Table 1 Information of grey system... | Emerald Publishing

Table 1

Information of grey system models used for comparison

Model name and reference	Model structure	Time response equation
GM (Liu and Forrest, 2010)	${\begin{array}{c} X^{(0)} (k) + a z^{(1)} (k) = b \\ z^{(1)} (k) = \frac{1}{2} [X^{(1)} (k) + X^{(1)} (k + 1)] \end{array}$	${\hat{X}}^{(1)} (k + 1) = (X^{(0)} (1) - \frac{b}{a}) e^{- a k} + \frac{b}{a}$
DGM (Xie et al., 2018)	$X^{(1)} (k + 1) = β_{1} X^{(1)} (k) + β_{2}$	${\hat{X}}^{(1)} (k + 1) = {β_{1}}^{k} (X^{(0)} (1) - \frac{β_{2}}{1 - β_{1}}) + \frac{β_{2}}{1 - β_{1}}$
NDGM (Xie et al., 2013)	$X^{(1)} (k + 1) = β_{1} X^{(1)} (k) + β_{2} k + β_{3}$	${\hat{X}}^{(1)} (k + 1) = {β_{1}}^{k} X^{(1)} (1) + β_{2} \sum_{j = 1}^{k} j {β_{1}}^{k - j} + \frac{1 - {β_{1}}^{k}}{1 - β_{1}} β_{3}$
FGM (Wu et al., 2013)	${\begin{array}{c} X^{(r)} (k) - X^{(r)} (k - 1) + β_{1} z^{(r)} (r) = β_{2} \\ z^{(r)} (k) = \frac{1}{2} [X^{(r)} (k) + X^{(r)} (k + 1)] \end{array}$	${\hat{X}}^{(r)} (k + 1) = (X^{(0)} (1) - \frac{b}{a}) e^{- a k} + \frac{b}{a}$
FDGM (Wu et al., 2014)	$X^{(r)} (k + 1) = β_{1} X^{(r)} (k) + β_{2}$	${\hat{X}}^{(r)} (k + 1) = {β_{1}}^{k} (X^{(0)} (1) - \frac{β_{2}}{1 - β_{1}}) + \frac{β_{2}}{1 - β_{1}}$
FNGM (Duan et al., 2018)	${\begin{array}{c} X^{(r)} (k) - X^{(r)} (k - 1) + β_{1} z^{(r)} (k) = β_{2} k \\ z^{(r)} (k) = \frac{1}{2} [X^{(r)} (k) + X^{(r)} (k + 1)] \end{array}$	${\hat{X}}^{(r)} (k + 1) = [X^{(0)} (1) - \frac{β_{2}}{β_{1}} + \frac{β_{2}}{{β_{1}}^{2}}] e^{- β_{1} k} + \frac{β_{2}}{β_{1}} (k + 1) + \frac{β_{2}}{{β_{1}}^{2}}$
FNDGM (Wu et al., 2014)	$X^{(r)} (k + 1) = β_{1} X^{(r)} (k) + β_{2} k + β_{3}$	${\hat{X}}^{(r)} (k + 1) = {β_{1}}^{k} X^{(0)} (1) + β_{2} \sum_{j = 1}^{k} j {β_{1}}^{k - j} + \frac{1 - {β_{1}}^{k}}{1 - β_{1}} β_{3}$
NIPGM (Wu et al., 2015)	${\begin{array}{c} X^{(1)} (k) - X^{(1)} (k - 1) + β_{1} z^{(1)} (k) = β_{2} \\ z^{(1)} (k) = \frac{1}{2} [X^{(1)} (k) + X^{(1)} (k + 1)] \end{array}$	${\hat{X}}^{(1)} (k + 1) = {β_{1}}^{k} (X^{(0)} (1) - \frac{β_{2}}{1 - β_{1}}) + \frac{β_{2}}{1 - β_{1}}$
NIPDGM (Zhou et al., 2017)	$X^{(1)} (k + 1) = β_{1} X^{(1)} (k) + β_{2}$	${\hat{X}}^{(1)} (k + 1) = {β_{1}}^{k} (X^{(0)} (1) - \frac{β_{2}}{1 - β_{1}}) + λ \frac{β_{2}}{1 - β_{1}}$
NIPNGM (Xia et al., 2020)	${\begin{array}{c} X^{(1)} (k) + β_{1} z^{(1)} (k) = β_{2} k \\ z^{(1)} (k) = \frac{1}{2} [X^{(1)} (k) + X^{(1)} (k + 1)] \end{array}$	${\hat{X}}^{(1)} (k + 1) = [X^{(0)} (1) - \frac{β_{2}}{β_{1}} + \frac{β_{2}}{{β_{1}}^{2}}] e^{- β_{1} k} + \frac{β_{2}}{β_{1}} (k + 1) + \frac{β_{2}}{{β_{1}}^{2}}$
NIPNDGM (Xie and Liu, 2008)	$X^{(1)} (k + 1) = β_{1} X^{(1)} (k) + β_{2} k + β_{3}$	${\hat{X}}^{(1)} (k + 1) = {β_{1}}^{k} X^{(1)} (1) + β_{2} \sum_{j = 1}^{k} j {β_{1}}^{k - j} + \frac{1 - {β_{1}}^{k}}{1 - β_{1}} β_{3}$
BernoulliGM (Chen et al., 2008)	${\begin{array}{c} X^{(1)} (k) - X^{(1)} (k - 1) + α z^{(1)} (k) = β_{1} {(z^{(1)} (k))}^{m} \\ z^{(1)} (k) = \frac{1}{2} [X^{(1)} (k) + X^{(1)} (k + 1)] \end{array}$	${\hat{X}}^{(1)} (k + 1) = [[({X^{(0)} (1)}^{(1 - m)} - \frac{β_{2}}{β_{1}}] e^{- β_{1} (1 - m) k} + \frac{β_{2}}{β_{1}}]$
FBernoulliGM (Wu et al., 2019)	${\begin{array}{c} X^{(r)} (k) - X^{(r)} (k - 1) + β_{1} z^{(r)} (k) = β_{2} {(z^{(r)} (k))}^{m} \\ z^{(r)} (k) = \frac{1}{2} [X^{(r)} (k) + X^{(r)} (k + 1)] \end{array}$	${\hat{X}}^{(r)} (k + 1) = {[({X^{(r)} (1)}^{(1 - m)} - \frac{β_{2}}{β_{1}}) e^{- β_{1} (1 - m) k} + \frac{β_{2}}{β_{1}}]}^{\frac{1}{1 - m}}$
NIPBernoulliGM (Xiang et al., 2020)	${\begin{array}{c} X^{(1)} (k) - X^{(1)} (k - 1) + η_{1} z^{(1)} (k) = η_{2} {(z^{(λ)} (k))}^{m} \\ z^{(1)} (k) = \frac{1}{2} [X^{(1)} (k) + X^{(1)} (k + 1)] \end{array}$	${\hat{X}}^{(1)} (k + 1) = {[({X^{(0)} (1)}^{1 - m} - \frac{β_{2}}{β_{1}}) e^{- β_{1} (1 - m) (k)} + \frac{β_{2}}{β_{1}}]}^{\frac{1}{1 - m}}$