveness of the first entry of the original series by GM(1, 1) is researched in this paper. The results show that the modelling values and forecasts are independent of the first entry of the original series. The grey prediction model presented in this paper is called first-entry GM(1, 1), abbreviated as FGM(1, 1), which is based on the existing GM(1, 1) but modelled with data including the first-entry's messages of the original series. A proof concerning this subject has been presented by other authors. However, the algorithm of their direct proof is too complicated. A more compact algorithm is presented in this paper to prove the first entry of the original series ineffective to the modelling values and forecasts by GM(1, 1). Then, an arbitrary number can be inserted in the front of the original series to extract the messages from its first entry. Only a few data (usually fewer than ten) are used for model building. This paper deals with the effectiveness of the first entry of the original series by GM(1, 1). © 2009 Elsevier Ltd. All rights reserved.
Tien, T. L. (2009). A new grey prediction model FGM(1, 1). Mathematical and Computer Modelling, 49(7–8), 1416–1426. https://doi.org/10.1016/j.mcm.2008.11.015