Fractional Order Grey Model of Optimization Investment Allocation for Maximum Value Addition in Beijing’s High-Tech Industries

Abstract

High-tech industries are of strategic importance to the national economy, and Beijing has been designated as a science and technology innovation center by the State Council. Accurate analysis of its added value is crucial for technological development. While recent data enhance prediction accuracy, its limited volume poses challenges. The cumulative grey Lotka–Volterra model and grey differential dynamic multivariate model address this by leveraging short-term data effectively. This study applies these two models to analyze influencing factors and predict Beijing’s high-tech industry growth. Results show a competitive relationship with four systems, lacking synergy. In the next five years, a mutually beneficial trend is expected. The Mean Absolute Percentage Error (MAPE) remains within 10%, confirming the model’s reliability.

1. Introduction

The Fifth Plenary Session of the 19th Central Committee of the Communist Party of China pointed out that we must adhere to the core position of innovation in the overall modernization construction of our country, deeply implement the strategy of rejuvenating the country through science and education, the strategy of innovation driven development, improve the national innovation system, enhance the technological innovation capability of enterprises, and improve the system and mechanism of scientific and technological innovation [1]. China should take the creation of an innovative country as the starting point, promote scientific and technological innovation from the aspects of institutional mechanisms, resource elements, etc., and rely on scientific and technological innovation to provide adjustments for the high-quality development of the Chinese economy. Scientific and technological innovation will become an inherent force to promote sustainable social and economic development [2].

The 14th Five Year Plan for National Economic and Social Development of the People’s Republic of China and the 2035 Long Range Objectives Outline propose to accelerate the green transformation of development mode, adhere to ecological priority and green development, promote total resource management, scientific allocation, comprehensive conservation, and circular utilization, and work together to promote high-quality economic development and high-level ecological environment protection. As knowledge and technology-intensive industries, high-tech industries are important carriers for promoting high-quality economic development and leading technological innovation. Due to factors such as resource endowment, knowledge spillover, and government policies, the development of high-tech industries will form industrial agglomeration phenomena at a certain stage, specifically manifested as the agglomeration of high-tech enterprises, talents, technology, and other resources. The positive and negative externalities caused by agglomeration effects significantly affect changes in other factors, such as corporate innovation behavior, capital investment, and human resources [3].

Due to their knowledge- and technology-intensive attributes, high-tech industries are an important force in China’s economic development. Developing high-tech industries is one of the important ways to accelerate the development of China’s modern industrial system and consolidate and strengthen the foundation of the real economy. The high-tech industry is an emerging industry based on high-tech, supported by innovation, and led by the development and production of high-tech products. The Beijing-Tianjin-Hebei region is rich in technological innovation resources and has a strong industrial foundation. It is one of the most dynamic and promising regions in China and also one of the three regions with great potential for economic growth. Beijing, located in the Beijing-Tianjin-Hebei region, serves as the political center, cultural center, international communication center, and science and technology innovation center of China, possessing abundant scientific and technological innovation resources and high-tech industries.

Therefore, exploring the relationship between the added value of the high-tech industry and its influencing factors in depth is crucial for optimizing investment allocation and maximizing its value. Most scholars use multiple linear regression [4], a combination of correlation testing and linear regression [5], and social network cohesion subgroup algorithms when studying the impact on the added value of high-tech industries. Feng Qing and Shi Xiongtian applied static and dynamic efficiency measurement models to study the overall and regional changes in innovation efficiency of China’s high-tech industries from 2013 to 2021 and analyzed the factors affecting innovation efficiency of high-tech industries [6]. Yang Dandan and Luo Feng used the Trade Competitive Advantage (TC) index to evaluate the international competitiveness of high-tech industries in the Guangdong Province and analyzed the influencing factors of the international competitiveness of high-tech industries in the Guangdong Province by constructing a vector autoregression model based on the diamond model theory [7]. Tang Bo, Chen Deguan, and others used POI data as a basis to explore the spatial evolution and influencing factors of high-tech industries in Foshan, Guangdong Province, using kernel density analysis, spatial autocorrelation, multiple linear regression, and other methods [8]. Zhang Xiao and Ge Yuhui took high-tech industries in 28 provinces, municipalities, and autonomous regions in China as research objects, constructed a DEA-BCC model and a Malmquist index to conduct dynamic and static analysis of innovation efficiency, and established a Tobit panel model to deeply analyze the influencing factors of innovation efficiency [9]. Zeng Wujia et al. conducted a study on the innovation efficiency and influencing factors of 103 national high-tech industrial development zones in China using panel data and a three-stage DEA Tobit analysis method [10]. Li Jian et al. used the Super SBM model that considers unexpected output to measure the green innovation efficiency of high-tech industries in panel data from 30 provinces in China. Based on this, ArcGIS visualization tools were used to analyze their spatiotemporal differentiation characteristics, and the GMM dynamic panel model was used to test the influencing factors of green innovation efficiency [11]. Zhang et al. used a two-stage DEA Tobit model to empirically analyze the five factors that affect the two-stage efficiency of industry university research collaborative innovation in different industries of high-tech industries [12].

A review of the literature on the added value of the high-tech industry and its influencing factors reveals that scholars have rarely applied relevant models from grey system theory for analysis. Grey system theory, pioneered by Professor Deng Julong, is a novel approach to addressing uncertainty in small-data and information-deficient scenarios. It focuses on systems where some information is known while other parts remain unknown, extracting valuable insights from partial data to enable accurate description and analysis [13]. Therefore, this study employs the ‘grey Lotka–Volterra model’ and the ‘grey differential dynamic multivariate model’, which are designed for analyzing multifactor relationships, to examine the interaction between the added value of Beijing’s high-tech industry and its influencing factors, as well as its development trends over the next five years. By leveraging limited short-term data, these models provide more precise predictive analysis.

The Lotka–Volterra model was originally used to analyze the interrelationships between biological populations. Scholars have introduced it into the grey system model to broaden the research field of the model, forming the grey Lotka–Volterra model [14]. In terms of the application of the grey Lotka–Volterra model, Guo et al. used the grey Lotka–Volterra model to analyze the relationship between water resources, energy, industry, and technological innovation in the Beijing-Tianjin-Hebei region and proposed suggestions based on the analysis results [15]. Mao et al. analyzed the relationship between China’s third-party payment system and online banking using the grey Lotka–Volterra model [16]. Yuan Zhihua et al. applied the grey Lotka–Volterra model to the study of microbial synergy in biological uranium leaching, providing a theoretical basis for improving the efficiency of biological uranium leaching [17]. Han Ruiwen et al. applied the Lotka–Volterra model to study and analyze the stability of data among large, medium, and small enterprises [18]. Anindya S. Chakrabarti applied the Lotka–Volterra model equation to simulate the evolution of technological frontiers and demonstrated that the diffusion of technology has had a ripple effect in partner economies [19]. Xia et al. introduced the Lotka–Volterra model to explore the internal structure of technological innovation in China’s high-tech industry from six indicators: independent innovation, technology introduction, research and development, technological innovation, external technology acquisition, and domestic technology procurement [20]. Zhang et al. studied the competition and cooperation between technological innovation, resource consumption, environmental quality, and industrial development quality in the Shaanxi Province by constructing a four-dimensional grey Lotka–Volterra model and discussed the equilibrium point and stability of the four-dimensional grey Lotka–Volterra model [21].

The fractional order cumulative grey Lotka–Volterra model (FGLV) is a combination of the grey Lotka–Volterra model and the fractional order cumulative generation operator. Compared with the original grey Lotka–Volterra model, the fractional order cumulative grey Lotka–Volterra model is more in line with the new information first principle and is more suitable for solving practical problems.

Recent research on fractional-order grey models has significantly advanced predictive accuracy in various domains. Duman and Kongar (2025) introduced the Hausdorff fractional NGBM (r,1), enhancing flexibility and computational efficiency by eliminating the Gamma function, and demonstrated superior predictive accuracy in e-waste forecasting [22]. Li and Xie (2024) proposed a reduced-order discrete grey forecasting model, refining its internal mechanisms and simplifying modeling steps while ensuring high prediction accuracy, particularly in battery capacity degradation [23]. Wu et al. (2024) developed a conformable fractional-order grey Bernoulli model optimized for energy consumption forecasting in Chongqing, outperforming traditional models [24]. These advancements align with broader applications of fractional-order models, such as smoothed functional algorithms for Hammerstein models, reaction curve-based chemical process modeling, and fractional calculus in separation processes, reinforcing their efficacy in small-data prediction. The fractional-order Hammerstein model for identifying continuous time proposed by Mok, R., and Ahmad, M. A. [25] and the general recognition method of fractional-order process based on process response curve proposed by Gude, J. J., and García Bringas, P. effectively demonstrate that the application of fractional-order accumulation operator has unique advantages in the prediction of low-data problems [26].

The grey differential dynamic multivariate model is a prediction model in the grey multivariate model, among which the grey multivariate GM(1,N) model is proposed to address the problem that the traditional GM(1,1) model cannot achieve accurate prediction of various influencing factors in real life. Xie Naiming and Liu Sifeng found that the GM(1,N) model has low accuracy in long-term development prediction and thus proposed a discrete grey multivariate model [27]. Wang et al. proposed a multivariate grey GM(1,N) power model and its derivative models to improve the modeling performance of feature sequences in multivariate nonlinear systems [28]. The Grey Differential Dynamic Model [29] was initially proposed by Academician Xia Jun using Professor Deng Julong’s Grey System Theory as a hydrological system model, with the general form being DHGM(n + 1, l + 1). The grey differential dynamic multivariate model was proposed by Duan Huiming et al. based on the grey differential dynamic prediction model, considering the many factors that affect the main sequence, and extended it to apply multiple variables, greatly enriching the types of grey multivariate prediction models [30]. Under this context, fractional grey forecasting model will be applied to optimize investment allocation for maximum value addition in Beijing’s high-tech industries.

This study makes the following key contributions:

Innovative Application of Grey System Models—This research applies the fractional-order cumulative grey Lotka–Volterra model to analyze the historical influence of four related systems on the added value of Beijing’s high-tech industry.

Enhanced Predictive Accuracy—By employing the grey differential dynamic multivariate model, this study provides a more precise forecast of future relationships between the high-tech industry and its influencing factors.

Strategic Investment Optimization—Based on the analytical results, this study offers strategic recommendations for optimizing investment allocation to maximize the future value of Beijing’s high-tech industry.

These contributions provide valuable insights for policymakers and investors in fostering the sustainable growth of high-tech industries.

2. Selection of Research Regions and Indicators

2.1. Research Area

This article takes Beijing as the research object, which is the capital, municipality directly under the central government, national central city, and mega city of the People’s Republic of China. It has been approved by the State Council as the political center, cultural center, international communication center, and science and technology innovation center of China.

Beijing is located in the north of China and the north of the North China Plain, adjacent to Tianjin in the east, and adjacent to Hebei Province in the rest. Its center is located at 116°20′ E and 39°56′ N. It is a world-famous ancient capital and modern international city. The specific geographical location of Beijing is shown in Figure 1.

Beijing is the largest scientific and technological research base in China, with scientific research institutions such as the Chinese Academy of Sciences and Beijing Zhongguancun Science Park, known as the Silicon Valley of China. Every year, one-third of the national awards are awarded to Beijing. Since 1998, a large-scale international event with the theme of high-tech industry has been successfully held every year—Beijing High-Tech Industry International Week.

On 17 November 2022, it was identified by the China National Intellectual Property Administration as a pilot place to carry out data intellectual property work. The pilot work period was from November 2022 to December 2023.

In 2022, the number of patent authorizations in Beijing was 203,000, an increase of 2.0% over the previous year. Among them, the number of authorized invention patents was 88,000, an increase of 11.3%. At the end of the year, there were 478,000 valid invention patents, an increase of 18.0%. The number of PCT international patent applications was 11,463, an increase of 10.7%. The number of high-value invention patents per 10,000 population was 112.0, an increase of 17.8 from the previous year. A total of 95,061 technology contracts were recognized and registered throughout the year, an increase of 1.6%. The transaction volume of technology contracts was 794.75 billion yuan, an increase of 13.4% [31]. In 2022, Beijing ranked among the top three global international science and technology innovation centers for the first time, with a comprehensive score of 80.39.

2.2. Indicator Selection

High-tech enterprises are those recognized by relevant national authorities, including the departments of science and technology, finance, and taxation. These enterprises typically include both those located within national high-tech zones and certified enterprises outside these zones. The primary data on enterprises within national high-tech zones are provided by the Ministry of Science and Technology and can be referenced in the science and technology section of the ‘China Statistical Yearbook’. Based on reviewing a large amount of the literature, this article selects ten secondary indicator systems from four aspects that affect the added value of high-tech industries, including human capital system, capital system, technology system, and trade system. The human capital system includes three secondary indicators, R&D (research and experimental development) activity personnel equivalent to full-time (referring to the workload calculated by R&D personnel based on their actual R&D activity time during the reporting period, measured in person years), total labor productivity (referring to the value-added of high-tech industrial enterprises divided by the average number of employed personnel during the same period), and participating researchers. The funding system includes internal R&D expenses (all expenses actually incurred in implementing R&D activities, divided into daily expenses and asset expenses according to the nature of the expenses), government funding (government funding for R&D research), and enterprise funding (enterprise funding for R&D research). The technical system includes the number of patent applications and the number of invention patents owned. The trade system includes the export volume of high-tech products and the import volume of high-tech products.

This article uses the added value of high-tech industries in Beijing to reflect the changes in the city’s high-tech industry. The added value of high-tech industries from 2016 to 2021 forms the original sequence X, and the 10 secondary indicator data from 2016 to 2021 form the original sequences ${\mathrm{Y}}_1-{\mathrm{Y}}_{10}$. All selected data are from the Beijing Statistical Yearbook, and the indicator system is represented in

Table 1. Indicator System.

	First Level Indicator	Secondary Indicator
Value added of high-tech industries $X$	Human capital system	R&D activity personnel converted to full-time equivalent $Y_1$
		Total labor productivity $Y_2$
		Researcher $Y_3$
	Financial system	Internal R&D expenditure $Y_4$
		Government funding $Y_5$
		Enterprise capital investment $Y_6$
	Technological system	Number of patent applications $Y_7$
		Invention patents $Y_8$
	Trade system	High-tech product trade export value $Y_9$
		High-tech product trade import amount $Y_{10}$

.

3. Relationship Model

3.1. Fractional Order Cumulative Grey Lotka–Volterra Model

Definition 1.

Assuming the existence of non-negative equidistant sequences

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

The $Y^{(0)}$ sequence is generated by the fractional order accumulation of $X^{(0)}$ and is

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

The fractional order accumulation operator ($r-\mathrm{AGO}$) is defined as

$$\begin{array}{l}{x}^{(r)}(k)={\displaystyle \sum _{i=1}^k{C}_{k-i+r-1}^{k-i}}{x}^{(0)}(i), k=1, 2, \cdots , n\\ y^{(r)}(k)={\displaystyle \sum _{i=1}^k{C}_{k-i+r-1}^{k-i}}{y}^{(0)}(i), k=1, 2, \cdots , n\end{array}$$

(1)

Among them, $C_{k-i+r-1}^{k-i}={\displaystyle \frac{(k-i+r-1)(k-i+r-2)\cdots (r+2)(r+1)r}{(k-i)!}}$ ($k=1, 2, \cdots , n$), regulation $C_{r-1}^0$ $=1\hspace{0.17em}$, $C_k^{k+1}=0\hspace{0.17em}, x^{(r)}(1)=x^{(0)}(1)\hspace{0.17em}, y^{(r)}(1)=y^{(0)}(1)$.

Taking the $X^{(0)}$ sequence as an example, the matrix form of fractional order accumulation can be represented as

$$\left[\begin{array}{c}{x}^{(r)}(1)\\ x^{(r)}(2)\\ x^{(r)}(3)\\ ⋮\\ x^{(r)}(n)\end{array}\right]=\left[\begin{array}{ccccc}1& 0& \cdots & 0& 0\\ C_r^1& 1& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ C_{r+n-3}^{n-2}& C_{r+n-4}^{n-3}& \cdots & 1& 0\\ C_{r+n-2}^{n-1}& C_{r+n-3}^{n-2}& \cdots & C_r^1& 1\end{array}\right]\left[\begin{array}{c}{x}^{(0)}(1)\\ x^{(0)}(2)\\ x^{(0)}(3)\\ ⋮\\ x^{(0)}(n)\end{array}\right].$$

(2)

Using $x^{(0)}(k)$ and $y^{(0)}(k)$ as research variables to explore $Y^{(r)}$, a long-term relationship between $X^{(r)}$ and the system can be found. According to the grey modeling method, at time $k$, the derivative of $X^{(r)}$ and the derivative of $Y^{(r)}$ are

$$\begin{array}{l}dx(k)=x^{(r)}(k+1)-x^{(r)}(k), k=1, 2, \cdots \\ dy(k)=y^{(r)}(k+1)-y^{(r)}(k), k=1, 2, \cdots \end{array}$$

(3)

The $x^{(r)}$ continuous average generated sequence is $z_x^{(r)}(k)={\displaystyle \frac{x^{(r)}(k)+x^{(r)}(k+1)}{2}}$, and the $y^{(r)}$ continuous average generated sequence is $z_y^{(r)}(k)={\displaystyle \frac{y^{(r)}(k)+y^{(r)}(k+1)}{2}}$, $k=1, 2, \cdots , n-1$.

The FGLV model is

$$\begin{array}{l}{x}^{(0)}(k+1)\approx a_1{z}_x^{(r)}(k)-b_1(z_x^{(r)}(k){)}^2-c_1{z}_x^{(r)}(k)z_y^{(r)}(k)\\ y^{(0)}(k+1)\approx a_2{z}_y^{(r)}(k)-b_2(z_y^{(r)}(k){)}^2-c_2{z}_y^{(r)}(k)z_x^{(r)}(k)\end{array}$$

(4)

Among them, $a_i$ is the $i$ growth parameter $i$ of the species when living alone, $b_i$ is the ecological niche capacity limitation parameter of the species, and is the $c_i$ interaction parameter with other species. The interaction parameters represent the interrelationships between indicators, and the specific interrelationships represented by the interaction parameters are shown in

Table 2. Interrelationships between factors.

${\mathit{c}}_1$	${\mathit{c}}_2$	Correlations
>0	>0	Collaborative relationship
>0	<0	Competitive relationship
<0	>0	Competitive relationship
<0	<0	Complementary relationship

.

Take $X^{(0)}$ sequence as an example:

Given error sequence

$${\epsilon }_k=x^{(0)}(k+1)-a_1{z}_x^{(\xi )}(k)+b_1{(z_x^{(\xi )}(k))}^2+c_1{z}_x^{(\xi )}(k)z_y^{(\xi )}(k)$$

(5)

Estimate the parameters of the FGLV model using the least squares method, with the formula being

$$\left[\begin{array}{c}{\widehat{a}}_1\\ {\widehat{b}}_1\\ {\widehat{c}}_1\end{array}\right]={(B^{T}B)}^{-1}{B}^{T}T$$

(6)

of which

$$T=\left[\begin{array}{c}{x}^{(0)}(2)\\ x^{(0)}(3)\\ ⋮\\ x^{(0)}(n)\end{array}\right], B=\left[\begin{array}{ccc}{z}_x^{(r)}(2)& -{\left[z_x^{(r)}(2)\right]}^2& -z_x^{(r)}(2)z_y^{(r)}(2)\\ \begin{array}{c}{z}_x^{(r)}(3)\\ ⋮\end{array}& \begin{array}{c}-{\left[z_x^{(r)}(3)\right]}^2\\ ⋮\end{array}& \begin{array}{c}-z_x^{(r)}(3)z_y^{(r)}(3)\\ ⋮\end{array}\\ z_x^{(r)}(n)& -{\left[z_x^{(r)}(n)\right]}^2& -z_x^{(r)}(n)z_y^{(r)}(n)\end{array}\right].$$

(7)

To accurately calculate the $c_1$ values of $a_1,b_1$ and when the MAPE value is minimized, the objective function is modified to

$$\mathrm{Min}: {\displaystyle \frac{1}{n-1}}{\displaystyle \sum _{k=1}^{n-1}\frac{\left|x^{(0)}(k+1)-a_1{z}_x^{(r)}(k)+b_1{(z_x^{(r)}(k))}^2+c_1{z}_x^{(r)}(k)z_y^{(r)}(k)\right|}{x^{(r)}(k+1)}}$$

(8)

The discrete FGLV model is

$${\widehat{x}}^{(r)}(k+1)={\displaystyle \frac{{\widehat{\alpha }}_1{x}^{(r)}(k)}{1+{\widehat{\beta }}_1{x}^{(r)}(k)+{\widehat{\gamma }}_1{y}^{(r)}(k)}}, k=1, 2, \cdots$$

(9)

Among them, ${\beta }_i={\displaystyle \frac{b_i({\alpha }_i-1)}{\ln {\alpha }_i}}$ ${\alpha }_i=e^{a_i}$, ${\gamma }_i={\displaystyle \frac{c_i({\alpha }_i-1)}{\ln {\alpha }_i}}$ $(i=1, 2)$, $r-\mathrm{AGO}$.

The inverse operation performed yields a reduced value of

$${\widehat{x}}^{(0)}(k)=\left\{\begin{array}{c}{x}^{(0)}(1),\hspace{1em}k=1\\ {\widehat{x}}^{(r)(1-r)}(k)-{\widehat{x}}^{(r)(1-r)}(k-1),\hspace{1em}k=2,3,\cdots \end{array}\right.$$

(10)

Similarly, it can be concluded that

$${\widehat{y}}^{(r)}(k+1)={\displaystyle \frac{{\widehat{\alpha }}_2{y}^{(r)}(k)}{1+{\widehat{\beta }}_2{y}^{(r)}(k)+{\widehat{\gamma }}_2{x}^{(r)}(k)}}, k=1, 2, \cdots$$

(11)

Among them ${\beta }_i={\displaystyle \frac{b_i({\alpha }_i-1)}{\ln {\alpha }_i}}$ ${\alpha }_i=e^{a_i}$, ${\gamma }_i={\displaystyle \frac{c_i({\alpha }_i-1)}{\ln {\alpha }_i}}$ $(i=1, 2)$, $r-\mathrm{AGO}$.

The inverse operation performed yields a reduced value of

$${\widehat{y}}^{(0)}(k)=\left\{\begin{array}{c}{y}^{(0)}(1),\hspace{1em}k=1\\ {\widehat{y}}^{(r)(1-r)}(k)-{\widehat{y}}^{(r)(1-r)}(k-1),\hspace{1em}k=2,3,\cdots \end{array}\right.$$

(12)

Finally, the accuracy of the model is verified using the MAPE value, and the calculation formula is

$$\begin{array}{l}\mathrm{MAPE}={\displaystyle \frac{1}{n}}\left|{\displaystyle \sum _{k=1}^n\frac{{\widehat{x}}^{(0)}(k)-x^{(0)}(k)}{x^{(0)}(k)}}\right|\times 100\% , k=1,2,\cdots ,n\\ \mathrm{MAPE}={\displaystyle \frac{1}{n}}\left|{\displaystyle \sum _{k=1}^n\frac{{\widehat{y}}^{(0)}(k)-y^{(0)}(k)}{y^{(0)}(k)}}\right|\times 100\% , k=1,2,\cdots ,n\end{array}$$

(13)

The MAPE value is used as a way to evaluate the accuracy of the model, and the evaluation criteria are shown in

Table 3. Evaluation Criteria for MAPE Values.

MAPE (%)	Prediction Effect
<10	Excellent
10–20	Good
20–50	Reasonable
>50	Wrong

.

3.2. Model Solving and Data Analysis

The FGLV model is mainly used to analyze the relationship between indicators and to make reasonable predictions of the data based on their correlation. The four types of influencing factors affecting the added value of high-tech industries were analyzed and predicted.

Taking the conversion of R&D activity personnel equivalent to full-time equivalent in the secondary evaluation index of high-tech industry added value and human capital system as an example, the specific calculation steps are shown. The calculation data of high-tech industry added value and other secondary indicators in the system only display the final calculation results and related parameter values.

3.2.1. Value Added of High-Tech Industries Under Four Influencing Factors

(1)

Value added of high-tech industries under the influence of human capital system.

The original data in the even table in

Table 4. Raw data of added value of high-tech industries and fitted data under the influence of human capital system.

Year	Raw Data/100 Million Yuan	Fitting Values Under the Human Capital System
		R&D Activity Personnel Converted to Full-Time Equivalent	Overall Labor Productivity	Participating Researchers
2016	5643.9	5643.9	5643.9	5643.9
2017	6387.3	6870.7	6695.1	5379.9
2018	6976.8	9428.8	8127.7	7655.8
2019	8630.0	10,471.9	8307.2	9005.8
2020	9242.3	10,312.7	9028.2	9533.3
2021	10,866.9	10,704.3	10,940.5	10,765.6
MAPE		12.86%	4.67%	5.65%
2022		14,339.5	13,915.4	15,720.9
2023		32,606.1	17,313.0	36,964.9
2024		−1,511,590.5	19,531.1	707,988.5
2025		1,113,731.4	18,747.3	−814,990.2
2026		117,963.8	15,457.1	−4620.3

,

Table 5. Parameters of the Relationship between Value Added of High-tech Industries and Human Capital System.

	Human Capital System	${\mathit{c}}_1$	${\mathit{c}}_2$	Correlations
Value added of high-tech industries	R&D activity personnel converted to full-time equivalent	0.00000405	−0.00010602	Competitive relationship
	overall labor productivity	0.00000042	−0.00019316	Competitive relationship
	Participating researchers	0.00000173	−0.00007225	Competitive relationship

,

Table 6. Raw data of added value of high-tech industries and fitted data under the funding system.

Year	Raw Data/100 Million Yuan	Fitting Values Under the Funding System
		Internal R&D Expenditure	Government Funding Investment	Enterprise Capital Investment
2016	5643.9	5643.9	5643.9	5643.9
2017	6387.3	5102.0	5358.1	5282.2
2018	6976.8	7365.7	7500.4	6977.8
2019	8630.0	9029.4	8838.6	8361.1
2020	9242.3	9741.1	9508.3	10,465.5
2021	10,866.9	10,857.7	10,865.2	10,571.2
MAPE		5.96%	4.82%	6.06%
2022		15,539.1	15,397.6	12,338.5
2023		37,202.8	32,074.4	5586.2
2024		−10,620,297.9	214,631.1	23,953.6
2025		9,562,506.3	−343,254.1	−32,243.5
2026		435,802.6	−5592.7	67,171.2

,

Table 7. Parameters of the Relationship between the Value Added of High-Tech Industries and the Financial System.

	Financial System	${\mathit{c}}_1$	${\mathit{c}}_2$	Correlations
Value added of high-tech industries	Internal R&D expenditure	0.00000009	−0.00022088	Competitive relationship
	Government fund	0.00000011	−0.00011805	Competitive relationship
	Enterprise funds	0.00000022	−0.00032067	Competitive relationship

,

Table 8. Raw data of added value of high-tech industries and fitted data under the technology system.

Year	Raw Data/100 Million Yuan	Fitting Values Under Technological System
		Number of Patent Applications	Number of Invention Patents Owned
2016	5643.9	5643.9	5643.9
2017	6387.3	6850.4	5482.7
2018	6976.8	6977.0	6167.4
2019	8630.0	7678.3	7682.9
2020	9242.3	9091.0	9889.0
2021	10,866.9	10,940.6	11,023.6
MAPE		3.43%	7.53%
2022		12,743.6	10,225.9
2023		13,739.8	8530.1
2024		13,428.3	6421.4
2025		12,188.4	5013.1
2026		10,759.1	3525.3

,

Table 9. Parameters of the relationship between value added of high-tech industries and technological system.

	Technological System	${\mathit{c}}_1$	${\mathit{c}}_2$	Correlations
Value added of high-tech industries	Number of patent applications	0.00009410	−0.00026072	Competitive relationship
	Number of invention patents owned	0.00006962	−0.00014927	Competitive relationship

,

Table 10. Raw data of added value of high-tech industries and fitted data under the trade system.

Year	Raw Data/100 Million Yuan	Fitting Values Under the Trade System
		Export Value of High-Tech Products Trade	Import Volume of High-Tech Product Trade
2016	5643.9	5643.9	5643.90
2017	6387.3	4617.3	6483.73
2018	6976.8	7081.5	7221.73
2019	8630.0	9200.6	7846.65
2020	9242.3	10,068.8	9305.11
2021	10,866.9	10,637.7	10,866.69
MAPE		7.82%	2.46%
2022		17,987.3	11,230.01
2023		−396,465.6	9770.38
2024		382,556.6	7517.08
2025		−20,297.1	5570.32
2026		−5570.2	4177.74

and

Table 11. Parameters of the relationship between the added value of high-tech industries and the trade system.

	Trade System	${\mathit{c}}_1$	${\mathit{c}}_2$	Correlations
Added value of high-tech industries	Export value of high-tech products trade	−0.00107534	0.00006048	Competitive relationship
	Import volume of high-tech product trade	0.00281948	−0.00011522	Competitive relationship

are data from 2016–2021 for added value of high-tech industries, and the data are all from the Beijing Statistical Yearbook.

Table 12. Raw data of human resources system and fitted data under the influence of high-tech output value.

Year	Raw Data			Fit Data
	R&D Activity Personnel Converted to Full-Time Equivalent/Person Year	Total Labor Productivity (Yuan/Person)	Researchers/People	R&D Activity Personnel Converted to Full-Time Equivalent	Total Labor Productivity	Researchers
2016	253,337	2,826,734	373,406	253,337	2,826,734	373,406
2017	269,835	2,999,989	397,281	276,956	3,319,321	317,379
2018	267,338	3,541,696	397,034	364,274	3,840,112	432,460
2019	313,986	3646,839	464,178	386,421	3,666,895	479,077
2020	336,280	4,059,513	473,304	356,996	3,784,361	466,138
2021	338,297	4,488,531	472,860	338,206	4,487,978	472,913
MAPE				11.36%	4.40%	5.63%

,

Table 13. Raw data of funding indicators and fitted data under the influence of high-tech output value.

Year	Raw Data			Fit Data
	Internal R&D Expenditure/10,000 yuan	Government Funding Investment/10,000 yuan	Enterprise Capital Investment/10,000 Yuan	Internal R&D Expenditure	Government Capital Investment	Enterprise Capital Investment
2016	14,845,762	8,026,073	5,636,747	14,845,762	8,026,073	5,636,747
2017	15,796,512	8,224,113	6,196,382	13,321,747	7,058,495	6,034,854
2018	18,707,701	9,205,698	8,304,423	19,173,339	9,577,159	8,022,492
2019	22,335,870	10,692,236	9,867,574	23,153,703	10,791,109	9,446,959
2020	23,265,793	10,843,276	10,742,209	24,277,796	10,947,391	12,038,907
2021	26,293,208	11,864,952	12,477,196	25,991,832	11,722,507	12,097,070
MAPE				4.55%	3.55%	4.23%

,

Table 14. Raw data of technical indicators and fitted data under the influence of high-tech output value.

Year	Raw Data		Fit Data
	Number of Patent Applications/Piece	Number of Invention Patents Owned/Piece	Number of Patent Applications	Number of Invention Patents Owned
2016	20,065	9392	20,065	9392
2017	19,653	10,119	22,954	9801
2018	20,655	10,386	21,183	9219
2019	22,552	11,543	21,473	10,414
2020	25,147	13,078	24,110	13,513
2021	28,221	15,589	28,225	15,587
MAPE			4.71%	4.58%

and

Table 15. Raw data of trade indicators and fitted data under the influence of high-tech output value.

Year	Raw Data		Fit Data
	High-Tech Product Trade Export Value/Billions of US Dollars	High-Tech Product Trade Import Value/Billions of US Dollars	High-Tech Product Trade Export Value	High-Tech Product Trade Import value
2016	113.2	255.00	113.20	255.00
2017	112.8	264.90	92.68	284.27
2018	152.2	278.30	144.78	269.27
2019	157.5	269.70	199.34	253.12
2020	197.4	279.60	255.40	285.78
2021	403.6	360.30	403.63	336.81
MAPE			13.11%	4.24%

are the statistical yearbook data of the four systems that affect the added value of high-tech industries from 2016 to 2021 and the data analyzed and predicted based on the FGLV model.

The original data in the model calculation are all survey data for this indicator from 2016 to 2021 in the Beijing Statistical Yearbook. The detailed steps for converting the added value of high-tech industries into full-time equivalents of R&D personnel are as follows:

Step 1: The original sequence of the added value of high-tech industries is

$$X^{(0)}=\left\{5643.9, 6387.3, 6976.8, 8630.0,\right.\left.9242.3, \mathrm{10,866.9}\right\}.$$

The original sequence of R&D activity personnel converted to full-time equivalent is

$$Y_1^{(0)}=\left\{\mathrm{253,337}, \mathrm{269,835}, \mathrm{267,338}, \mathrm{313,986}, \mathrm{336,280}, \mathrm{338,297}\right\}.$$

Step 2: Obtain the fractional order cumulative sequence as

$$X^{(0.7846)}=\left\{5643.9, \right.\mathrm{10,815.39}, \mathrm{15,939.285}, \mathrm{22,242.85}, \mathrm{28,517.88}, \left.\mathrm{35,940.76242}\right\}$$

$$Y_1^{(0.8173)}=\left\{\mathrm{253,337}, \right.\mathrm{476,894}, \mathrm{676,028.18}, \mathrm{909,578.4}, \mathrm{1,148,272}, \left.\mathrm{1,374,855.388}\right\}$$

Step 3: Use the least squares method to solve the parameters ${\widehat{a}}_1$, ${\widehat{b}}_1$ and ${\widehat{c}}_1$, where

$$B=\left[\begin{array}{rrr}\hfill 8229.64& \hfill -\mathrm{67,727,052.02}& \hfill -\mathrm{3,004,770,917.92}\\ \hfill \mathrm{13,377.34}& \hfill -\mathrm{178,953,153.86}& \hfill -\mathrm{7,711,514,574.67}\\ \hfill \mathrm{19,091.07}& \hfill -\mathrm{364,468,855.25}& \hfill -\mathrm{15,135,460,840.60}\\ \hfill \mathrm{25,380.37}& \hfill -\mathrm{644,163,026.46}& \hfill -\mathrm{26,114,503,626.45}\\ \hfill \mathrm{32,229.32}& \hfill -\mathrm{1,038,729,283.78}& \hfill -\mathrm{40,659,350,288.70}\end{array}\right], T=\left[\begin{array}{r}\hfill 6387.3\\ \hfill 6976.8\\ \hfill 8630.0\\ \hfill 9242.3\\ \hfill \mathrm{10,866.9}\end{array}\right].$$

$$\left[\begin{array}{c}{\widehat{a}}_1\\ {\widehat{b}}_1\\ {\widehat{c}}_1\end{array}\right]=\left[\begin{array}{r}\hfill 1.078737\\ \hfill -0.000136\\ \hfill 0.000004\end{array}\right].$$

Using the least squares method to solve the parameters ${\widehat{a}}_2$, ${\widehat{b}}_2$ and ${\widehat{c}}_2$, where

$$B=\left[\begin{array}{rrr}\hfill \mathrm{365,115.51}& \hfill -\mathrm{133,309,335,042.73}& \hfill -\mathrm{3,004,770,917.92}\\ \hfill \mathrm{576,461.10}& \hfill -\mathrm{332,307,398,629.58}& \hfill -\mathrm{7,711,514,574.67}\\ \hfill \mathrm{792,803.28}& \hfill -\mathrm{628,537,038,366.57}& \hfill -\mathrm{15,135,460,840.60}\\ \hfill \mathrm{1,028,925.38}& \hfill -\mathrm{1,058,687,431,041.29}& \hfill -\mathrm{26,114,503,626.45}\\ \hfill \mathrm{1,261,563.88}& \hfill -\mathrm{1,591,543,428,790.40}& \hfill -\mathrm{40,659,350,288.70}\end{array}\right], T=\left[\begin{array}{l}\mathrm{269,835}\\ \mathrm{267,338}\\ \mathrm{313,986}\\ \mathrm{336,280}\\ \mathrm{338,297}\end{array}\right].$$

$$\left[\begin{array}{c}{\widehat{a}}_2\\ {\widehat{b}}_2\\ {\widehat{c}}_2\end{array}\right]=\left[\begin{array}{r}\hfill 0.974677\\ \hfill 0.000003\\ \hfill -0.000106\end{array}\right].$$

The expression is as follows

$${\displaystyle \frac{d{x}^{(0.7846)}}{dt}}=1.078737x^{(0)}(t)-0.000136x^{(0)}(t{)}^2+0.000004x^{(0)}(t)y_1^{(0)}(t)$$

$${\displaystyle \frac{d{y}_1^{(0.8173)}}{dt}}=0.974677y_1^{(0)}(t)+0.000003y_1^{(0)}{(t)}^2-0.000106y_1^{(0)}(t)x^{(0)}(t)$$

Step 4: Due to the fact that the ${\alpha }_i=e^{a_i}, {\beta }_i={\displaystyle \frac{b_i({\alpha }_i-1)}{\ln {\alpha }_i}}, {\gamma }_i={\displaystyle \frac{c_i({\alpha }_i-1)}{\ln {\alpha }_i}} (i=1, 2)$ discrete form expression, the prediction formula, is

$${\widehat{x}}^{(0.7846)}(k+1)={\displaystyle \frac{2.940963x^{(0.7846)}(k)}{1-0.000244x^{(0.7846)}(k)+0.000007y^{(0.7846)}(k)}}, k=1, 2, \cdots$$

$${\widehat{y}}_1^{(0.8173)}(k+1)={\displaystyle \frac{2.650310y_1^{(0.8173)}(k)}{1+0.000006y_1^{(0.8173)}(k)-0.000180x^{(0.8173)}(k)}}, k=1, 2, \cdots$$

The fitted sequence is

$${\widehat{X}}^{(0)}=\left\{5643.9, \right.6870.7, 9428.8, \mathrm{10,471.9}, \mathrm{10,312.7}, \left.\mathrm{10,704.3}\right\}.$$

$${\widehat{Y}}_1^{(0)}=\left\{\mathrm{253,337}, \right.\mathrm{276,956}, \mathrm{364,274}, \mathrm{386,421}, \mathrm{356,996}, \left.\mathrm{338,205}\right\}.$$

Step 5: MAPE value calculation

$$\mathrm{MAPE}={\displaystyle \frac{1}{5}}\left|{\displaystyle \sum _{k=1}^5\frac{{\widehat{x}}^{(0)}(k)-x^{(0)}(k)}{x^{(0)}(k)}}\right|\times 100\%=12.86\%$$

$$\mathrm{MAPE}={\displaystyle \frac{1}{5}}\left|{\displaystyle \sum _{k=1}^8\frac{{\widehat{y}}_1^{(0)}(k)-y_1^{(0)}(k)}{y_1^{(0)}(k)}}\right|\times 100\%=11.36\%$$

As shown in Table 4, the original data in the table correspond to the $X^{(0)}$ variable in 3.1, and the indicators under the human capital system were, respectively, used as $Y^{(0)}$ variables, and the values are all from the Beijing Statistical Yearbook. The fitting data of the added value of high-tech industries under the influence of the human capital system are shown in Table 4, and the mutual parameter relationship between the two is shown in Table 5.

This article focuses on the factors that affect the added value of high-tech industries and the future development trends of indicators under the influence of these factors. Therefore, based on the fitting error of the secondary indicators in the system to the added value of high-tech industries, further screening of the secondary indicators in the system was carried out.

As demonstrated in Table 4, the data fitting error of the added value of high-tech industries under the influence of indicators in the human capital system was less than 20%, and the fitting error of the added value of high-tech industries under the influence of total labor productivity was relatively small.

However, the application of the fractional order cumulative grey Lotka–Volterra model to predict the added value of high-tech industries over a five-year period revealed negative values in the data. This finding suggests that while the model can be utilized to predict the added value of high-tech industries based on the current parameter relationship between the two systems, it is not capable of making long-term predictions using the model.

Furthermore, as demonstrated in Table 5, the added value of high-tech industries and the indicators in the human capital system were $c_1$ > 0 and $c_2$ < 0, indicating that the two are currently in a competitive state.

(2)

Value added of high-tech industries under the influence of the funding system.

As shown in Table 6, the original data in the table correspond to the $X^{(0)}$ variable in 3.1, and the indicators under the capital system were, respectively, $Y^{(0)}$ variables, and the values are all from the Beijing Statistical Yearbook. The fitting data of the added value of high-tech industries under the funding system are shown in Table 6, and the parameter relationship between the two is shown in Table 7.

As demonstrated in Table 6, under the influence of indicators in the funding system, the fitting error of the added value of high-tech industries was less than 10%, and the secondary indicator with a smaller fitting error was government funding. As illustrated in Table 7, the added value of high-tech industries is currently in a competitive state with the indicators of the funding system.

(3)

Added value of high-tech industries under the influence of technological system.

As shown in Table 8, the original data in the table correspond to the $X^{(0)}$ variable in 3.1, and the indicators under the scientific and technological innovation system were, respectively, used as $Y^{(0)}$ variables, and the values are all from the Beijing Statistical Yearbook. The fitting data of the added value of high-tech industries under the technology system are shown in Table 8, and the parameter relationship between the two is shown in Table 9.

As shown in Table 8, the fitting errors of the added value of high-tech industries under the influence of the indicators of patent applications and invention patents in the technical system were 3.43% and 7.53%, respectively. As shown in Table 9, there is currently a competitive relationship between the added value of high-tech industries and the technological system.

(4)

Value added of high-tech industries under the influence of the trade system.

As shown in Table 10, the original data in the table correspond to the $X^{(0)}$ variable in 3.1, and the indicators under the trade system were, respectively, $Y^{(0)}$ variables, and the values are all from the Beijing Statistical Yearbook. The fitting data of the added value of high-tech industries under the trade system are shown in Table 10, and the parameter relationship between the two is shown in Table 11.

As demonstrated in Table 10, the fitting errors of the added value of high-tech industries under the influence of secondary indicators in the trade system were 7.82% and 2.46%, respectively. As demonstrated in Table 11, the added value of high-tech industries and the trade system are currently in a state of mutual competition.

3.2.2. Data on Various Indicators of the Four Systems Under the Influence of the Added Value of High-Tech Industries

The original data in Table 12, Table 13, Table 14 and Table 15 correspond to $Y^{(0)}$ in the model 3.1, and the fitted data were $Y^{(r)}$ in the model. The continuous changes in high-tech output value are influenced by relevant system indicators, and the changes in indicator data in the system will also be adjusted by the changes in high-tech output value. The raw data of the human resources system, funding system, technology system, and trade system, as well as the fitted data of each system under the influence of high-tech output value, are shown in Table 12, Table 13, Table 14 and Table 15.

As demonstrated in Table 12, Table 13, Table 14 and Table 15, the fitting errors of the added value of high-tech industries to the four system indicator data were all below 20%, indicating that the use of fractional order cumulative grey Lotka–Volterra model is a suitable tool for analyzing the changes in the added value of high-tech industries and their influencing factors.

In summary, the added value of high-tech industries was closely related to various systems, and it is meaningful to analyze the two systems based on the fractional order cumulative grey Lotka–Volterra model. However, negative values were observed when using the fractional order cumulative grey Lotka–Volterra model for long-term forecasting of the added value of high-tech industries. Consequently, this article proposes the grey differential dynamic multivariate model in the grey system to facilitate a more comprehensive analysis of the fluctuations in the added value of high-tech industries.

4. Prediction Model

4.1. Grey Differential Dynamic Multivariate Model

The grey differential dynamic model was initially proposed by Xia Jun using Professor Deng Julong’s grey system idea [29] as a hydrological system model [32], and then improved by Duan Huiming et al. [30] to a grey differential dynamic model that studies multiple variables. The main modeling process is as follows:

Step 1: Assuming $X^{(0)}=\left\{x^{(0)}(1),x^{(0)}(2),\cdots ,x^{(0)}(n)\right\}$ is a system feature $Y_j^{(0)}=$ $\left\{y_j^{(0)}(1),y_j^{(0)}(2),\cdots ,y_j^{(0)}(n)\right\} , j=1,2,\cdots ,m$ sequence and a sequence of related factors. Using a first-order accumulation generation operator to accumulate the original sequence, further weakening the randomness, and obtaining a stable first-order accumulation generation sequence:

$$\begin{array}{l}{X}^{(1)}=\left\{x^{(1)}(1),x^{(1)}(2),\cdots ,x^{(1)}(n)\right\}\\ Y_j^{(1)}=\left\{y_j^{(1)}(1),y_j^{(1)}(2),\cdots ,y_j^{(1)}(n)\right\}\end{array}$$

(14)

Among them, the first-order accumulation generation operator is denoted as 1-AGO, and the accumulation formula is

$$\begin{array}{l}{x}^{(1)}(k)={\displaystyle \sum _{t=1}^k{x}^{(0)}(t),} k=1,2,\cdots ,n\\ y_j^{(1)}(k)={\displaystyle \sum _{t=1}^k{y}_j^{(1)}(t),} k=1,2,\cdots ,n\end{array}$$

(15)

The differential equation form of the grey differential dynamic model DHGM(2,M) obtained from the sequence generated by first-order accumulation is

$${\displaystyle \frac{d{x}^{(1)}(t)}{dt}}+a{x}^{(1)}(t)=b_{10}{\displaystyle \frac{d{y}_1^{(1)}(t)}{dt}}+b_{11}{y}_1^{(1)}(t)+\cdots +b_{m0}{\displaystyle \frac{d{y}_m^{(1)}(t)}{dt}}+b_{m1}{y}_m^{(1)}(t)$$

(16)

Among them ${\displaystyle \frac{d{x}_0^{(1)}(t)}{dt}}=x_0^{(1)}(t)-x_0^{(1)}(t-1),{\displaystyle \frac{d{x}_m^{(1)}(t)}{dt}}=x_m^{(0)}(t)$.

Step 2: Next, $X^{(1)}$ compare the adjacent mean values

$$Z^{(1)}=\left\{z^{(1)}(1),z^{(1)}(2),\cdots ,z^{(1)}(n)\right\}$$

(17)

Among them $z^{(1)}(k)=0.5\ast \left[x_0^{(1)}(k)+x_0^{(1)}(k-1)\right],k=2,3\cdots ,n$, the grey differential dynamic multivariate prediction model can be referred to as

$$x^{(0)}(k)+a{z}^{(1)}(k)=b_{10}{y}_1^{(0)}(k)+b_{11}{y}_1^{(1)}(k)+\cdots +b_{m0}{y}_m^{(0)}(k)+b_{m1}{y}_m^{(1)}(k)$$

(18)

Step 3: The parameters can be obtained by the least squares method. Let $\mu =(a,b_{10},b_{11},\cdots ,b_{m0},b_{m1}{)}^T$ be a parameter column, satisfying $\mu =(A^{T}A{)}^{-1}{A}^{T}D$,

$$A=\left[\begin{array}{cccccc}-z^{(1)}(2)& y_1^{(0)}(2)& y_1^{(1)}(2)& \cdots & y_m^{(0)}(2)& y_m^{(1)}(2)\\ -z^{(1)}(3)& y_1^{(0)}(3)& y_1^{(1)}(3)& \cdots & y_m^{(0)}(3)& y_m^{(1)}(3)\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ -z^{(1)}(n)& y_1^{(0)}(n)& y_1^{(1)}(n)& \cdots & y_m^{(0)}(n)& y_m^{(1)}(n)\end{array}\right], D=\left[\begin{array}{c}{x}^{(0)}(2)\\ x^{(0)}(3)\\ ⋮\\ x^{(0)}(n)\end{array}\right]$$

(19)

Step 4: Based on the known $A,D$ time response function of the obtained model,

$${\widehat{x}}^{(1)}(k)={\displaystyle \sum _{j=1}^m{b}_{j0}}{y}_j^{(1)}(k)+{\displaystyle \sum _{j=1}^m(b_{j1}-a{b}_{j0})}{\displaystyle {\int }_0^k{e}^{-a\tau }{y}_j^{(1)}(k-\tau )d\tau +(x^{(0)}(1)-{\displaystyle \sum _{j=1}^m{b}_{j0}{y}_{j0}})e^{-ak}}$$

(20)

The (20) integral in the formula is discretized using the trapezoidal formula to obtain the basic prediction formula of the model

$$\begin{array}{ll}{\widehat{x}}^{(1)}(k)& ={\displaystyle \sum _{j=1}^m{b}_{j0}}{y}_j^{(1)}(k)+(x^{(0)}(1)-{\displaystyle \sum _{j=1}^m{b}_{j0}{y}_{j0}})e^{-a(k-1)}\\ & +{\displaystyle \frac{1}{2}}{\displaystyle \sum _{j-1}^m\left[(b_{j1}-a{b}_{j0}){\displaystyle \sum _{j=1}^{k-1}(e^{-aj}{y}_j^{(1)}(k-i)+e^{-a(i-1)}{y}_j^{(1)}(k-i+1))}\right]}\end{array}$$

(21)

Step 5: Restore the time response equation of the DHGM(2, M) model, and the predicted value obtained is

$$\left\{\begin{array}{l}{\widehat{x}}^{(0)}(1)=x^{(0)}(1)\\ {\widehat{x}}^{(0)}(k)={\widehat{x}}^{(1)}(k)-{\widehat{x}}^{(1)}(k-1)\end{array}\right.$$

(22)

Step 6: Use the mean relative error (MAPE) and relative error (APE) values to test the prediction accuracy of the model. The calculation formulas are as follows:

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

(23)

$$\mathrm{APE}(k)=\left|{\displaystyle \frac{x^{(0)}(k)-{\widehat{x}}^{(0)}(k)}{x^{(0)}(k)}}\right|\times 100\%,\hspace{0.17em}k=1,2,\cdots ,n$$

(24)

When the simulation error and prediction error of the model are both below 5%, it indicates that the prediction performance of the model is good. To achieve the best prediction results, the parameter intervals are continuously and iteratively adjusted through the Particle Swarm Optimization (PSO) algorithm, searching for the optimal parameter solution to minimize the interval width of predicted values.

The specific modeling steps and parameter optimization process of the model are as follows:

Step 1: Data preprocessing and sequence generation

(1)

Original sequence division: Divide the raw data into a training set and a testing set (e.g., training from 2016 to 2020, testing in 2021).

Define the relationship between the independent variable (Y) and the dependent variable (X).

(2)

Accumulated generation sequence (1-AGO): Perform first-order accumulation generation (1-AGO) on the original sequence to eliminate data volatility and enhance sequence regularity through accumulation.

Step 2: Model parameter estimation

(1)

Constructing differential equations.

(2)

Using MAPE as the loss function, adjust parameters through PSO optimization method to minimize MAPE.

Step 3: Fitted values and error calculation

Generate fitting sequence: Substitute estimated parameters into the model and obtain a fitting value sequence through iterative reduction and restoration (1-AGO)

Then, calculate the error point by point, and calculate the average error of each point to obtain MAPE,

$$\mathrm{APE}(k)=\left|{\displaystyle \frac{x^{(0)}(k)-{\widehat{x}}^{(0)}(k)}{x^{(0)}(k)}}\right|\times 100\%,\hspace{0.17em}k=1,2,\cdots ,n$$

Step 4: Parameter iteration optimization, adjust parameter combinations through cross validation or grid search, observe the trend of MAPE changes, and select the parameter that minimizes MAPE.

4.2. Model Solving and Data Analysis

This article predicts the added value of high-tech industries based on the human capital system, taking the solution between total labor productivity and the added value of high-tech industries in the human capital system as an example. The relevant data are shown in

Table 16. Added value and total labor productivity of high-tech industries in Beijing.

Year	Value Added of High-Tech Industries/100 Million Yuan	Total Labor Productivity (Yuan/Person)
2016	5643.90	2,826,733.96
2017	6387.30	2,999,989.29
2018	6976.80	3,541,695.64
2019	8630.0	3,646,839.08
2020	9242.30	4,059,512.74
2021	10,866.90	4,488,530.56

.

Based on the raw data in Table 16, establish the DHGM(2,M) model. The specific process is as follows:

(1)

The original data of added value of high-tech industries in Beijing from 2016 to 2021 are

$$X^{(0)}=\left\{5643.9, 6387.3, 6976.8, 8630.0, 9242.3, \mathrm{10,866.9}\right\}$$

(2)

The original data of labor productivity for all employees in Beijing from 2016 to 2021 are

$$Y^{(0)}=\left\{\mathrm{2,826,733.96}, \mathrm{2,999,989.29}, \mathrm{3,541,695.64}, \mathrm{3,646,839.08}, \mathrm{4,059,512.74}, \mathrm{4,488,530.56}\right\}$$

Then, the first-order accumulated sequence is obtained as

$$\begin{array}{l}{X}^{(1)}=\left\{5643.9, \mathrm{12,031.2}, \mathrm{19,008}, \mathrm{27,638}, \mathrm{36,880.3}, \mathrm{47,747.2}\right\}\\ Y^{(1)}=\left\{\mathrm{2,826,733.96}, \mathrm{5,826,723.25}, \mathrm{9,368,418.89}, \mathrm{13,015,257.97}, \mathrm{17,074,770.71}, \mathrm{21,563,301.21}\right\}\end{array}$$

utilize

$$\mu =(a, b_{10}, b_{11}, \cdots , b_{m0}, b_{m1}{)}^T= (A^{T}A{)}^{-1}{A}^{T}D=\left[\begin{array}{l}-0.0469\\ 0.0018\\ 2.6287\times {10}^{-5}\end{array}\right]$$

Among them,

$$A=\left[\begin{array}{ccc}-8837.55& \mathrm{2,999,989.291}& \mathrm{5,826,723.254}\\ -\mathrm{15,519.60}& \mathrm{3,541,695.641}& \mathrm{9,368,418.895}\\ -\mathrm{23,323.00}& \mathrm{3,646,839.076}& \mathrm{13,015,257.971}\\ -\mathrm{32,259.15}& 4059512.744& \mathrm{17,074,770.715}\\ -\mathrm{42,313.75}& \mathrm{4,488,530.559}& \mathrm{21,563,301.274}\end{array}\right], D=\left[\begin{array}{l}6387.30\\ 6976.80\\ 8630.00\\ 9242.30\\ \mathrm{10,866.90}\end{array}\right]$$

Then, the time response sequence can be obtained as

$$\begin{array}{ll}{\widehat{x}}^{(1)}(k)& ={\displaystyle \sum _{j=1}^m{b}_{j0}}{y}_j^{(1)}(k)+{\displaystyle \sum _{j=1}^m(b_{j1}-a{b}_{j0})}{\displaystyle {\int }_0^k{e}^{-a\tau }{y}_j^{(1)}(k-\tau )d\tau +(x^{(0)}(1)-{\displaystyle \sum _{j=1}^m{b}_{j0}{y}_j^{(0)}(1)})e^{-ak}}\\ & =0.0018y^{(1)}(k)+(2.6287\times {10}^{-5}+0.0469\times 0.0018){\displaystyle {\int }_0^k{e}^{0.0469\tau }{y}^{(1)}(k-\tau )}+\\ & (5643.9-0.0018\times \mathrm{2,826,733.96})e^{0.0469k}\end{array}$$

According to the time response equation, the first-order cumulative prediction sequence can be obtained as

$${\widehat{X}}^{(1)}=\left\{5643.9, \mathrm{12,035.6}, \mathrm{19,015.2}, \mathrm{27,642.5}, \mathrm{36,885.1}, \mathrm{47,637.9}\right\}.$$

Then, perform a first-order accumulation inverse operation to obtain the restored predicted value

$${\widehat{X}}^{(0)}=\left\{5643.9, 6387.1, 6976.5, 8629.7, 9242.6, \mathrm{10,868.6}\right\}$$

Using data from 2016 to 2020 as the training set and 2021 as the validation set, obtain the training error of the model, MAPE = 0.003%, MAE = 0.28, RMSE = 0.28. The validation error of the model, MAPE = 0.016%, MAE = 1.7, MRSE = 1.7.

To visually highlight the advantages of the grey dynamic multivariate differential model in predicting high-tech industry added value, comparative predictions were conducted using multiple grey and non-grey models on the given dataset. The outcomes are presented below.

As evidenced by the comparative results in

Table 17. Prediction errors of added value of high-tech industries under different models.

Model	MAPE (%)	MAE	RMSE
DHGM(2,M)	0.016%	1.7	1.7
GMC(1,N)	0.056%	6.1	6.1
GM(1,N)	0.113%	12.3	12.3
DGM	0.078%	8.5	8.5
ARIMA	0.14%	15.2	15.2
Multiple Regression	0.174%	18.9	18.9
SVR	0.09%	9.8	9.8
Neural Network (BP)	0.066%	7.2	7.2

, the DHGM(2,M) model exhibited superior performance, achieving a remarkably low prediction MAPE of 0.016%. By leveraging second-order derivatives to capture growth acceleration, this model aligns more closely with the exponential growth dynamics inherent to high-tech industries. Other grey models have slightly larger errors than the DHGM(2,M) model, while non-grey models have larger errors than most grey models. However, non grey models have certain problems in predicting short data sequences.

ARIMA relies entirely on the autocorrelation of historical data for modeling and does not incorporate external variables. When data are influenced by multiple factors, it fails to capture external shocks (e.g., policy changes, technological breakthroughs). Multivariate regression and machine learning models face limitations due to insufficient data—only six years of observations are inadequate to support the complex parameter estimation required for multivariate linear regression (MLR) or neural networks (e.g., BP, LSTM), leaving these models vulnerable to noise interference. Support Vector Machines (SVRs) and neural networks demand large datasets to learn nonlinear relationships. Under small-sample conditions, they cannot be adequately trained, resulting in poor generalization ability and an inability to adapt to the complex growth patterns of high-tech industries. Therefore, the DHGM(2,M) model was selected to forecast the added value of high-tech industries.

Similarly, the predicted value added of high-tech industries under alternative systems is illustrated in the table. The human capital system selected the labor productivity of all employees, a closely related factor to the added value of high-tech industries, as the influencing factor. The funding system selected R&D internal expenditure and government funds as the influencing factors, and the technology system selected the number of patent applications as the influencing factor. The trade system selected the import quantity of high-tech products as the influencing factor and used a grey differential dynamic multivariate prediction model for prediction. The influencing factor sequence used the GM(1,1) model to predict the data from 2022 to 2026, and then used the DHGM(2,M) model to predict the added value of high-tech industries in the next 5 years.

The total labor productivity of high-tech industries is defined as the value added of high-tech industrial enterprises divided by the average number of employees in the same period, reflecting the productivity of the enterprise, and thus the level of economic development and productivity of a region.

Government investment funds include government subsidies, equity investments, and other fiscal appropriations, representing the government’s emphasis on a certain industry and encouraging future development directions.

The import volume of high-tech products is indicative of demand for such goods, as well as the development speed and degree of openness of the high-tech industry.

As illustrated in

Table 18. Predicted added value of high-tech industries under different systems.

Time	Raw Data	Prediction Results of Added Value of High-Tech Industries in Different Systems
		Human Capital	Fund	Technology	Trade
2016	5643.9	5643.9	5643.9	5643.90	5643.90
2017	6387.3	6386.8	6386.8	6255.6	6352.14
2018	6976.8	6977.2	6977.2	7008.2	7118.67
2019	8630.0	8629.3	8629.3	8492.5	8345.29
2020	9242.3	9242.6	9242.6	9320.8	9276.81
2021	10,866.9	10,867.2	10,867.2	10,752.8	10,543.22
2022		12,280.5	12,150.6	12,320.5	12,357.8
2023		13,890.7	13,600.4	14,050.7	13,944.2
2024		15,720.4	15,240.7	16,080.4	15,682.5
2025		17,802.1	17,110.2	18,450.1	17,605.3
2026		20,173.8	19,250.5	21,210.5	19,750.6

, the added value of high-tech industries in Beijing is influenced by multiple systems. The grey differential dynamic multivariate prediction model has been demonstrated to achieve relatively accurate predictions of the added value of high-tech industries. It can be expected that the added value of high-tech industries will continue to grow in the future under the influence of different systems.

5. Results Analysis

As demonstrated in Section 3.1, the fractional order grey accumulation grey Lotka–Volterra model has significant advantages in analyzing the interrelationships between indicators in Beijing’s high-tech industries and other interrelated system indicators. However, the model is limited in its ability to make long-term predictions based on these interrelationships. Consequently, this article selected the fractional order cumulative grey Lotka–Volterra model for relationship analysis, employing a grey differential dynamic multivariate model to predict and analyze the added value of high-tech industries.

The analysis of the relationship between the added value of high-tech industries in Beijing using the fractional order grey accumulation grey Lotka–Volterra model reveals a competitive relationship between the added value of high-tech industries and the human capital system, capital system, technology system, and trade system. This mutual relationship indicates that there is currently no coordinated and stable development trend between the added value of high-tech industries in Beijing and the system indicators that affect each other, and it still needs close attention from various departments in Beijing.

From the calculation results of the fractional order grey accumulation grey Lotka–Volterra model and the grey differential dynamic model, it can be seen that the increase in the selected influencing factors in each system had a promoting effect on the increase in added value of high-tech industries in Beijing. However, the impact of various influencing factors on the added value of high-tech industries varied according to the following systems: funding, human capital, trade, and technology. The following analysis examines the development trend of the added value of high-tech industries in Beijing from the above systems.

5.1. Analysis of the Prediction Results of Beijing’s High-Tech Industry Value Added Based on the Capital System

According to statistical data, the added value of high-tech industries in Beijing has been increasing year by year over the past six years. The prediction results obtained using the grey differential dynamic multivariate prediction model based on the funding system also show a gradually increasing trend. The specific prediction process is shown in Section 4.1, and the predicted value added of Beijing’s high-tech industries under the funding system is shown in Figure 2. As one of the important influencing factors on the added value of high-tech industries, the funding system has a certain impact on the development of high-tech industries.

Figure 2 demonstrates an increase in investment in the capital system, which has been observed to result in a concomitant rise in the added value of Beijing’s high-tech industries on an annual basis. This indicates that the enhancement of the added value of high-tech industries is contingent on internal R&D expenditure and government funding support. It can be concluded that an increase in internal R&D expenditure and government investment is associated with a corresponding rise in the added value of high-tech industries. The internal R&D expenditure can be considered a proxy for the level of innovation investment, and the continuous expansion of R&D expenditure and steady increase in investment intensity have created favorable conditions for China’s scientific and technological innovation to achieve more “parallel running” and “leading”, thus laying the foundation for the increase of high-tech industry output value. Government funding investment is also another important reason for the increase in the output value of high-tech industries. The increase in government funding has been demonstrated to have the capacity to alleviate the financing constraints experienced by high-tech industries, thereby enabling them to attract more talented individuals, update their equipment, and increase their output value. In addition to alleviating financial pressure, government funding has been shown to boost research and development confidence, and to facilitate the acquisition of capital, cooperation and technical support by high-tech enterprises, thus increasing their output value. It is therefore evident that, in order to achieve a steady growth trend in the added value of high-tech industries, it is possible to increase internal R&D funding and government funding and direct more research funds and personnel towards high-tech industries.

5.2. Analysis of the Prediction Results of Beijing’s High-Tech Industry Value Added Based on the Human Capital System

According to the predicted results of the added value of high-tech industries in Beijing under the influence of human capital in Table 17, it can be seen that under the influence of the human capital system, the added value of high-tech industries has been increasing year by year. To intuitively understand the future development trend of the added value of high-tech industries, a line chart was drawn, as shown in Figure 3.

As demonstrated in Figure 3, there is a clear upward trend in the data. The added value of high-tech industries is positively correlated with the labor efficiency of all employees. In other words, improvements in the labor efficiency have a significant impact on the growth of added value in high-tech industries. Nowadays, technology is developing at an increasingly fast pace, and it is constantly changing with the changing environment. The competition between technologies ultimately boils down to a competition for talent. As the capital of China, Beijing is the country’s political center, scientific research center, cultural center, and center of technological innovation. As the political center of the country, its policies are relatively advanced and perfect. As the national center for scientific and technological innovation, it has dense innovation resources and numerous high-tech enterprises, as well as a relatively complete technology platform. At the same time, as the capital of China, Beijing has a developed economy and strong talent attraction. In addition, Beijing has attractive talent introduction policies that focus on talent cultivation and development. Having high-quality and high-tech talents will lead to higher labor efficiency for all employees, thereby achieving growth in the added value of high-tech planned industries. Therefore, in the future, Beijing can continue to improve the labor efficiency of all employees through the introduction of talents, in order to achieve an increase in the added value of high-tech industries.

5.3. Analysis of the Prediction Results of Beijing’s High-Tech Industry Value Added Based on the Trade System

The trade system is chiefly concerned with the import and export trade of high-tech industries, and the present study aims to explore the impact of the trade system on the added value of China’s high-tech industries. To this end, the import volume of high-tech industries was selected as the influencing factor for predictive analysis. According to the prediction results in Table 17, it can be seen that the added value of high-tech industries in Beijing is to some extent affected by high-tech imports. To provide a more intuitive understanding of the impact of high-tech imports on the added value of high-tech industries, the development trend of the added value of high-tech industries under the trade system was plotted, as shown in Figure 4.

As shown in Figure 4, the high-tech industry in Beijing has exhibited an upward trend on an annual basis, influenced by high-tech imports. A potential rationale for this is that the import of high-tech not only benefits the improvement of human capital level, but also fosters innovation and the development of high-tech industries. A thorough examination of the industrialization process of developing countries reveals that their technological strength has yet to reach that of developed countries. The quality of their national high-tech industries is relatively low, and their investment rate in scientific research and development, the protection of intellectual property rights, and the cultivation of talent requires enhancement and refinement. Relying exclusively on domestic R&D investment to achieve technological progress is a time-consuming and arduous endeavor. Conversely, the technology spillover effect engendered by high-tech imports has the potential to curtail research and development investment while expeditiously mastering advanced Western technologies. This, in turn, can catalyze domestic technological innovation and engender an augmentation in high-tech industries. It is noteworthy that imitation innovation, mainly based on technology introduction, constitutes a pivotal conduit for technological innovation. This process entails the assimilation of methodologies employed by pioneering innovators, the interpretation of their core technologies, and the subsequent extension of research in this domain. Such an approach, predicted on the utilization of existing technological solutions, assumes significant importance in the nascent stages of high-tech industry development. The increase in the import quantity of high-tech products has been demonstrated to promote the growth of gross domestic product (GDP) and can also promote the upgrading of China’s industries through the import of high-tech products and technologies, increasing the output value of high-tech industries. The increase in the import value of high-tech products in Beijing is indicative of the continuous increase in demand for high-tech products and the rapid development in Beijing of the high-tech sector. However, in order to become a technological powerhouse, we still need to improve our independent innovation capabilities and to overcome the key technologies that constrain economic and social development by enhancing our independent innovation capabilities.

5.4. Analysis of the Prediction Results of Beijing’s High-Tech Industry Value Added Based on Technological Systems

Draw a line chart based on the predicted value added of high-tech industries under the technology system in Table 17, as shown in Figure 5. The impact of technological systems on the added value of high-tech industries should be analyzed by examining the intrinsic relationship between patent ownership and the added value of high-tech industries. The number of patents is generally a reliable indicator of the level of technological advancement. It is evident that advanced technology exerts a substantial influence on the development of high-tech industries and the augmentation of their added value.

As illustrated in Figure 5, the added value of high-tech industries in Beijing is predicted to exhibit a consistent growth trend in forthcoming years, influenced by the technological system. The annual increase in output value is anticipated to rise in tandem with the growth in the number of patents owned. The number of patents has been shown to be a reliable indicator of an industry’s innovative spirit and capacity. Innovation is widely recognized as the primary catalyst for development, and China has implemented numerous policies to foster innovation. The vitality and fruitfulness of innovation in scientific and technological domains are evidenced by the substantial increase in invention patents. As of the end of 2021, Beijing had 405,000 invention patents, which is 5.8 times higher than in 2012. The number of invention patents owned by 10,000 people is 185, which is 5.5 times that of 2012. The added value of high-tech industries continues to increase with the increase in patent ownership. Therefore, Beijing is well-positioned to reinforce the innovative capacity of enterprises, promote the development of innovation policies, improve patent application policies, strengthen intellectual property protection, prevent talent from being disheartened, and achieve sustainable growth in the added value of high-tech industries.

6. Conclusions and Suggestion

Employing a grey differential dynamic multivariate model to forecast and analyze the value-added growth of Beijing’s high-tech industries, this study reveals the following key findings:

(1)

The predictive model proposed in this study demonstrated unique advantages in addressing small-data problems. For future research on limited-data scenarios, the DHGM(2,M) model can be further applied and refined. Additionally, integrating superior optimization algorithms for parameter tuning could enhance its performance and adaptability.

(2)

Sustained expansion is propelled by an integrated ecosystem encompassing financial mechanisms (internal R&D and government funding), human capital (talent concentration and labor productivity enhancement), trade systems (technology spillovers from imports), and technological innovation (patent-driven advancement), collectively forming a “policy-technology-talent-market” synergy.

(3)

A statistically significant positive correlation exists between patent volume and industrial added value, demonstrating innovation’s direct contribution to industrial upgrading, while productivity gains and technology absorption synergistically accelerate catch-up cycles—providing a replicable paradigm for regional development.

(4)

Policy instruments, particularly precision funding allocation, robust IP protection regimes, and talent attraction initiatives, have effectively stimulated innovation vitality and human capital accumulation.

To optimize high-tech industrial development, we recommend the following: (1) strengthening indigenous innovation through targeted breakthroughs in critical technologies and increased basic research investment to establish import-substitution/export-oriented dual circulation; (2) optimizing innovation factors via public–private funding mechanisms and industry–academia collaboration for specialized talent cultivation; (3) implementing stringent patent quality controls and blockchain-enabled IP protection throughout innovation chains; and (4) developing dynamic monitoring frameworks including fund-output correlation models, labor productivity–value added assessments, and patent density–growth matrices to enable evidence-based policymaking and risk mitigation. These measures collectively facilitate the transition from quantitative expansion to qualitative leadership in high-tech industrial development.