Grey Model Prediction Enhancement via Bernoulli Equation with Dynamic Polynomial Terms

Abstract

The grey prediction model is designed to characterize systems comprising both partially known information (referred to as white) and partially unknown dynamics (referred to as black). However, traditional GM(1,1) models are based on linear differential equations, which limits their capacity to capture nonlinear and non-stationary behaviors. To address this issue, this paper develops a generalized grey differential prediction approach based on the Bernoulli equation framework. We incorporate the Bernoulli mechanism with a nonlinear exponent n and a dynamic polynomial-driven term. In this work, we propose a new model designated as BPGM(1,1). Another key innovation of this work is the adoption of a nonlinear least squares direct parameter identification strategy to calculate the exponent and polynomial parameters in the Bernoulli equation, which achieves a higher degree of freedom in parameter selection and effectively circumvents the model distortion issues caused by traditional background value estimation. Furthermore, the Euler discretization method is utilized for numerical solving, reducing the reliance on traditional analytical solutions for linear structures. Numerical experiments indicate that BPGM(1,1) surpasses GM(1,1), NFBM(1,1), and their improved versions. By leveraging the synergistic mechanism between Bernoulli-type nonlinear regulation and polynomial-driven external excitation, this framework significantly enhances prediction accuracy for systems characterized by non-stationary behaviors and multi-scale trends.

1. Introduction

The grey system theory was originally proposed by Professor Deng [1,2], with its core idea centered on integrating deterministic structures (the white part) with empirical inference (the black part) to model and analyze systems characterized by partially known and partially unknown information. Specifically, grey prediction models construct differential equations with well-defined mathematical structures as the white component, while incorporating unknown information inferred from observed data as the black component. This approach enables the discovery of essential system patterns from limited information. The most well-known model in grey system theory is GM(1,1).

Let an original non-negative and uniformly spaced function be

$$X^{\left(0\right)}=\{x^{\left(0\right)}\left(1\right),x^{\left(0\right)}\left(2\right),\cdots ,x^{\left(0\right)}\left(n\right)\}.$$

(1)

The classical GM(1,1) model employs an accumulated generating operation (AGO) on the original data sequence to reduce randomness and produce a distinctly monotonic increasing 1-AGO sequence, denoted as $X^{\left(1\right)}$. Based on this transformed sequence, a first-order differential equation with a step-degree term is formulated. An exact solution is then derived for the equation containing unknown parameters. The least squares approach estimates the unknown parameters, while numerical methods solve the differential equation. Afterward, the derived parameters are substituted into the analytical solution to formulate the grey forecasting equation based on observed data. Ultimately, the inverse accumulated generating operation (IAGO) restores the original dataset, facilitating both reconstruction and future trend prediction.

The 1-AGO sequence $X^{\left(1\right)}$ is given as follows

$$X^{\left(1\right)}=\{x^{\left(1\right)}\left(1\right),x^{\left(1\right)}\left(2\right),\dots ,x^{\left(1\right)}\left(n\right)\},$$

(2)

as   

$$x^{\left(1\right)}\left(k\right)=\sum _{i=0}^k{x}^{\left(0\right)}\left(i\right)=x^{\left(1\right)}(k-1)+x^{\left(0\right)}\left(k\right),x^{\left(1\right)}\left(0\right):=0,k=1,2,\dots ,n.$$

(3)

By this equation, the 1-AGO transformed sequence maintains an increasing trend, which leads us to express its mathematical form as follows

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

(4)

a denotes the grey developmental coefficient, and b represents the grey action quantity, both parameters (a and b) need to be estimated. By incorporating the initial condition $x^{\left(1\right)}\left(1\right)=x^{\left(0\right)}\left(1\right)$, the analytical form of the differential equation’s solution can be obtained. The specific form is as follows

$$x^{\left(1\right)}\left(t\right)=[x^{\left(1\right)}\left(1\right)-{\displaystyle \frac{b}{a}}]e^{-a(t-1)}+{\displaystyle \frac{b}{a}}.$$

(5)

Thus, determining the original grey prediction model hinges on computing a and b. Hence, performing the integral operation on $[k,k+1],k=1,2,\dots ,n-1$, the original differential equation yields the following expression

$${\int }_k^{k+1}{\displaystyle \frac{d{x}^{\left(1\right)}\left(t\right)}{dt}}dt+a{\int }_k^{k+1}{x}^{\left(1\right)}\left(t\right)dt=b,$$

(6)

which is   

$$x^{\left(1\right)}(k+1)-x^{\left(1\right)}\left(k\right)+a{\int }_k^{k+1}{x}^{\left(1\right)}\left(t\right)dt=b.$$

(7)

As we know, the difference between two consecutive terms is $x^{\left(0\right)}\left(k\right)$; therefore, the above equation can be rewritten as

$$x^{\left(0\right)}(k+1)+a{\int }_k^{k+1}{x}^{\left(1\right)}\left(t\right)dt=b.$$

(8)

For simplicity of expression, we can define $z^{\left(1\right)}(k+1)={\int }_k^{k+1}{x}^{\left(1\right)}\left(t\right)dt$. Thus, the information of the first term in the equation has already been provided by the original data. By determining the value of $z^{\left(1\right)}(k+1)$ for different values of k, and treating a and b as the least squares coefficients, we can apply the least squares method, and use the formula to obtain the estimated values of a and b.

However, the difficulty lies in the fact that the computation of $z^{\left(1\right)}(k+1)$ depends on the specific values of a and b, leading to circular reasoning. Therefore, we consider using numerical methods to approximate the values of $z^{\left(1\right)}(k+1)$. In GM(1,1), the computation of $z^{\left(1\right)}(k+1)$ relies on linear polynomial interpolation $L\left(t\right)=(k+1-t)x^{\left(1\right)}\left(k\right)+(t-k)x^{\left(1\right)}(k+1)$ to obtain the numerical form of $x^{\left(1\right)}\left(t\right)$, and is computed using integration methods, with the specific formula as follows

$$z^{\left(1\right)}(k+1)\approx {\int }_k^{k+1}L\left(t\right)dt={\displaystyle \frac{1}{2}}[x^{\left(1\right)}\left(k\right)+x^{\left(1\right)}(k+1)].$$

(9)

At this point, the numerical solution of the integral is substituted back into Equation (8). For different values of k, we assume $z_k$ and $y_k$ as the dependent variable, and a and b as the least squares fitting coefficients. A least squares fitting equation is established, where the normal equations are as follows

$$\left(\begin{array}{c}a\\ b\end{array}\right)={\left(G^{T}G\right)}^{-1}{G}^{T}X,$$

(10)

where   

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

(11)

Substituting $\widehat{a},\widehat{b}$ into Equation (5), and obtaining the final solution to the differential equation for the 1-AGO sequence, the specific form is as follows

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

(12)

Finally, by subtracting the estimated values of two consecutive terms in the 1-AGO sequence, the estimated values of the original data are derived, as expressed in the following formula

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

(13)

Theoretical analysis indicates that the estimation of the background value has a significant impact on the prediction performance of the model, which has led many researchers to focus their attention on the numerical integration methods for background value calculation. Li and Dai [3] proposed using a higher-order Newton interpolation polynomial. Tang and Xiang [4] established a piecewise quadratic interpolation polynomial to resolve this dilemma. Wang et al. [5], Yang et al. [6], and Zhu et al. [7] applied different cubic spline interpolation techniques for high-precision background value estimation. In addition to improving background value estimation, many scholars have also focused on refining the grey action quantity. Xi et al. [8] innovatively reconstructed the grey action quantity as a multi-parameter hyperbolic delay function. In [9], a dual-parameter polynomial structure was designed. Both [10,11] incorporate polynomial terms into their models to enhance nonlinear modeling capabilities; however, due to computational constraints, the polynomial order is limited to three in both cases. In [12], a composite grey action quantity with dynamic frequency response was developed, surpassing the traditional linear assumption; however, it introduced a significant computational burden.

The grey models generally used include the GM(1,1) [13], grey model with the first order equation and N variables (GM(1,N)) [14], and grey model with Nth order equation and one variable (GM(N,1)) [15]. At the same time, various optimized algorithms have been derived from grey models, such as unbiased grey model (UGM(1,1)) [16], fractional time-delayed grey model (FTDGM(1,1)) [17], fractional grey model (FGM(1,1)) [18], and the nonlinear Bernoulli model (NGBM(1,1)) [19]. Among them, NGBM(1,1) has great potential for improvement and optimization. Zhang and Chen [20] improved modeling accuracy by refining background values and initial conditions. However, during the background value optimization process, while expressing the curve as a nonhomogeneous exponential function, the exponential component of the time response was overlooked. Ding et al. [21] proposed an improved stepwise Bernoulli model, utilizing the Simpson algorithm to enhance the estimation of $z^{\left(1\right)}(k+1)$ and employing the particle swarm optimization algorithm for parameter computation. Zhou et al. [22] optimized both the exponent and initial conditions and achieved a certain level of accuracy improvement. In [23], an innovative application of game theory in the parameter optimization of a novel two-stage Nash NGBM(1,1) was proposed. Zeng et al. [24] developed a matrix-based NGBM(1,1) for interval sequences, deriving recursive prediction formulas using Cramer’s rule. In [25], an optimization scheme based on the Fourier series was introduced, first computing error values using NGBM(1,1), followed by Fourier series filtering for secondary computation to achieve high-quality prediction. In [26], NGBM(1,1) with linearly time-varying parameters was introduced for predicting non-equidistant time series. In [27], Ma et al. established a multivariate NGBM(1,1), incorporating nonlinear equations and transforming them into a linear form, extending the computational approach of NGBM(1,1). In [28], the paper focused on the physical background of data, replacing traditional time-series accumulation with physical operators, enabling high-quality prediction in real-world contexts. In [29], based on the Bernoulli framework, Cheng and Bin proposed a new model in which the grey action quantity is improved to a quadratic polynomial and designed a least squares algorithm for parameter estimation.

However, GM (1,1), based on the Bernoulli equation mentioned above, only includes specified lower-order polynomials asan improvement of the grey action quantity, so the generalization ability of these resulting models is limited. Therefore, this work aims to incorporate unrestricted adaptive-order polynomials into the Bernoulli equation framework to achieve high-quality nonlinear fitting of the data. Additionally, nonlinear least squares estimation is employed instead of the normal equation method to reduce computational load and improve accuracy. The original differential equation is also discretized, eliminating the dependence on analytical solutions for prediction computation, which significantly reduces computational complexity.

In Section 2, a novel Bernoulli model with polynomial-driven algorithm (BPGM(1,1)) will be established. In Section 3, numerical examples will be compared and analyzed. In Section 4, we present our conclusions.

2. Establish BPGM(1,1)

A nonlinear differential equation of the following form is known as a Bernoulli equation

$${\displaystyle \frac{dy}{dx}}+P\left(x\right)y=Q\left(x\right)y^n.$$

(14)

For the given Bernoulli equation, dividing both sides by $y^n$ yields

$$y^{-n}{\displaystyle \frac{dy}{dx}}+P\left(x\right)y^{1-n}=Q\left(x\right).$$

(15)

By introducing the substitution $z=y^{1-n}$, the Bernoulli equation can be transformed into the following form

$$z^{\prime }+(1-n)P\left(x\right)z=(1-n)Q\left(x\right).$$

(16)

Based on this, we introduce the Bernoulli equation into the grey prediction model and modify it as follows

$$\left\{\begin{array}{c}y=x^{\left(1\right)}\left(t\right),\hfill \\ x=t,\hfill \\ P\left(x\right)=a,\hfill \\ Q\left(x\right)=b_0+b_{1}t+\cdots +b_m{t}^m.\hfill \end{array}\right.$$

(17)

Thus, we obtain the improved Bernoulli equation-based grey prediction model (BPGM(1,1)), which is expressed as follows

$${\displaystyle \frac{d{x}^{\left(1\right)}\left(t\right)}{dt}}+a{x}^{\left(1\right)}\left(t\right)=(b_0+b_{1}t+\cdots +b_m{t}^m){\left[x^{\left(1\right)}\left(t\right)\right]}^n.$$

(18)

This Bernoulli equation integrates both the endogenous growth and exogenous driving mechanisms of the system. The polynomial term represents the time-varying external excitation, where the degree of the polynomial reflects the level of nonlinear fluctuations in the external influence. Meanwhile, ${\left[x^{\left(1\right)}\left(t\right)\right]}^n$ captures the system’s nonlinear self-feedback.

When n = 0 and m = 0, the model reduces to GM(1,1), see the references [1,2]. When m = 0, the model reduces to NGBM(1,1), see [19].

Therefore, BPGM(1,1) possesses greater flexibility in solution formulation and lays the foundation for high-precision forecasting.

Through a similar derivation and computation process, the analytical solution of the above differential equation can be obtained in the following form

$$x^{\left(1\right)}\left(t\right)={[e^{-(1-n)at}((1-n)\sum _{j=0}^m{\displaystyle \frac{b_j}{{\left[(1-n)a\right]}^{j+1}}}\sum _{i=0}^j{(-1)}^i{\displaystyle \frac{j!}{i!}}{t}^{j-i}+\mathit{C})]}^{{\displaystyle \frac{1}{1-n}}},$$

(19)

the value of the constant C can be determined from the initial condition $x^{\left(0\right)}\left(1\right)=x\left(1\right)$.

At this point, the key issue lies in estimating the values of the parameters. Traditional methods integrate the original grey prediction equation, estimate the background value, and use the least squares method to solve the normal equation to obtain the estimated parameter values. Unlike traditional approaches, this work employs a nonlinear least squares estimation method to estimate $a,b_0,b_1,\dots ,b_m,n$. This method provides a more robust parameter estimation for handling nonlinear data and data noise. We define the following formula as the residual expression

$$mi{n}_{a,\left\{b_i\right\},n}\sum _{k=1}^h{\left(x_{\mathrm{pred}}\left(k\right)-x_{\mathrm{data}}\left(k\right)\right)}^2.$$

(20)

To solve this nonlinear least squares problem, the Levenberg–Marquardt (LM) algorithm [30] is introduced. This is a numerical method that combines Gradient Descent and Newton’s Method. The specific formulation is as follows

$${\theta }^{(k+1)}={\theta }^{\left(k\right)}-{\left[J^{T}J+\lambda I\right]}^{-1}{J}^{T}r,$$

(21)

where $\theta$ represents the parameter vector, J is the Jacobian matrix of the residual function, r is the residual vector, and $\lambda$ is a damping parameter that adjusts the balance between the Gradient Descent and Newton’s Method.

By substituting the estimated parameters into the solution of the differential equation, the predicted values can then be obtained as follows

$${\widehat{x}}^{\left(1\right)}\left(t\right)={[e^{-(1-\widehat{n})\widehat{a}t}((1-\widehat{n})\sum _{j=0}^m{\displaystyle \frac{\widehat{b_j}}{{\left[(1-\widehat{n})\widehat{a}\right]}^{j+1}}}\sum _{i=0}^j{(-1)}^i{\displaystyle \frac{j!}{i!}}{t}^{j-i}+C)]}^{{\displaystyle \frac{1}{1-\widehat{n}}}}.$$

(22)

Since the value of n is unknown beforehand, directly substituting the estimated n obtained from the nonlinear least squares estimation into the equation may lead to cases where n is very close to 1. This would cause $1-n$ to approach 0, creating a risk of exponential explosion when substituted into the exact solution, thereby significantly affecting prediction accuracy.

Moreover, the original analytical solution is highly complex, requiring two nested iterative computations for each predicted value, leading to high computational complexity. Therefore, in this work, we do not use the exact analytical solution. Instead, we discretize the original differential equation into the following form

$${\displaystyle \frac{d{x}^{\left(1\right)}\left(t\right)}{dt}}=-a{x}^{\left(1\right)}\left(t\right)+(b_0+b_{1}t+\cdots +b_m{t}^m){\left(x^{\left(1\right)}\left(t\right)\right)}^n.$$

(23)

The equation can be rewritten as

$$x^{\left(1\right)}(k+1)=x^{\left(1\right)}\left(k\right)+[-a{x}^{\left(1\right)}\left(k\right)+(b_0+b_1{t}_k+\cdots +b_m{t}_k^m)]{\left[x^{\left(1\right)}\left(k\right)\right]}^n.$$

(24)

Therefore, substituting the estimated parameters obtained from nonlinear least squares estimation into Equation (22) improves the data utilization efficiency while reducing computational complexity and avoiding issues such as exponential explosion. Once the $x^{\left(1\right)}\left(t\right)$ time series is obtained, the original $x^{\left(0\right)}\left(t\right)$ sequence can be recovered through cumulative subtraction, thereby completing the forecasting process.

At this point, we have fully constructed BPGM(1,1). Summarizing the above discussion, we give the following algorithm 1 for establishing BPGM(1,1).

Algorithm 1 The implementation process of BPGM(1,1)

Input: original non-negative and uniformly-spaced sequence $X^{\left(0\right)}=\{x^{\left(0\right)}\left(1\right),x^{\left(0\right)}\left(2\right),\cdots ,x^{\left(0\right)}\left(n\right)\}$. Output: grey prediction formula ${\widehat{x}}^{\left(0\right)}\left(k\right),k=1,2,\cdots$.   1: Compute the 1-AGO sequence $X^{\left(1\right)}=\{x^{\left(1\right)}\left(1\right),x^{\left(1\right)}\left(2\right),\cdots ,x^{\left(1\right)}\left(n\right)\}$ by Equation (3);   2: Set initial values for parameters $\theta =(a,b_0,b_1,\cdots ,b_m,n)$;   3: Define residual function based on Equation (20);   4: Solve nonlinear least squares optimization by LM algorithm;   5: Extract estimated parameters $\widehat{a}$, $\widehat{b_0},\widehat{b_1},\dots ,\widehat{b_m}$, and $\widehat{n}$;   6: Compute the predicted sequence ${\widehat{x}}^{\left(1\right)}\left(k\right)$ iteratively by Equation (24);   7: Obtain the predicted values of the original sequence ${\widehat{x}}^{\left(0\right)}\left(k\right)$ by Equation (13).

3. Numerical Experiment

In this section, we will present several examples to demonstrate that the BPGM(1,1) achieves higher predictive accuracy compared to GM(1,1) and other improved methods. In the following examples, the relative error is computed by

$$\epsilon ={\displaystyle \frac{\left|{\widehat{x}}^{\left(0\right)}\left(k\right)-x^{\left(0\right)}\left(k\right)\right|}{x^{\left(0\right)}\left(k\right)}}\times 100\%,k=1,2,\dots ,h.$$

(25)

the average error is computed by

$$\overline{ϵ}={\displaystyle \frac{{\sum }_{k=2}^h\left|{\widehat{x}}^{\left(0\right)}\left(k\right)-x^{\left(0\right)}\left(k\right)\right|}{h-1}}.$$

(26)

Example 1.

In this example, we adopt the grey forecasting data dataset from [7], which contains syphilis patient health statistics in China from 2000 to 2010 (unit: millions). The original work in [7] achieved significant error reduction by employing cubic monotonicity-preserving interpolation splines for high-precision background value estimation. As shown in 

Table 1. Numerical results for Example 1.

${\mathit{x}}^{\left(0\right)}$	GM(1,1)		The Model in [7]		BPGM(1,1)
	Predicted Value	$\mathbf{ϵ}$ (%)	Predicted Value	$\mathbf{ϵ}$ (%)	Predicted Value	$\mathbf{ϵ}$ (%)
5.08	5.08	0.00	5.08	0.00	5.08	0.00
4.80	4.25	11.38	4.24	11.51	4.97	3.60
4.67	5.26	12.55	5.25	12.57	4.11	12.08
4.50	6.49	44.33	5.51	22.44	5.05	12.18
7.12	8.03	12.72	8.05	13.09	7.05	0.99
9.67	9.92	2.55	9.97	3.05	9.66	0.09
12.80	12.25	4.27	12.33	3.64	12.67	1.05
15.88	15.14	4.65	15.26	3.88	15.95	0.41
19.19	18.71	4.00	18.89	3.07	19.44	0.28
23.07	23.12	0.22	23.38	1.34	23.09	0.10
26.86	28.57	6.36	28.93	7.72	26.90	0.13
$\overline{ϵ}$ (%)		9.03		7.48		2.81

, the BPGM(1,1) demonstrates a significant improvement in forecasting accuracy compared to both GM(1,1) and the model in [7]. These results confirm that when simulating complex epidemiological time series, the optimized Bernoulli model can better capture the trend of the data. The numerical results of GM(1,1) and BPGM(1,1) are shown in Table 1. The comparison of the fitting performance of both methods on the original data is illustrated in Figure 1. At this point, m = 2, n = 0.1154.

Example 2.

In this example, we utilize the market clearing price (MCP) dataset from [9] as the forecasting target to conduct a comprehensive comparison among three prediction frameworks: (i) GM(1,1), (ii) the OFOPGM model proposed in [9], and (iii) BPGM(1,1) in this paper. Through systematic parameter optimization, the polynomial degree m = 9 is identified as the optimal configuration that minimizes prediction errors. Although [9] modifies the grey action quantity into two adaptive exponential forms, providing a stronger fitting capability for nonlinear data compared to GM(1,1), the model introduced in this work, based on high-order polynomials and the Bernoulli equation, can effectively capture the inherent multi-scale variations in highly fluctuating and non-stationary time series, thereby significantly improving fitting accuracy. It also demonstrates that BPGM(1,1) performs excellently in long-term sample forecasting. The numerical results of GM(1,1) and BPGM(1,1) are shown in 

Table 2. Numerical results for Example 2.

${\mathit{x}}^{\left(0\right)}$	GM(1,1)		The Model in [9]		BPGM(1,1)
	Predicted Value	$\mathbf{ϵ}$ (%)	Predicted Value	$\mathbf{ϵ}$ (%)	Predicted Value	$\mathbf{ϵ}$ (%)
2024.25	2024.25	0.00	2024.25	0.00	2024.25	0.00
1986.57	2751.61	38.51	1734.92	12.67	1954.01	1.64
1935.58	2774.73	43.35	1935.58	0.00	2024.30	4.58
1918.93	2798.04	45.81	2179.64	13.59	1873.85	2.35
1968.74	2821.55	43.32	2436.27	23.75	1899.01	3.54
2199.92	2845.26	29.33	2687.44	22.16	2221.96	1.00
2719.32	2869.17	5.51	2921.80	7.45	2762.95	1.60
3285.38	2893.28	11.93	3132.11	4.67	3368.91	2.54
3854.09	2917.59	24.30	3313.71	14.02	3871.30	0.45
4330.08	2942.10	32.05	3463.76	20.01	4136.56	4.47
4162.72	2966.82	28.73	3580.63	13.98	4119.36	1.04
3768.91	2991.75	20.62	3663.54	2.80	3882.59	3.02
3395.73	3016.89	11.16	3712.29	9.32	3563.01	4.93
3415.93	3042.24	10.94	3727.10	9.11	3302.79	3.31
3249.56	3067.80	5.59	3708.48	14.12	3187.91	1.90
3142.49	3093.58	1.56	3657.11	16.38	3220.85	2.49
3486.03	3119.57	10.51	3573.78	2.52	3330.86	4.45
3375.06	3145.78	6.79	3459.36	2.50	3412.14	1.10
3212.71	3172.22	1.26	3314.74	3.18	3378.26	5.15
3219.66	3198.87	0.65	3140.83	2.45	3216.89	0.09
3172.57	3225.75	1.68	2938.51	7.38	3011.93	5.06
2924.39	3252.85	11.23	2708.67	7.38	2880.56	1.50
2614.87	3280.19	25.44	2452.13	6.22	2762.60	5.65
2113.71	3307.75	56.49	2169.69	2.65	1993.39	5.69
$\overline{ϵ}$ (%)		19.45		9.09		2.81

. The comparison of the fitting performance of both methods on the original data is illustrated in Figure 2. At this point, m = 9, n = 0.5034.

Example 3.

This example utilizes the annual electricity consumption data of China from 2004 to 2017 (unit: trillion kWh) provided in [11] to construct the test set. A systematic comparison was conducted to evaluate the performance of three grey prediction models: (i) GM(1,1), (ii) the improved GM(2,1) model based on differential polynomials proposed in [11], (iii) BPGM(1,1) in this paper. Through parameter sensitivity analysis, it was found that when the polynomial order was set to m = 7, the MAPE of BPGM(1,1) decreased to 0.63%, which was significantly better than GM(1,1) (MAPE = 4.30%) and the model in [11] (MAPE = 1.88%). This optimization result indicates that, under the synergistic effect of the exponential parameter of the Bernoulli model and the adaptive polynomial, the model effectively captures the nonlinear and multi-scale variation characteristics in electricity consumption data. Moreover, it successfully avoids the issues in [11], such as the computational complexity of double accumulation and the limitation of polynomial order to three, achieving a simpler and more efficient computation. The numerical results of the GM(1,1) and BPGM(1,1) are shown in 

Table 3. Numerical results for Example 3.

${\mathit{x}}^{\left(0\right)}$	GM(1,1)		The Model in [11]		BPGM(1,1)
	Predicted Value	$\mathbf{ϵ}$ (%)	Predicted Value	$\mathbf{ϵ}$ (%)	Predicted Value	$\mathbf{ϵ}$ (%)
21,971.37	21,971.37	0.00	21,971.37	0.00	21,971.37	0.00
24,940.32	28,566.58	14.54	24,940.32	0.00	24,935.22	0.02
28,587.97	30,703.53	7.40	28,511.11	0.27	28,552.91	0.12
32,711.81	33,000.34	0.88	32,042.62	2.05	32,872.42	0.49
34,541.35	35,468.97	2.69	35,443.53	2.61	34,100.09	1.28
37,032.14	38,122.27	2.94	38,703.49	4.51	37,589.60	1.51
41,934.49	40,974.05	2.29	41,932.67	0.00	41,755.69	0.43
47,000.88	44,039.15	6.30	45,309.84	3.60	46,441.53	1.19
49,762.64	47,333.55	4.88	48,950.38	1.63	50,573.10	1.63
54,203.41	50,874.39	6.14	52,780.55	2.63	53,806.96	0.73
56,383.69	54,680.11	3.02	56,531.47	0.26	56,198.50	0.33
58,019.97	58,770.52	1.29	59,903.10	3.25	58,346.91	0.56
61,297.09	63,166.91	3.05	62,817.39	2.48	61,055.42	0.39
64,820.97	67,892.18	4.74	65,570.70	1.16	64,904.37	0.13
$\overline{ϵ}$ (%)		4.30		1.88		0.63

. The comparison of the fitting performance of both methods on the original data is illustrated in Figure 3. At this point, m = 7, n = $-0.0669$.

Example 4.

In this example, we use the 1994–1999 production value of optoelectronic components in Taiwan’s optoelectronics industry, as presented in [19]. The work in [19] was the first to propose a grey prediction equation based on the Bernoulli model. As shown in 

Table 4. Numerical results for Example 4.

${\mathit{x}}^{\left(0\right)}$	GM(1,1)		The Model in [19]		BPGM(1,1)
	Predicted Value	$\mathbf{ϵ}$ (%)	Predicted Value	$\mathbf{ϵ}$ (%)	Predicted Value	$\mathbf{ϵ}$ (%)
20.30	20.30	0.00	20.30	0.00	20.30	0.00
29.50	26.19	11.23	29.50	0.01	29.50	0.00
31.10	31.20	0.33	30.81	0.92	31.10	0.00
37.20	37.18	0.06	36.06	3.05	37.20	0.00
38.20	44.30	15.96	43.92	14.97	38.20	0.00
57.50	52.78	8.21	54.53	5.16	57.50	0.00
$\overline{ϵ}$ (%)		7.16		4.82		0.00

, when the parameter is set to m = 3, BPGM(1,1) demonstrates exceptional performance in fitting accuracy, achieving an almost one-to-one fit within the permissible error range. This result indicates that the introduction of the polynomial term has an outstanding impact, significantly enhancing the model’s capability to capture variations in time series data. The comparison of the fitting performance of both methods on the original data is illustrated in Figure 4. At this point, m = 3, n = 0.7090.

Example 5.

In this example, we use the 2010–2019 global net electricity generation data from [21] as the test set. The work in [21] proposed an optimized Bernoulli model, which improves background value calculation accuracy using the Simpson rule and enhances stability by employing the Particle Swarm Optimization (PSO) algorithm for automatic parameter tuning. As shown in 

Table 5. Numerical results for Example 5.

${\mathit{x}}^{\left(0\right)}$	GM(1,1)		The Model in [21]		BPGM(1,1)
	Predicted Value	$\mathbf{ϵ}$ (%)	Predicted Value	$\mathbf{ϵ}$ (%)	Predicted Value	$\mathbf{ϵ}$ (%)
33.32	33.32	0.00	33.32	0.00	33.32	0.00
66.16	90.73	37.14	56.62	14.42	66.16	0.00
103.83	117.91	13.56	93.40	10.05	103.84	0.01
145.28	153.24	5.48	139.11	4.25	145.17	0.08
198.91	199.15	0.12	194.98	1.98	199.33	0.21
251.71	258.82	2.82	262.84	4.42	250.71	0.40
328.33	336.36	2.45	345.10	5.11	329.80	0.45
439.19	437.14	0.47	444.85	1.29	437.74	0.33
565.94	568.11	0.38	565.94	0.00	566.84	0.16
700.48	738.32	5.40	713.14	1.81	700.10	0.05
$\overline{ϵ}$ (%)		7.53		4.81		0.17

, BPGM(1,1) significantly enhances the model’s accuracy in capturing fluctuation patterns in complex systems, demonstrating outstanding predictive performance. The comparison of the fitting performance of both methods on the original data is illustrated in Figure 5. At this point, m = 6, n = $-0.6925$.

Example 6.

In this example, the test set is constructed based on the historical data of annual water consumption in Wuhan from 2005 to 2012, as presented in [25]. The work in [25] proposed an entirely new Bernoulli model, which first calculates the prediction error using the traditional NGBM(1,1). Then, this approach employs Fourier expansion analysis on the error sequence in the frequency domain, selecting low-frequency components for secondary computation to achieve a more precise fitting effect. As shown in 

Table 6. Numerical results for Example 6.

${\mathit{x}}^{\left(0\right)}$	GM(1,1)		The Model in [25]		BPGM(1,1)
	Predicted Value	$\mathbf{ϵ}$ (%)	Predicted Value	$\mathbf{ϵ}$ (%)	Predicted Value	$\mathbf{ϵ}$ (%)
367,204	367,204	0.00	367,204	0.00	367,204	0.00
369,355	360,128	2.50	369,355	0.00	369,352	0.00
363,883	365,835	0.54	363,883	0.00	363,880	0.00
360,467	371,634	3.10	360,467	0.00	360,463	0.00
377,944	377,523	0.11	377,944	0.00	377,939	0.00
379,346	383,507	1.10	379,346	0.00	379,338	0.00
397,345	389,585	1.95	397,345	0.00	397,334	0.00
395,713	395,760	0.01	395,713	0.00	395,702	0.00
$\overline{ϵ}$ (%)		1.16		0.00		0.00

, within the two-decimal-place error range, both BPGM(1,1) and the model in [25] achieve an astonishing 0% prediction error. Considering that the model in [25] introduces a complex Fourier series and requires two iterative calculations, the computational cost associated with the Fourier series and its matrix operations increases significantly, especially when dealing with long-term sequence data. In contrast, BPGM(1,1) is computationally efficient and easy to implement, making it more competitive for real-world applications. The comparison of the fitting performance of both methods on the original data is illustrated in Figure 6. At this point, m = 13, n = 0.7167.

Example 7.

In this example, the test data are selected from the average observed weekday traffic flow values from 29 May to 4 June 2017, as presented in [28], to evaluate the model’s effectiveness in traffic flow prediction. In [28], a novel Bernoulli optimization model was developed, introducing the physical conservation Cusum operator to ensure that the model adheres to the physical conservation principle, thereby providing a more accurate representation of the underlying dynamic patterns in the original time series. As shown in 

Table 7. Numerical results for Example 7.

${\mathit{x}}^{\left(0\right)}$	GM(1,1)		The Model in [28]		BPGM(1,1)
	Predicted Value	$\mathbf{ϵ}$ (%)	Predicted Value	$\mathbf{ϵ}$ (%)	Predicted Value	$\mathbf{ϵ}$ (%)
13.20	13.20	0.00	11.68	11.52	13.20	0.00
13.80	25.81	87.03	13.86	0.43	13.43	2.71
16.40	25.91	57.99	16.25	0.91	16.04	2.20
18.20	26.02	42.97	18.81	3.35	18.29	0.49
19.40	26.13	34.69	21.48	10.72	21.01	8.32
23.60	26.23	11.14	24.17	2.42	24.11	2.17
28.60	26.34	7.90	26.79	6.33	27.20	4.90
32.60	26.44	18.90	29.24	10.31	29.93	8.20
29.20	26.55	9.08	31.40	7.53	32.09	9.89
34.00	26.66	21.59	33.18	2.41	33.62	1.13
34.60	26.77	22.63	34.51	0.26	34.54	0.17
35.20	26.88	23.64	35.33	0.37	34.95	0.70
32.80	26.99	17.71	35.61	8.57	34.96	6.57
35.40	27.10	23.45	35.36	0.11	34.64	2.15
35.00	27.21	22.26	34.62	1.09	34.05	2.71
34.80	27.32	21.49	33.44	3.91	33.21	4.57
31.60	27.43	13.20	31.90	0.95	32.10	1.57
29.80	27.54	7.58	30.08	0.94	30.66	2.90
28.40	27.65	2.64	28.07	1.16	28.87	1.66
25.80	27.76	7.60	25.93	0.50	26.70	3.50
25.00	27.88	11.52	23.75	5.00	24.18	3.26
22.40	27.99	24.96	21.58	3.66	21.43	4.34
18.40	28.10	52.72	19.47	5.82	18.66	1.39
15.80	28.22	78.62	17.46	10.51	16.22	2.65
$\overline{ϵ}$ (%)		25.89		4.12		3.26

, BPGM(1,1) achieves the lowest average error, indicating that it maintains exceptionally high computational efficiency even when applied to long-term traffic flow prediction. The comparison of the fitting performance of both methods on the original data is illustrated in Figure 7. At this point, m = 6, n = 0.6303.

Combining the results of the seven numerical examples above, along with the error comparison between BPGM(1,1) and GM(1,1) shown in Figure 8, the average error of the seven numerical examples decreases by 89%, and the fitting accuracy is significantly improved, demonstrating outstanding predictive capability. Compared with examples 1, 2, and 3, BPGM(1,1) exhibits an outstanding performance over methods that adjust background values, refine the grey action quantity, and optimize the model into GM(2,1). Similarly, in comparison with examples 4, 5, 6, and 7, BPGM(1,1) achieves extremely high optimization accuracy relative to NGBM(1,1) and its improved variants, while avoiding excessive computational complexity. The numerical experiments confirm that introducing higher-order polynomials does not lead to the Runge phenomenon. Instead, when combined with the Bernoulli model, it effectively enhances the model’s ability to approximate data, enabling high-precision forecasting for nonlinear and fluctuating datasets.

4. Conclusions

In this work, we have improved the classical GM(1,1) through a dual-innovation mechanism, placing the grey prediction model within the broader framework of the Bernoulli equation and employing a data-driven polynomial grey action quantity, thereby formulating BPGM(1,1). Based on the Bernoulli model when $n>1$, the model exhibits superlinear feedback capability, which can accommodate scenarios such as the Allee effect in population growth within ecological models and autocatalytic explosive growth in chemical reactions. When $0<n<1$, the model represents sublinear feedback, making it suitable for fitting saturation growth under resource constraints, as well as sub-growth infection rates in disease transmission. When $n<0$, the model exhibits inhibitory feedback, allowing it to capture the inverse regulatory effects of policy interventions on growth in economic models. By incorporating a time-varying externally driven adaptive-order polynomial, BPGM(1,1) effectively addresses nonlinear dynamic forecasting problems in complex systems. Moreover, we employ the nonlinear least squares LM algorithm to directly optimize the parameter vector, thereby avoiding the high computational cost associated with traditional algorithms that estimate the background value and solve parameters through matrix inversion. Benefiting from the superior performance of the LM algorithm, the computational complexity is significantly reduced. Furthermore, in the final solving stage, the numerical solution based on first-order explicit Euler discretization is used to replace the analytically derived double-nested summation, which also substantially lowers the time cost of computation. This facilitates the application of BPGM(1,1) to complex real-world scenarios that demand high model dynamism and real-time performance.

Future research could explore integrating other uncertainty modeling methods or concepts from machine learning to further improve prediction accuracy and enhance the model’s ability to generalize to complex scenarios. For instance, it might be valuable to consider adaptively selecting the initial parameters of the iterative process based on the characteristics of the data, to avoid falling into local minima. Additionally, the improved model could be applied to real-time data scenarios in dynamic systems, such as energy demand forecasting or environmental monitoring, to validate its timeliness.