# An unified framework for fractional grey system models: Memory effects
perspective

Wanli Xie  Institute of EduInfo Science and Engineering, Nanjing Normal
University, Nanjing 210097, China  Wen-Ze Wu  School of Economics and Business
Administration, Central China Normal University, Wuhan 430079, China
<EMAIL_ADDRESS>Chong Liu  School of Science, Northeastern
University, Shenyang 110819, China<EMAIL_ADDRESS>
###### Abstract

As an essential characteristics of fractional calculus, the memory effect is
served as one of key factors to deal with diverse practical issues, thus has
been received extensive attention since it was born. By combining the
fractional derivative with memory effects and grey modeling theory, this paper
aims to construct an unified framework for the commonly-used fractional grey
models already in place. In particular, by taking different kernel and
normalization functions, this framework can deduce some other new fractional
grey models. To further improve the prediction performance, the four popular
intelligent algorithms are employed to determine the emerging coefficients for
the UFGM(1,1) model. Two published cases are then utilized to verify the
validity of the UFGM(1,1) model and explore the effects of fractional
accumulation order and initial value on the prediction accuracy, respectively.
Finally, this model is also applied to dealing with two real examples so as to
further explain its efficacy and equally show how to use the unified framework
in practical applications.

###### keywords:

Fractional derivative, Memory effect, Grey modeling technique, Unified
framework , Intelligent algorithm

##  1 Introduction

Grey system theory, an effective approach to addressing the issues in
uncertain systems, was proposed by Deng [1]. The grey system theory can fully
dig out the internal rule of systems with limited data, this enables it has
very important application prospects and successfully solved many practical
problems. Grey prediction model is the most essential model in the grey system
theory, and it has become a research hotspot of scholars in recent years. To
be specific, GM(1,1) [2], DGM(1,1) [3] are two kinds of commonly-used grey
models, there are many published papers focusing on them [4, 5, 6, 7, 8]. At
the same time, the studies related to grey system forecasting models have been
still emerged, and the prediction performance of grey forecasting models will
be continuously enhanced as well [9, 10]. In order to further improve the
accuracy and application range of the grey models already in place, many
scholars have proposed several new types of grey prediction models. For
example, Lang et al. [11] proposed a new time-lag grey model to forecast
photovoltaic power generation in the Asia-Pacific region. Zhou et al. [12] put
forward a new discrete grey model for natural gas prediction in Jiangsu
Province, China. Wu et al. [13] constructed a new nonlinear grey model for
short-term natural gas prediction. Zhou et al. [14] established a novel
discrete grey seasonal model and used several practical examples to verify the
effectiveness of their model. Wang and Jv [15] combined the grey prediction
model and quantile regression so as to design a combination for forecasting
time-series sequence. Xiao et al. [16] proposed an optimized nonlinear grey
model for biomass energy consumption prediction. Li et al. [17] searched the
optimal fractional accumulation order for improved seasonal grey model by
virtue of the particle swarm optimization (PSO). Zeng et al. [18] optimized
the structure of the grey Verhulst model. Wei and Xie[19] proposed a method
family for nonlinear parameter estimation of grey forecasting models, their
numerical results show that the proposed method has advantages of high
accuracy and robustness. Zeng [20] proposed a novel discrete grey Riccati
model and verified its effectiveness in a range of practical application
scenarios. Xiao et al. [21] put forward a grey Riccati Bernoulli model and
successfully applied it to deal with clean energy consumption. Ye et al. [7]
proposed a novel time-delay multivariate grey model And used in China’s carbon
dioxide analysis.

Admittedly, the above models achieve satisfactory results and address many
practical problems, however, the accumulation order in these models are all
integer-order, which is prone to impair the prediction performance. In other
words, the accumulation order should dynamically alter in accordance with
modeing situations. To this end, Wu et al. [10] proposed the fractional gray
model in 2013. Subsequently, various extensive forms of fractional models were
proposed [22, 23]. At the same time, several other types of fractional
accumulation operators have also been proposed, each with its own advantages.
Ma et al. [24] proposed a conformable fractional accumulation (denoted as CFA)
and built a conformable fractional grey prediction model. Chen et al. [25]
incorporated Fractional Hausdorff derivative into the modeling procedure of
grey model so as to construct a new method. Liu et al. [26] proposed the
damping accumulated generating operator and constructed a new fractional grey
model. With the continuous development of fractional grey models, fractional
derivatives are applied on the differential equation of grey models [27.28] .

For example, Mao et al. [29] proposed a fractional grey model based on non-
singular exponential kernel.

$\frac{1}{{1-r}}\int_{0}^{t}{{x^{\prime}}}(\tau)\exp\left[{-\frac{{r(t-\tau)}}{{1-r}}}\right]d\tau+a{x^{(r)}}(t)=b;{\text{
}}r>0,t>0.$ (1)

where ${x^{(r)}}(t)$ is the $r$-order accumulation [10] of the original
sequence ${x^{({\text{0}})}}(t)$,and literature [28] proposed a fractional
prediction model with conformable derivatives. The research status of
fractional grey model can be seen in Table 1. Using different fractional
derivatives in the gray model can make full use of different types of
derivatives to dig out potential laws among data. Fractional calculus has
become a research hotspot of scholars [29]. The famous Riemann-Liouville
fractional integral with $r\in\mathbf{R}$ order [30] by

$I_{{t_{0}},r}^{RL}f(t)=\frac{1}{{\Gamma(r)}}\int_{t_{0}}^{t}{{{(t-\tau)}^{r-1}}}f(\tau)d\tau={\Phi_{r}}*f(t);{\text{
}}r>0,x>{t_{0}},$ (2)

motivated by the Cauchy integral

$\int_{t_{0}}^{t}d{\tau_{1}}\int_{t_{0}}^{{\tau_{1}}}d{\tau_{2}}\cdots\int_{t_{0}}^{{\tau_{n-1}}}f\left({{\tau_{n}}}\right)d{\tau_{n}}=\frac{1}{{\Gamma(n)}}\int_{t_{0}}^{t}{{{(t-\tau)}^{n-1}}}f(\tau)d\tau.$
(3)

Where * is the convolution operator and ${\Phi_{r}}$ is the power function
${\Phi_{r}}(\tau)=\frac{1}{{\Gamma(r)}}\tau_{+}^{r-1}$. Zhao and Luo [30]
proposed a generalized fractional operator, which proved that several types of
calculus expressions are special cases of this operator.

Table 1: Research findings of the different fractional grey models in recent years. Author’s name(year) | Approach | Abbreviation | FAGO |  FD
---|---|---|---|---
Wu (2013)[10] |  Grey model with the fractional
order accumulation | FGN(1,1) | ✓ | -
Mao (2016)[28] |  Fractional grey system model | FGM(q,1) | ✓ | ✓
Yang (2016)[11] |  Continuous fractional-order
grey model | GM(q,1)/GM(q,N) | - | ✓
Yang (2018)[31] | Interval grey modelling based on fractional calculus | Interval GM(1,1) |  | -
Wu (2018)[32] |  The GMC(1,n) model with
fractional order accumulation | FGMC(1,n) | ✓ | -
Wu (2018)[33] |  Fractional order accumulation
grey model | FAGMO(1,1,k) | ✓ | -
Ma (2019)[34] |  Fractional discrete multivariate
grey model | FDGM(1,n) | ✓ | -
Chen (2019)[35] |  Time-delayed polynomial
fractional order grey model | TDPFOGM(1,1) | ✓ | -
Ma (2019)[24] | Conformable fractional grey model | CFGM(1,1) | ✓ | -
Wang (2019)[36] |  Fractional calculus function of
grey prediction model | FGM(1,1) | ✓ | -
Ma (2019)[37] |  Fractional time delayed grey
model | FTDGM(1,1) | ✓ | -
Zhu (2019)[38] |  Self-adaptive fractional
weighted grey model | SFOGM(1,1) | - | -
Yan (2020)[25] |  Fractional Hausdorff grey model | FHGM(1,1) | ✓ | -
Xie (2020)[39] |  Continuous grey model with conformable fractional derivative | CCFGM(1,1) | ✓ | ✓
Wu (2020)[13] |  conformable fractional non-homogeneous grey model | CFNGM(1,1) | ✓ | -
Kang(2020)[40] |  Fractional derivative multivariable grey model | CFGMC(q,N) | ✓ | ✓
Gao(2020) [41] |  fractional grey Riccati model | FGRM(1,1) | ✓ | -
Mao (2020)[29] |  Fractional grey model based on non-singular exponential kernel | EFGM(q,1) | ✓ | ✓
Liu (2021)[26] |  The damping accumulated grey model | DAGM | ✓ | -
Liu (2021)[42] |  Optimized fractional grey model-based variable background value | OFAGM(1,1) | ✓ | -
Zhang (2021)[43] |  Fractal derivative fractional grey Riccati model | FDFGRM | ✓ | ✓

Based on the above acknowledge, it is concluded that the fractional grey model
has been widely used in diverse fields and a large number of fractional grey
models has been constantly emerged. However, in return, there exist a variety
of fractional grey models with various constructions of fractional accumulated
generating operator, which is not conducive for beginners to understand and
grasp the relevant modeling mechanism; at the same time, it is difficult for
participants to identify which model can provide satisfactory results in
specific scenarios, thus restraining the further methodological developments
and practical applications.

To this end, this paper develops an unified framework for commonly-used
fractional grey models by combining general fractional derivative with memory
effects and grey modeling theory. The main contributions can be outlined as
follows. (i) In accordance with the GC and GRL derivatives, the unified
framework for diverse fractional grey model is developed, this framework also
derive other new fractional grey models by taking different kernel functions
and normalization functions. (ii) The exact solution (often called time
response function) and the corresponding estimates of the structural
parameters are, in detail, deduced from this proposed framework. (iii) The
four popular intelligent algorithms are employed to determine the emerging
coefficients $r$, $\alpha$. (iv) Two published cases are utilized to verify
the effectiveness of the UFGM(1,1) model and especially explore the effects of
fractional accumulation order and initial value on the prediction performance.
(v) The UFGM(1,1) model is also applied in two real cases to explain how to
apply the proposed approach.

The rest of paper is organized as follows: Section 2 briefly introduces
general fractional derivatives with memory effects. Section 3 discusses the
modeling procedure of the unified framework. Section 4 verifies the validity
of the UFGM(1,1) model. Section 5 applies this method in dealing with Henan’s
and Chongqing’s water supply production capacity and Section 6 concludes.

##  2 General fractional derivatives with memory effects

In this section, we will give a brief review of general fractional derivatives
with memory effects and, in detail, deduce two important properties for these
two kinds of general fractional derivative, which is helpful in constructing
the unified framework for general fractional grey models in the following
section.

######  Definition 1 (See [30])

Given order $r\in(0,1),t>{t_{0}}$, the general fractional derivative with
memory effects in the Caputo sense is presented as

$D_{{t_{0}},r}^{GC}f(t)=N(r)\int_{t_{0}}^{t}k(t-\tau,r)\frac{{df(\tau)}}{{d\tau}}d\tau,$
(4)

where $G$ and $C$ represent the general definition and Caputo sense,
respectively. $k(t-\tau,r)$ and $N(r)$ denote kernel function and
normalization one, respectively.

Eq.(4) can be rewritten in the form of convolution as follows,

$D_{{t_{0}},r}^{GC}f(t)=N(r)k(\tau,r)*\frac{{df(t)}}{{dt}}$ (5)

By taking different kernel functions and normalization ones, Eq.(5) can be
transferred into diverse fractional derivatives, for example

$\displaystyle D_{{t_{0}},r}^{{{Caputo}}}f(t)$ $\displaystyle=$
$\displaystyle\frac{1}{{\Gamma(1-r)}}\int_{t_{0}}^{t}{{{(t-\tau)}^{-r}}}\frac{{df(\tau)}}{{d\tau}}d\tau$
(6) $\displaystyle=$
$\displaystyle\frac{1}{{\Gamma(1-r)}}{{\text{t}}^{-r}}{\text{*}}\frac{{df(t)}}{{dt}}$

Eq.(9) is the basic form of the Caputo fractional derivative.

######  Definition 2 (See [30])

Given $r\in(0,1),t>{t_{0}}$, the general fractional derivative with memory
effects in the Riemann$-$Liouville sense is represented as

$D_{{t_{0}},r}^{GRL}f(t)=\frac{d}{{dt}}N(r)\int_{t_{0}}^{t}k(t-\tau,r)f(\tau)d\tau,$
(7)

where $RL$ denotes the Riemann-Liouville sense, $k(t-\tau,r)$ and $N(r)$ are
the same as in Definition 1.

Similar to Eq.(5), Eq.(7) can be expressed in the form of convolution as
follows,

$D_{{t_{0}},r}^{GRL}f(t)=\frac{d}{{dt}}N(r)k(\tau,r)*f(t)$ (8)

Meanwhile, Eq.(8) can be reduced to other fractional derivatives by taking
different kenel functions and normalization ones, for example, the Riemann-
Liouville fractional derivative is calculated as

$\displaystyle D_{{t_{0}},r}^{{{RL}}}f(t)$ $\displaystyle=$
$\displaystyle\frac{{df(t)}}{{dt}}\frac{1}{{\Gamma(1-r)}}\int_{t_{0}}^{t}{{{(t-\tau)}^{-r}}}f(\tau)d\tau$
(9) $\displaystyle=$
$\displaystyle\frac{{df(t)}}{{dt}}\frac{1}{{\Gamma(1-r)}}{t^{-r*}}f(t)$

Each fractional derivative has its own advantages and limitations, in
addition, other kinds of the fractional derivative with diverse kernel
functions and normalization ones are also shown in
Ref.[zhao2019representations] and references therein.

##  3 Unified representation of fractional grey model

In this section, we present a unified framework for the general fractional
grey models based on the above findings. In addition to analyze the
characteristics of this framework, another focus is to deduce its time
response function and structural parameters.

Assume $X^{(0)}=\left\\{x^{(0)}(1),x^{(0)}(2),\cdots,x^{(0)}(n)\right\\},n\geq
4$ to be a nonnegative sequence, and the $\alpha$-order accumulated generating
operation ($\alpha-FAGO$) sequence is given as
$X^{(\alpha)}=\left\\{x^{(\alpha)}(1),x^{(\alpha)}(2),\cdots,x^{(\alpha)}(n)\right\\}$.
then consider the general fractional grey model in the Caputo and Riemann-
Liouville senses, namely UFGM, one has

$N(r)\int_{t_{0}}^{t}k(t-\tau,r)\frac{{d{x^{(\alpha)}}(\tau)}}{{d\tau}}d\tau+a{x^{(\alpha)}}(t)=\psi(t)+c,$
(10)

and

$\frac{d}{{dt}}N(r)\int_{t_{0}}^{t}k(t-\tau,r){x^{(\alpha)}}(\tau)d\tau+a{x^{(\alpha)}}(t)=\psi(t)+c$
(11)

respectively, where $\psi(t)$ is the unknown function with respect to $t$ and
the discrete form of ${x^{(\alpha)}}(t)$ gives

${x^{(\alpha)}}(k)=\sum\limits_{i=1}^{k}\binom{k-i+\alpha-1}{k-i}{x^{(0)}}(i),k=1,2,\ldots,n,$
(12)

where
$\binom{k-i+\alpha-1}{k-i}=\frac{(\alpha+k-i-1)(\alpha+k-i-2)\cdots(\alpha+1)\alpha}{(k-i)!}$.
The fractional accumulation plays an important role in the construction of
grey models. For more details on the accuracy analysis of fractional
accumulation, see [10] for a review.

For simplicity, $\psi(t)$ is taken as $\psi(t)=bt+c$ and, Eqs.(10)-(11) become

$N(r)\int_{t_{0}}^{t}k(t-\tau,r)\frac{{dx^{(\alpha)}(\tau)}}{{d\tau}}d\tau+a{x^{(\alpha)}}(t)=bt+c,$
(13)

and

$\frac{d}{{dt}}N(r)\int_{t_{0}}^{t}k(t-\tau,r)x^{(\alpha)}(\tau)d\tau+a{x^{(\alpha)}}(t)=bt+c.$
(14)

Next, using the trapezoid formula gives the discrete-time equation of Eq.(13)

$\nabla
D_{{t_{0}},r}^{GC}{x^{(\alpha)}}({\text{k}})+\frac{a}{2}\left\\{{{x^{(\alpha)}}(k)+{x^{(\alpha)}}(k-1)}\right\\}=bk+c,k=1,2,3,...n$
(15)

For the structural parameter estimation, set
$\bm{\varphi}={\left({a,b,c}\right)^{\text{T}}}$ and
$f\left(x^{(\alpha)}(k),\bm{\varphi}\right)=-\frac{a}{2}\left(x^{(\alpha)}(k)+x^{(\alpha)}(k-1)\right)+bk+c$,
one has

$\displaystyle\bm{\varphi}={\left({a,b,c}\right)^{\text{T}}}=\left(\bm{A}^{\text{T}}\bm{A}\right)^{-1}\bm{A}^{\text{T}}\bm{Y}$
(16)

where

$A=\left[{\begin{array}[]{*{20}{c}}{\frac{{\partial
f\left({{x^{(\alpha)}}(2),\varphi}\right)}}{{\partial a}}}&{\frac{{\partial
f\left({{x^{(\alpha)}}(2),\varphi}\right)}}{{\partial b}}}&{\frac{{\partial
f\left({{x^{(\alpha)}}(2),\varphi}\right)}}{{\partial c}}}\\\ {\frac{{\partial
f\left({{x^{(\alpha)}}(3),\varphi}\right)}}{{\partial a}})}&{\frac{{\partial
f\left({{x^{(\alpha)}}(3),\varphi}\right)}}{{\partial b}}}&{\frac{{\partial
f\left({{x^{(\alpha)}}(3),\varphi}\right)}}{{\partial c}}}\\\
\vdots&\vdots&\vdots\\\ {\frac{{\partial
f\left({{x^{(\alpha)}}(n),\varphi}\right)}}{{\partial a}}}&{\frac{{\partial
f\left({{x^{(\alpha)}}(n),\varphi}\right)}}{{\partial b}}}&{\frac{{\partial
f\left({{x^{(\alpha)}}(n),\varphi}\right)}}{{\partial
c}}}\end{array}}\right],Y=\left[{\begin{array}[]{*{20}{c}}{\nabla
D_{{t_{0}},r}^{GC}{x^{(\alpha)}}(2)}\\\ {\nabla
D_{{t_{0}},r}^{GC}{x^{(\alpha)}}(3)}\\\ \vdots\\\ {\nabla
D_{{t_{0}},r}^{GC}{x^{(\alpha)}}(n)}\end{array}}\right],$ (17)

and $\nabla D_{{t_{0}},r}^{GC}{x^{(\alpha)}}(k)$ represents the discrete form
of $D_{{t_{0}},r}^{GC}{x^{(\alpha)}}(k)$. It is worth noting that the
structural parameter estimation of Eq.(14) has been omitted because of their
high similarity.

###### Theorem 1

Set ${\text{
}}{\mathcal{L}^{-1}}\left\\{{\frac{1}{{\left[{N(r)K(s,r)s+a}\right]}}}\right\\}=\Theta(r,t,a)$
and $j$ indicates the discrete variable corresponding to $t$, the exact
solution of Eq.(13) is expressed as

${{\hat{x}}^{(\alpha)}}(j)=\Theta(r,j,a)*jb+\Theta(r,j,a)*c+{{\hat{x}}^{(\alpha)}}(0)N(r)k(j,r)*\Theta(r,j,a),$
(18)

where

${{\hat{x}}^{(\alpha)}}(0)=\frac{{{x^{(\alpha)}}(1)-\Theta(r,{\text{1}},a)*b+\Theta(r,{\text{1}},a)*c}}{{N(r)k({\text{1}},r)*\Theta(r,{\text{1}},a)}},$
(19)

${\mathcal{L}^{-1}}\left\\{.\right\\}$ denotes the inverse Laplace transform.
${{\hat{x}}^{(r)}}(j)$ is the output value by the prediction model.

Proof. By applying the Laplace transform of general fractional derivative
mentioned in [30], one has

$N(r)\left\\{sX(s)-{x^{(\alpha)}}(0)\right\\}K(s,r){\text{ +
a}}X(s)=\frac{b}{{{s^{2}}}}+\frac{c}{s},$ (20)

Let $\frac{1}{{\left\\{{N(r)K(s,r)s{\text{ + }}a}\right\\}}}=\Xi(r,s,a)$,
Eq.(20) becomes

$X(s)=\frac{b}{{{s^{2}}}}\Xi(r,s,a)+\frac{c}{s}\Xi(r,s,a)+{x^{(\alpha)}}(0)K(s,r)N(r)\Xi(r,s,a),$
(21)

Subsequently, through the inverse Laplace transform, Eq.(21) yields

$\displaystyle{x^{(\alpha)}}(t)$ $\displaystyle=$
$\displaystyle{\mathcal{L}^{-1}}\\{\Xi(r,s,a)\\}*tb+{\mathcal{L}^{-1}}{\\{\Xi(r,s,a)\\}^{*}}c$
(22) $\displaystyle+$
$\displaystyle{x^{(\alpha)}}(0)N(r){\mathcal{L}^{-1}}{\\{K(s,r)\\}^{*}}{\mathcal{L}^{-1}}\\{\Xi(r,s,a)\\}$

where $\mathcal{L}\left\\{{{{\hat{x}}^{(\alpha)}}(t)}\right\\}=X(s)$. Then
substituting Eq.(22) into Eq.(20), one can write

${{\hat{x}}^{(\alpha)}}(t)=\Theta(r,t,a)*tb+\Theta(r,t,a)*c{\text{ +
}}{{\hat{x}}^{(\alpha)}}(0)N(r)k(t,r)*\Theta(r,t,a).$ (23)

Set ${{\hat{x}}^{(\alpha)}}(1)={x^{(\alpha)}}(1)$ and $j=t$, this completes
the proof. $\Box$

###### Theorem 2

The exact solution of Eq.(14) is expressed as

${\hat{x}^{(\alpha)}}(t)=\Theta(r,t,a)*tb+\Theta{(r,t,a)^{*}}c+{x^{(\alpha)}}(0)N(r)k{(x,r)^{*}}\Theta(r,t,a)$
(24)

where
$\widehat{x}^{(\alpha)}(0)=\frac{x^{(\alpha)}(1)-\Theta(r,1,a)*b+\Theta(r,1,a)^{*}c}{N(r)k(x,r)^{*}\Theta(r,1,a)}$
and
$\mathcal{L}^{-1}\left\\{\frac{1}{N(r)K(s,r)s+x^{(\alpha)}(0)+a}\right\\}=\Theta(r,t,a)$.

Proof. By applying the link between the GC and GRL derivatives in [30]

$D_{r,{t_{0}}}^{GC}f(t)=D_{r,{t_{0}}}^{GRL}x^{(\alpha)}(t)-N(r)x^{(\alpha)}(0)k(x,r)$
(25)

Thus, Eq.(14) is rearranged as

$D_{r,{t_{0}}}^{GC}x^{(\alpha)}(t)+N(r)x^{(\alpha)}(0)k(x,r)+ax^{(\alpha)}(t)=bt+c$
(26)

By applying the laplace transform on both sides of Eq. (26), we obtain

$N(r)\left\\{sX(s)-x^{(\alpha)}(0)\right\\}K(s,r)+x^{(\alpha)}(0)X(s)+aX(s)=\frac{b}{s^{2}}+\frac{c}{s}$
(27)

Set
$\frac{1}{{\left\\{{N(r)K(s,r)s+{x^{(\alpha)}}(0)+a}\right\\}}}=\mathcal{W}(r,s,a)$,
it follows that

$X(s)=\mathcal{W}(r,s,a)\frac{b}{{{s^{2}}}}+\mathcal{W}(r,s,a)\frac{c}{s}+{x^{(\alpha)}}(0)N(r)K(s,r)\mathcal{W}(r,s,a)$
(28)

and

$\displaystyle\widehat{x}^{(\alpha)}(t)$ $\displaystyle=$
$\displaystyle\mathcal{L}^{-1}\left\\{\mathcal{W}(r,s,a)\right\\}*tb+\mathcal{L}^{-1}\left\\{\mathcal{W}(r,s,a)\right\\}*c$
(29) $\displaystyle+$ $\displaystyle
x^{(r)}(0)N(r)\mathcal{L}^{-1}\left\\{K(s,r)\right\\}^{*}\mathcal{L}^{-1}\left\\{\mathcal{W}(r,s,a)\right\\}$

where $\mathcal{L}^{-1}\left\\{\mathcal{W}(r,s,a)\right\\}=\Theta(r,t,a)$.
Finally, set ${\hat{x}^{(\alpha)}}(1)={x^{(\alpha)}}(1)$ and $s=t$, the exact
solution (often called time response function) of Eq.(24) can be easily
obtained. $\Box$

###### Remark 1

Set $k(t-\tau,r)=e^{-\frac{r}{1-r}x}$ and $N(r)=1$ [29], a novel fractional
grey model is presented as

$_{0}^{\exp}D_{t}^{r}{x^{(\alpha)}}(t)+a{x^{(\alpha)}}(t)=bt+c.$ (30)

By applying the laplace transformation on Eq.(30), which yields that

$\mathcal{L}\left\\{{{}_{0}^{\exp}D_{t}^{r}{x^{(\alpha)}}}\right\\}+\mathcal{L}\left\\{{a{x^{(\alpha)}}(t)}\right\\}=\mathcal{L}\left\\{{bt}\right\\}+\mathcal{L}\left\\{c\right\\}$

and

$\displaystyle\frac{{sX(s)-{x^{(\alpha)}}(0)}}{{s+r(1-s)}}+{a}X(s)=\frac{{{b}}}{{{s^{2}}}}+\frac{{{c}}}{s}$
(31)

By further solving Eq.(31), one can write

$X(s)=\frac{{cr{s^{2}}+brs-crs-c{s^{2}}-{x^{(\alpha)}}(0){s^{2}}-br-
bs}}{{{s^{2}}\left({ars-ar-as-s}\right)}}$ (32)

The continuous solution to Eq.(32) can be calculated by the inverse laplace
transformation operation, which is as follows,

${\widehat{x}^{(\alpha)}}({\text{t}})=\frac{{b\left({\frac{{art}}{{ar-a-1}}-1}\right)}}{{{a^{2}}r}}+\frac{{bt(ar-a-1)+{e^{\frac{{art}}{{ar-a-1}}}}(-ax+c)+c(ar-a-1)}}{{(ar-a-1)a}}$
(33)

Set ${\widehat{x}^{(\alpha)}}(1)={x^{(\alpha)}}(1)$ and $t=k$, the discrete
time response function of can be given as

${\hat{x}^{(\alpha)}}(k)=\left({{x^{(\alpha)}}(1)-\frac{b}{a}-\frac{c}{a}+\frac{b}{{{a^{2}}r}}}\right){e^{\frac{{-ar(k-1)}}{{1-ar+a}}}}+\frac{b}{a}k+\frac{c}{a}-\frac{b}{{{a^{2}}r}}$
(34)

###### Remark 2

Inspired by the idea in [28], the discrete forma of Eq.(30) is
${x^{(\alpha-r)}}(k)+\frac{a}{2}\left\\{{{x^{(\alpha)}}(k-1)+{x^{(\alpha)}}(k)}\right\\}=bk+c$.
Subsequently we estimate the structural parameters by constructing a cost
function expressed as

${\chi^{2}}={\sum\limits_{k=2}^{n}{\left\\{{{x^{(\alpha-r)}}(k)+a{z^{(\alpha)}}(k)-bk-c}\right\\}}^{2}}$
(35)

And by minimizing ${\chi^{2}}$, one has a set of equations with respect to
partial function expressed as

$\left\\{{\begin{array}[]{*{20}{l}}{\frac{{\partial{\chi^{2}}}}{{\partial
a}}=\sum\limits_{k=2}^{n}{2{z^{(\alpha)}}(k){x^{(\alpha-r)}}(k)+2a{{\left\\{{{z^{(\alpha)}}(k)}\right\\}}^{2}}-2{z^{(\alpha)}}(k)bk-2{z^{(\alpha)}}(k)c}=0,}\\\
\begin{gathered}\frac{{\partial{\chi^{2}}}}{{\partial
b}}=\sum\limits_{k=2}^{n}{-2k{x^{(\alpha-r)}}(k)-2ka{z^{(\alpha)}}(k)+2{k^{2}}b+2kc}=0,\hfill\\\
\frac{{\partial{\chi^{2}}}}{{\partial
c}}=\sum\limits_{k=2}^{n}{-2{x^{(\alpha-r)}}(k)-2a{z^{(\alpha)}}(k)+2bk+2c=0.}\hfill\\\
\end{gathered}\end{array}}\right.$ (36)

which yields that

$\left\\{{\begin{array}[]{*{20}{l}}{\mathop{\sum}\limits_{k=2}^{n}a{{\left\\{{{z^{(\alpha)}}(k)}\right\\}}^{2}}-\mathop{\sum}\limits_{k=2}^{n}{z^{(\alpha)}}(k)bk-\mathop{\sum}\limits_{k=2}^{n}{z^{(\alpha)}}(k)c=\mathop{-\sum}\limits_{k=2}^{n}{z^{(\alpha)}}(k){x^{(\alpha-r)}}(k),}\\\
\begin{gathered}-\mathop{\sum}\limits_{k=2}^{n}ka{z^{(\alpha)}}(k)+\mathop{\sum}\limits_{k=2}^{n}{k^{2}}b+\mathop{\sum}\limits_{k=2}^{n}kc=\mathop{\sum}\limits_{k=2}^{n}k{x^{(\alpha-r)}}(k),\hfill\\\
-\mathop{\sum}\limits_{k=2}^{n}a{z^{(\alpha)}}(k)+\mathop{\sum}\limits_{k=2}^{n}bk+\mathop{\sum}\limits_{k=2}^{n}c=\mathop{\sum}\limits_{k=2}^{n}{x^{(\alpha-r)}}(k).\hfill\\\
\end{gathered}\end{array}}\right.$ (37)

and then, using the least squares, the structural parameter estimates give in
matrix form expressed as

$\displaystyle\left(a,b,c\right)^{\text{T}}=\left(\mathcal{O}^{\text{T}}\mathcal{O}\right)^{-1}\mathcal{O}^{\text{T}}{\rm
H}$ (38)

where

$\mathcal{O}=\begin{pmatrix}\sum_{k=2}^{n}\left\\{z^{(\alpha)}(k)\right\\}^{2}&-\sum_{k=2}^{n}z^{(\alpha)}(k)k&-\sum_{k=2}^{n}z^{(\alpha)}(k)\\\
-\sum_{k=2}^{n}kz^{(\alpha)}(k)&\sum_{k=2}^{n}k^{2}&\sum_{k=2}^{n}k\\\
-\sum_{k=2}^{n}z^{(\alpha)}(k)&\sum_{k=2}^{n}k&\sum_{k=2}^{n}\end{pmatrix}$

${\rm H}=\begin{pmatrix}-\sum_{k=2}^{n}z^{(\alpha)}(k)x^{(\alpha-r)}(k)\\\
-\sum_{k=2}^{n}kx^{(\alpha-r)}(k)\\\
-\sum_{k=2}^{n}x^{(\alpha-r)}(k)\end{pmatrix}$

Substituting the estimates of the structural parameters and initial value
$\widehat{x}^{(\alpha)}(1)=x^{(0)}(1)$ into Eq.(34), the restored values of
the original sequence can be given by then using the inverse fractional
accumulated generating operation

$\displaystyle{\hat{x}^{(0)}}(k)=\sum\limits_{i=1}^{k}{\left({\begin{array}[]{*{20}{c}}{k-i+1-\alpha-1}\\\
{k-i}\end{array}}\right)}{\hat{x}^{(\alpha)}}(i)-\sum\limits_{i=1}^{k-1}{\left({\begin{array}[]{*{20}{c}}{k-1-i+1-\alpha-1}\\\
{k-i-1}\end{array}}\right)}{\hat{x}^{(\alpha)}}(i)$ (43)

We recall the above remarks, where Remark 1 shows that the unified model has
the flexibility to present the continuous fractional grey models and integer-
order grey models. For instance, if fractional accumulation order $\alpha$ is
taken as “1”, Eq.(34) is simplified as the basic form of NHGM(1,1)
[XIE20135059], whose time response function is presented as
${\hat{x}^{(\alpha)}}(k)=\left({{x^{(\alpha)}}(0)-\frac{b}{a}-\frac{c}{a}+\frac{b}{{{a^{2}}}}}\right){e^{-a(k-1)}}+\frac{b}{a}k+\frac{c}{a}-\frac{b}{{{a^{2}}}}$.
Remark 2, in detail, introduces the calculation process of the least squares
in the special example, which help readers better understand the relevant
calculation mechanism. Remarks 1-2 also introduce the possibility of
developing some other novel fractional models.

##  4 Simulation studies

For the purpose of numerically analyzing the prediction performance of the
unified model (i.e., UFGM(1,1)), we carry out the experiments from the diverse
two aspects: One is to compare the prediction results by UFGM(1,1) with those
by other benchmarks including the GM(1,1) [1], DGM(1,1) [3], FGM(1,1) [10] and
ANN [45] models; another is to explore the impact of initial value and
fractional accumulation order on the prediction accuracy of UFGM(1,1).

###  4.1 Validity of the UFGM(1,1) model

Three error-value metrics are chosen to evaluate the accuracy of UFGM(1,1),
which embrace the mean absolute percentage error (MAPE), mean square error
(MSE) and mean absolute error (MAE), which are defined, respectively, as
follows,

$\displaystyle MAPE$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{k=1}^{n}\frac{\left|\widehat{x}^{(0)}(k)-x^{(0)}(k)\right|}{x^{(0)}(k)}\times
100\%$ (44) $\displaystyle MSE$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{k=1}^{n}\left(\widehat{x}^{(0)}(k)-x^{(0)}(k)\right)^{2}$
(45) $\displaystyle MAE$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{k=1}^{n}\left|\widehat{x}^{(0)}(k)-x^{(0)}(k)\right|$
(46)

For the experimental design, we consider 20 data sets, collected from the
National Bureau of Statistics of China (http://www.stats.gov.cn/), as the
inputs to calibrate models, which describes the electricity consumption in 20
provinces of China from 2014 to 2019. In UFGM(1,1), the emerging coefficients
(i.e., $r$, $\alpha$) plays an important role in the modeling process and more
importantly, has a directly impact on the prediction accuracy. Thus we utilize
the four commonly-used intelligent techniques, namely the grey wolf optimizer
(GWO) [46], whale optimization algorithm (WOA) [47], particle swarm
optimization (PSO) [48] and ant lion optimization (ALO) [49], to determine the
optimal emerging coefficients for UFGM(1,1). For intuition, the fitting
results produced by UFGM(1,1) with the help of the different algorithms are
plotted in Figure 1.

Figure 1: Prediction performance of the different algorithms-based UFGM(1,1)
model

Figure 1 shows that there is no fixed algorithm that enables UFGM(1,1) to
produce the highest accuracy in all data sets, the reason for this situation
is that each algorithm has its own advantages and limitations, thus it is
possible to obtain better solutions for different problems. In the following
analysis, it is necessary to evaluate the fitness value of these algorithms in
each experiment, and the optimal emerging coefficients are chosen for model
calibration. By reference with the current study and other published papers
related to benchmarks, their comparative results are shown in Table 2. As
table shows, the MAPE and MAE values of UFGM(1,1) are all smaller than those
of other competitors, although its MSE values are slightly higher than
traditional fractional grey model in several experiments, inferring that the
UFGM(1,1) model outperforms other benchmarks in terms of model’s prediction
performance.

Table 2: Errors by the five competitive models in 20 data sets. Case | Region | Metrics | GM(1,1) | DGM(1,1) | ANN | FGM(1,1) | UFGM(1,1)
---|---|---|---|---|---|---|---
1 | Beijing | MAPE | 4.4274 | 4.4237 | 3.362543 | 1.6294 | 1.4879
MSE | 1046 | 1047.3 | 826.1047 | 210.14 | 235.57
MAE | 27.941 | 27.86 | 24.12434 | 11.186 | 10.614
2 | Tianjin | MAPE | 4.4274 | 6.1891 | 3.132174 | 3.8301 | 2.404
MSE | 1046 | 1312.1 | 495.232 | 533.62 | 646.51
MAE | 27.941 | 28.989 | 18.21007 | 20.381 | 15.52
3 | Hebei | MAPE | 10.76 | 10.854 | 4.81092 | 7.9273 | 4.0727
MSE | 54708 | 54986 | 14763.99 | 29176 | 33992
MAE | 188.23 | 188.5 | 108.4177 | 149.38 | 104.99
4 | Shanxi | MAPE | 10.596 | 10.669 | 6.432003 | 6.8308 | 4.5779
MSE | 17969 | 18005 | 9898.418 | 8890.9 | 9452.1
MAE | 113.37 | 113.58 | 87.11586 | 81.838 | 65.915
5 | Liaoning | MAPE | 5.7687 | 4.88467 | 3.135815 | 4.0297 | 3.0828
MSE | 10219 | 8775.161 | 6730.438 | 4893.4 | 6243.1
MAE | 77.149 | 75.33215 | 66.32129 | 59.219 | 48.428
6 | Heilongjiang | MAPE | 2.4237 | 2.4298 | 2.524523 | 2.054 | 1.6633
MSE | 510.6 | 510.81 | 511.9586 | 291.74 | 260.93
MAE | 16.947 | 16.962 | 18.7526 | 14.031 | 12.103
7 | Shanghai | MAPE | 7.5002 | 7.5101 | 2.947381 | 2.9327 | 2.1168
MSE | 6967.2 | 6971.7 | 1679.166 | 1439.8 | 999.77
MAE | 72.783 | 72.7 | 33.71344 | 31.477 | 23.533
8 | Zhejiang | MAPE | 10.609 | 10.652 | 3.280667 | 5.1743 | 3.1869
MSE | 47904 | 48199 | 9964.663 | 13238 | 14965
MAE | 183.32 | 182.89 | 85.00789 | 100.01 | 84.197
9 | Anhui | MAPE | 5.6001 | 5.6303 | 5.0247 | 4.2216 | 2.125
MSE | 3372 | 3449 | 3805.964 | 2134.8 | 1828.6
MAE | 44.015 | 43.871 | 53.12803 | 35.507 | 24.597
10 | Fujian | MAPE | 6.5022 | 6.5479 | 4.896582 | 3.9373 | 2.0374
MSE | 5766.2 | 5839.6 | 5161.624 | 2369.6 | 2138.1
MAE | 56.337 | 56.409 | 57.57798 | 41.634 | 27.153
11 | Jiangxi | MAPE | 6.1053 | 6.1617 | 4.425646 | 3.7039 | 2.7821
MSE | 1254.7 | 1281.8 | 1303.201 | 574.3 | 641.04
MAE | 27.619 | 27.768 | 28.03915 | 19.493 | 16.943
12 | Shandong | MAPE | 7.1293 | 7.1613 | 5.36149 | 3.0638 | 2.9972
MSE | 31808 | 32177 | 36099.57 | 16175 | 15496
MAE | 148.94 | 148.89 | 160.1942 | 91.306 | 90.343
13 | Henan | MAPE | 9.5074 | 9.5784 | 4.283333 | 5.791 | 3.7368
MSE | 34638 | 34792 | 9638.134 | 15935 | 21391
MAE | 155.34 | 155.53 | 85.30263 | 104.06 | 86.514
14 | Hubei | MAPE | 5.9726 | 6.018 | 4.077524 | 4.4207 | 2.6483
MSE | 5783 | 5809.9 | 3683.539 | 3511.3 | 2765.1
MAE | 60.408 | 60.544 | 49.47484 | 47.658 | 34.901
15 | Hunan | MAPE | 7.1714 | 7.2392 | 5.419488 | 4.7514 | 3.7075
MSE | 5898.6 | 5922.9 | 4106.629 | 3018.3 | 3426.1
MAE | 62.315 | 62.555 | 55.92601 | 45.21 | 41.214
16 | Guangxi | MAPE | 7.3815 | 7.4647 | 4.084182 | 5.8011 | 3.3829
MSE | 4063.5 | 4111 | 1579.356 | 2460.7 | 2418.8
MAE | 50.793 | 50.964 | 35.79186 | 41.435 | 30.92
17 | Hainan | MAPE | 6.6034 | 6.6737 | 4.084182 | 3.5861 | 2.3553
MSE | 65.205 | 69.117 | 1579.356 | 25.571 | 26.381
MAE | 6.4182 | 6.4535 | 35.79186 | 4.2047 | 3.6773
18 | Chongqing | MAPE | 4.9022 | 4.9783 | 3.78875 | 4.5603 | 2.9425
MSE | 1036.4 | 1048.1 | 726.5574 | 796.52 | 775.06
MAE | 22.673 | 22.88 | 22.4828 | 21.756 | 17.092
19 | Sichuan | MAPE | 5.9849 | 6.0382 | 4.378797 | 4.1841 | 2.6998
MSE | 8572.9 | 8615 | 6206.165 | 4955.6 | 3959.6
MAE | 71.844 | 72.127 | 65.22631 | 58.053 | 42.153
20 | Yunnan | MAPE | 8.4035 | 8.5588 | 5.935477 | 7.3737 | 3.5109
MSE | 7395.2 | 7500.6 | 6465.405 | 5041.9 | 4563.6
MAE | 59.009 | 59.653 | 58.08669 | 58.986 | 36.112

###  4.2 Effects of fractional accumulation order and initial value on
modelling

In order to explore the effects of fractional accumulation order and initial
value on modelling, the initial value is sampled at every interval of 200000
in the range of [1, 800000], and then $r$, $\alpha$, $a$, $b$, $c$ are all
controlled within [0, 1], of which the step is 0.2. Consider the data on total
population at the end of the year ($10^{4}$ persons) from the National Bureau
of Statistics of China (http://www.stats.gov.cn/). The pseudo-code for the
relevant calculations are shown in Algorithm 1.

Input: Raw data
$X^{(0)}=\left\\{x^{(0)}(1),x^{(0)}(2),\ldots,x^{(0)}(n)\right\\}$

Output: Simulation value ${\hat{x}^{(0)}}(k)$

1 Initialize ${\text{ MAPE}}{{\text{ }}_{\min}}=+{\text{ inf }}$ for
_$Initial$ $value=1$ to $800000$ $Step=200000$_ do

2 for  _$\alpha=0.1$ to $1$ $Step=0.2$_ do

3 for  _$r=0.1$ to $1$ $Step=0.2$_ do

4

5 for  _$a=0.1$ to $1$ $Step=0.2$_ do

6 for  _$b=0.1$ to $1$ $Step=0.2$_ do

7 for  _$c=0.1$ to $1$ $Step=0.2$_ do

8 Calculating
${X^{(\alpha)}}=\left({{x^{(\alpha)}}(1),{x^{(\alpha)}}(2),\ldots,{x^{(\alpha)}}(n)}\right)$
using the Eq.(12);

9 Calculating $\hat{x}^{(\alpha)}(k)$ using the Eq.(34);

10 Calculate the fitting results $\hat{x}^{(0)}(k)$ using Eq.(43);

11 Calculate the error of this model using the Eq.(44);

12

13

14

15

16

17 return $\hat{x}^{(0)}(k)$;

18

Algorithm 1 Effects of the fractional accumulation order and initial value on
the prediction performance. Figure 2: Effects of the fractional order
accumulation order $r,\alpha$ on the UFGM(1,1) model in the four initial value
scenarios: (a)$initial\ value=1$; (b)$initial\ value=200001$; (c)$initial\
value=400001$; (d)$initial\ value=600001$;

Figure 2 shows that with the increase of the initial condition value, the MAPE
values get larger when the other conditions are fixed, for example, $r=0.1$
and $\alpha=0.1$, inferring the predicted values by the UFGM(1,1) model
deviate from the actual ones. In return, in the small initial value scenario,
the smaller the fractional accumulation order, the lower the fitting error
(i.e., MAPE); with the increase of the initial value, the fractional
accumulation order should properly increase so as to better mine the
characteristics hidden in the original sequence, for example, $r=\alpha=0.3$
enables this model to provide the more accurate forecasts rather than that of
$r=\alpha=0.1$ in the large initial value scenario (i.e., $initial\
value=800000$).

##  5 Application

Foresight of water supply capacity is of great significance for the
sustainable development of society, is conducive to the rational distribution
of water resources, and plays an important role in the development of the
national economy. In this section, we further apply the UFGM(1,1) model to
deal with the data on water supply production capacity (ten thousand cubic
meters/day) in Henan and Chongqing, respectively. The prediction performance
of the UFGM(1,1) model will be compared with the above benchmarks in terms of
MAPE (elaborated on in Section 4.1). The total amount of water supply
production capacity is collected from the National Bureau of Statistics of
China (http://www.stats.gov.cn/), as shown in Tables 3-4.

Case 1. (Forecasting Henan’s water supply production capacity) We consider the
total amount of water supply production capacity from 2004 to 2019, in which
the data from 2004 to 2015 are used for model calibration, and the left four
samples are used to examine the accuracy. For the purpose of obtaining the
optimal emerging coefficients for the UFGM(1,1) model, the four algorithms are
carried out in this experiments, thus the track of searching process by the
different algorithms is graphed in Figure LABEL:fig:search. Subsequently, the
minimum MAPE corresponding to the algorithm is applied to determine the
emerging coefficients.

Table 3: Simulated and predicted values by the five competitive models in Case 1. Raw data | GM(1,1) | Error(%) | DGM(1,1) | Error(%) | ANN | Error(%) | FGM(1,1) | Error(%) | UFGM(1,1) | Error(%)
---|---|---|---|---|---|---|---|---|---|---
In-sample
1038.31 | 1038.31 | 0.000 | 1003.8 | 3.32367 | 1038.31 | 0.0000 | 1038.31 | 0.000 | 1038.31 | 0.000
1026.51 | 1003.7 | 2.222092 | 1003.8 | 2.212351 | 1026.51 | 0.0000 | 1026.8 | 0.028251 | 1022.9 | 0.351677
1023.7 | 1011 | 1.240598 | 1011.1 | 1.230829 | 1023.7 | 0.0000 | 1021.8 | 0.185601 | 1014.9 | 0.859627
1039.85 | 1018.4 | 2.062798 | 1018.5 | 2.053181 | 1039.85 | 0.0000 | 1019.3 | 1.976247 | 1011.5 | 2.726355
1013.91 | 1025.9 | 1.182551 | 1026 | 1.192414 | 1047.962 | 3.358434 | 1018.7 | 0.472429 | 1011.9 | 0.198242
1007.79 | 1033.4 | 2.541204 | 1033.5 | 2.551127 | 1042.344 | 3.42871 | 1019.8 | 1.191717 | 1015.9 | 0.804731
1010.34 | 1041 | 3.034622 | 1041 | 3.034622 | 1039.884 | 2.924204 | 1023.1 | 1.262941 | 1023.3 | 1.282737
1037.56 | 1048.6 | 1.064035 | 1048.6 | 1.064035 | 1031.243 | 0.608813 | 1029.3 | 0.796099 | 1033.9 | 0.352751
1042.31 | 1056.3 | 1.342211 | 1056.3 | 1.342211 | 1034.192 | 0.778837 | 1039.8 | 0.240811 | 1047.6 | 0.507527
1047.26 | 1064.1 | 1.608006 | 1064 | 1.598457 | 1042.544 | 0.450299 | 1056.4 | 0.872754 | 1064.5 | 1.646201
1083.62 | 1071.9 | 1.08156 | 1071.8 | 1.090788 | 1055.567 | 2.588786 | 1082.1 | 0.140271 | 1084.3 | 0.062753
1121.39 | 1079.7 | 3.717707 | 1079.6 | 3.726625 | 1071.847 | 4.417999 | 1121.4 | 0.000892 | 1107.1 | 1.274311
MAPE |  | 1.9183 |  | 1.9171 |  | 2.319511 |  | 0.65119 |  | 0.91541
Out-of-sample
1180.32 | 1087.6 | 7.855497 | 1087.5 | 7.863969 | 1093.27 | 7.375093 | 1181.1 | 0.066084 | 1132.8 | 4.026027
1150.37 | 1095.6 | 4.761077 | 1095.5 | 4.76977 | 1132.02 | 1.595139 | 1271.4 | 10.52096 | 1161.3 | 0.950129
1166.64 | 1103.7 | 5.39498 | 1103.5 | 5.412124 | 1157.322 | 0.798721 | 1407.6 | 20.65419 | 1192.6 | 2.225194
1281.52 | 1111.7 | 13.25145 | 1111.5 | 13.26706 | 1181.88 | 7.77515 | 1612.6 | 25.83495 | 1226.7 | 4.277733
MAPE |  | 7.8142 |  | 7.8292 |  | 4.386027 |  | 14.26904 |  | 2.8704

From Table 3, it is found that the MAPE value of UFGM(1,1) in the in-sample
period is 0.91541%, while that of competitive models, namely GM(1,1),
DGM(1,1), ANN and FGM(1,1), are 1.9183%, 1.9171%, 2.319511%, and 0.65119%,
respectively, indicating the FGM(1,1) and UFGM(1,1) models have the higher
accuracy in this stage because their MAPE values are closer to zeros. As for
the out-of-sample period, the MAPE values of the current model and benchmarks
are 2.8704%, 7.8142%, 7.8292%, 4.386027% and 14.2690%, respectively. It is
obvious that the MAPE value of the UFGM(1,1) model is far less than others,
this means the UFGM(1,1) model achieves the precise prediction in this
experiment. The same finding can be also obtained from Figure 4.

Figure 4: Prediction performance of the five competitors in Case 1:
(a)Simulated and predicted values by the competitors; (b)Errors by the
competitors.

Case 2. (Forecasting Chongqing’s water supply production capacity.) Similar to
case one, the raw data is divided into two groups, one, training set, from
2004 to 2015 are used for calibrating structural parameters; another, testing
set, from 2016 to 2019 are used to evaluate the prediction performance. First
and foremost, according to the performance of the four algorithms on the
determination of emerging coefficients shown in Figure 5, it is found that the
fitness value by PSO is smallest, this means the emerging coefficients of
UFGM(1,1) model will be determined by this algorithm. Subsequently, the
prediction results by these competitive models are obtained by reference with
the current study and references therein, as shown in Table 4. Meanwhile,
these values in Table 4 are vividly displayed in Figure 6.

Figure 5: $r$, $\alpha$ and the fitness value of the UFGM(1,1) model in each trial by using the four algorithms in Case 2. Table 4: Simulated and predicted results by the five competitive models in Case 2 Raw data | GM(1,1) | Error(%) | DGM(1,1) | Error(%) | ANN | Error(%) | FGM(1,1) | Error(%) | UFGM | Error(%)
---|---|---|---|---|---|---|---|---|---|---
In-sample
373.65 | 373.65 | 0.000 | 373.65 | 0.000 | 373.65 | 0.000 | 373.65 | 0.000 | 373.65 | 0.000
405.61 | 381.59 | 5.921945 | 381.79 | 5.872636 | 405.61 | 0.000 | 380.87 | 6.099455 | 403.23 | 0.586771
391.7 | 392.78 | 0.275721 | 392.96 | 0.321675 | 391.7 | 0.000 | 389.02 | 0.684197 | 396.63 | 1.258616
419.2 | 404.31 | 3.552004 | 404.46 | 3.516221 | 419.2 | 0.000 | 398.22 | 5.004771 | 393.74 | 6.073473
418.16 | 416.17 | 0.475894 | 416.29 | 0.447197 | 431.3829 | 3.162163 | 408.6 | 2.286206 | 398.61 | 4.675244
420.35 | 428.38 | 1.910313 | 428.47 | 1.931724 | 435.0692 | 3.501649 | 420.32 | 0.007137 | 409.54 | 2.571666
412.3 | 440.94 | 6.946398 | 441 | 6.960951 | 441.3492 | 7.045642 | 433.55 | 5.154014 | 424.63 | 2.990541
429.27 | 453.88 | 5.732989 | 453.9 | 5.737648 | 437.3996 | 1.89382 | 448.47 | 4.472709 | 442.55 | 3.093624
447.83 | 467.2 | 4.325302 | 467.18 | 4.320836 | 446.0885 | 0.388882 | 465.31 | 3.903267 | 462.45 | 3.264632
491.22 | 480.9 | 2.100892 | 480.85 | 2.11107 | 459.6022 | 6.436593 | 484.32 | 1.404666 | 483.83 | 1.504418
506.89 | 495.01 | 2.343704 | 494.92 | 2.361459 | 493.1399 | 2.712648 | 505.78 | 0.218982 | 506.36 | 0.104559
529.92 | 509.53 | 3.847751 | 509.4 | 3.872283 | 516.7286 | 2.489319 | 529.99 | 0.01321 | 529.85 | 0.01321
MAPE |  | 3.4031 |  | 3.4052 |  | 3.45384 |  | 2.6586 |  | 2.3759
Out-of-sample
566.12 | 524.48 | 7.355331 | 524.3 | 7.387126 | 543.2325 | 4.042864 | 557.32 | 1.554441 | 554.15 | 2.114393
599.87 | 539.87 | 10.00217 | 539.64 | 10.04051 | 574.3076 | 4.26133 | 588.16 | 1.95209 | 579.16 | 3.452415
616.99 | 555.7 | 9.93371 | 555.43 | 9.977471 | 607.9011 | 1.473103 | 622.96 | 0.967601 | 604.82 | 1.972479
627.76 | 572.01 | 8.880782 | 571.68 | 8.93335 | 633.7528 | 0.954632 | 662.24 | 5.492545 | 631.08 | 0.528865
MAPE |  | 9.043 |  | 9.0848 |  | 2.682982 |  | 2.4922 |  | 2.0169
Figure 6: Prediction performance of the five competitors in Case 2:
(a)Simulated and predicted values by the competitors; (b)Errors by the
competitors.

Firstly, we observe from Table 4 and Figure 6 that for the in-sample period
all of the competitors have a relatively satisfactory performance because
their outputs of simulating and predicting Chongqing’s water supply production
capacity are close to actual ones. However, in the out-of-sample period, the
predicted results by the GM(1,1) and DGM(1,1) models deviate from the actual
ones although it fall within the acceptable range. Among these models, the
UFGM(1,1) model have a higher accuracy because the corresponding prediction
values are closer to the actual amount of Chongqing’s water supply production
capacity.

Next, the error-value indicator, i.e. MAPE, is applied to further analyze the
prediction performance. In the simulation stage, the MAPE values of the
UFGM(1,1) and other competitive models are 2.3759%, 3.4031%, 3.4052%, 3.45384%
and 2.6586%, respectively; those of competitors are 2.01691%, 9.0431%,
9.08481%, 2.6829821% and 2.49221%, respectively, for the prediction stage. It
is concluded that all of models are suitable for forecasting Chongqing’s water
supply production capacity in accordance with Lewis standard [50]. To be
specific, the GM(1,1) and DGM(1,1) models have a relatively poor prediction
accuracy due to their higher MAPEs, the UFGM(1,1) has a superiority over
others because its MAPEs are smaller either in the in-sample or out-of-sample
period. It is worth mentioned that the FGM(1,1) model has a second best
performance due to its second smallest MAPE. In consequence, the UFGM(1,1)
will be treated as an optimal tool for forecasting Chongqing’s water supply
production capacity in this experiment, and the FGM(1,1) model should be
considered as a sub-optimal technique in dealing with Chongqing’s water supply
production capacity.

##  6 Conclusion and summary

In this study, an unified framework for fractional grey models is presented by
combining general fractional derivative with memory effects and grey modeling
theory. This unified framework (denoted as UFGM(1,1) for short) not just
covers commonly-used fractional grey models already in place, but also can
derived other new ones. The main conclusions are summarized as follows.

  * 1.

Based on the GR and GRL fractional derivatives, the framework for published
fractional grey models is developed. By taking different kernel functions and
normalization ones, some other new fractional grey models can be deduced, a
new fractional grey model, for example, with $k(x,r)=e^{-\frac{r}{1-r}x}$ and
$N(r)=1$, is expressed as
${}_{0}^{\exp}D_{t}^{r}{x^{(\alpha)}}(t)+a{x^{(\alpha)}}(t)=bt+c.$

  * 2.

The commonly-used intelligent algorithms, namely GWO, PSO, WOA and ALO, are
employed to search for the optimal emerging coefficients, i.e., $r$, and
$\alpha$, for the UFGM(1,1) model in the current study.

  * 3.

Two published cases are utilized to verify the superiority of the UFGM(1,1)
model and explore the effects of fractional accumulation order and initial
value on the prediction performance.

  * 4.

The UFGM(1,1) model is also applied to dealing with Henan’s and Chongqing’s
water supply production capacity so as to explain the validity of the proposed
method. Numerical results show that the UFGM(1,1) model outperforms a range of
benchmarks.

In addition, the unified framework is not without limitations, for example,
this framework is focusing on continuous-fractional grey models, potentially
neglecting the discrete-fractional grey model (e.g. conformable fractional
grey model [24]), this is a primary research direction of our following study.
As stated in [51], the data-preprocessing techniques (e.g., data grouping
[52], rolling mechanism [53], etc.) should be also worth exploring in the
coming work.

Acknowledgements

This work was supported by the Postgraduate Research & Practice Innovation
Program of Jiangsu Province (KYCX20_1144) and the Fundamental Research Funds
for the Central Universities of China (Grant no. 2019YBZZ062).

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