Research on the Construction of Digital Economy Index System Based on K-means-SA Algorithm

The digital economy is developing rapidly worldwide, which is of great strategic significance in leading and driving the process of high-quality and high-efficiency development of the economy. It has gradually become a new engine to promote China’s economic growth. Therefore, it is vital to establish a digital economy indicator system. In this context, according to the connotation of digital economy, comprehensively consider the four dimensions of digital foundation, digital application, digital innovation and digital benefit, and build a full-scale digital economy indicator system. However, some previous literature proposed operating the K-means clustering algorithm in optimizing the index system. The algorithm, susceptible to the initial selection of cluster center, generates completely different clustering results with other random seed points and is thus unsuitable for optimizing the index system. Based on this, the advantages of GRA, SA, and K-means algorithms are combined to propose a K-means-SA algorithm that can obtain the global optimal solution. Then, by combining the K-means SA algorithm and the rough set algorithm, the constructed index system is further optimized. Ultimately, it establishes a set of all-round, multi-system and multi-dimensional digital economy index systems, which is of great reference significance for formulating relevant policies, provides scientific index support for the subsequent extension of digital economy theory, and promotes the benign growth of the worldwide economy. Plain language summary The importance of constructing and optimizing the digital economy indicator system In today’s rapid development of the digital economy, a set of scientific, feasible, and practical application values of the digital economy evaluation index system is constructed to evaluate better and promote the development of the digital economy. Based on the four dimensions of digital foundation, digital application, digital innovation, and digital benefit, this paper constructs a comprehensive evaluation index system for the development level of the digital economy. In addition, by combining the advantages of GRA, SA, and K-means algorithms, a K-means SA algorithm can obtain the global optimal solution proposed. Then, the K-mean-SA algorithm is combined with the rough set algorithm to optimize the constructed index system further. Finally, a comprehensive, multisystem, and multi-dimensional evaluation index system for the development level of the digital economy is established, which provides scientific support for the development of digital economy theory and promotes the healthy development of the digital economy. Although this paper collects indicators related to the digital economy as comprehensively as possible based on existing studies, there are still some limitations in the selection of needles due to the strong integration of the digital economy.


Introduction
With the continuous development and popularization of digital technology, the digital economy has evolved into a new engine of the international economy, which has a significant position in fostering economic growth, social welfare, industrial upgrading, cultural exchanges, and other fields, and has gradually become the center force leading China's economic growth, industrial change and the construction of a new international pattern (D.Shi, 2021;J. Wang et al., 2021).At the G20 Hangzhou Summit held in 2016, the leaders of the participating countries jointly signed the G20 Initiative on Digital Economy Development and Cooperation, and the digital economy was listed as an essential issue in the G20 Blueprint for Innovative Growth for the first time.Since then, the digital economy has evolved as a focus of awareness for all countries worldwide.Since the 18th Party Congress, the government has attached grand matter to developing the digital economy.In 2017, the word digital economy appeared for the first time in the report on the work of our government.2019 to 2022, been noted in the national work report for four straight years and successively put forward to grow the digital economy, build a unique advantage in the digital economy, accelerate digital development, and form a digital China to promote the growth of the digital economy, strengthen the overall layout of the build of digital China and other development plans.In the past 5 years, the ranking of China's digital economy has thrived, effectively hedging the economic downward pressure in the context of the epidemic.It has become one of the primary ambitious forces of financial increase and the core competitiveness of China to gain economic conversion and change the pattern of global competition.The digital economy has evolved a fresh focus of international attention and is the first significant new impetus to lead highquality economic development (Cao, 2018;W. Sun & Liu, 2022).The 20th Party Congress also put forward more clear requirements for the digital economy, such as concentrating on promoting the integration of the digital economy with the actual economy and realizing the digital accurate twin technology (X.Jiang & Jin, 2022).According to the relevant information from the China Communications Research Institute, China's digital economy gets 45.5 trillion yuan in 2021, up 16.2% yearon-year, indicating that China's digital economy industry resumes to increase, the process of industrial digitization is accelerating, and the digital economy is showing a heightened rate of growth (Z.Li & Liu, 2021).
However, against the background of the overall trend of China's digital economic growth, China's digital economic development is still meeting the issue of the ''digital divide,'' and there is still a polarization of the station of digital economic expansion between regions.Therefore, In the background of the development of the digital economy, it is necessary to construct a set of comprehensive evaluation index systems for China's digital economy (Pan et al., 2021;M. Zhang, 2017).To accurately estimate China's digital economy growth and then accurately grasp the regional differences and nonequilibrium characteristics.There are moderately more studies on the indicator system of the digital economy, but the controversial differences are still prominent (Williams, 2021).At home and abroad, there have been many studies on the definition of the digital economy, statistical accounting, and the index system for measuring development.However, a uniformly recognized system of statistical indicators still needs to be (Cong et al., 2022;Guo et al., 2022).Furthermore, at present, the research ideas of academics and governmental departments are divided into two kinds: one is to directly measure the volume of scale created by a particular defined region at a specific point in term; the other is to, through the establishment of a set of multi-dimensional indicators to evaluate the system, to evaluation of development of a defined region at a specific point in term (Q.Y. Xu et al., 2018), such as the OECD, the European Union, the United States, Tencent, Saidi and other nations and related organizations have a statistical digital economy indicator system, but the difference between each other is relatively significant.
Given this, based on the research of existing scholars, this study offers a new K-means-SA algorithm by combining the advantages of GRA, SA, and K-means algorithms.Further, it optimizes the constructed index system by connecting with the rough set algorithm to build a set of multi-system, multi-dimensional evaluation index systems on the development of the digital economy, to help decision-makers better understand the expansion of the digital economy, to formulate more scientific and reasonable policies, and to infiltrate new kinetic power for the country's economic growth.

Literature Review
This section first classifies the applications and effects of the digital economy, secondly organizes and reviews the present condition of study on evaluation indicators of the digital economy, and finally reviews and summarizes the evaluation and measurement strategies for the digital economy.

Research on the Application and Impact of the Digital Economy
The digital economy is a new type of economy based on digitization and networking, which uses advanced data technology and network technology to attain rapid access, processing and transmission of information and has an essential impact on the conversion and upgrading of enterprises (X.Jiang, 2020;Savina, 2018;Sturgeon, 2021).The digital economy involves many areas, including e-commerce, digital finance, cloud computing, significant data, and additional emerging economic fields (L.Shi & Chen, 2023).It has broad connotations and extensions (Xia et al., 2023).Regarding the application and effect of the digital economy, the publication of the 2017 China Digital Economy Index has made essential contributions to developing the digital economy.Subsequent scholars have provided insights on the evaluation of digital economy index from various angles or levels such as digital economy theory (Yao, 2020), macro and micro (Jing & Sun, 2019), value creation sources and economic scale (Pei et al., 2018), consumer rights and interests (Qi et al., 2020), and the implementation of digital economy in China's small and medium-sized enterprises (Ren, Lee, & Hu, 2023;Y. Wang, 2021).Some scholars have also explored the connection with the digital economy in terms of the degree of industrial integration (Xin et al., 2023), the growth of regenerative energy (Hwang, 2023;B. Wang et al., 2023;Zheng & Wong, 2024), new government-business relations (Y.Chen et al., 2023), global energy transition (Onifade et al., 2021;Shahbaz et al., 2022), and reduce carbon emissions (Z.Li & Wang, 2022;Yan et al., 2022).Most scholars have specifically analyzed the application and influence of the digital economy from the perspectives of highquality development, networking, and industrial conversion and upgrading (Han et al., 2022;Ren, Wang, & Liu 2023;X. Sun et al., 2022;Zhang, Zhao, & Cheng).However, more studies must evaluate the digital economy, especially in forming a digital economy index system.Therefore, this paper focuses on selecting and constructing an index for assessing digital economic development.

Research on the Index System for the Level of Development of the Digital Economy
In designing the index system for estimating digital economic growth, some students have attributed it to the digital scale and used the digital economy development index as a reference to determine the corresponding evaluation index (P.Chen, 2022; Z. Li & Liu, 2021;Sidorov & Senchenko, 2020).Domestic and foreign professionals, scholars, and organizations have guided numerous explorations on the level of digital economic development and have constructed an indicator system for evaluating the level of digital economic development from multiple dimensions (Belanova et al., 2020).Foreign scholars primarily study the development trends and mechanisms of the digital economy to construct an indicator system for evaluating various aspects such as government digital transformation, innovation projects, and the development of smart cities (Basol & Yalcın, 2021;Ivanova et al., 2019).Another aspect is that domestic scholars focus more on establishing a more scientifically grounded indicator system to assess digital economy development.Founded on the Organization for Economic Co-operation and Development (OECD) digital economy research framework, Xiang and Wu (2018) used officially published foreign data to estimate the digital economy and set forward suggestions.J. Liu et al. (2020) discussed the relationship between information, networking, and digital transactions and constructed eight secondary and 17 tertiary indicators.They operated the SAR ideal to measure China's digital economic expansion from 2015 to 2018.Pan et al. (2021) constructed 20 evaluation indicators from four levels and used the entropy method to comprehensively score China's digital effect level from 2012 to 2019.Y. Sun et al. (2022) combed through the current evaluation theories and modes of the digital economy and constructed an evaluation system consisting of seven dimensions and 26 primary indicators.Based on this, Scholars have analyzed the evaluation index system of the digital economy and high-quality development and green innovation by constructing the evaluation index system and empirically analyzing the role mechanism between the two (Luo et al., 2023;Tao et al., 2022;W. Zhang et al., 2021;Zhao et al., 2019).Zhou and Liu (2022) constructed 15 secondary and 35 tertiary indicators from five dimensions of innovation, governance, economy, security, and service and analyzed the regional digital competitiveness development trend.L. Wang (2022) constructed an indicator system consisting of 19 primary and 44 secondary indicators from six dimensions of the digital economy.Lian et al. (2022) constructed five primary indicators and 20 secondary indicators from the viewpoints of digital governance, digital infrastructure, digital invention, industrial digitization, and digital automation and measured the digital economic development level of 19 Chinese cities using methods such as the Dagum Gini coefficient.Du, Huang, & Dong (2022) constructed four primary indexes and 22 secondary indexes from the perspective of the coordinated coupling of rural revitalization and the digital economy.Infrastructure development is essential.That is, the development of the digital economy must be reinforced by a digital foundation and fueled by digital innovation (Onifade, Ay, et al., 2020).The China Academy of Information and Communications Technology (ICT) estimated the ranking of China's digital economy from multiple perspectives (Wu & Wang, 2012).Some scholars have also constructed a digital economy development indicator system established on the inter-provincial level (Z.Wang & Shi, 2021).
Through the above analysis, it is not difficult to find that domestic and foreign scholars, when constructing the index system for the digital economy, pay more attention to considering the demand side or using a single dimension to estimate the level of development and lack a comprehensive measurement of the digital economy.Based on existing research, this article adopts the literature analysis approach, based on the frequency of evaluation indexes, screening and integrating the existing evaluation indexes, and initially constructs a set of more complete and feasible digital economy index systems.

Research on the Evaluation Method of Digital Economy Development
In terms of evaluating the level of development of the digital economy, most scholars operate the K-means algorithm to optimize the indicator system, and some scholars use the entropy weight mode, gray correlation analysis, and factor analysis method (Y.Xu & Li, 2022;Y. Yang & Dong, 2022;S. Yang & He, 2022).The Kmeans algorithm is a widely used clustering algorithm.After years of research, the K-means algorithm has been employed in multiple areas, such as text data mining, genetic data recognition, and speech recognition (Abdullah et al., 2021;Dong & Zhu, 2020;O ¨zo¨g˘€ ur Aky€ uz et al., 2023).However, in practical applications, different cluster centers can lead to fluctuations in the clustering outcomes, and the number of groups is challenging to determine.When implementing the K-means algorithm, researchers operated the most straightforward explanation and randomly selected the initial cluster center points several times to find the local optimum solution.However, later studies found that the initial cluster number of the traditional K-means algorithm is determined randomly, and the algorithm is sensitive to cluster center abnormalities, resulting in clustering results that are different for different cluster centers (F.Li & Nong, 2013;H. Du et al., 2018;M. Li et al., 2020;X. Wang & Bai, 2016).In other words, the K-means algorithm is a greedy algorithm that tends to obtain stuck in local optimal solutions (Pham et al., 2004).To solve these shortcomings, scholars have proposed various improved algorithms, such as the DMK-Kmeans (H.Jiang et al., 2018), K-means-PSO-SVR (P.Liu et al., 2022), DIC-DOC-K-means algorithm (Lakshmi & Baskar, 2019).DMK-Kmeans algorithm strives to unravel the issue of local optimality of clustering results generated by the traditional K-means algorithm's random initial cluster center selection.The K-means-PSO-SVR algorithm is a local modeling method, and the DIC-DOC-K-means algorithm is established on the dissimilarity of initial centroids selection to enhance reader document clustering.With the development of deep learning and deep learning, scholars have proposed a CMFGN clustering algorithm based on multi-layer features and graph attention net and have demonstrated the efficiency of the CMFGN algorithm through many experiments (Hou et al., 2023).In addition, H. Du et al. (2018) used the improved density peak algorithm (DPC) to select the initial clustering center.Some scholars have replaced the traditional Euclidean distance with other distances to calculate the space between sample points and have demonstrated through experiments that the enhanced k-means algorithm has a more acceptable clustering result and intersection than the traditional one (Zhu et al., 2021).Although scholars have improved the traditional Kmeans algorithm from different angles, most improvements are limited to the random selection of initial cluster centers and do not fundamentally solve the problem of getting stuck in locally optimal solutions.
In summary, current research on the digital economy's development level by both domestic and foreign scholars has yielded fruitful results.However, there are also the following areas for improvement.Firstly, these traditional research models only analyze the development of the digital economy from a one-sided and localized perspective through theory or other aspects.They cannot effectively combine the indicator system with the model.Secondly, The evaluation indicators for the level of digital economic development currently have a narrow scope of coverage, and most studies are considered from the demand side, resulting in a limited evaluation of the level of digital economic development.Finally, Some scholars use the K-means algorithm to screen digital economy indicators.However, the K-means algorithm has inherent flaws and cannot find globally optimal solutions, so it is unsuitable for optimizing the evaluation indicator system of digital economic development.

Construction of the Evaluation Index System
In this paper, the steps for selecting the evaluation index system of the digital economy are ''literature collectionindex selection-preliminary construction of the index system.''Based on the regulations of scientific, representative, orientation, and comparability, it aims to construct an index system that combines the existing research and the wisdom of authoritative experts and scholars so that the constructed evaluation index system tends to be reasonable and universally applicable (Onifade, Cxevik, et al., 2020;Y. Sun & Chen, 2018).
Consider the conditions and application environment of the digital economy in the context of sustainable development scenarios based on the connotations and elements of the digital economy (Jin et al., 2019).This paper takes ''digital economy indicator system'' as the keyword.It manually screens out 40 documents (the selected documents are all published in well-known and authoritative journals or publishers, with high citation counts and high academic level and value, including 28 documents in Chinese and 12 in foreign languages).The indicators contained therein were collected manually, and after collating, summarizing, and counting, Table 1 displays the indicators with a frequency share more significant than 7% To provide the comprehensiveness and fullness of the selection of indicators, this study selected 19 indicators with a frequency greater than or equal to 8 times and added the indicators that ranked lower in frequency and were not filtered for the first time, resulting in the selection of 28 indicators.Indicators are classified based on existing research.The development of the digital economy relies on ''four needs,'' that is, the need for the growth and advancement of the digital basis, the need for the evolution and deepening of digital applications, the need for the increase and service of digital innovation, and the need for the conversion and upgrade of the digital enterprise, thereby achieving the enhancement of the benefits of the whole industry and society.Thus, based on the connotation of the digital economy and concerning the evaluation index system for the level of development of the digital economy established by scholars and third-party institutions at home and abroad, four first-level indicators, namely, digital foundation, digital application, digital innovation, and digital benefit, were constructed from the four levels of infrastructure, application, innovation, and benefit (Ji & Zhu, 2022;Shen & Zhou, 2023;X. Xu & Zhang, 2020;X. Zhang & Jiao, 2017).
Digital infrastructure is an essential element in the digital age and is the prerequisite for developing and improving the digital economy.It is also the foundation for realizing a digital society.Referring to studies by scholars such as Lv and Fan (2023), Huang et al. (2023), Wan and Luo (2022), and Y. Li et al. (2023).Segment the digital infrastructure into device infrastructures such as mobile, fixed, and network.Digital application refers to various business processes and information management through digital technology to achieve efficient, convenient, accurate, and sustainable business operations.Referring to research by Wen et al. (2020)  innovation into two indicators: innovation input and innovation output, injecting new innovative power into enterprise development.Digital benefits are achieved through digital technology and tools, increasing operational efficiency, reducing costs, and enhancing innovation capabilities, according to studies by Shen and Zhou (2023) and Y. Sun et al. (2022).Segment the digital benefits into economic benefits and social benefits.Based on the analysis above, this research is an constructed an evaluation indicator system for the digital economy development, including four primary, nine secondary, and 28 tertiary indicators (Table 2).

K-means-SA Algorithm
The immediate purpose of this paper is to construct a set of indicators that accurately measure the level of digital economic development through indicator screening, clustering, and optimization.The purpose of clustering is, on the one hand, to find out whether the information of the selected set of indicators is overlapping and to eliminate it, and on the other hand, to perform an accurate cluster analysis for each set of indicators after removing one secondary indicator at a time.Currently, the clustering algorithm commonly used in academic research is the K-means algorithm.However, despite its many advantages, it affects the clustering results due to its sensitivity to the initial clustering center and anomalous data and its tendency to fall into a local optimum.
While some studies have also found that the K-means algorithm has deficiencies, and academics have made different degrees of improvement, most scholars only improve the K-means algorithm initial clustering center randomly selected and did not fundamentally solve the defects that easily fall into the local optimum.The final effect of constructing indicators in this paper is to seek the optimal result.That is, the existing improved Kmeans algorithm does not apply to the screening of indicators in this paper.However, the SA algorithm is an improved optimization algorithm based on the hillclimbing algorithm, and its most prominent feature is that it satisfies the Monte Carlo criterion and can obtain the global optimal solution.If the benefits of the SA and K-means algorithms are combined efficiently and used for the dynamic clustering process, it could overcome the early clustering occurrence witnessed in the conventional K-means algorithm.At this point, the procedure and the fitness function are designed based on the actual situation of different clustering problems so that the algorithm converges to the global optimal solution more efficiently and avoids falling into the local optimum.The rough set algorithm estimates the established indicator system, creating a more precise one.
In this paper, we use the idea of K-means clustering to determine the objective function.That is, divide the sample data into K clusters, each with its own centroid (center of mass), and minimize the sum of the distances between the sample data and the corresponding center of mass points to find an optimal clustering scheme, which determines the objective function.Since the gray correlation coefficient formula is highly similar to the distance formula, and the larger the gray correlation coefficient is, the better the result is, the paper sets the distance formula as the inverse of the gray correlation coefficient.The weighted sum of the inverse of the gray correlation coefficient is the objective function.
The K-means-SA algorithm is a search process for new solutions based on the initial solution.That is, under the premise of specifying the initial solution of the original problem, it randomly generates new solutions within the proximity of the initial solution and carries out the subsequent optimization process on top of it.The specific implementation steps of the K-means-SA algorithm are as follows: Step 1: Set the algorithm parameters and initialize the solution vector.
Set the initial temperature to T 0 = 1 3 10 1 , the current temperature to T = T 0 , the terminal temperature to T end = 1 3 10 25 , the cooling rate to q = 0.9, LL = 100 represents the number of iterations, the current iteration number K = 1, assume that the current global optimal solution is X = X 0 , and import the number of clusters as num, which has different value ranges due to different numbers of indicators.
Step 2: Construct a new solution.
Based on the existing solution, randomly search for a new solution X new in the neighboring area and check whether the new solution X new meets the conditions.If not, it keeps looping and searching for a new solution again until it gets a feasible one.
Step 3: Comparison and judgment Calculate the objective function values of the initial solution X 0 and the new solution X new , represented as F(X 0 ) and F(X new ), respectively.If F(X new ).F(X) = F(X 0 ), set X 0 = X = X new , and jump to Step 5. Otherwise, jump to Step 4.
Step 4: Metropolis criterion Calculate the cost of the new solution PF, let PF = F(X 0 )-F(X new ), and accept X new as the current new solution with a probability P(A) = EXP(2PF/T).
Step 5: The number of iterations K = K + 1.If K .1, jump to Step 6.Otherwise, return to Step 2.
Step 6: Temperature decreases, and the current temperature decreases to T = T 3 q.If T\T end , the algorithm ends, the optimization process terminates, and the optimal solution X, the optimal clustering result determined by the number of clusters, is obtained.Otherwise, return to Step 2.

Model
Combining the K-means-SA algorithm, we select evaluation indicators from five aspects: constructing a scoring matrix, indicator normalization, gray dynamic clustering analysis, calculating F-statistics, and rough set indicator reduction (Su et al., 2022).
(1) Constructing a scoring matrix We are inviting seven experts engaged in research in the field of digital economy to hold a symposium on the constructed digital economy indicator system, including three who worked in different companies related to digital technology and four senior professors or associate professors from different universities' digital economy research fields.The scoring matrix is the data source for system optimization.
(2) Indicator normalization Because the indicators involved in this paper are benefit type due to the standard 0 to 1 transformation of the initial sample data without loss, high degree of reduction, and the method in various types of research in a wide range, this paper uses this method to achieve the normalization of the data (G.Chen et al., 2022;Hong & Huang, 2021;F. Liu et al., 2021;Wu et al., 2022;L. Zhang & Gao, 2020), the standardization of the formula as the Appendix of the formula 1.
(3) gray dynamic clustering analysis For the attribute set obtained after the first screening (Table 2), the gray Relational Analysis (GRA) algorithm can overcome the limitations of the number and distribution patterns of research objects and has a significant advantage in representing information overlap using a gray relational matrix.Equation 2 in the Appendix is the formula for the gray correlation coefficient, and Equations 3 and 4 in the Appendix is the formula for the gray correlation matrix.
(4) Calculation of F-statistic The large number of clusters generated during the clustering process can lead to inconsistent clustering results, necessitating a scientific method for finding the optimal number of classifications.As is known from mathematical statistics, the value of F in the F-statistic directly represents the distance between two categories.The greater the value, the greater the distance between different categories, and the closer the classification is to being optimal, resulting in a more perfect classification performance.Therefore, the advantage of using the Fstatistic is that it encourages more scientific and reasonable classification results.This paper calculates the optimal number of classifications using the F-statistic, calculated as Equation 5 in the Appendix.
(5) Rough set index reduction The attribute reduction process in this paper compares the initial attribute set and the clustering results after successive deletion of attributes.Suppose there is a difference between the two clustering results.In that case, it indicates that the deleted attribute significantly impacts the clustering type of the attribute set and is a necessary index, so this attribute should be retained.Conversely, exclude the attribute.
Based on the detailed description of the index screening process in the previous section, a flowchart for index screening can be drawn as shown (Figure 1).

K-means-SA Empirical Analysis
Based on the research design.Scores were given for ''typicality, scientific, systematic, and sustainability,'' with 25 points assigned to each category and a perfect score of 100.The expert scoring data is shown (Table 3).Furthermore, the expert scoring data was normalized using formula 1, shown (Table 4).
Based on normalized data, the screening of indicators shown in Table 2 with the three-level indicators C 11 to C 13 and C 21 to C 22 as examples, the screening process of the rest of the indicators is the same as this process, and the gray correlation matrix C of the scoring data according to Equations 24 is established.
The type of clustering can affect the final clustering results.The optimal number of classifications needs to be determined to obtain a more scientific clustering result.Formula 5 was used in this paper to calculate the F-values for each category.According to Table 4, when the number of clusters is 3, the F-value is the largest, suggesting that the best classification is {y 1 ,y 4 },{y 2 ,y 5 ,y 6 ,y 7 }, and {y 3 }.On this basis, the indicators have rough sets to approximate them.Precisely, remove one indicator at a time, and the K-means-SA algorithm was operated to search for the optimal clustering results for three classifications and compared with the original clustering results to remove indicators that did not affect the clustering results.The set of all attributes is C. Clustering by deleting C 11 ;C 13 and C 21 ;C 22 .The clustering results are shown (Table 6).
From the dynamic clustering results of Table 5, we can see that the optimal clustering results after removing any indicator are different from the clustering results of all indicators.Explain that all indicators need to be retained in their entirety and not abbreviated.Similarly, we can reduce the third-level indicators belonging to other secondary indicators through the above steps.The specific results are as follows: the clustering results after removing A 14 and A 15 in Mobile Device Foundation A 1 are the same as the clustering results of all indicators, indicating that A 14 and A 15 have no impact on the comprehensive evaluation results, so these two indicators should be simplified; the clustering results after removing A 32 in Network Equipment Foundation A 3 are the same as the clustering results of all indicators, indicating that A 32 should be simplified; the clustering results after removing D 14 in Economic Benefits D 1 are the same as the clustering results of all indicators, indicating that D 14 should be simplified; the clustering results after removing D 21 in Social Benefits D 2 are the same as the clustering results of all indicators, indicating that D 21 should be simplified.For other secondary indicators, the optimal clustering results differ from all indicators' clustering results after removing any indicator, so all indicators need to be retained without reduction.
The K-means SA algorithm optimizes the indicator system (Figure 1).Finally, an indicator system with four  dimensions, nine secondary indicators, and 22 tertiary indicators comes into being, which evaluates the digital economy as more rational, comprehensive, and holistic and provides a scientific basis and reference for the government, enterprises, and society, which is conducive to the planning of policies and strategies related to the digital economy and the promotion of the high-quality development of the digital economy.The optimized digital economic evaluation indicator system is shown (Table 7).

Theoretical Contribution
The primary goal of this paper is to eliminate the overlap between attributes, making the constructed indicator system more precise.The main contributions of this article are the following: (1) Combining GRA, SA, and Kmeans algorithms, we propose a K-means-SA algorithm that can obtain the global optimal solution.It not only overwhelms the defects of the original K-means algorithm's random selection of initial cluster centers but also solves the inherent defects of single algorithm convergence to local optimum and premature, thus making the clustering results more accurate and stable.This supplements and extends the theoretical basis of relevant algorithms.
(2) By combing and reviewing the indicator system in the existing literature, we comprehensively consider the four needs for the development of the digital economy, namely, the need for the development and improvement of the digital foundation, the need for the expansion and deepening of the digital application, the need for the increase and service of digital innovation, and the need for the change and upgrade of the digital enterprise, which will in turn lead to the enhancement of the benefits of the whole industry and the whole society.
Based on the four dimensions of digital foundation, digital application, digital innovation, and digital benefit, a set of all-round, multi-system, and multi-dimensional indicator systems is built.It aims to comprehensively evaluate the development level of the digital economy, enriching and promoting the research on the evaluation index system of the digital economy.

Limitations and Future Research
This paper also needs to include the following shortcomings that warrant further research in the future.(1) Combine and extend the advantages of the K-means-SA algorithm based on the digital economy evaluation index system formed in this article.Although the construction of the indicator system in this paper is relatively comprehensive, more is needed at the digital economic development applications, such as digital cities, digital education, and digital livelihoods, and there are fewer indicators at the level of digital governance.Subsequent research can continue to use relevant methods to streamline and supplement the indicators and replace the more difficult-to-measure indexes.It will also use China's inter provincial time-series panel data to conduct empirical analyses to guarantee that the data can remember the actual case of digital economic development.
(2) Although the proposed K-means-SA algorithm effectively optimizes the indicator system, it inevitably needs to be improved.The following research step will optimize and improve the algorithm to adapt to a more complex and variable digital economy development environment and improve the indicator system's accuracy and comprehensiveness.In addition, it enriches the accumulation of actual data and cases to improve the indicator system's applicability and provide more scientific, accurate, and feasible theoretical support for the fundamental operation and management of the digital economy.

Conclusion and Policy Recommendation
The digital economy is an essential part of the current direction of economic development., and assessing the level of digital economic development can deliver a scientific foundation and guidance for country departments to design digital economic policies, financial institutions to make investment decisions, and enterprises to make management decisions.Continue to promote the high-quality regional development of the digital economy.We are following the laws of development and the connotations of the digital economy.Using digital foundation, digital application, digital innovation, and digital benefit as the dimensions of analysis, it constructed a digital economy indicator system containing 9 secondary and 28 tertiary indicators.It is of outstanding academic and practical value to optimize the indicator system through the K-means-SA algorithm and rough set approximation to reduce the overlap to enhance the precision and strength of the digital economic development index system.

Policy Recommendation
The following policy recommendations are provided: (1) The digital economy is a new form of economy accompanied by continuous innovation in digital technologies.
With the widespread use of big data, blockchain, artificial intellect, and 5G, the profound integration of the digital economy with the whole of economizing has made everything data-enabled, and data has become a new element of display after ground, capital, technology, and knowledge.With the full development of the digital economy and the in-depth development of the value of data elements, the value of data will occupy a paramount share of the digital economy in the future.Therefore, to further deepen the understanding of the digital economy, it is vital to increase research on the definition of the concept of the digital economy, industry classification, value-added accounting, the building of a comprehensive index system and other aspects of the digital economy, and to expand the authoritative classification and accounting standards, thereby unifying the caliber of the digital economy accounting and clarifying the scope of the digital economy accounting.(2) In constructing the indicator system, comprehensive consideration from 4 dimensions, namely, digital foundation, digital application, digital innovation, and digital benefits, is given.It lays a basis for the study of the digital economy theoretically and practically.It provides a referable indicator system for evaluating the consequence of China's digital economy in this background to better exert the role and support of the digital economy for the high-quality and highly efficient development of the economy.Government departments should exercise their leading role and join forces with digital economy enterprises, universities, and research institutes to form a working group to conduct exchanges and discussions and analyze the current problems in measuring the digital economy.
(3) The existing indicator system is relatively biased toward the supply side and pays less attention to the demand side.From the experience of different economies at home and abroad over the past few decades, the statistical indicator systems constructed for the digital economy and its related economic forms are mainly supplyside indicators favoring the industrial dimension, while the construction of demand-side indicators corresponding to consumption, investment, and export is very preliminary whether it is the level and secondary indicators of the digital economy indicator system constructed by the US, OECD, or the primary and second-level indicators of the EU's Digital Economy and Society Index, as well as the indicators of the new economy, knowledgebased economy, information economy or digitization.
The Digital Economy and Statistical Indicator System, released by China's National Bureau of Statistics, are all obviously classified from the supply side of the different industries and sectors dimensions to define and count.Therefore, the construction of demand-side consumption, investment, and export indicators corresponding to supply-side industries is also an essential direction for improving the statistical indicator system of the digital economy in the present and future. Appendix Where Y={y 1 (1),y 2 (2),..,y m (n)}: the rating series; a ik : correlation coefficient of evaluation object y i to y k at the jth indicator.
F ¼ ð P r i¼1 n i jj y i À yjjÞ=ðr À 1Þ ð P r i¼1 P n i j¼1 y i À y 2 Þ=ðn À rÞ ð5Þ Where n i : number of evaluation subjects in category i (i = 1,2,..,n); y i À y = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P n h = 1 ( y i h À y) q : euclidean distance between the centre of clustering in class i and the centre of clustering of the overall sample; y i j À y i = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P n h = 1 (y i jh À y i h ) q : euclidean distance between the normalised data and the cluster centre of its class.

Table 1 .
Zhang, Dong, & Wang et alDu, Zhang, &Xia (2022)al application was divided into personal and enterprise applications, laying the foundation for promoting digital innovation.Digital innovation is based on digital applications and explores new business, commercial, and management models.It is the key to achieving digital benefits.Referring to research byZhang, Dong, & Wang et al. (2022),Zhou and Liu  (2022), andDu, Zhang, &Xia (2022).Segment the digital Frequency Statistics.
Source.Summarized by the author.

Table 2 .
Evaluation Index System for the Development Level of Digital Economy.

Table 7 .
The Optimized Digital Economic Indicator System.