A Unified Grey Riccati Model

Abstract

The grey Riccati model (GRM) is a generalization of the grey Verhulst model (GVM). Both the GRM and GVM generally perform well in simulating and forecasting the raw sequences with a bell-shaped or single peak feature. Although there are several methods to solve the Riccati differential equation, the existing time response functions of the GRM are generally complicated. In order to reduce the computational complexity of the time response function, this study attempts to transform the Riccati equation into a Bernoulli equation with the help of a known particular solution. Then, a unified time response function of the GRM is obtained by the proposed methodology. The simulation results demonstrate that the proposed unified grey Riccati model performs the same as the grey generalized Verhulst model (a kind of grey Riccati model) and is better than the traditional grey Verhulst model. The fact also reveals that the newly developed grey Riccati model is reasonable and effective.

1. Introduction

The grey forecasting model is one of the most important fields of grey system theory and is generally implemented by a GM(1,1) model, i.e., a first-order univariate grey model [1]. Based on current knowledge of the first-order linear ordinary differential equation and the least-squares method, the grey model could employ a simple calculation process to estimate the behavior of unknown systems with an ideal forecasting effect. Furthermore, especially in contrast to the traditional prediction techniques requiring a large amount of historical data, the GM(1,1) model only requires a limited amount of statistical data without knowing their statistical distribution [2]. Due to the GM(1,1) model having a high forecasting accuracy, it has been successfully applied to numerous fields, such as agricultural, economic, industrial and social studies [3,4,5,6].

The GM(1,1) model and its variants, the grey Verhulst model [7,8] and nonlinear grey Bernoulli model [9,10], are the most widely used models in grey forecasting applications. Generally speaking, the GM(1,1) model is suitable for data observed with an exponential distributing behavior, the grey Verhulst model (GVM) is adequate for data observed with a bell shape, and the nonlinear grey Bernoulli model (GBM) is valid for data observed with oscillatory distributing types [9,11]. Note, however, that the traditional GM(1,1) and grey Verhulst models are both special forms of the nonlinear grey Bernoulli model, with the power exponent r being 0 and 2, respectively. The background value at the time instant $k$, i.e., $z^{\left(1\right)}\left(k\right)$ [12,13], the initial value [9,11,14], and the power exponent r [15,16] are important parameters affecting the prediction accuracy of the grey model. Until now, many pieces of literature proposed different parameter optimization methodologies to improve the forecasting precision of grey models [9,11,12,13,14,15,16]. Aside from the optimization of model parameters, the modification of the model structure is another available scheme that could correct the residual errors so as to enhance the forecasting performance of grey models [2,17,18,19].

While introducing a new constant $c$ into the grey action quantity of the grey Verhulst model, the control quantity will become $b{\left(z^{\left(1\right)}\left(k\right)\right)}^2+c,$ and the resulting univariate grey model is the so-called grey Riccati model (GRM) [20,21,22]. Aside from the GM(1,1) model, the grey Verhulst model and the grey Riccati model are two other basic but important grey forecasting models. Generally speaking, these two grey models could perform well in simulating and forecasting the raw sequence with a bell-shaped or single peak characteristic or a one-time accumulated generating sequence with the “S” distributing (i.e., saturated distributing) shape. On the other hand, the grey Riccati model takes the Riccati equation with constant coefficients as the whitenization differential equation. There are several methods to solve the Riccati differential equation. However, the existing time response functions of the grey Riccati model are generally complicated. Take the response of the grey generalized Verhulst model (GGVM) given in the next section, for example [20]. It is dependent on the discriminant of the corresponding whitenization differential equation. As is well known, there are three causes for the discriminant: positive (two distinct roots), zero (exactly one real root), and negative (no real roots). Each cause in the GGVM further corresponds to one or two distinct time response expressions. As can be seen, the representation of the GGVM’s response is slightly complicated. In order to reduce the computational complexity of the time response function, this study first attempts to transform the Riccati equation into a Bernoulli equation with the help of a given particular solution. Then, the Bernoulli equation can be further converted to a linear equation using the change of dependent variable approach. The new equation is a first-order linear differential equation and then can be solved explicitly. That is to say, while a particular solution of the Riccati equation is given, the time response function of the corresponding grey Riccati model can be represented in a closed-form solution. This is the central idea of the proposed unified grey Riccati model. Two numerical examples were used to verify the performance of the unified grey Riccati model, and the results showed that the proposed grey Riccati model performs the same as the GGVM and is better than the traditional grey Verhulst model.

The remainder of this study is organized as follows. Section 2 briefly describes some basic knowledge of the grey models, including the GM(1,1) model, the grey Verhulst model and the grey Riccati model. The proposed unified grey Riccati model, the corresponding time response and the modeling procedure are given in Section 3. Section 4 presents the simulation results of the newly developed grey model and compares it with the traditional grey Verhulst model and the grey generalized Verhulst model [20]. Finally, Section 5 presents the conclusions of this study.

2. Grey Models

This section briefly introduces an overview of three basic grey models and a grey model variant. Let $x^{\left(0\right)}=\left(x^{\left(0\right)}\left(1\right),x^{\left(0\right)}\left(2\right),\dots ,x^{\left(0\right)}\left(n\right)\right)$, where $x^{\left(0\right)}\left(k\right)\ge 0$, $k=1,2,\dots ,n$, be the raw data sequence. Define $x^{\left(1\right)}$ as the 1-AGO (accumulated generating operation) sequence of $x^{\left(0\right)}$. That is, $x^{\left(1\right)}=\left(x^{\left(1\right)}\left(1\right),x^{\left(1\right)}\left(2\right),\dots ,x^{\left(1\right)}\left(n\right)\right)$, where $x^{\left(1\right)}\left(k\right)={{\displaystyle \sum }}_{i=1}^k{x}^{\left(0\right)}\left(i\right)$, $k=1,2,\dots ,n$. In addition, the background value of the grey model is generally defined by

$$z^{\left(1\right)}\left(k\right)=\lambda x^{\left(1\right)}\left(k\right)+(1-\lambda )x^{\left(1\right)}\left(k-1\right),k=2,3,\dots ,n,$$

(1)

where $\lambda \in \left[0, 1\right]$ and $\lambda$ are usually specified as 0.5.

2.1. GM(1,1) Model

The grey differential equation of the GM(1,1) model is given as:

$$x^{\left(0\right)}\left(k\right)+a{z}^{\left(1\right)}\left(k\right)=b,k=2,3,\dots ,n,$$

(2)

where $a$ and $b$ represent the development coefficient and the grey action quantity, respectively [1]. The corresponding whitenization function is

$$\frac{d}{dt}{x}^{\left(1\right)}+a{x}^{\left(1\right)}=b.$$

(3)

Then, the time response function of (2) with an initial condition $x^{\left(1\right)}\left(1\right)=x^{\left(0\right)}\left(1\right)$ is

$${\widehat{x}}^{\left(1\right)}\left(k\right)=\left(x^{\left(0\right)}\left(1\right)-\frac{b}{a}\right)e^{-a\left(k-1\right)}+\frac{b}{a},k=1,2,\dots ,n,$$

(4)

where ${\widehat{x}}^{\left(1\right)}\left(k\right)$ represents the forecasting value of $x^{\left(1\right)}\left(k\right)$. The restored value of the GM(1,1) model can be obtained by applying the inverse accumulated generating operation (IAGO) to ${\widehat{x}}^{\left(1\right)}\left(k\right)$ as follows:

$${\widehat{x}}^{\left(0\right)}\left(k\right)={\widehat{x}}^{\left(1\right)}\left(k\right)-{\widehat{x}}^{\left(1\right)}\left(k-1\right),k=2,3,\dots ,n.$$

(5)

Substituting (4) into (5) yields

$${\widehat{x}}^{\left(0\right)}\left(k\right)=\left(1-e^a\right)\left(x^{\left(0\right)}\left(1\right)-\frac{b}{a}\right)e^{-a\left(k-1\right)}$$

(6)

where $k=2,3,\dots ,n,$ and ${\widehat{x}}^{\left(0\right)}\left(1\right)=x^{\left(0\right)}\left(1\right)$. It can be seen from (6) that the GM(1,1) model is essentially a homogenous exponential model. Finally, the model parameters $a$ and $b$ can be determined by using the ordinary least-squares method as follows:

$${\left[a, b\right]}^T={\left(B^{T}B\right)}^{-1}{B}^{T}Y,$$

(7)

where

$$B=\left[\begin{array}{cc}\begin{array}{c}-z^{\left(1\right)}\left(2\right)\\ -z^{\left(1\right)}\left(3\right)\end{array}& \begin{array}{c}1\\ 1\end{array}\\ \begin{array}{c}⋮\\ -z^{\left(1\right)}\left(n\right)\end{array}& \begin{array}{c}⋮\\ 1\end{array}\end{array}\right]Y=\left[\begin{array}{c}\begin{array}{c}{x}^{\left(0\right)}\left(2\right)\\ x^{\left(0\right)}\left(3\right)\end{array}\\ \begin{array}{c}⋮\\ x^{\left(0\right)}\left(n\right)\end{array}\end{array}\right]$$

(8)

2.2. Grey Verhulst Model

The grey differential equation of the grey Verhulst model is defined as:

$$x^{\left(0\right)}\left(k\right)+a{z}^{\left(1\right)}\left(k\right)=b{\left(z^{\left(1\right)}\left(k\right)\right)}^2,k=2,3,\dots ,n,$$

(9)

and the corresponding whitenization function is

$$\frac{d}{dt}{x}^{\left(1\right)}+a{x}^{\left(1\right)}=b{\left(x^{\left(1\right)}\right)}^2,$$

(10)

where $a$ and $b$ are the model parameters and the power exponent of the non-homogeneous term $x^{\left(1\right)}$ is 2 [7,8]. As can be seen, the grey action quantity of the grey Verhulst model is $b{\left(z^{\left(1\right)}\left(k\right)\right)}^2$, which is not just a constant $b$ as is seen in the GM(1,1) model. Let the initial value ${\widehat{x}}^{\left(1\right)}\left(1\right)=x^{\left(1\right)}\left(1\right)$, then solving (10) could obtain the time response function of the grey Verhulst model as

$${\widehat{x}}^{\left(1\right)}\left(k\right)=\frac{a{x}^{\left(1\right)}\left(1\right)}{b{x}^{\left(1\right)}\left(1\right)+\left(a-b{x}^{\left(1\right)}\left(1\right)\right)e^{a\left(k-1\right)}},k=1,2,\dots ,n.$$

(11)

Subsequently, the restored value ${\widehat{x}}^{\left(0\right)}\left(k\right)$ can be obtained by (5). On the other hand, the parameters estimation of the grey Verhulst model is also used as the least-squares method, that is, ${\left[a, b\right]}^T={\left(B^{T}B\right)}^{-1}{B}^{T}Y$, but the data matrices $B$ and Y become

$$B=\left[\begin{array}{cc}\begin{array}{c}-z^{\left(1\right)}\left(2\right)\\ -z^{\left(1\right)}\left(3\right)\end{array}& \begin{array}{c}{\left(z^{\left(1\right)}\left(2\right)\right)}^2\\ {\left(z^{\left(1\right)}\left(3\right)\right)}^2\end{array}\\ \begin{array}{c}⋮\\ -z^{\left(1\right)}\left(n\right)\end{array}& \begin{array}{c}⋮\\ {\left(z^{\left(1\right)}\left(n\right)\right)}^2\end{array}\end{array}\right],Y=\left[\begin{array}{c}\begin{array}{c}{x}^{\left(0\right)}\left(2\right)\\ x^{\left(0\right)}\left(3\right)\end{array}\\ \begin{array}{c}⋮\\ x^{\left(0\right)}\left(n\right)\end{array}\end{array}\right].$$

(12)

2.3. Grey Riccati Model

The grey differential equation of the grey Riccati model is defined as:

$$x^{\left(0\right)}\left(k\right)+a{z}^{\left(1\right)}\left(k\right)=b{\left(z^{\left(1\right)}\left(k\right)\right)}^2+c,k=2,3,\dots ,n,$$

(13)

and the corresponding whitenization function is

$$\frac{d}{dt}{x}^{\left(1\right)}+a{x}^{\left(1\right)}=b{\left(x^{\left(1\right)}\right)}^2+c,$$

(14)

where $a$, $b$ and $c$ are the model parameters [20,21,22]. Herein, the grey action quantity contains two terms. One is $b{\left(z^{\left(1\right)}\left(k\right)\right)}^2$, which is the same as the term given in the grey Verhulst model. The other is an adjustment coefficient $c$, which is used to eliminate the drift phenomenon caused by the grey Verhulst model [20]. According to (13), the model parameters can be estimated by the least-squares method as follows:

$${\left[a, b,c\right]}^T={\left(B^{T}B\right)}^{-1}{B}^{T}Y,$$

(15)

where

$$B=\left[\begin{array}{ccc}\begin{array}{c}-z^{\left(1\right)}\left(2\right)\\ -z^{\left(1\right)}\left(3\right)\end{array}& \begin{array}{c}{\left(z^{\left(1\right)}\left(2\right)\right)}^2\\ {\left(z^{\left(1\right)}\left(3\right)\right)}^2\end{array}& \begin{array}{c}1\\ 1\end{array}\\ \begin{array}{c}⋮\\ -z^{\left(1\right)}\left(n\right)\end{array}& \begin{array}{c}⋮\\ {\left(z^{\left(1\right)}\left(n\right)\right)}^2\end{array}& \begin{array}{c}⋮\\ 1\end{array}\end{array}\right],Y=\left[\begin{array}{c}\begin{array}{c}{x}^{\left(0\right)}\left(2\right)\\ x^{\left(0\right)}\left(3\right)\end{array}\\ \begin{array}{c}⋮\\ x^{\left(0\right)}\left(n\right)\end{array}\end{array}\right]$$

(16)

According to the whitenization differential Equation (14), the corresponding discriminant can be determined from the univariate polynomial equation $b{x}^2-ax+c=0$. Then, based on the initial value ${\widehat{x}}^{\left(1\right)}\left(1\right)=x^{\left(1\right)}\left(1\right)$, the time response expression of the GGVM is represented as follows [20]:

Case 1.

$a^2-4bc>0$

$${\widehat{x}}^{\left(1\right)}\left(k\right)=\left\{\begin{array}{cc}\frac{a}{2b}-\frac{\lambda }{2b}\left(1+\frac{2}{e^{\lambda \left(k-\Phi \right)}-1}\right),& if \frac{2b{x}^{\left(0\right)}\left(1\right)-a-\lambda }{2b{x}^{\left(0\right)}\left(1\right)-a+\lambda }>0\\ \frac{a}{2b}-\frac{\lambda }{2b}\left(1-\frac{2}{e^{\lambda \left(k-\Phi \right) }+1}\right),& if \frac{2b{x}^{\left(0\right)}\left(1\right)-a-\lambda }{2b{x}^{\left(0\right)}\left(1\right)-a+\lambda }<0\end{array}\right.$$

(17)

where $\lambda =\sqrt{a^2-4bc}$, $\Phi =1-\frac{1}{\lambda }\ln \left|\frac{2b{x}^{\left(0\right)}\left(1\right)-a-\lambda }{2b{x}^{\left(0\right)}\left(1\right)-a+\lambda }\right|$and $k=1,2,\dots ,n$.

Case 2.

$a^2-4bc=0$

$${\widehat{x}}^{\left(1\right)}\left(k\right)=\left\{\begin{array}{cc}\frac{1}{b}\left(\sqrt{bc}-\frac{1}{k-\Phi }\right), \Phi =1+\frac{1}{b{x}^{\left(0\right)}\left(1\right)-\sqrt{bc}},& if a=\sqrt{4bc}, \\ -\frac{1}{b}\left(\sqrt{bc}+\frac{1}{k-\Phi }\right), \Phi =1+\frac{1}{b{x}^{\left(0\right)}\left(1\right)+\sqrt{bc}},& if a=-\sqrt{4bc}, \end{array}\right.$$

(18)

where $k=1,2,\dots ,n.$

Case 3.

$a^2-4bc<0$.

$${\widehat{x}}^{\left(1\right)}\left(k\right)=\frac{\lambda }{2b}\tan \frac{\lambda \left(k-\Phi \right)}{2}+\frac{a}{2b},$$

(19)

where $\lambda =\sqrt{4bc-a^2}$, $\Phi =\left(1-\frac{2}{\lambda }\right){\tan }^{-1}\frac{2b{x}^{\left(0\right)}\left(1\right)-a}{\lambda }$ and $k=1,2,\dots ,n$.

It is obvious that the time response expression of the grey Riccati model, (17)–(19), is more complex than that of the grey Verhulst model, as given in (11).

2.4. New Grey Verhulst Model: N_Verhulst Model

Let $y^{\left(0\right)}=\left(y^{\left(0\right)}\left(1\right),y^{\left(0\right)}\left(2\right),\dots ,y^{\left(0\right)}\left(n\right)\right)$ be the reciprocal sequence of $x^{\left(0\right)}$, i.e., $y^{\left(0\right)}\left(k\right)=1/x^{\left(0\right)}\left(k\right)$, $k=1,2,\dots ,n$. According to [7], the new grey Verhulst model is defined as follows:

$$\frac{d}{dt}{y}^{\left(1\right)}+a{y}^{\left(1\right)}=\frac{1}{2}\left(2t-1\right)b+c.$$

(20)

where $a$, $b$ and $c$ are the undermined parameters. Then, solving (20) with the initial condition $y^{\left(1\right)}\left(1\right)=y^{\left(0\right)}\left(1\right)$ will obtain

$$y^{\left(1\right)}\left(t\right)=C_1{e}^{-a\left(t-1\right)}+\frac{b}{a}t-\frac{b}{a^2}-\frac{b}{2a}+\frac{c}{a},$$

(21)

where $C_1=y^{\left(0\right)}\left(1\right)-\frac{b}{a}+\frac{b}{a^2}+\frac{b}{2a}-\frac{c}{a}$. Based on (21), the restore value ${\widehat{y}}^{\left(0\right)}\left(k\right)$ can be obtained by

$${\widehat{y}}^{\left(0\right)}\left(k\right)={\widehat{y}}^{\left(1\right)}\left(k\right)-{\widehat{y}}^{\left(1\right)}\left(k-1\right)=C_1\left(1-e^a\right)e^{-a\left(k-1\right)}+\frac{b}{a},k=1,2,\dots ,n.$$

(22)

Then,

$${\widehat{x}}^{\left(0\right)}\left(k\right)=\frac{1}{{\widehat{y}}^{\left(0\right)}\left(k\right)}=\frac{a}{a{C}_1\left(1-e^a\right)e^{-a\left(k-1\right)}+b},k=1,2,\dots ,n.$$

(23)

The previous equation is the time response function of the N_Verhulst model.

Rather than directly estimating the model parameters $a$, $b$ and $c$ by the least-squares method, the N_Verhulst model calculates the undermined parameters according to the following simplified function. Let ${\sigma }_1=e^{-a}$, ${\sigma }_2=\frac{b}{a}\left(1-e^{-a}\right)$ and ${\sigma }_3=\frac{b}{a}+\left(\frac{c}{a}-\frac{b}{a^2}-\frac{b}{2a}\right)\left(1-e^{-a}\right)$. Then, (21) can be rewritten as

$$y^{\left(1\right)}\left(t\right)={\sigma }_1{y}^{\left(1\right)}\left(t-1\right)+{\sigma }_{2}t+{\sigma }_3.$$

(24)

Consequently, the undermined parameters ${\sigma }_1$, ${\sigma }_2$ and ${\sigma }_3$ can be obtained by

$${\left[{\sigma }_1, {\sigma }_2,{\sigma }_3\right]}^T={\left({\tilde{B}}^T\tilde{B}\right)}^{-1}{\tilde{B}}^T\tilde{Y},$$

(25)

where

$$\tilde{B}=\left[\begin{array}{ccc}\begin{array}{c}\begin{array}{c}{y}^{\left(1\right)}\left(1\right)\\ y^{\left(1\right)}\left(2\right)\end{array}\\ \begin{array}{c}⋮\\ y^{\left(1\right)}\left(n-1\right)\end{array}\end{array}& \begin{array}{c}\begin{array}{c}1\\ 2\end{array}\\ \begin{array}{c}⋮\\ n-1\end{array}\end{array}& \begin{array}{c}\begin{array}{c}1\\ 1\end{array}\\ \begin{array}{c}⋮\\ 1\end{array}\end{array}\end{array}\right],\tilde{Y}=\left[\begin{array}{c}\begin{array}{c}{y}^{\left(1\right)}\left(2\right)\\ y^{\left(1\right)}\left(3\right)\end{array}\\ \begin{array}{c}⋮\\ y^{\left(1\right)}\left(n\right)\end{array}\end{array}\right].$$

(26)

Once the undermined parameters ${\sigma }_1$, ${\sigma }_2$ and ${\sigma }_3$ are determined, the model parameters $a$, $b$ and $c$ can be obtained as follows:

$$a=-\ln {\sigma }_1,b=\frac{a{\sigma }_2}{1-{\sigma }_1},c=\frac{a{\sigma }_3-b}{1-{\sigma }_1}+\frac{b}{a}+\frac{b}{2}.$$

(27)

Then, the restore value ${\widehat{x}}^{\left(0\right)}\left(k\right)$ can be obtained by (23).

3. Unified Grey Riccati Model

In order to reduce the computational complexity of the time response expression of the grey generalized Verhulst model, this study transforms the Riccati Equation (14) into a first-order linear differential equation by use of a given particular solution and the change of dependent variable approach. Then, the new equation becomes a first-order linear differential equation and, therefore, can be solved explicitly. The final solution is the time response function of the grey Riccati model and is generally represented in a closed-form solution. Such a grey Riccati model is, therefore, termed the unified grey Riccati model in this study.

3.1. Time Response Function

Consider the whitenization function given in (14) and suppose that the particular solution is $x_p$. Since $x_p$ is a solution of (14), the following equation is true.

$$\frac{d}{dt}{x}_p+a{x}_p=b{x}_p^2+c.$$

(28)

In addition, since $c$ is a constant, the particular solution $x_p$ can be assumed to be a constant, say $K$. Substituting this into (28) will obtain $b{K}^2-aK+c=0$. Then, the particular solution can take any value of $K$, i.e.,

$$x_p=\frac{a\pm \sqrt{a^2-4bc}}{2b}$$

(29)

Once the model parameters are determined by (15), the particular solution $x_p$ can be obtained by (29). If $a^2-4bc\ge 0$, then the equation has at least one real root; otherwise, the equation has no real roots.

When the particular solution $x_p$ is known and real, the general solution of (14) is given by

$$x^{\left(1\right)}=x+x_p.$$

(30)

By substituting (28) and (30) into (14), we obtain

$$\frac{d}{dt}x+\left(a-2b{x}_p\right)x=b{x}^2.$$

(31)

Case 1.

${\mathit{a}}^2-4\mathit{b}\mathit{c}>0$

If $a^2-4bc>0$, then (31) is a Bernoulli Equation with the power exponent being 2. Make one more substitution with $v=x^{-1}$, divide the Bernoulli Equation (31) by $x^2$ and rewrite it in terms of$v$:

$$\frac{d}{dt}v-\left(a-2b{x}_p\right)v=-b.$$

(32)

The last equation is a linear ordinary differential equation, and the corresponding general solution is

$$v\left(t\right)=-C{e}^{\left(a-2b{x}_p\right)t}+\frac{b}{a-2b{x}_p},$$

(33)

where $C$ is an undetermined coefficient. Since $x=v^{-1}$, the general solution for $x$ is written as follows:

$$x\left(t\right)=\frac{a-2b{x}_p}{-C\left(a-2b{x}_p\right)e^{\left(a-2b{x}_p\right)t}+b}.$$

(34)

According to (30), the complete time response function of the grey Riccati model can be represented by

$$x^{\left(1\right)}\left(t\right)=\frac{a-2b{x}_p}{-C\left(a-2b{x}_p\right)e^{\left(a-2b{x}_p\right)t}+b}+x_p.$$

(35)

Subsequently, if the initial condition is $x_1=x^{\left(0\right)}\left(1\right)$, then the undetermined coefficient $C$ is obtained by

$$C=\frac{b\left(x_1+x_p\right)-a}{(x_1-x_p) \left(a-2b{x}_p\right)}{e}^{-\left(a-2b{x}_p\right)}.$$

(36)

Substituting (36) into (35) obtains

$${\widehat{x}}^{\left(1\right)}\left(t\right)=\frac{\left(a-2b{x}_p\right)\left(x_1-x_p\right)}{\left(a-b{x}_1-b{x}_p\right)e^{\left(a-2b{x}_p\right)\left(t-1\right)}+b\left(x_1-x_p\right)}+x_p.$$

(37)

Finally, let $\alpha =a-2b{x}_p$ and $\beta =x^{\left(0\right)}\left(1\right)-x_p$. Then, (37) can be reduced to

$${\widehat{x}}^{\left(1\right)}\left(k\right)=\frac{\alpha \beta }{\left(\alpha -b\beta \right)e^{\alpha \left(k-1\right)}+b\beta }+x_p,k=2,3,\dots ,n.$$

(38)

Note that substituting (29) into (38) could yield the complete response of the grey Riccati model, where the particular solution $x_p$ can take any value of (29). As can be seen, (38) is a closed-form solution and simpler than (17).

Case 2.

${\mathit{a}}^2-\mathit{4}\mathit{b}\mathit{c}=0$

If $a^2-4bc=0$, then $x_p=a/2b$ and (31) can be rewritten as

$$\frac{d}{dt}x=b{x}^2.$$

(39)

The general solution of (39) is

$$x\left(t\right)=-\frac{1}{bt+C}.$$

(40)

Analogously, the corresponding complete time response function of the grey Riccati model is

$$x^{\left(1\right)}\left(t\right)=-\frac{1}{bt+C}+x_p.$$

(41)

If the initial condition is $x_1=x^{\left(0\right)}\left(1\right)$, then the undetermined coefficient $C$ is obtained by

$$C=\frac{1}{x_p-x_1}-b.$$

(42)

Substituting (42) into (41) yields

$${\widehat{x}}^{\left(1\right)}\left(t\right)=\frac{x_1-x_p}{1+b\left(x_p-x_1\right)\left(t-1\right)}+x_p,$$

(43)

where $x_p=a/2b$. Obviously, (43) is also simpler than (18).

Case 3.

${\mathit{a}}^2-4\mathit{b}\mathit{c}<0$

If the particular solution $x_p$ is not a real root, then the restored values ${\widehat{x}}^{\left(0\right)}\left(k\right)$, $k=2,3,\dots ,n$, are directly obtained by the following grey differential equation:

$${\widehat{x}}^{\left(0\right)}\left(k\right)=-a{z}^{\left(1\right)}\left(k\right)+b{\left(z^{\left(1\right)}\left(k\right)\right)}^2+c,k=2,3,\dots ,n.$$

(44)

where the model parameters $\left[a, b, c\right]$ are also determined by (15).

Assume that the raw data sequence is $x^{\left(0\right)}=\left(x^{\left(0\right)}\left(1\right),x^{\left(0\right)}\left(2\right),\dots ,x^{\left(0\right)}\left(n\right)\right)$, where $x^{\left(0\right)}\left(k\right)\ge 0$, $k=1,2,\dots ,n$. Then, the modeling procedure of the unified grey Riccati model can be s briefly described as follows:

Step 1: Determine the 1-AGO sequence $x^{\left(1\right)}$ by $x^{\left(1\right)}\left(k\right)={{\displaystyle \sum }}_{i=1}^k{x}^{\left(0\right)}\left(i\right)$, $k=1,2,\dots ,n$, and the background values $z^{\left(1\right)}\left(k\right)$, $k=2,3,\dots ,n$, by (1).

Step 2: Estimate the model parameters ${\left[a, b,c\right]}^T$ by using the least-squares method given in (15) and (16).

Step 3: Calculate the particular solution $x_p$ by (29).

Step 4: Obtain the time response function of the unified grey Riccati model ${\widehat{x}}^{\left(1\right)}\left(k\right)$ by (38) if $a^2-4bc>0$, or (43) if $a^2-4bc=0$.

Step 5: Find the restored value of ${\widehat{x}}^{\left(0\right)}\left(k\right)$ by (5) if $a^2-4bc\ge 0$, or (44) if $a^2-4bc<0$.

3.2. Relationship between Unified Grey Riccati Model and GGVM

The proposed unified grey Riccati model and the grey generalized Verhulst model originated from the same whitenization function (14), but their solution representations are quite different. Theoretically, their modeling and forecasting values must be identical. The fact will be verified in the following.

If $a^2-4bc>0$, then the whitenization function of (14) has two distinct real roots, i.e., $x_p=\left(a-\lambda \right)/2b$ and $x_p=\left(a+\lambda \right)/2b$, where $\lambda =\sqrt{a^2-4bc}$. Take the solution $x_p=\left(a-\lambda \right)/2b$, i.e., $\lambda =a-2b{x}_p$, for example. Let $x_1=x^{\left(0\right)}\left(1\right)$ and substituting $\lambda =a-2b{x}_p$ into the term $e^{\lambda \left(k-\mathsf{\Phi }\right)}$ given in (17) yields

$$\begin{gathered} e^{\lambda \left(k-\mathsf{\Phi }\right)}=e^{\lambda k}{e}^{-\lambda \left(1-\frac{1}{\lambda }\ln \left|\frac{2b{x}^{\left(0\right)}\left(1\right)-a-\lambda }{2b{x}^{\left(0\right)}\left(1\right)-a+\lambda }\right|\right)}=e^{\lambda \left(k-1\right)}{e}^{\ln \left|\frac{2b{\mathrm{x}}_1-a-\lambda }{2b{\mathrm{x}}_1-a+\lambda }\right|} \\ =\left|\frac{b{x}_1-a+b{x}_p}{b{x}_1-b{x}_p}\right|e^{\left(a-2b{x}_p\right)\left(k-1\right)}. \end{gathered}$$

(45)

If $\frac{2b{x}_1-a-\lambda }{2b{x}_1-a+\lambda }<0$, i.e., $\left|\frac{2b{x}_1-a-\lambda }{2b{x}_1-a+\lambda }\right|=-\frac{b{x}_1-a+b{x}_p}{b{x}_1-b{x}_p}$, then the corresponding time response expression is

$$\begin{gathered} {\widehat{x}}^{\left(1\right)}\left(k\right)=\frac{a}{2b}-\frac{\lambda }{2b}\left(1-\frac{2}{e^{\lambda \left(k-\mathsf{\Phi }\right)}+1}\right)=\frac{a-\lambda }{2b} \\ +\frac{\lambda }{2b}\frac{2}{-\frac{b{x}_1-a+b{x}_p}{b{x}_1-b{x}_p}{e}^{\left(a-2b{x}_p\right)\left(k-1\right)}+1}=x_p+ \\ \frac{a-2b{x}_p}{2b}\frac{2\left(b{x}_1-b{x}_p\right)}{-\left(b{x}_1-a+b{x}_p\right)e^{\left(a-2b{x}_p\right)\left(k-1\right)}+\left(b{x}_1-b{x}_p\right)}=\frac{\left(a-2b{x}_p\right)\left(x_1-x_p\right)}{\left(a-b{x}_1-b{x}_p\right)e^{\left(a-2b{x}_p\right)\left(k-1\right)}+b\left(x_1-x_p\right)}+x_p. \end{gathered}$$

(46)

If $\frac{2b{x}_1-a-\lambda }{2b{x}_1-a+\lambda }>0$, i.e., $\left|\frac{2b{x}_1-a-\lambda }{2b{x}_1-a+\lambda }\right|=\frac{b{x}_1-a+b{x}_p}{b{x}_1-b{x}_p}$, then the time response expression is

$$\begin{gathered} {\widehat{x}}^{\left(1\right)}\left(k\right)=\frac{a}{2b} \\ -\frac{\lambda }{2b}\left(1+\frac{2}{e^{\lambda \left(k-\mathsf{\Phi }\right)}-1}\right)=\frac{a-\lambda }{2b}-\frac{\lambda }{2b}\frac{2}{\frac{b{x}_1-a+b{x}_p}{b{x}_1-b{x}_p}{e}^{\left(a-2b{x}_p\right)\left(k-1\right)}-1}=x_p- \\ \frac{a-2b{x}_p}{2b}\frac{2\left(b{x}_1-b{x}_p\right)}{\left(b{x}_1-a+b{x}_p\right)e^{\left(a-2b{x}_p\right)\left(k-1\right)}-\left(b{x}_1-b{x}_p\right)}= \\ \frac{\left(a-2b{x}_p\right)\left(x_1-x_p\right)}{\left(a-b{x}_1-b{x}_p\right)e^{\left(a-2b{x}_p\right)\left(k-1\right)}+b\left(x_1-x_p\right)}+x_p. \end{gathered}$$

(47)

It is obvious that (46) and (47) are identical to (38).

If $a^2-4bc=0$, then the whitenization function of (14) has a repeated real root, i.e., $x_p=a/2b$. If $a=\sqrt{4bc}$, i.e., $\sqrt{bc}=a/2$, the corresponding time response expression given in (18) is

$$\begin{gathered} {\widehat{x}}^{\left(1\right)}\left(k\right)=\frac{1}{b}\left(\sqrt{bc}-\frac{1}{k-\mathsf{\Phi }}\right)=\frac{1}{b}\left(\frac{a}{2}-\frac{1}{k-1-\frac{1}{b{x}_1-0.5a}}\right)=\frac{a}{2b}- \\ \frac{1}{b}\frac{b{x}_1-\frac{a}{2}}{\left(k-1\right)\left(b{x}_1-\frac{a}{2}\right)-1}=\frac{a}{2b}-\frac{x_1-\frac{a}{2b}}{b\left(k-1\right)\left(x_1-\frac{a}{2b}\right)-1}=\frac{x_1-x_p}{1+b\left(x_p-x_1\right)\left(k-1\right)}+x_p, \end{gathered}$$

(48)

where $\mathsf{\Phi }=1+1/\left(b{x}^{\left(0\right)}\left(1\right)-\sqrt{bc}\right)$. If $a=-\sqrt{4bc}$, i.e., $\sqrt{bc}=-a/2$, then the time response expression is

$$\begin{gathered} {\widehat{x}}^{\left(1\right)}\left(k\right)=-\frac{1}{b}\left(\sqrt{bc}+\frac{1}{k-\mathsf{\Phi }}\right)=-\frac{1}{b}\left(\frac{a}{2}+\frac{1}{k-1- \frac{1}{b{x}_1-0.5a}}\right)=\frac{a}{2b}- \\ \frac{1}{b}\frac{b{x}_1-\frac{a}{2}}{\left(k-1\right)\left(b{x}_1-\frac{a}{2}\right)-1}=\frac{x_1-x_p}{1+b\left(x_p-x_1\right)\left(k-1\right)}+x_p, \end{gathered}$$

(49)

where $\mathsf{\Phi }=1+1/\left(b{x}^{\left(0\right)}\left(1\right)+\sqrt{bc}\right)$. Obviously, (48) and (49) are identical to (43).

To sum up, the proposed unified grey Riccati model is equivalent to the grey generalized Verhulst model if $a^2-4bc\ge 0$. In addition, the former is simpler than the latter in terms of the time response expression. On the other hand, the grey Verhulst model and the grey Riccati model are generally suitable for the raw sequence with a bell-shaped or single peak feature or its 1-AGO sequence with the “S” distributing trend [9,11,23]. However, it can be seen from (19) that the time response expression of the grey generalized Verhulst model, in the case of $a^2-4bc<0$, is a shifted tangent function with the range being of ${\widehat{x}}^{\left(1\right)}\left(k\right)\in \left(-\infty ,-\left|\lambda /2b\right|\left]\cup \right[\left|\lambda /2b\right|,+\infty \right)$. The fact reveals that the grey generalized Verhulst model is inappropriate for modeling and forecasting the bell- or S-shaped sequences if $a^2-4bc<0$. This is the main reason why the restored value ${\widehat{x}}^{\left(0\right)}\left(k\right)$ of the proposed unified grey Riccati model does not derive from the whitenization function of (14) but originates directly from the grey differential Equation (13) if $a^2-4bc<0$.

4. Simulation Results

This section applies two numerical examples to verify the performance of the proposed unified grey Riccati model by comparing it with the traditional Verhulst model, the N_Verhulst model [7] and the grey generalized Verhulst model [20]. Although the data may be outdated, the main purpose of the given examples is for a performance comparison of the different models for the simulation and prediction of an approximately saturated S-shaped sequence. In addition, the absolute percentage error $\Delta \left(k\right)$ and MAPE (mean absolute percentage error) $\overline{\Delta }$ were employed to measure the simulation/forecasting performance as the following definitions [24]:

$$\Delta \left(k\right)=\frac{\left|x^{\left(0\right)}\left(k\right)-{\widehat{x}}^{\left(0\right)}\left(k\right)\right|}{x^{\left(0\right)}\left(k\right)}\times 100\%,$$

(50)

$$\mathrm{MAPE}\overline{\Delta }=\left(\frac{1}{n-1}{{\displaystyle \sum }}_{k=2}^n\frac{\left|x^{\left(0\right)}\left(k\right)-{\widehat{x}}^{\left(0\right)}\left(k\right)\right|}{x^{\left(0\right)}\left(k\right)}\right)\times 100\%.$$

(51)

4.1. Expenditures on the Research of a Certain Kind of Torpedo

Table 1. Actual and simulative results of accumulated expenditures on the research of a certain type of torpedo (Unit: million Yuan).

Year	Actual ${\mathit{x}}^{\left(1\right)}\left(\mathit{k}\right)$	Grey Riccati Model				Grey Verhulst Model
		Unified Grey Riccati Model		Grey Generalized Verhulst Model		Traditional Grey Verhulst Model		N_Verhulst Model
		${\widehat{\mathit{x}}}^{\left(1\right)}\left(\mathit{k}\right)$	$\Delta \left(\mathit{k}\right)$	${\widehat{\mathit{x}}}^{\left(1\right)}\left(\mathit{k}\right)$	$\Delta \left(\mathit{k}\right)$	${\widehat{\mathit{x}}}^{\left(1\right)}\left(\mathit{k}\right)$	$\Delta \left(\mathit{k}\right)$	${\widehat{\mathit{x}}}^{\left(1\right)}\left(\mathit{k}\right)$	$\Delta \left(\mathit{k}\right)$
1995	496	496	---	496	---	496	---	496	---
1996	1275	1326.97	4.07%	1326.97	4.07%	1119.11	12.22%	1279.39	0.34%
1997	2462	2390.05	2.92%	2390.05	2.92%	2116.01	14.05%	2413.37	1.98%
1998	3487	3348.45	3.97%	3348.45	3.97%	3177.48	8.87%	3467.09	0.57%
1999	3975	3975.56	0.01%	3975.56	0.01%	3913.73	1.54%	4077.57	2.58%
2000	4230	4304.57	1.76%	4304.57	1.76%	4286.18	1.32%	4339.57	2.59%
2001	4387	4457.36	1.60%	4457.36	1.60%	4444.80	1.31%	4437.33	1.15%
2002	4497	4524.29	0.60%	4524.29	0.60%	4507.36	0.23%	4471.88	0.56%
2003	4584	4552.85	0.67%	4552.85	0.67%	4531.27	1.15%	4483.85	2.18%
2004	4663	4564.91	2.10%	4564.91	2.10%	4540.31	2.63%	4487.97	3.75%
$\mathrm{MAPE}(\%)\overline{\Delta }$		1.97%		1.97%		4.81%		1.75%

lists the accumulated expenditures on the research of a certain type of torpedo [7]. Since the data trend shows a typical saturated S-shape characteristic, the data is suitable for modeling and forecasting with the grey Verhulst and grey Riccati models. While dealing with the sigmoid-type sequence, the raw data is generally regarded as the 1-AGO sequence $x^{\left(1\right)}$ and its 1-IAGO sequence is regarded as $x^{\left(0\right)}$ to construct a grey Verhulst model for simulating or forecasting the sequence $x^{\left(1\right)}$. That is, this example utilizes the sequence $x^{\left(0\right)}$ = (496, 779, 1187, 1025, 488, 255, 157, 110, 87, 79) to establish the grey model.

According to the modeling procedure given in Section 3, the model parameters of the grey Riccati model obtained by the least-squares method (15) and (16) are

$${\left[a, b,c\right]}^T=\left[\begin{array}{c}-0.72276\\ -0.00017422\\ 338.7214\end{array}\right].$$

Then, solving the equation $b{x}^2-ax+c=0$ will yield the particular solution as $x_p=4573.6244$ or $x_p=-425.08909$. Given $x^{\left(0\right)}\left(1\right)=496$ and let $x_p=4573.6244$, then

$$\alpha =a-2b{x}_p=0.8709,$$

$$\beta =x^{\left(0\right)}\left(1\right)-x_p=-4077.6244.$$

According to (30), the time response function of the proposed unified grey Riccati model is

$${\widehat{x}}^{\left(1\right)}\left(k\right)=\frac{-3551.139}{0.1605e^{0.8709\left(k-1\right)}+0.7104}+4573.6244,k=1,2,\dots ,n.$$

(52)

The unified grey Riccati model and the grey generalized Verhulst model have the same model parameters, i.e., $a=-0.72276$, $b=-0.00017422$ and $c=338.7214$. Since $a^2-4bc=0.8709>0$ and $\left(2b{x}^{\left(0\right)}\left(1\right)-a-\lambda \right)/\left(2b{x}^{\left(0\right)}\left(1\right)-a+\lambda \right)=-0.2259<0$, the time response function of the GGVM is then determined by (17), i.e.,

$${\widehat{x}}^{\left(1\right)}\left(k\right)=2074.2677+2499.3567\left(1-\frac{2}{e^{0.8709\left(k-2.7083\right)}+1}\right),$$

(53)

where $k=1,2,\dots ,n$. By neglecting the effect of the truncation error, it can be shown that (52) is identical to (53).

On the other hand, the model parameters of the traditional grey Verhulst model obtained by (12) are:

$${\left[a, b\right]}^T=\left[\begin{array}{c}-0.98079\\ -0.00021576\end{array}\right]$$

Substituting the above model parameters and the initial condition $x^{\left(1\right)}\left(1\right)=496$ into (11) will obtain the time response function of the grey Verhulst model as follows:

$${\widehat{x}}^{\left(1\right)}\left(k\right)=\frac{-486.47184}{-0.10702-0.87378e^{-0.98079\left(k-1\right)}},k=1,2,\dots ,n.$$

(54)

As far as the N_Verhulst model is considered, the undermined parameters ${\sigma }_1$, ${\sigma }_2$ and ${\sigma }_3$ obtained by (25) are

$${\left[{\sigma }_1, {\sigma }_2,{\sigma }_3\right]}^T=\left[\begin{array}{c}0.34289445\\ 0.00014634\\ 0.00196008\end{array}\right].$$

According to (27), we have the following model parameters: $a=1.070332$, $b=0.000238$ and $c=0.003171$. In addition, rather than using $x^{\left(0\right)}\left(1\right)=496$ as the initial value $C_0$, the initial value of the N_Verhulst model optimized by the least-squares method is

$$C_0=\frac{{{\displaystyle \sum }}_{k=2}^n\left[\left(y^{\left(0\right)}\left(k\right)-\frac{b}{a}\right)e^{-ak}\right]}{{{\displaystyle \sum }}_{k=2}^n\left[\left(1-e^a\right)e^{-2ak}\right]}=-0.002481$$

(55)

As reported in [7], the time response function of the N_Verhulst model is:

$${\widehat{x}}^{\left(0\right)}\left(k\right)=\frac{1}{0.0007816\times {0.3428945}^{k-2}+{{\displaystyle \sum }}_{i=1}^{k-2}0.0001463\times {0.3428945}^{i-1}},$$

(56)

where $k=2,3,\dots ,n$.

On the basis of the above time response functions (52), (53), (54) and (56), the corresponding simulated values are also given in Table 1. As can be seen, the proposed unified grey Riccati model and the grey generalized Verhulst model have the same forecasting results. In addition, the MAPEs of the proposed grey Riccati model, the traditional grey Verhulst model, and the N_Verhulst model were 1.97%, 4.81%, and 1.75%, respectively. That is to say, the proposed unified grey Riccati model performs as well as the N_Verhulst model and is better than the traditional grey Verhulst model. Note, however, that the initial value of the N_Verhulst model $C_0$ is optimized by the least-squares method, as given in (55). However, the undetermined coefficient of the unified grey Riccati model $C$, given in (36), is directly obtained from $x^{\left(0\right)}\left(1\right)$, not the optimal value. Despite this, the unified grey Riccati model performs almost the same as the N_Verhulst model. The simulation results also reveal that the proposed approach actually reduces the residual error of the traditional grey Verhulst model, and the proposed unified grey Riccati model performs a good modeling ability for the S-shaped sequences. In particular, the unified grey Riccati model is simpler than the GGCM in the computational complexity of solving the time response function.

4.2. Large and Medium-Sized Agricultural Tractors

The number of large and medium-sized agricultural tractors used in Henan Province from 1978 to 1983 is depicted in

Table 2. Actual and simulative numbers of large and medium-sized agricultural tractors used in Henan Province (in ten thousand).

Year	Actual ${\mathit{x}}^{\left(1\right)}\left(\mathit{k}\right)$	Grey Riccati Model				Grey Verhulst Model
		Unified Grey Riccati Model		Grey Generalized Verhulst Model		Traditional Grey Verhulst Model		N_Verhulst Model
		${\widehat{\mathit{x}}}^{\left(1\right)}\left(\mathit{k}\right)$	$\Delta \left(k\right)$	${\widehat{\mathit{x}}}^{\left(1\right)}\left(\mathit{k}\right)$	$\Delta \left(\mathit{k}\right)$	${\widehat{\mathit{x}}}^{\left(1\right)}\left(\mathit{k}\right)$	$\Delta \left(\mathit{k}\right)$	${\widehat{\mathit{x}}}^{\left(1\right)}\left(\mathit{k}\right)$	$\Delta \left(\mathit{k}\right)$
In-sample (Simulation)
1978	4.1299	4.1299	---	4.1299	---	4.1299	---	4.1299	---
1979	5.2382	5.1780	1.14%	5.1780	1.14%	5.2605	0.42%	5.2276	0.20%
1980	5.9666	5.9655	0.02%	5.9655	0.02%	5.9255	0.68%	6.0450	1.31%
1981	6.4590	6.2807	2.76%	6.2807	2.76%	6.2494	3.24%	6.3202	2.15%
1982	6.3160	6.3741	0.91%	6.3741	0.91%	6.3927	1.21%	6.4014	1.35%
$\mathrm{MAPE}(\%)\overline{\Delta }$		1.21%		1.21%		1.39%		1.25%
Out-of-sample (Prediction)
1983	6.4389	6.3991	0.61%	6.3991	0.61%	6.4786	0.62%	6.4244	0.23%

[7,25]. According to the suggestion given in [7], the data from 1978 to 1982 are regarded as the training samples to construct the grey models, while the data in 1983 is used to verify the forecasting performance of the models. The characteristic of the original data is similar to a sigmoid curve, such that the given data is suitable for modeling and forecasting with the grey Verhulst model. Analogously, the raw data is regarded as the 1-AGO sequence $x^{\left(1\right)}$ and its 1-IAGO sequence as $x^{\left(0\right)}$ in the simulation and prediction. With this suggestion, the sequence used to establish the grey model is $x^{\left(0\right)}$ = (4.1299, 1.1083, 0.7284, 0.4924, −0.1430).

In this example, the model parameters of the unified grey Riccati model estimated by the least-squares method are:

$${\left[a, b,c\right]}^T=\left[\begin{array}{c}-3.99680\\ -0.41743\\ -8.47130\end{array}\right].$$

Once the model parameters are determined, the particular solutions of the whitenization function, obtained by (29), are $x_p=6.4079$ and $x_p=3.1670$. In addition, given $x^{\left(0\right)}\left(1\right)=4.1299$ and let $x_p=6.4079$, we have $\alpha =1.3528$ and $\beta =-2.2780$. Then, the time response function of the proposed unified grey Riccati model is:

$${\widehat{x}}^{\left(1\right)}\left(k\right)=\frac{-3.0818}{0.4019e^{1.3528\left(k-1\right)}+0.9509}+6.4079,k=1,2,\dots ,n.$$

(57)

Furthermore, it can be seen from the previous model parameters that $a^2-4bc=1.8302>0$ and $\left(2b{x}^{\left(0\right)}\left(1\right)-a-\lambda \right)/\left(2b{x}^{\left(0\right)}\left(1\right)-a+\lambda \right)=-0.4227<0$. Therefore, the time response function of the GGVM is still determined by (17), i.e.,

$${\widehat{x}}^{\left(1\right)}\left(k\right)=4.7875+1.6205\left(1-\frac{2}{e^{1.3528\left(k-1.6365\right)}+1}\right),$$

(58)

where $k=1,2,\dots ,n$. Analogously, (58) is identical to (57).

As far as the traditional grey Verhulst model is considered, the model parameters obtained by (12) are

$${\left[a, b\right]}^T=\left[\begin{array}{c}-0.89165\\ -0.13726\end{array}\right].$$

According to (11), the time response function of the grey Verhulst model is

$${\widehat{x}}^{\left(1\right)}\left(k\right)=\frac{-3.6824}{-0.5669-0.3248e^{-0.89165\left(k-1\right)}},k=1,2,\dots ,n.$$

(59)

All the simulated values are also given in Table 2, where the information about the N_Verhulst model is obtained from [7]. Again, the unified grey Riccati model and the grey generalized Verhulst model have the same modeling (in-sample) and forecasting (out-of-sample) results. Moreover, it can be seen that MAPEs of the unified grey Riccati model, the N_Verhulst model and the traditional grey Verhulst model are 1.21%, 1.25% and 1.39%, respectively, on the in-sample simulation. The proposed unified grey Riccati model performs slightly better than the N_Verhulst and the grey Verhulst models. Table 2 also lists the predicted value for the number of large and medium-sized tractors used in agriculture in Henan Province in 1983. The predicted value determined by the proposed grey Riccati model is 6.3991. The corresponding absolute percentage error is 0.61%. In other words, the forecasting accuracy reached 99.39%, and the unified grey Riccati model could attain a good forecasting ability. The traditional grey Verhulst model also performs a good forecasting ability with an accuracy of 99.38%, while the N_Verhulst model has the best forecasting accuracy of 99.77%.

4.3. Discussions

The numerical results of the previous two examples demonstrate that the proposed unified grey Riccati model performs the same as the GGVM and is slightly better than the traditional grey Verhulst model on the in-sample simulation and out-of-sample prediction. The fact that the unified grey Riccati model performs the same as the GGVM is also identical to the theoretical analysis given in Section 3.2, but the former is simpler than the latter on the computational complexity of solving the time response function.

The initial value of the N_Verhulst model $C_0$ is optimized by the least-squares method, but the undetermined coefficient of the unified grey Riccati model $C$ is determined by the initial condition $x^{\left(0\right)}\left(1\right)$ only. Even so, the proposed unified grey Riccati model performs almost the same as the N_Verhulst model, as shown in Section 4.1. In addition, the numerical results given in Section 4.2 reveal that the unified grey Riccati model outperforms the N_Verhulst model on the in-sample simulation, but the former is worse than the latter on the out-of-sample prediction. That is to say, for these two grey models, each has its own strengths.

To sum up, the unified grey Riccati model performs the same as the GGVM, but the proposed grey model has a low computational complexity in solving the time response function. Moreover, the proposed unified grey Riccati model also outperforms the traditional grey Verhulst model and performs as well as the N_Verhulst model on the given two numerical examples.

5. Conclusions

This study proposes the unified grey Riccati model to reduce the computational complexity of the time response expression of the GGVM. Although the solution representations of the unified grey Riccati model and GGVM are quite different, their modeling and forecasting values are identical. The numerical results of the two examples showed that the proposed unified grey Riccati model performs the same as the GGVM, better than the traditional grey Verhulst model, and performs as well as the N_Verhulst model. The fact also reveals that the proposed approach actually reduces the residual error of the traditional grey Verhulst model. Furthermore, especially under the same modeling and forecasting performances, the unified grey Riccati model is simpler than the GGVM on the computational complexity of solving the time response function.

Some future works of this study may concern the following issues in the near future. The first issue is how to further enhance the modeling and forecasting performances of the unified grey Riccati model. The issue may be solved by optimizing the undetermined coefficient $C$ as the N_Verhulst model does or other optimization methods. The second one is applied to the real-time time series forecasting problem to demonstrate the practical application ability of the proposed grey model.