Local governments are closely related to economy, society and ecology, and are closely related to people’s lives. As an important institution for the implementation of national political management, local governments are the specific implementers of national will ( Jordan & Boody, 2007 ). Government behavior refers to the administrative activities conducted by the government to manage social public affairs in accordance with the relevant policies formulated by the government in order to safeguard public interests and improve the public foundation ( Chen & Hsieh, 2011 ). It is of public welfare and publicity, providing universal public goods and services for people, such as solving people’s problems in education, medical care, health and housing, improving the social security system, developing charity and other public welfare behaviors ( Geng et al., 2013 ).

In industrial development, government behavior plays an important role. The government is not only the maker of rules, but also the manager, supporter and supervisor of the market.

1) Formulate favorable policies and create a favorable environment. Through favorable policies on industry, foreign investment and talents, local governments can create a good external environment for industrial development, effectively improve the initiative of industrial subjects, induce and promote the industry to cultivate its own industrial innovation ability, promote the training and introduction of high-quality talents, and accelerate industrial development ( Zhen & Bi, 2011 ).

2) Guide capital investment and activate the industrial market. Industrial development requires a large number of funds, and local governments formulate preferential tax subsidy policies and overall plans for industrial development, which are conducive to foreign investment entering the industrial market, expanding financing channels, promoting the research and development and application of industrial technology, and reducing possible risks in industrial development ( Yang, 2012a ).

3) Establish a regulatory system to maintain market stability. The government is the regulatory department in industrial development, and the formulation of a series of supervision and management systems matching the industrial development is an important basis for industrial development. By establishing an operating mechanism combining government supervision and self-regulatory supervision, and constructing and improving the legal system of industrial supervision mechanism, the government can strengthen the supervision efficiency in the industrial field, prevent and resolve industrial development risks, and ensure the scientific and stable operation of the industrial market ( Zhang et al., 2007 ).

The sports industry belongs to the tertiary industry in the national economy and is an industry aiming at the production of sports products and sports services. According to the economic theory of the tertiary industry, after the input of production factors such as manpower and material resources, the sports industry does not have any output, but produces non-physical sports service products in the form of sports ( Zhang et al., 2008 ). The sports service products produced by the sports industry have both value and use value. In the market economy system, sports service products can be used for exchange, which is the result of the hard work of sports workers and has value. The monetary expression of value is the price of sports service products. At the same time, it also has non-physical form of use value, and is an important part of people’s physical fitness, aesthetic communication and other entertainment and leisure activities ( Guo & Fan, 2009 ). Therefore, the sports industry is a service industry with both input and output.

Government behavior refers to the activities that the local government and its staff carry out social public management and provide social services under the guidance of the governing philosophy in accordance with relevant policies and legal systems in a specific period ( He, Liao, & Wang, 2012 ). It is closely related to the concept of government, which is often based on the ruling ideology of the government, and government behavior is influenced and guided by the concept of government, which is the external manifestation of the concept of government ( Deng, 2010 ).

The performance of government behavior refers to the achievements and effects achieved by the government under certain investments in the process of public management, which is the measurement and feedback of the government’s governance ability ( Bao & Wang, 2012 ). According to the achievements of management efficiency, service quality and public satisfaction of different regional governments, the performance reflected in the evaluation can understand the behavioral performance of different regional governments. According to different levels of government, government behavior performance can be divided into Central Government performance and local government performance ( Ji, 2007 ). Local government behavior performance can be further divided into provincial government behavior performance, municipal government behavior performance, county government behavior performance and township government behavior performance. After an in-depth investigation, the research group of government behavior performance combined with the design ideas and methods of relevant index systems at home and abroad, summarized the analysis results of relevant experts, and put forward the performance index system of local government in China. The system includes three levels of government function index, government influence index and government potential index. Among them, government function indicators mainly include social management, economic regulation of public services and market supervision, etc. Government influence indicators mainly include economic growth, improvement of people’s quality of life and social development, etc. Government potential indicators mainly include personnel allocation, personnel quality and work efficiency ( Gong et al., 2014 ).

The performance of government behavior of regional sports industry refers to the achievements and effects achieved by governments at all levels in the process of public management and development of the sports industry in a province (municipality directly under the Central Government or autonomous region) as a unit, which reflects the development level of the sports industry in the region and is often closely related to local culture, economy and policies. It has strong regional characteristics.

The theoretical basis of this study mainly includes the theory of government behavior, the principle of industrial economics, the theory of regional industry, the theory of public policy and the basic theory of sports industry. This paper analyzes the connotation and decisive factors of government behavior affecting regional sports industry, and establishes the performance evaluation model of regional sports industry government behavior. This study draws on the ideas and methods of the relevant index design research at home and abroad, consults the relevant experts in economics and sports science, combines the actual level of China, and follows the principles of comprehensiveness, scientificity and feasibility of the research to determine the evaluation index system suitable for the performance of the government behavior of China’s regional sports industry (see Figure 1 ). It includes the gross national product, the proportion of the tertiary industry in the gross national product, the proportion of the sports industry expenditure in the financial expenditure, the proportion of government funds in the sports industry investment, the Engel coefficient, the sports lottery income, the number of international and national sports events, the number of sports of sports There are 16 indicators, including the number of mobilization, the

bureaus above the county (city), the number of sports schools, and the number number of stadiums, the number of employees in the sports system, the proportion of administrative personnel with bachelor’s degree or above, the number of sports-related policies formulated by the provincial (municipal) government, and the number of sports-related documents issued by the provincial (municipal) sports bureau.

This study combines logical analysis with empirical analysis, qualitative analysis with quantitative analysis. It mainly involves the principal component analysis in the multi-index comprehensive evaluation method, and the software used is SPSS 22.0.

Principal component analysis is also an advanced multivariate statistical analysis method. The basic idea is to summarize the comprehensive index that can summarize the information of the original index by analyzing the correlation between the original indicators and using the idea of dimensionality reduction.

Let’s say X₁, X₂, ..., X_m total m indicators, to use Z₁, Z₂, ..., Z_m several comprehensive indicators summarize the information of the m original indicators, then the following mathematical model can be established:

$\{\begin{array}{c}{Z}_1=a_{11}{X}_1+a_{12}{X}_2+\cdots +a_{1m}{X}_m\\ Z_2=a_{21}{X}_1+a_{22}{X}_2+\cdots +a_{2m}{X}_m\\ ⋮⋮\\ Z_m=a_{m1}{X}_1+a_{m2}{X}_2+\cdots +a_{mm}{X}_m\end{array}$

where Z_i is the original index X₁, X₂, ... The ith principal component of X_m, Z₁, is the linear combination of the original index with the largest difference, so Z₁ is called the first principal component. The principal components are independent of each other, Z₂ is X₁, X₂, other than Z₁... The linear combination of X_m has the largest difference. Since the purpose of principal component analysis is to reflect the information of all original indicators with fewer indicators, principal component analysis will reduce the number of original indicators.

1) Since the original indicators use different dimensions and the units are not uniform, it is necessary to standardize each original indicator before analysis ( Wang et al., 2014b )

${X^{\prime }}_{ij}=\frac{X_{ij}-{\bar{X}}_j}{S_j},j=1,2,3,\cdots m$

The original index is standardized, and then the standardized index $X^{\prime }$ is analyzed by the principal component method.

2) Find the correlation matrix R of the standardized index $X^{\prime }$ and analyze the correlation of the original index.

$R=\left[\begin{array}{cccc}{r}_{11}& r_{12}& \cdots & r_{1m}\\ r_{21}& r_{22}& \cdots & r_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ r_{m1}& r_{m2}& \cdots & r_{mm}\end{array}\right]=\left[\begin{array}{cccc}1& r_{12}& \cdots & r_{1m}\\ r_{21}& 1& \cdots & r_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ r_{m1}& r_{m2}& \cdots & 1\end{array}\right]$

3) Find the eigenvalues and eigenvectors of the correlation matrix

Since R is a semi-definite matrix, it can be determined by the eigenequation of R

$\left|R-\lambda I\right|=0$

The eigenvalues λ of m correlation matrices are obtained, and λ are all non-negative.

4) Determine the number of principal components. Generally, not all principal components are required, and the number of principal components retained is usually determined according to the following criteria: a) The principal component is determined by the cumulative contribution rate: When the cumulative contribution rate of the current k principal components reaches a certain value (generally greater than 80%), the k principal components are retained. b) Principal components are determined by eigenvalues: that is, only principal components whose eigenvalues are greater than 1 are retained ( Bao & Wang, 2012 ).

5) Calculation of factor load and factor load matrix. Factor load is the product of the square root of the eigenvalue and the coefficient of the original index, which reflects the closeness and direction of the relationship between the principal component and the original index ( Han et al., 2012 ). The matrix composed of factor loads is called factor load matrix

$Q=\left[\begin{array}{cccc}\sqrt{{\lambda }_1}{a}_{11}& \sqrt{{\lambda }_1}{a}_{12}& \cdots & \sqrt{{\lambda }_1}{a}_{1m}\\ \sqrt{{\lambda }_2}{a}_{21}& \sqrt{{\lambda }_2}{a}_{22}& \cdots & \sqrt{{\lambda }_2}{a}_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ \sqrt{{\lambda }_m}{a}_{m1}& \sqrt{{\lambda }_m}{a}_{m2}& \cdots & \sqrt{{\lambda }_m}{a}_{mm}\end{array}\right]$

6) Calculation of each principal component score. The principal component score can be used to evaluate and infer the sample ( Lin & Du, 2013 ).

Based on the theory of government behavior, combined with the regional sports industry theory and the existing performance evaluation index system of government behavior, relevant reports were consulted to collect the performance indicators of government behavior in the sports industry of 24 provinces (municipalities directly under the Central Government and autonomous regions) in China in the past five years from 2014 to 2018. Some data are difficult to obtain, so Beijing is not selected as the research object in this study. And perform logical verification on the data to ensure the authenticity and reliability of the data. The data collected in this paper are mainly from China Statistical Yearbook, local statistical yearbook of various regions, government summary reports, government-related websites and sports industry development reports in the past five years.

Standardization is an equivalent linear transformation, which can eliminate the influence of different index dimensions, maintain the relative comparability of indicators, and retain the correlation of variables. Due to the different dimensions of the regional sports industry government behavior performance evaluation indicators selected in this study, the different dimensions of the indicators have a great impact on the results of principal component analysis, so it is necessary to adopt standardization method for all the indicators before analysis Data conversion of the original indicator. By standardizing raw data metrics, ${X^{\prime }}_{ij}=\left(X_{ij}-{\bar{X}}_j\right)/S_j$ convert to standardized Z-scores.

It can be seen from the correlation coefficient matrix of government behavior performance evaluation indicators ( Table 1 ) that there is a certain correlation among all indicators, among which the gross national product (X₁) and the number of stadiums (X₁₂) are highly correlated with each indicator. The proportion of government investment in the sports industry (X₄) has a strong positive correlation with the number of sports schools (X₁₀), the number of athletes (X₁₁) and the number of stadiums (X₁₂), and the Engel coefficient (X₅) has a negative or weak correlation with most of the indicators, with strong correlation among the indicators and more overlapping information. Therefore, principal component analysis can be used to reduce the dimension of indicators, and the original indicators can be transformed into several unrelated principal components, so as to better evaluate the government behavior performance of provinces (municipalities and autonomous regions).

From the characteristic values of correlation matrix of sports industry government behavior performance evaluation index ( Table 2 ), it can be seen that the characteristic values of the first four principal components are all greater than 1 (6.562, 3.487, 1.744, 1.340, respectively), and the cumulative contribution rate is 82.074%. Therefore, it is more appropriate to extract four principal components in this study and synthesize most of the information of government behavior performance evaluation indicators.

In order to further understand the correlation between the original index and the principal component, the factor load matrix reflecting the correlation coefficient between the original index and the principal component can be obtained, as shown in Table 3 . The professional significance of the four principal components in the unrotated factor load matrix is difficult to explain, and the integration effect of the original data information is not clear, so that each principal component is not representative and has no obvious practical significance. Therefore, the data is rotated, and the orthogonal rotation method of variance maximum is used in this study to obtain the rotated factor load matrix.

As can be seen from Table 3 , the first principal component after the rotation of the factor load matrix is closely related to the gross national product (X₁), the proportion of the tertiary industry in the gross national product (X₂), the Engel coefficient (X₅) and the sports lottery income (X₆), which mainly reflects the local economic level. The second main component is closely related to the proportion of sports industry expenditure in fiscal expenditure (X₃), the proportion of government investment in sports industry investment (X₄), the number of sports schools (X₁₀), the number of athletes (X₁₁) and the number of sports venues (X₁₂), which mainly reflects the degree of investment in local sports industry. The third component is closely related to the number of international and national sports events hosted (X₇), the number of MEDALS won in international and national sports events (X₈), the number of sports-related policies formulated by the provincial (municipal) government (X₁₅) and the number of sports-related documents issued by the provincial (municipal) sports bureau (X₁₆), which mainly reflects the sports management policies of local governments. The fourth component is closely related to the number of sports bureaux above the county (city) (X₉), the number of employees in the sports system (X₁₃) and the proportion of administrative personnel with bachelor’s degree or above (X₁₄), which mainly reflects the local government’s sports industry service ability. Therefore, the four principal components extracted mainly reflect the original number. ( Figure 2 )

After principal component analysis, the original index reflecting the performance evaluation of sports industry government behavior is reduced to four independent principal components, which summarize the economic level, capital investment, management policy and service ability of the original data. According to the sports industry government behavior performance evaluation index factor score coefficient matrix (see Table 4 ), the expression of four principal components can be listed as follows:

Economic level:

$Z_1=0.536X_1-0.339X_2+0.422X_3+0.558X_4+\cdots +0.077X_{16}$

Capital input:

$Z_2=0.192X_1+0.089X_2+0.070X_3+0.071X_4+\cdots +0.240X_{16}$

Management Policy:

$Z_3=0.071X_1-0.049X_2+0.209X_3+0.073X_4+\cdots +0.416X_{16}$

Service capability:

$Z_4=0.088X_1+0.123X_2+0.138X_3+0.083X_4+\cdots +0.120X_{16}$

1) Comprehensive score of government behavior performance evaluation. Using the expression of each principal component and the variance contribution rate of each factor as the weight, we can calculate the government behavior performance ranking of all regions in China. Publicity can be expressed as:

$Z\text{synthesis}=Z_1\times 34.027\%+Z_2\times 25.559\%+Z_3\times 12.770\%+Z_4\times 9.719\%$

As can be seen from the scores of government behavior performance evaluation of provinces (municipalities directly under the Central Government and autonomous regions) ( Table 5 ), the comprehensive scores of government behavior performance evaluation of economically developed regions such as Guangdong, Shanghai and Jiangsu are relatively high, while the comprehensive scores of government behavior performance evaluation of some economically underdeveloped regions (such as Guizhou, Gansu and Qinghai) have been at a low level in the past five years.

2) The growth rate of the comprehensive score of government behavior performance evaluation. According to the comprehensive score of government behavior performance evaluation of each province (municipality directly under the Central Government, autonomous region), the growth rate of the comprehensive score of performance evaluation of each region can be calculated, and the calculation formula is as follows.

As can be seen from Table 6 , the comprehensive scores of government behavior performance evaluation of all provinces (municipalities directly under the Central Government and autonomous regions) have increased in the past five years, among which Sichuan Province has the fastest growth rate, followed by Hunan, Anhui and Tianjin, with growth rates of more than 100%. Gansu had the lowest growth rate in the past five years.

3) Analysis of government behavior performance evaluation score composition. The provinces and cities with the highest and lowest comprehensive scores and growth rates of government behavior performance evaluation were selected to analyze their score composition. As can be seen from Figure 3 , Guangdong Province has a higher economic level and capital input, a more balanced proportion of each component, and the highest comprehensive score in the performance evaluation of government behavior; The economic level, capital investment, management policy and service ability of Qinghai Province are all low, and all parts are extremely unbalanced, capital investment occupies a large proportion, and the comprehensive score of government behavior performance evaluation is the lowest. The government behavior performance of Sichuan province is mainly government management policy and capital input, and the comprehensive score increases the fastest; In Gansu Province, the investment of each part is small, the proportion of management policy and capital investment is low, and the comprehensive score of government behavior performance evaluation has the slowest growth rate. It can be seen that in the performance of government behavior,

capital investment is the core of the development of the sports industry, management policy is the key to the development of the sports industry, economic level is the leading role of the development of the sports industry, and service ability is the auxiliary of the development of the sports industry. The four complement each other and are indispensable.