Supply–Demand Imbalance in School Land: An Eigenvector Spatial Filtering Approach

Abstract

The spatial flows of school-age children and educational resources have been driven by such factors as regional differences in population migration and the uneven development of the education quality and living standards of residents in urban and rural areas. This phenomenon further leads to a supply–demand imbalance between the area of school land and the number of school-age children in the geographical location of China. The georeferenced data characterizing supply–demand imbalance presents an obvious spatial autocorrelation. Therefore, a spatial data analysis technique named the Eigenvector Spatial Filtering (ESF) approach was employed to identify the driving factors of the supply–demand imbalance of school land. The eigenvectors generated by the geographical coordinates of all primary schools were selected and added into the ESF model to filter the spatial autocorrelation of the datasets to identify the driving factors of the supply–demand imbalance. To verify the performance of the technique, it was applied to a county in the southwest of Shandong Province, China. The results from this study showed that all the georeferenced indicators representing population migration and education quality were statistically significant, but no indicator of the living standards of residents showed statistical significance. The eigenvector spatial filtering approach can effectively filter out the positive spatial autocorrelation of the datasets. The findings of this research suggest that a sustainable school-land-allocation scheme should consider population migration and the possible preference for high-quality education.

1. Introduction

Rapid urbanization and increased population migration have resulted in mobile resources such as capital, industries, skills, and various support services becoming concentrated in cities. A large socio-economic gap between urban and rural areas caused by regional differences in population migration and the uneven development of the education service quality and living standards of residents have been identified in developing countries [1] In the 1990s, promoted by the education policy to expand access to education in China, there was one primary school in every village [2]. Given the loss of school-age children in rural areas, a school mapping adjustment was conducted in 2001 for purchasing efficiency based on the economies of scale theory; however, this process ceased in 2012 to avoid an increasingly long road to school [3] School land refers to the entire land allocated or acquired by schools, including the site occupied by buildings, playgrounds, greenspaces, etc. As a component of construction land, the effective use of school land is an issue that needs to be solved as construction land becomes scarce [4]. Due to population migration and parents’ preference for high-quality schools, the land for high-quality schools in urban areas tends to be undersupplied, while the land for lower-quality schools in rural areas is oversupplied in China [4]. When the difference between the supply of and demand for school land exceeds a specific range, which is commonly determined according to the actual situation, a supply–demand imbalance in school land occurs [4]. Scarce land resources are wasted in oversupplied schools, and a sharp decrease in school-age children influences the normal operation of schools, while schools where land is in short supply are at risk of declining educational quality. The continuous population migration trend is worsening such imbalances in school land. The imbalance in school land will have an impact on public education services because the supply–demand balance in school land is a basic guarantee of normal education. It is of great importance to ensure that every citizen receives equivalent and high-quality public services by formulating an optimized school-land-allocation scheme so as to promote the sustainable development of both urban and rural areas [4]. Addressing the problem of land use for public facilities in the process of land consolidation is an effective way to resolve the imbalanced development of public services.

Under the premise of limited land resources, understanding the mechanisms of the supply–demand imbalance in school land and identifying the driving factors are useful for reducing the unsustainable use of land resources and guaranteeing the right of citizens to high-quality education [4]. The unlimited expansion of urban and rural construction land has adverse effects on food security and environmental sustainability [4]. A flexible and sustainable school-land-allocation scheme should take the main driving factors of the supply–demand imbalance in school land into consideration and achieve intensive land use. Nevertheless, there is still a lack of quantitative research to identify these driving factors and to explore the impacts of these factors on changes in school land supply and demand.

Regional differences in population migration and the uneven development of the education service quality and living standards of residents affect the spatial flows of school-age children and mobile educational resources, which leads to an imbalance between available school land and the number of school-age children [1,5,6]. This paper aimed to capture the driving factors of the supply–demand imbalance in school land and provide valuable suggestions in formulating an optimized and sustainable school-land-allocation scheme against the background of rapid urbanization. Georeferenced indicators characterizing three driving factors of the supply–demand imbalance in school land were identified and collected based on the existing literature and regional government reports. Eigenvectors were generated through a spatial weight matrix built up based on the coordinates of primary schools. Candidate eigenvectors were extracted by constructing a mathematical relationship between the spatial weight matrix and Moran’s index. The eigenvectors that were selected by the adjusted $R^2$ as explanatory variables were added into the regression model to filter the influence of the spatial autocorrelation. The feasibility of this approach was demonstrated in an empirical example. The findings of this research are expected to be applicable in the practical organization of educational facilities and to provide valuable suggestions for decision-makers in creating a sustainable school-land-allocation scheme.

This paper is organized as follows: Section 2 identifies georeferenced indicators based on the mechanisms of supply–demand imbalance in school land. Section 3 presents the methodology and describes how to apply the ESF approach to identify the driving factors of the supply–demand imbalance of school land. The ESF approach is applied to an empirical example in Section 4, which is followed by Section 5, which discusses the results of the empirical example. Finally, Section 6 provides the main conclusions.

2. Mechanisms of Supply–Demand Imbalance in School Land in China

The mechanisms of the supply–demand imbalance in school land refer to the process by which the influencing factors drive the supply of and demand for school land from balance to imbalance. Although few studies have discussed the driving factors of the supply–demand imbalance in school land directly, the existing studies have performed much work in relation to the following three aspects: the education of migrating workers’ children [7,8] parents’ preference for high-quality schools [5,9], and the intention of populations flowing into cities [10]. Historical evidence from the research related to these three aspects found that the driving factors originate from population migration, education quality [9,11,12], and the living standards of residents [1,13,14]. Ten literature works closely related to these three aspects and published between 2017 to 2022 are summarized. Each main driving factor is composed of several components. The three driving factors and their 16 components were extracted from the references related to these factors. The components of each driving factor were sorted according to the frequency of citation and are summarized in

Table 1. Summary of the driving factors, sub-factors, and their citation frequency in the literature.

Driving Factor	Sub-Factor	Literature *										Frequency of Citation
		1	2	3	4	5	6	7	8	9	10
Population migration	School-age children	√	√		√	√	√	√	√		√	8
	Migrant workers			√	√	√		√	√	√	√	7
	Labor force			√	√						√	3
	Aged people			√								1
	Left-behind children										√	1
Quality of education service	High-quality teachers		√		√	√			√			4
	Buildings areas		√		√							2
	Fixed assets				√	√						2
	Mobile assets		√									1
	Green space						√					1
Living standard of residents	Income of residents	√	√		√			√	√	√		6
	Public facilities								√	√	√	3
	Enterprises or jobs		√	√					√	√		4
	Output value	√								√		2
	Municipal facilities								√			1
	Arable land	√										1

* 1: [1]; 2: [11]; 3: [15]; 4: [9]; 5: [5]; 6: [16]; 7: [13]; 8: [7]; 9: [17]; 10: [8].

. The components whose frequency of citation was less than one were filtered out.

Population migration directly influences the supply–demand imbalance in school land by changing the distribution of school-age children [9]. This can be reflected by the proportions of specific populations such as workers, school-age children, and migrant workers in the total population. The quality of education, which is represented by the quality of teachers and the quantities of buildings and fixed assets, directly affects the supply–demand imbalance in school land through its impact on parental choice [9]. The living standards of residents are the root cause of the supply–demand imbalance in school land as a high living standard drives an increase in internal migration [1,13], which in turn indirectly affects the supply–demand imbalance in school land [11]. Residents’ living standards can be reflected by such aspects as the number and value of enterprises, the income of residents, and the structure of public facilities [13,17].

Population migration impacts the spatial distribution of school-age children directly through the migration of workers who have reached the age of marriage and are raising children [10]. This can be represented by using age structure statistical data [15]. Migrant workers move into cities and towns to seek better jobs and living and educational environments [15], and they get married and have children in the cities where they work. This phenomenon is changing the age structures of residential settlements in cities and villages [8,15]. Because of the dual urban–rural household registration systems and the high threshold for obtaining an urban household registration, only some migrant workers are able to become permanent urban residents, and a considerable number of them are temporary residents in China [18]. Due to the dual occupational identities of farmers and workers, the temporary residents reside seasonally in urban and rural areas [9]. The children of these seasonal migrant workers have the right to receive education in the city of their work according to central government policies [19]. The permanent and temporary changes in the numbers of migrant workers and their children’s schooling have affected the spatial distribution of school-age children [5]. The permanent population refers to the population that has no household registration in a region, but has lived and worked in that region for more than six months; the registered residents mean people who have registered in China’s permanent residence registry system; the temporary residents indicate those who have no household registration in a region and have lived and worked in that region for less than six months [20]. The statistics for the registered population and the permanent population can reflect the spatial distribution of migrant workers to a certain extent [18].

Education quality is the direct cause of the supply–demand imbalance in school land as it is the primary consideration for most parents when choosing a school for their children [7]. High-quality education can be represented by a good built environment, fixed assets accumulated with sufficient financial support, and high-quality teachers [11]. This has been confirmed in prior studies, which indicate that the school environment [21], financial investment [22], and quality of teachers [11] affect the educational environment of schools and the learning performance of students. Buildings, assets, and teachers differ significantly between schools in cities and villages [5]. Educational resources such as funds and high-quality teachers have been prioritized for urban schools through the formulation and implementation of public policies and the operation of market power [7]. Although the national and local governments have proposed strategies to increase education funds to disadvantaged areas in recent years, it is not an easy task to make up for the long-term underinvestment [5].

The living standards of residents are also identified as a driving factor of the school land supply–demand imbalance. Although residents’ living standards do not directly influence the supply of or demand for school land, they impact the school land imbalance through various complex linkages such as inflows and outflows of migrant populations [1,13], educational funding supply [11], and the availability of high-quality teachers [11]. The living standards of residents comprise economic conditions and living environments. The number of jobs available is subject to the number and the value of enterprises, which reflect the economic conditions [15]. The living environment influences people’s quality of life and satisfaction with life [23]. Urban life is attractive because it enables access to various public facilities that are not available in many villages. The income that residents earn through different ways (e.g., working on farms or in factories) is a necessary condition for improving their living environments [18]. In China, the collective economy and the individual economy are the main sources of residents’ income, of which the collective-economy income is comprised of agricultural and collective-enterprises’ income [24]. The collective economy plays an essential role in the development of rural areas and accounts for a considerable proportion of the GDP in China as it makes possible product manufacturing with relatively low land costs [24]. Population, land, and industry are important factors that systematically measure the level of urbanization and reflect differences in regional development [13]. Although the statistical data that can reflect living standards are limited and not very accurate, there are still some data that can reflect the living standards from different perspectives. For example, the annual collective income represents the level of economic development in the statistical unit from the perspective of collective economic development [24].

With rapid urbanization, the decline of rural areas is inevitable [25]. However, from the perspective of sustainable development, the revitalization of rural areas and co-dependence between villages and cities are effective means of future development. In particular, two-way flows of resources between urban and rural areas are the basis for realizing China’s rural revitalization [26], as its population and mobile educational resources such as teachers and public finances have been moving from rural areas to urban areas [1]. The above-described spatial imbalance mechanism of school land indicates that three driving factors influence the imbalance in school land, which has led to the following hypothesis in this research:

Population migration and the quality of education are the direct driving factors affecting the supply–demand imbalance in school land, while the living standards of residents are not a direct driving factor.

3. Eigenvector Spatial Filtering Approach

3.1. Eigenvector Spatial Filtering Approach to Deal with Spatial Autocorrelation

The datasets of school land resources comprise observational units that are related to each other in specific ways, such as being linked members in a subgroup of a population or being neighbors in a geographical distribution [27]. Although both geographically weighted regression and Bayesian spatially varying coefficients’ regression can be used to handle georeferenced data, they are used to solve the regional variation or non-stationarity existing in spatial data, instead of the spatial autocorrelation [28]. Moreover, the results of geographically weighted regression are limited by the algorithm, especially for the control of coefficient smoothness, and the computation process for the Bayesian spatially varying coefficients model entails a huge computational burden [29].

The ESF approach can deal with the spatial autocorrelation problem by introducing a candidate subset of eigenvectors extracted from a spatial weight matrix into a regression model specification as control variables [27]. Additionally, the ESF approach can obtain a better result with a smaller computational burden through constructing an eigenfunction using Moran’s index [29]. For example, it was used to explore the potential relationships between high-speed rail accessibility and county-level regional development [30]. Due to the spatial autocorrelated datasets related to high-speed rail stations, the ESF approach can effectively solve the spatial autocorrelation of these spatial datasets and ensure the accuracy of the result. An ESF-based spatial varying coefficient approach was developed to estimate ground the PM2.5 concentration [31]. Similarly, eigenvectors are used to filter the autocorrelation of spatial datasets related to observation points. Traditional analysis methods (e.g., multiple linear regression) are not suitable for processing georeferenced data as they cannot solve the geographical information contained in the datasets in a targeted manner [27]. Classic ESF, which is a spatial autocorrelation modeling method, was put forward by Griffith to deal with potential spatial effects [32]. Based on this classic ESF, Murakami and Griffith proposed a random effects ESF, which can estimate local impacts [29]. The linear combination of the selected eigenvectors successfully filters out the spatial autocorrelation [27]. The ESF approach has been widely used across demography, ecology, economics, geographic information science, and regional science disciplines related to spatial data [33].

3.2. Moran Eigenvectors

The ESF approach, which was adopted here to identify the driving factors of school land supply–demand matching, relies on the Moran eigenvectors of a double-centered spatial weight matrix $MWM$ [32]. The eigenvectors are extracted through the following eigen-decomposition:

$$MWM=E\Lambda E^T$$

(1)

where $M=(\mathrm{I}-{\mathbf{11}}^{\mathrm{T}}/n)$ is the centering matrix and $W$ is an $n\times n$ spatial weight matrix constructed based on the geographical coordinates of all $n$ schools in the research area.$T$ is the matrix transposition operation; $\mathbf{1}$ indicates an $n$ × $1$ summing vector of response variables, whose value for every entry is always $\mathbf{1}$; $I$ is an $n$ × $n$ identity matrix. ${E=[\mathit{e}}_{\mathbf{1}}\dots ,{\mathit{e}}_{\mathit{n}}]$ is an eigenvector matrix, and ${\mathit{e}}_{\mathit{l}}={(e_{l1}\dots ,e_{li},{\dots ,e}_{ln})}^{\mathrm{T}}$ is the $l^{th}$ eigenvector. $\Lambda$ is an $n$ × $n$ diagonal matrix with entries taken from the corresponding eigenvalues $({{\lambda }_1,{\lambda }_2,\dots ,\lambda }_n$).

The eigenvectors are interpretable in terms of Moran’s index, which is a diagnostic statistic of spatial autocorrelation. Moran’s index for a variable vector z is defined as follows [29]:

$$MI\left[\mathit{z}\right]=\frac{n}{\mathbf{1}W{\mathbf{1}}^T} \frac{{\mathit{z}}^{T}MWM\mathit{z}}{{\mathit{z}}^{T}M\mathit{z}}$$

(2)

MI[z] takes a positive value in the presence of positive spatial autocorrelation, whereas a negative value means negative spatial autocorrelation. Moran’s index $MI\left[{\mathit{e}}_{\mathit{l}}\right]$ of the eigenvector ${\mathit{e}}_{\mathit{l}}$ has the following expression [32]:

$$MI\left[{\mathit{e}}_{\mathit{l}}\right]=\frac{n}{\mathbf{1}W{\mathbf{1}}^T}{\lambda }_l$$

(3)

Equation (3) implies that the eigenvectors corresponding to a positive eigenvalue explain positive spatial autocorrelation [32]. The eigenvectors with a Moran’s index greater than the threshold are ordered from large to small and form an $n$ × $k$ matrix ${E^{\prime }=[\mathit{e}\mathbf{\prime }}_{\mathbf{1}}\dots ,{{\mathit{e}}^{\mathbf{\prime }}}_{\mathit{l}}\dots ,{{\mathit{e}}^{\mathbf{\prime }}}_{\mathit{k}}], (k<n)$. The eigenvector ${\mathit{e}\mathbf{\prime }}_1$ corresponding to the largest eigenvalue also corresponds to the largest Moran’s index value; the eigenvector ${\mathit{e}\mathbf{\prime }}_2$ corresponding to the second-ranked eigenvalue also corresponds to the second-ranked Moran’s index value, and so on [34]. If the ordering property of eigenvectors is reflected on a map, it will portray a set of gradient-changing spatial autocorrelation patterns. For the eigenvectors with the largest, medium, and smallest Moran’s indexes, the largest, middle-sized, and smallest clusters of eigenvector values represent the global, regional, and local spatial autocorrelation features, respectively [32].

3.3. Eigenvector Spatial Filtering

In the regression analysis of this study, the eigenvectors were used to eliminate residual spatial autocorrelation, which can lead to erroneous inference. A response variable $Y_i$ for this study was determined before constructing the spatial filtering equation. A supply–demand imbalance, which is comprised of oversupply and undersupply, means that the difference between supply and demand exceeds a reasonable or acceptable range. The range always depends on the actual situation of the supply–demand of school land. To be expressed more accurately, the relationship between supply and demand can be quantified as the supply–demand ratio. The land supply–demand ratio for school $i$, $Y_i$, is defined as follows:

$$Y_i=\frac{S_i-D_i}{S_i}\times 100\%$$

(4)

where $S_i$ is the land supply of school $i$ and $D_i$ represents school $i$’s land demand. The demand for school land is the function of the number of students in school $i$ and the school land area per student regulated by construction standards. The Chinese government has different construction standards to control school land-use in urban and rural areas. The planning and construction of rural schools are regulated by the Construction Standard for Rural General Middle Schools and Primary Schools [35], while urban schools are controlled by the Standard for Urban Public Service Facilities Planning [36]. A reasonable range for the supply–demand balance can be established by setting an upper and a lower threshold for the supply–demand ratio according to the actual supply–demand situation. Thus, the supply–demand imbalance in school land means that the supply–demand ratio is beyond this reasonable range.

After extracting the candidate eigenvectors and determining the response variable, ESF was constructed to identify the driving factors of the supply–demand imbalance by adding the selected eigenvectors in order to filter the spatial autocorrelation of the dataset of the response variable. The selected eigenvectors $E^{″}$ are an $n$ × $g$ matrix $(g<k)$, which is obtained from $E^{\prime }$. Using a forward stepwise principle, the $g$ eigenvectors in $E^{\prime }\prime$ were selected as follows. First, each of the $k$ eigenvectors obtained from $E^{\prime }$ was put into Equation (5) to conduct a linear regression analysis with all independent variables and the response variable. The eigenvector with the maximum adjusted $R^2$ was selected and added to the list of influencing indicators. This process continued for the remaining eigenvectors and finished by regressing stepwise until the value of the adjusted $R^2$ could not be increased compared to the adjusted $R^2$ in the previous step. ESF exploring the driving factors of the supply–demand imbalance in school land based on the selected eigenvectors $E^{″}$ was formulated as below:

$$Y_i=X\mathit{\beta }+E^{″}\mathit{\gamma }+\mathit{\epsilon }, \mathit{\epsilon } ~ N(0, {\sigma }^{2}I)$$

(5)

where $Y_i$ is the supply–demand ratio and $X$ is an $n$ by $x$ matrix of the explanatory variables representing the factors that may affect the response variable. $\mathit{\beta }$ is the $x$ × $1$ vector of the estimated regression parameters. $\mathit{\epsilon }$ is an $n\times 1$ error vector whose elements are normal random variables. $\mathit{\epsilon }$ obeys a normal distribution $N(0, {\sigma }^{2}I)$, where ${\sigma }^2$ is the variance of the normal distribution. $\mathit{\gamma }=[{\gamma }_1,\dots , {\gamma }_g]$ is a $g\times 1$ fixed coefficient vector, which balances the model’s accuracy and parsimony using stepwise eigenvector selection [27].

4. An empirical Example

4.1. Establishment of Georeferenced Indicators Based on Local Statistical Data

Yuncheng County in the southwest of Shandong Province, China, which is experiencing rapid urbanization, is a typical representative of the supply–demand imbalance of school land. At the end of 2018, its urbanization rates in relation to the permanent and the registered populations were 50.1% and 39.4%, respectively [37]. There are 686 residential settlements in Yuncheng County, including villages in the rural areas and communities in the urban areas. The imbalanced development of society, the economy, and the environment is prevailing in Yuncheng County. The 686 residential settlements are divided into 266 school districts, which are the statistical units in this study. Each school district is served by one public primary school.

The supply of and demand for school land are imbalanced in quantity. The land supply–demand ratios for the 266 schools were calculated via Equation (4), which reflects the quantitative imbalance between the supply of and demand for school land. Schools with a supply–demand ratio greater than or equal to 20% account for more than 63% of the total. If the upper and lower thresholds of the supply–demand ratio representing the supply–demand balance are set as −20% and 20%, respectively, there are 28 undersupplied schools and 171 oversupplied schools that are in an unbalanced state of supply–demand in school land. If the range of the supply–demand balance is defined as [−50%, 50%], there are 15 schools short of land and 69 schools with excess land, respectively.

Georeferenced indicators that reflect the drivers of the supply–demand imbalance in school land were constructed and calculated in this study mainly on the basis of the existing 2018 statistical data from government departments in Yuncheng County. The sources included: the Education Department for detailed data on each school; the Population and Social Security Department for demographic data on administrative divisions; the Land Resources Department for basic data on the arable land use of each administrative village; the Housing and Urban–Rural Construction Departments for data on construction land use and economic development. The establishment of the georeferenced indicators mainly considered population migration, education quality, and living standards, which lead to school land imbalance. From an age structure perspective, the existing statistical data on the proportions of children aged 7–12 years represent school-age children and labor force sub-indicators, respectively. The ratio of permanent residents to the household registered population reflects the number of migrant workers to some extent. The student–teacher ratio with the number of teachers with a Bachelor’s degree or above, fixed assets per student, and school building area per student characterize the education quality from the perspectives of high-quality teachers, building areas, and fixed assets. The existing statistical data, which can reflect enterprises, the output value, the incomes of residents, and public facilities are the number of enterprises, collective income per year, annual income per capita, and public facility structure, respectively. The georeferenced indicators that affect the supply–demand imbalance in school land are summarized in

Table 2. Summary of the georeferenced indicators that may affect the supply–demand imbalance in school land.

Driving Factor	Georeferenced Indicator	Variable
Population migration	Proportion of children aged 7–12 years	$x_1$
	Proportion of adults aged 19–40 years	$x_2$
	Ratio of permanent residents to household registered population	$x_3$
Education quality	Student–teacher ratio with teachers with a Bachelor’s degree or above	$x_4$
	Fixed assets per student	$x_5$
	School building area per student	$x_6$
Living standards of residents	Number of enterprises	$x_7$
	Collective income per year	$x_8$
	Annual income per capita	$x_9$
	Public facility structure	$x_{10}$

. Unless otherwise specified, the statistical range of each georeferenced indicator refers to the school district, and the school refers to the primary school.

The analysis of the supply–demand imbalance involved all 10 georeferenced indicators that characterize the three driving factors. The mechanisms of the supply–demand imbalance in school land illustrate that population migration and the quality of education are the direct driving factors of the supply–demand imbalance in school land, and the living standards of residents are the indirect driving factor. If at least one indicator in a factor has statistical significance, it could be the case that this factor has a direct impact on the supply–demand imbalance in school land and drives it directly. If no indicator in a factor shows statistical significance, this factor cannot be identified as a direct driving factor. Therefore, only if at least one indicator in each of the first two factors is statistically significant and no indicators in the last factor show statistical significance is the hypothesis proposed in this research supported.

4.2. Application of Eigenvector Spatial Filtering in Yuncheng County

The ESF approach was applied to Yuncheng County to explore the driving factors of the supply–demand imbalance in school land. The dataset of the 266 schools was integrated into ArcGIS 10.7 for visualizing, analyzing, and compiling the geographical information data [28]. After importing the dataset of the independent variables and the response variable from ArcGIS into R Studio, the whole process of the ESF approach was achieved via an open-source package [38]. R Studio 2022.07.0 is a free software environment for statistical computing and data manipulation, and it is particularly suitable for calculations on arrays and matrices [39]. In this study, the spatial weight $w_{i,j}$ between any two schools $i$ and $j$ is expressed as an exponentially decaying function $w_{i,j}{=e}^{(-d_{ij}/r)}$, where $d_{ij}$ is the Euclidean distance between the two schools. Following [40], the maximum of the minimum spanning tree is the range parameter $r$.

After establishing the spatial weight matrix, the 266 eigenvectors and 266 eigenvalues of the 266 × 266 double-centered spatial weight matrix were generated using Equation (1) and expressed as ${[e}_1,e_2\dots ,e_{266}]$ and ${{(\lambda }_1,{\lambda }_2,\dots ,\lambda }_{266})$, respectively. According to Equation (3), Moran’s index of the $l^{th}$ eigenvector $MI\left[{\mathit{e}}_{\mathit{l}}\right]$ was equal to $n/{\left(\mathbf{1}W{\mathbf{1}}^T\right)\lambda }_l$. Once the spatial weight matrix $W$ was determined, the Moran’s index of the $l^{th}$ eigenvector was proportional to its corresponding eigenvalue. These 266 eigenvalues and their Moran’s indexes are illustrated in Figure 1, which fully demonstrates the ordering property of the eigenvalues. In this case study, the threshold for Moran’s index was set as 0. Thus, all 57 positive eigenvalues and their corresponding eigenvectors were extracted as candidate variables and added to the regression equation.

The 57 candidate eigenvectors were selected based on their contribution to the adjusted $R^2$ value. Each of these candidate eigenvectors was added one by one to the regression equation established with the supply–demand ratio as the response variable and the georeferenced indicators as the independent variables. The eigenvector with the highest contribution to the adjusted $R^2$ was selected. Of these 57 candidate eigenvectors, 20 eigenvectors were selected according to this forward stepwise principle, maximizing the adjusted $R^2$ value. Those 20 selected eigenvalues and their corresponding Moran’s indexes were both sorted according to their Moran’s index. The first of the 20 selected eigenvectors was the candidate eigenvector whose spatial autocorrelation ranked 25th. The second selected eigenvalue was ${\lambda }_{15}$, the 15th candidate eigenvector. This means the second-ranked eigenvector contributing to the adjusted $R^2$ was ranked 15th in terms of Moran’s index. Figure 2 summaries the influences of the 20 selected eigenvectors on the adjusted $R^2$ values. These values increased by adding the 20 selected eigenvalues one by one. The original adjusted $R^2$ value with only 10 independent variables was 0.4220. After adding the 20 selected eigenvectors into the regression equation, the value of adjusted $R^2$ was 0.5012.

4.3. Improved Accuracy of Independent Variable Estimation Parameters

The estimated parameters and performance parameters of the ESF approach are shown in

Table 3. Coefficients of eigenvector spatial filtering models for driver factors of supply–demand imbalance of school land.

Variable	Estimate	Standard Error	t-Value	p-Value	VIF
$x_1$	−0.01000	0.00113	−8.84927	0.00000 **	3.59908
$x_2$	0.17822	0.05712	3.11998	0.00203 **	1.13568
$x_3$	0. 23062	0.03355	6.87462	0.00000 **	3.37613
$x_4$	−0.00237	0.00069	−3.42078	0.00074 **	1.21507
$x_5$	0.09021	0.02421	3.72624	0.00024 **	1.33038
$x_6$	0.02044	0.00406	5.02951	0.00000 **	1.60896
$x_7$	−0.00040	0.00128	−0.31313	0.75446	1.14854
$x_8$	0.00001	0.00067	0.02005	0.98402	1.44116
$x_9$	−0.01945	0.02890	−0.67321	0.50148	1.20913
$x_{10}$	0.01030	0.00873	1.17948	0.23940	1.20833

** Significant at 0.01; adjusted $R^2$ = 0.50; Moran’s index (max) = 0.28.

. The estimated parameters present the quantitative relationships between the response variable and each independent variable. Among the five estimated parameters in Table 3, the estimated and standard error correspond to the coefficients $\mathit{\beta }$ and $\mathit{\epsilon }$ in Equation (5), respectively. VIF indicates whether there is collinearity among variables. The p-value, which is used to test whether variable coefficients are statistically significant, can be applied to set the significance cut-off at p = 0.01. According to p-values less than or equal to 0.01, six independent variables were statistically significant, including the proportion of children aged 7–12 years, the proportion of adults aged 19–40 years, the ratio of permanent residents to the household registered population, the student–teacher ratio with teachers with a Bachelor’s degree or above, the fixed assets per student, and the school building area per student. The first three variables are the population migration indicators, which show that the number of school-age children is related to the age structure and the migrant workers in the population. Their corresponding estimated parameters were −0.01000, 0.17822, and 0.23062, respectively. From the estimated values of the first statistically significant georeferenced indicators, it can be stated that school districts with a higher proportion of children aged 7–12 years have a lower supply–demand ratio. Unexpectedly, the second and third indicators showed that areas with a high proportion of laborers aged 19–40 years and a high number of migrant workers are areas that provide more jobs, but also have an oversupply of school land. This shows that a considerable number of young workers are still concentrated in the rural areas of Yuncheng County. Over the years, as this population moves into cities and towns, the increase in the number of urban school-age children will result in the shortage of supplying school land and the decrease in the number of rural school-age children will increase the school land excess further.

The last three variables are the education quality indicators. The estimated parameter values of the fifth and sixth georeferenced indicators mean that the higher the values of fixed assets per student and school building area per student, the greater the supply–demand ratio value is. This implies that the supply of fixed assets and the school buildings align with the supply of school land. If a school’s land is oversupplied, its fixed assets and buildings will also tend to be in surplus. Interestingly, in school districts with a higher student–teacher ratio with teachers with a Bachelor’s degree or above, their school land tends to be undersupplied. High-quality teachers are a critical factor that determines the quality of school education. They are also what the parents of school-age children prefer. Differences in teacher quality and imbalanced development are widespread between urban and rural areas of China [9]. The spatial distribution of high-quality teachers is not restricted by public policies, but is closely related to the living standard of a region under the action of market forces. The calculation results also showed that the school land, fixed assets, and school buildings as immobile resources have reverse development trends to that of skills resources. No indicator of the living standards of residents had statistical significance, which showed that the living standards of residents are not a direct driving factor of the supply–demand imbalance in school land.

The performance parameters of the ESF approach mainly include Moran’s index (max) and the adjusted $R^2$. The adjusted $R^2$ indicates the explanatory contribution of the independent variables to the response variable. The adjusted $R^2$ of the ESF approach was 0.50. Moran’s index (max), which equals the sum of the products of each selected eigenvector and its corresponding estimated parameter, denotes the scaled Moran’s index value of the map pattern of the response variable. The value of Moran’s index (max) was 0.28 in this case study, which suggests that the response variable has relatively local map patterns. All these numerical results specify that the hypothesis of this research is accepted.

5. Discussion

An urban–rural difference in school land utilization driven by population migration is one of the causes of the supply–demand imbalance in school land. The population migration in rapid urbanization results from migrant workers’ expectation of a better living standard [7]. The migrant workers further bring about the geographical redistribution of school-age children [7], shown in the decrease in villages and the increase in cities and towns specifically, which in turn results in an urban–rural difference in school land utilization. The outcome of the research showed that population migration drives the supply–demand imbalance in school land, which can help decision-makers formulate an optimized and sustainable school-land-allocation scheme in urban and rural areas, respectively. In urban areas, school land should be allocated according to the prediction of the increase of school-age children, which is an incremental planning in newly developed area and an inventory planning in built-up areas [41]. In rural areas, school land allocation should be a decremental planning with respect to the decrease of school-age children [42]. A school district difference in school land utilization is caused by parents’ preference for high-quality schools, which is another cause of the supply–demand imbalance in school land. Parents’ preference for high-quality schools is determined by their expectation that their children could achieve a higher standard of living [9]. Residents’ need for high-quality educational services concentrated in cities and towns further strengthens the oversupply of school land in these areas [43] The difference in the quality of educational services among primary schools broke the balance distribution of school-age children in adjacent school districts and presents a challenge for the school-land-allocation scheme. While there are other sub-factors, such as the provision of green space, impacting the study performance of school-age children, this paper focused on the three key indicators of education quality presented in the recent literature [16]. The paper proposes the implementation of policy instruments aimed at adapting to the dynamic shift of school-age children, including the consideration of flexible planning and effective use of construction land resources. The school land-use scheme should conform to the future development trends of local population migration, and the education service should achieve inclusive and equitable quality education [44]. Furthermore, research on inclusive urban–rural integrated development and intensive land use has continually stressed the need to solve the supply–demand imbalance in school land [23]. Identifying the driving factors of the supply–demand imbalance in school land provides an empirical basis for rational use and balanced supply–demand development of school land. From a sustainable development perspective, the revival of rural areas and integrated urban–rural development is imperative. In the short term, the supply–demand imbalance in school land needs to be addressed through public policies formulated by the government. In the long term, the fundamental solution to the imbalance issue depends on a regional balance of population migration and education quality, and the establishment of an institutional mechanism and policy system for urban–rural integration.

6. Conclusions

In this research, the application of the ESF approach in identifying the driving factors of the supply–demand imbalance in school land was examined. The ESF approach, including Moran eigenvector extraction and parameter estimation, was applied to find the driving factors of the supply–demand imbalance in school land. The empirical example of Yuncheng County was used. Candidate eigenvectors were extracted from a spatial weight matrix. The estimated parameters of the ESF approach were calculated after adding the selected eigenvectors into a regression model specification as the control variables. The influence of the positive spatial autocorrelation derived from the response variable was filtered by using the ESF approach.

The eigenvector spatial filtering approach was applied to Yuncheng County located in the southwest of Shandong Province, China, based on the spatial and feature data of population migration, education quality, and the living standards of residents. Georeferenced indicators characterizing the driving factors were established based on the existing statistical data from government reports. This procedure of identifying the driving factors using the ESF approach was performed and examined. A spatial weight matrix was established based on the geographical coordinates of the schools in the county. Of the 266 candidate eigenvectors generated from the spatial weight matrixes of the 266 schools, 57 eigenvectors were extracted according to the threshold of Moran’s index. Then, 20 eigenvectors were selected and added to the regression model according to the value of the adjusted R-squared. The results from this study showed that the ESF approach can effectively filter out the positive spatial autocorrelation of the dataset related to the school land imbalance and decrease the erroneous estimation. In the imbalance analysis involving the three driving factors, three georeferenced indicators representing population migration and three georeferenced indicators characterizing education quality presented statistically significant spatial associations with the supply–demand ratios. These six georeferenced indicators explain the supply–demand ratios in Yuncheng County at a level of 0.50. No indicator of the living standards of residents showed statistical significance. Therefore, the hypothesis of this research is accepted, and population migration and education quality are the direct driving factors of the supply–demand imbalance in school land. The role of the living standards of residents as a potential and indirect driving factor did not show statistical significance. This is mainly because the living standards of residents indirectly affect the imbalance of school land by acting on the spatial flows of populations and resources, but do not act directly on the supply of and demand for school land. However, from the results of the qualitative analysis, whether the living standards of residents are an indirect factor of the supply–demand imbalance in school land still needs further exploration.

Although the ESF approach improved the accuracy of the estimated parameters and the explanation magnitude of the response variable, there are still some gaps that need to be filled in further research. For example, the connection between the spatial autocorrelation of the independent variables and the response variable was not developed. When the independent variables with spatial autocorrelation have the same spatial pattern as the response variable, the ESF approach may not be needed. Moreover, as only positive autocorrelation is discussed in most existing research, the geospatial literature needs to include more examples of negative spatial autocorrelation [33]. For the sample data in this case study, schools with surplus land and schools with insufficient land show a supply–demand imbalance in school land under the influence of two main driving factors. As the statistical data of 266 schools is the only cross-sectional data available, there is still an inevitable accidental regression result for a specific independent variable. The limited sample size may also be the reason that a considerable number of georeferenced indicators showed no statistical significance.