Spatial autocorrelation analysis

Spatial autocorrelation refers to the potential interdependence among the observation data of several variables in the same distribution area. According to Tobler’s First Law, the phenomenon of spatial correlation is universal, and the correlation of close places is stronger than that of far ones [16]. Spatial autocorrelation is typically used to detect the spatial correlation and cluster of some variables, which is known referred to as spatial dependence. Compared with the traditional autocorrelation analysis, spatial autocorrelation introduces a spatial weight matrix to represent the degree of spatial correlation among regions. The spatial weight matrix is determined by the adjacency of the spatial unit or the distance among these units. Spatial autocorrelation can be divided into two categories: global and local spatial autocorrelation [17].

Global spatial autocorrelation

Global spatial autocorrelation is used to analyze whether cluster characteristics are present in the entire spatial range of geographical data. However, this category cannot accurately identify the specific cluster area. Global Moran’s I is a commonly used global spatial autocorrelation statistical index that is defined as

Local spatial autocorrelation

To study the heterogeneity of spatial autocorrelation, local spatial autocorrelation is typically used to test whether a cluster area is present in the local spatial units. This exploration makes up for the limitation of the global spatial autocorrelation, which cannot reflect local aggregation. The local Moran’s I index (i.e., local indicator of spatial association [LISA]) is proposed to measure the correlation between the spatial unit and its surrounding spatial units [18]. On the basis of this index, the Moran scatter plot can be constructed to study the local spatial heterogeneity. The scatter plot includes a Cartesian coordinate system, where the abscissa represents Z_i, ∑w_ijZ_j denotes the normalized value of the central target unit, and the ordinate is the space lag value.

In accordance with the attributes of the Cartesian coordinate system, four types of local spatial correlation can be obtained.

Spatial constant coefficient regression model

Compared with the ordinary least squares (OLS) model, the spatial constant coefficient regression model adds autocorrelation factors, and the spatial dependence and spatial weight are considered. The spatial constant coefficient regression model includes the spatial lag model (SLM) and spatial error model (SEM), which are estimated through the maximum likelihood method [19].

SLM

SLM is used to explore the spatial dependence caused by spatial diffusion and spatial spillover effects. A spatial autocorrelation is assumed to be present among the dependent variables in the study area and absent among the independent variables. This model can be mathematically expressed as



where Y is a vector with a dimension of n × 1, X is an independent variable matrix with a dimension of n × k, W₁ is a spatial weight matrix with a dimension of n × n, ρ represents spatial autocorrelation coefficient, βrepresents the independent variables coefficient of k × 1 dimension, ε represents random error items, σ² is the variance of ε.

SEM

SEM assumes that the error has a spatial autocorrelation. This model is defined as



where Y is a residual vector with a dimension of n × 1, λ represents the residual correlation coefficient, W₂ is a residual vector with a dimension of n × n, and the other parameters are the same as those of SLM.

Selection of the dependent variable

Population density is a direct representation and an important index of population distribution. This factor is the most commonly used index in social science research and the most recognized index in demography. Therefore, this study selects the county’s rural population density of the Poyang Lake ecological economic zone as an index for measuring the distribution of rural population [25].

Selection of the independent variables

Numerous factors affect the population distribution. In accordance with the requirement of this study, the influencing factors of rural population distribution are divided into two categories: climatic and other factors. The climatic factors include the AR, AS, AT, and AW. The values of ALT, AGR, ELE, and DI are selected in accordance with the existing studies and by comprehensively considering the situation of the Poyang Lake ecological economic zone [26–30]. The specific indicators are listed in Table 2.

Table 2

Index system of the rural population distribution.

Variables		Index	Abbreviation	Unit
Dependent Variable		Rural population density	RPD	people/square kilometer
Independent Variables	Climatic Factors	Annual Average rainfall	AR	millimeter
		Annual Average Sunshine	AS	hour
		Annual Average Temperature	AT	centigrade
		Annual Average Windspeed	AW	meter/second
	Other Factors	Average Altitude	ALT	meter
		Output value of agriculture, forestry, animal husbandry, and fishery per unit area	AGR	104 RMB/ hectare
		Rural electricity consumption per unit area	ELE	kilowatt-hour/square kilometer
		Per capita disposable income of rural residents	DI	RMB/people

Model building

To overcome the problem of heteroscedasticity in cross-sectional data, this study analyzes all variables logarithmically in accordance with the general method of econometrics (same treatment with SLM and SEM). Disregarding the spatial dependence, the density of rural population is considered as the dependent variable, and the OLS model is constructed using the Stata 14 software. The influence direction and degree of the independent variables on the density of rural population are also analyzed.

The final result is displayed in Table 3. During the process of stepwise regression, AT fails to satisfy the significance test, and thus is eliminated. The constant terms and seven independent variables passed the significance test; the goodness of fit and adjusted goodness of fit of the model are 0.8480 and 0.8071, respectively. From the result of the multiple collinearity test, the variance inflation factor (VIF) of all independent variables is less than 10, and no multiple collinearity is observed. The probability of the Breusch–Pagan heteroscedasticity test is 0.3265, which satisfies the significant test at 1% level and indicates the absence of heteroscedasticity. In conclusion, the fitting effect of the OLS model is satisfactory.

Table 3

OLS model parameter estimation and test results.

Variables	Coefficient	t-statistic	p-value	VIF
C	98.5615***	4.9380	0.0000
lnAR	−3.0027**	−2.6439	0.0137	1.6900
lnAS	−9.2576***	−3.1546	0.0040	4.5500
lnAW	0.8284**	2.1800	0.0385	4.3700
lnALT	−0.2530**	−2.6852	0.0125	4.0200
lnAGR	0.3093***	3.3395	0.0025	1.9100
lnELE	0.2435***	3.2372	0.0033	2.6900
lnDI	−0.3742*	−1.7552	0.0910	1.3900
R2	0.848		LogL	6.5306
R2adj	0.8071		AIC	2.9389
Pbp	0.3265		SC	15.1498

Note: The symbols *, **, and *** indicate significance at levels of 10%, 5%, and 1%, respectively.

OLS results analysis

The south area experiences large amounts of sunshine and rainfall. The larger the sunshine and the higher the amount of rainfall, the greater the density of rural population. However, the results of the OLS analysis show that the AS and AR coefficients are negative, and the remaining positive and negative coefficients are consistent with the expected results. Given that the OLS model does not consider the spatial dependence, the credibility of the influence of the correlation coefficient on the regional rural population should be further explored. The spatial econometric models (SLM and SEM) must be constructed, and the optimal model should be selected before drawing conclusions and interpretations.

Spatial econometric model (SLM and SEM) and analysis of the optimal model results.

Model construction and optimal model determination

On the basis of the OLS model, SLM and SEM are constructed using the Geoda software, and the weight with the spatial inverse distance is set. Table 4 indicates that the probabilities of the Breusch–Pagan heteroscedasticity test for SLM and SEM are 0.2604 and 0.3265, respectively. Both models satisfiy the significant test at 1% level, thereby indicating that no heteroscedasticity is present between them. Similarly, the constant terms and independent variables of the two models pass the significance test.

Table 4

Parameter estimation and test results of the spatial econometric models.

Variables	SLM			SEM
	Coefficient	Z-statistic	p-value	Coefficient	Z-statistic	p-value
C	95.6372***	5.1942	0.0000	92.4114***	6.4822	0.0000
lnAR	−2.5664**	−2.2779	0.0227	−3.2294***	−4.2519	0.0000
lnAS	−9.3730***	−3.6718	0.0002	−8.2041***	−3.7000	0.0001
lnAW	0.8532***	2.5881	0.0097	0.6881**	2.5312	0.0113
lnAlt	−0.2501***	−3.0590	0.0022	−0.2754***	−3.4294	0.0006
lnAGR	0.3028***	3.7414	0.0002	0.3013***	3.6106	0.0003
lnELE	0.2418***	3.6989	0.0002	0.2370***	3.5003	0.0005
lnDI	−0.3785**	−2.0464	0.0407	−0.3482**	−2.0240	0.0430
R2	0.8505			0.8601
Pbp	0.3577			0.3133
LogL	6.7650			7.2253
AIC	4.4699			1.5494
SC	18.2072			13.7603

Note: The symbols *, **, *** indicate significance at levels of 10%, 5%, and 1% respectively.

OLS model, SLM, and SEM are highly appropriate for analyzing the rural population distribution in the Poyang Lake. The rules of likelihood (LogL), Akaike information criterion (AIC), and Schwartz criterion (SC) are generally used to discriminate the optimal model. The larger the LogL and the smaller the AIC and SC, the better the model effect is [31]. The LogL of SEM is larger than those of SLM and OLS model, whereas the AIC and SC of the former are smaller than those of the latter two. This finding reveals that SEM is more satisfactory than the other two. The overall significance level of the SEM coefficient is higher than those of SLM and OLS model. The results suggest that SEM is the optimal model.

Analysis of the optimal model (SEM) results

By comparing the standard coefficients of each variable of the SEM, the degree of influence on the dependent variables from large to small is ranked as follows: AS, AR, AW, DI, AGR, ALT, and ELE. Given the logarithmic analysis of all variables, the coefficient represents the elasticity. The elasticity of the climatic factors is larger than that of the economic factors. This result indicates that the influence of the former on the distribution of rural population is greater than that of the latter. Generally speaking, economic factors play a leading role in the distribution of total population or urban population, while climate factors are more dominant than economic factors in the distribution of rural population. This also confirms that rural population is a vulnerable group to cope with climate change. The elasticity of the average elevation is small because the Poyang Lake ecological economic zone is mainly composed of plains. In addition, the negative influence of topographic factors, such as altitude, on the distribution of rural population in the region is small. The result of the comparison of the coefficients (absolute value) of the OLS model and SLM shows that in terms of climatic factors, the former underestimates the influence of rainfall on the distribution of rural population, overestimates the influence of sunshine and windspeed, and the values of the coefficient of other factors are not significantly different. According to the first theorem of geography, everything is related to everything else, but near things are more related to each other. This is especially true of climate factors, which have spatial correlation and spatial dependence in spatial distribution. The spatial econometric model considering spatial dependence can more accurately reflect the influence of climate factors on the distribution of rural population.

Moreover, all positive and negative signs corresponding to the OLS model and SEM are consistent. Therefore, the negative effects of rainfall and sunshine on the distribution of the regional rural population are confirmed. This study suggests that this negative relationship should be explained in combination with the actual situation of the Poyang Lake ecological economic zone. This zone, which is located in the middle and lower reaches of the Yangtze River, has a subtropical humid monsoon climate with abundant light and rainfall throughout the year. However, excessive sunshine and rainfall might have an excessive effect on the rural population distribution in the region. First, excessive amount of sunshine and rainfall will produce heat and water stresses on local rice cultivation [32, 33]. The Poyang Lake ecological economic zone is located in a typical flood area. Therefore, excessive rainfall during rice pollination stage will affect rice pollination and fruiting and increase the risk of natural disasters, such as flood and waterlogging. These factors will lead to the decline in rice yield per unit. Agricultural development is positively related to the distribution of rural population. Second, excessive levels of sunshine and rainfall are not conducive to the survival and development of rural population. The sumer heat in the Poyang Lake ecological economic zone is difficult to bear. Coupled with global warming and heat, excessive sunshine will increase the temperatures; elevated temperature is not conducive to the construction of rural ecology. As previously mentioned, excessive rainfall increases the risk of natural disasters, such as floods, and endangers the lives and property of the rural population. The density of rural population decreases with the increase in the levels of sunshine and rainfall. The results show that the elasticity of sunshine and rainfall is far beyond those other factors, thereby highlighting that the negative influence of sunshine and rainfall can not be ignored. In addition, the effect of average wind speed on the distribution of regional rural population is significantly positive, because the Poyang Lake ecological economic zone is hot in summer, the wind speed can reduce the human somatosensory temperature to a certain extent.