使用R进行空间自相关检验

「全局溢出」当一个区域的特征变化影响到所有区域的结果时，就会产生全局溢出效应。这甚至适用于区域本身，因为影响可以传递到邻居并返回到自己的区域（反馈）。具体来说，全球溢出效应影响到邻居、邻居到邻居、邻居到邻居等等。
「局部溢出」是指影响只落在附近或近邻的情况，在它们影响邻邻区域之前就消失了。

对应全局与局部溢出，存在全局与局部自相关检验。全局自相关检验指标主要有 moran'I 指数、Geary 指数 C 统计量以及 Getis-Ord global G 统计量；局部自相检验指标主要有局部moran指数、Local Getis-Ord Gi 和 Gi* 统计量。此外，也可以通过图示的方法来显示空间的依赖关系。

下面将对相关指标的计算以及可视化的方法进行演示(所用数据均可通过公共号后台回复「地图可视化R」获取)。

全局空间自相关检验

首先导入数据和矩阵，并进行适当转换

library(readxl)
library("spdep")
# 设置工作路径
setwd('E:\\空间计量专题\\R-空间计量')
# 导入经济变量数据
cdata <- read_xlsx("E:/空间计量专题/R-空间计量/cdata.xlsx")
# 导入自定义矩阵并做适当格式转换
w1 <- read_xlsx("w1.xlsx", col_names = FALSE)
w2 <- as.matrix(w1)
w2[1:5, 1:5]
# 转换格式并标准化
w <- mat2listw(w2, style="W")

导入shp文件，合并数据

library(rgdal)
# 导入shp文件
shpt <- readOGR("广东地级市.shp")
# 合并数据
cdatashpt <- merge(shpt, cdata, by = "city")

moran'I 指数

1.Moran’s I 统计量的计算

>>> moran(cdatashpt$gdp2017, listw=w, n=length(cdatashpt$gdp2017), S0=Szero(w))

$I
0.065546870128173
$K
2.99333822437931

I Moran’s I 统计量, K 变量的峰度

当数据中出现孤岛或者缺失值时，可通过下面的子选项进行调整：

zero.policy默认值为空，使用全局选项值；如果为真，则将0分配给没有邻居的区域的滞后值，如果为假，则分配为NA

NAOK如果 TRUE，则 x 中的任何 NA 或 NaN 或 Inf 值都将传递给外部函数。如果 FALSE ，则存在 NA 或 NaN 或 Inf 值将被视为错误。

2.Monte-Carlo simulation of Moran's I

>>> # Monte-Carlo simulation of Moran's I
>>> set.seed(12345)
>>> moran.mc(cdatashpt$gdp2017, listw = w, nsim = 999, alternative = 'greater')

	Monte-Carlo simulation of Moran I

data:  cdatashpt$gdp2017 
weights: w  
number of simulations + 1: 1000 

statistic = 0.065547, observed rank = 790, p-value = 0.21
alternative hypothesis: greater

3.moran散点图

>>> # Moran 散点图
>>> moran.plot(cdatashpt$gdp2017, w, zero.policy=NULL, spChk=NULL, labels=TRUE, xlab=NULL, ylab=NULL, quiet=NULL)

labels给具有高影响度量值的点添加字符标签，如果设置为FALSE，则不会为具有较大影响的点绘制标签

4.Moran’s I 空间自相关检验

>>> # Moran’s I test for spatial autocorrelation
>>> moran.test(cdatashpt$gdp2017, w)

	Moran I test under randomisation

data:  cdatashpt$gdp2017  
weights: w    

Moran I statistic standard deviate = 0.76045, p-value = 0.2235
alternative hypothesis: greater
sample estimates:
Moran I statistic       Expectation          Variance 
       0.06554687       -0.05000000        0.02308712 

Geary 检验

>>> geary.test(cdatashpt$gdp2017, listw=w, randomisation=TRUE, alternative="greater")

	Geary C test under randomisation

data:  cdatashpt$gdp2017 
weights: w 

Geary C statistic standard deviate = 1.2898, p-value = 0.09856
alternative hypothesis: Expectation greater than statistic
sample estimates:
Geary C statistic       Expectation          Variance 
       0.79614300        1.00000000        0.02498145 

Geary’s C 检验

1.计算 Geary’s C 统计量

>>> geary(cdatashpt$gdp2017, listw=w, n=length(w), n1=length(w)-1, S0=Szero(w))

$C
0.0796142995607693
$K
0.427619746339901

2.Monte-Carlo simulation of Geary's C

>>> geary.mc(cdatashpt$gdp2017, listw=w, nsim=999, alternative="greater")

	Monte-Carlo simulation of Geary C

data:  cdatashpt$gdp2017 
weights: w 
number of simulations + 1: 1000 

statistic = 0.79614, observed rank = 116, p-value = 0.116
alternative hypothesis: greater

3.模拟分布图

>>> set.seed(12345)
>>> gdpgeary <- geary.mc(cdatashpt$gdp2017, listw=w, nsim=999, alternative="greater")
>>> plot(gdpgeary, type='l', col='orange')
>>> gdpgeary.dens = density(gdpgeary$res)
>>> polygon(gdpgeary.dens, col="gray")
>>> abline(v=gdpgeary$statistic, col='orange',lwd=2)

Getis-Ord global G 检验

>>> globalG.test(cdatashpt$gdp2017, listw=w, alternative="greater")

	Getis-Ord global G statistic

data:  cdatashpt$gdp2017 
weights: w 

standard deviate = -1.1248, p-value = 0.8697
alternative hypothesis: greater
sample estimates:
Global G statistic        Expectation           Variance 
      4.948903e-02       5.000000e-02       2.063625e-07 

空间相关图

>>> w.nb <- w$neighbours
>>> spcorrI = sp.correlogram(w.nb, cdatashpt$gdp2017, order = 2, method = "I", style = "W", randomisation = TRUE)
>>> spcorrI

Spatial correlogram for cdatashpt$gdp2017 
method: Moran's I
        estimate expectation  variance standard deviate Pr(I) two sided
1 (21)  0.065547   -0.050000  0.023087           0.7605          0.4470
2 (21) -0.130212   -0.050000  0.017101          -0.6134          0.5396

>>> plot(spcorrI, main="Spatial correlogram of gdp2017")

局部空间自相关检验

局部moran检验

>>> localmoran(cdatashpt$gdp2017, listw=w, alternative = "greater")

A localmoran: 21 × 5 of type dbl
           Ii	        E.Ii	  Var.I          Z.Ii	       Pr(z > 0)
1	-0.01370494	-0.05	0.1146316	0.107200142	0.4573151
2	0.65060843	-0.05	0.4279122	1.071021124	0.1420800
3	-0.32033200	-0.05	0.4279122	-0.413256930	0.6602908
4	0.01710619	-0.05	0.4279122	0.102585331	0.4591460
5	0.05648861	-0.05	0.1146316	0.314522027	0.3765623
6	0.48390574	-0.05	0.1929517	1.215458873	0.1120956
7	-0.48582426	-0.05	0.1929517	-0.992172269	0.8394433
8	0.10421133	-0.05	0.1929517	0.351068574	0.3627685
9	-0.26267734	-0.05	0.1146316	-0.628158331	0.7350499
10	-0.44651083	-0.05	0.1929517	-0.902673594	0.8166504
11	-0.38816462	-0.05	0.2712719	-0.649270674	0.7419183
12	0.02013525	-0.05	0.1929517	0.159665854	0.4365722
13	-0.03877614	-0.05	0.1459596	0.029378254	0.4882815
14	0.41014395	-0.05	0.2712719	0.883469050	0.1884914
15	0.02782689	-0.05	0.8978331	0.082135687	0.4672694
16	0.11878306	-0.05	0.2712719	0.324060781	0.3729460
17	0.56506605	-0.05	0.2712719	1.180917005	0.1188178
18	0.47054422	-0.05	0.1929517	1.185040809	0.1180007
19	-0.05142908	-0.05	0.2712719	-0.002743819	0.5010946
20	0.06940159	-0.05	0.1929517	0.271822740	0.3928792
21	0.38968220	-0.05	0.1459596	1.150860065	0.1248949

Local Getis-Ord Gi 和 Gi*统计量

>>> localG(cdatashpt$gdp2017, listw=w)

 [1] -0.05909956  1.09552796 -0.05815482 -0.09536069 -1.69368469 -1.94012537
 [7]  0.69295805  0.34995778  0.96964875 -0.34538087 -0.15784709  0.51802708
[13]  0.20826225 -0.87609799 -0.19862319 -1.09970094 -1.06316301 -1.42853393
[19]  0.38484915  1.17379925 -1.43159536
attr(,"gstari")
[1] FALSE
attr(,"call")
localG(x = cdatashpt$gdp2017, listw = w)
attr(,"class")
[1] "localG"

基于残差项的moran检验

>>> ols <- lm(gdp2017 ~ kj2017 + l2017 + ks2017 + pe2017 + inex2017 + new_inc2017 + pri_en2017 + high_stu2017, data = cdatashpt)
>>> lm.morantest(ols, listw = w, alternative = "two.sided")

	Global Moran I for regression residuals

data:  
model: lm(formula = gdp2017 ~ kj2017 + l2017 + ks2017 + pe2017 +
inex2017 + new_inc2017 + pri_en2017 + high_stu2017, data = cdatashpt)
weights: w

Moran I statistic standard deviate = 0.95884, p-value = 0.3376
alternative hypothesis: two.sided
sample estimates:
Observed Moran I      Expectation         Variance 
     0.001873225     -0.123283278      0.017037907 

散点图的绘制

>>> moran.plot(ols$residuals, w)

局部moran检验

localmoran(ols$residuals, w)

A localmoran: 21 × 5 of type dbl
           Ii	        E.Ii	  Var.I          Z.Ii	       Pr(z > 0)
1	-0.0363243420	-0.05	0.1168494	0.040006912	0.48404381
2	0.8370573121	-0.05	0.4404798	1.336560700	0.09068304
3	-1.1222105809	-0.05	0.4404798	-1.615537694	0.94690285
4	0.2373930487	-0.05	0.4404798	0.433025295	0.33249820
5	-0.1380790378	-0.05	0.1168494	-0.257667335	0.60166817
6	-0.1081663716	-0.05	0.1977570	-0.130799494	0.55203304
7	0.1159985869	-0.05	0.1977570	0.373283232	0.35446883
8	0.7082098711	-0.05	0.1977570	1.704996632	0.04409753
9	0.0612709718	-0.05	0.1168494	0.325513260	0.37239632
10	0.4204181464	-0.05	0.1977570	1.057835549	0.14506521
11	0.3774028474	-0.05	0.2786646	0.809648516	0.20907111
12	-0.0868802622	-0.05	0.1977570	-0.082933137	0.53304765
13	0.1244368071	-0.05	0.1492124	0.451580986	0.32578544
14	0.1874982314	-0.05	0.2786646	0.449903626	0.32638997
15	-0.5517955762	-0.05	0.9259254	-0.521481405	0.69898427
16	-0.1211737778	-0.05	0.2786646	-0.134827702	0.55362595
17	-0.0030386117	-0.05	0.2786646	0.088961079	0.46455642
18	-0.4168336708	-0.05	0.1977570	-0.824903760	0.79528688
19	-0.0539797929	-0.05	0.2786646	-0.007539102	0.50300764
20	-0.3916838150	-0.05	0.1977570	-0.768348944	0.77886005
21	-0.0001822581	-0.05	0.1492124	0.128967879	0.44869153

其他相关检验类似，只需将变量替换为残差项即可，不再演示。此外，这里仅演示了各命令的常见用法，需要具体了解的话，可以点击「阅读原文」，这里给出了各命令的详细说明。

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