Hands-on Exercise 2.2 & 2.3: Global & Local Measures of Spatial Autocorrelation

Published

November 23, 2023

Modified

November 24, 2023

1. Learning Objectives

Learning how to compute Global and Local Measure of Spatial Autocorrelation (GLSA) by using spdep package, including:

  • import geospatial data using appropriate function(s) of sf package
  • import csv file using appropriate function of readr package
  • perform relational join using appropriate join function of dplyr package
  • compute Global Spatial Autocorrelation (GSA) statistics by using appropriate functions of spdep package
    • plot Moran scatterplot
    • compute and plot spatial correlogram using appropriate function of spdep packag
  • compute Local Indicator of Spatial Association (LISA) statistics for detecting clusters and outliers by using appropriate functions spdep package
  • compute Getis-Ord’s Gi-statistics for detecting hot spot or/and cold spot area by using appropriate functions of spdep package
  • visualise the analysis output by using tmap package.

2. Getting Started

In spatial policy, one of the main development objective of the local government and planners is to ensure equal distribution of development in the province. Our task in this study, hence, is to apply appropriate spatial statistical methods to discover if development are even distributed geographically. If the answer is No. Then, our next question will be “is there sign of spatial clustering?”. And, if the answer for this question is yes, then our next question will be “where are these clusters?”

In this case study, we are interested to examine the spatial pattern of a selected development indicator (i.e. GDP per capita) of Hunan Provice, People Republic of China (https://en.wikipedia.org/wiki/Hunan).

Datasets:

  • Geospatial: Hunan province administrative boundary layer at county level; in ESRI shapefile format.
  • Aspatial: Hunan_2012.csv containing selected Hunan’s local development indicators in 2012.

Ensure that spdep, sf, tmap and tidyverse packages are installed:

  • sf is use for importing and handling geospatial data in R,
  • tidyverse is mainly use for wrangling attribute data in R,
  • spdep will be used to compute spatial weights, global and local spatial autocorrelation statistics, and
  • tmap will be used to prepare cartographic quality chropleth map.
pacman::p_load(sf, spdep, tmap, tidyverse)

# -   Creates a package list containing the necessary R packages
# -   Checks if the R packages in the package list have been installed
# -   If not installed, will installed the missing packages & launch into R environment.

3. Getting the Data Into R Environment

In this section, you will learn how to bring a geospatial data and its associated attribute table into R environment. The geospatial data is in ESRI shapefile format and the attribute table is in csv fomat.

The code chunk below uses st_read() of sf package to import Hunan shapefile into R. The imported shapefile will be simple features Object of sf.

hunan <- st_read(dsn = "data/geospatial", 
                 layer = "Hunan")
Reading layer `Hunan' from data source 
  `C:\kytjy\ISSS624\Hands-on_Ex\Hands-on_Ex2\data\geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 88 features and 7 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 108.7831 ymin: 24.6342 xmax: 114.2544 ymax: 30.12812
Geodetic CRS:  WGS 84

Next, we will import Hunan_2012.csv into R by using read_csv() of readr package. The output is R data frame class.

hunan2012 <- read_csv("data/aspatial/Hunan_2012.csv")

The code chunk below will be used to update the attribute table of hunan’s SpatialPolygonsDataFrame with the attribute fields of hunan2012 dataframe. This is performed by using left_join() of dplyr package.

hunan <- left_join(hunan,hunan2012) %>%
  select(1:4, 7, 15)

Visualising Regional Development Indicator

Now, we are going to prepare a basemap and a choropleth map showing the distribution of GDPPC 2012 by using qtm() of tmap package.

basemap <- tm_shape(hunan) +
  tm_polygons() +
  tm_text("NAME_3", size=0.3)

gdppc <- qtm(hunan, "GDPPC")
tmap_arrange(basemap, gdppc, asp=1, ncol=2)

4. Global Spatial Autocorrelation

4.1 Computing Contiguity Spatial Weights

Before we can compute the global spatial autocorrelation statistics, we need to construct a spatial weights of the study area. The spatial weights is used to define the neighbourhood relationships between the geographical units (i.e. county) in the study area.

  • poly2nb() of spdep package to compute contiguity weight matrices for the study area.
  • This function builds a neighbours list based on regions with contiguous boundaries. If you look at the documentation you will see that you can pass a “queen” argument that takes TRUE or FALSE as options. If you do not specify this argument the default is set to TRUE, that is, if you don’t specify queen = FALSE this function will return a list of first order neighbours using the Queen criteria.
wm_q <- poly2nb(hunan, 
                queen=TRUE)
summary(wm_q)
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 448 
Percentage nonzero weights: 5.785124 
Average number of links: 5.090909 
Link number distribution:

 1  2  3  4  5  6  7  8  9 11 
 2  2 12 16 24 14 11  4  2  1 
2 least connected regions:
30 65 with 1 link
1 most connected region:
85 with 11 links

Interpretation:

  • There are 88 area units in Hunan.
  • Most connected area unit has 11 neighbours.
  • There are two area units with only one neighbour.

4.2 Row-standardised weights matrix

  • Assign weights to each neighboring polygon. In our case, each neighboring polygon will be assigned equal weight (style=“W”).

  • This is accomplished by assigning the fraction 1/(# of neighbors) to each neighboring county then summing the weighted income values.

  • While this is the most intuitive way to summaries the neighbors’ values it has one drawback in that polygons along the edges of the study area will base their lagged values on fewer polygons thus potentially over- or under-estimating the true nature of the spatial autocorrelation in the data.

  • Style=“W” option used for this example for simplicity’s sake but more robust options are available, notably style=“B”.

    • Styles:
      • W: row standardised (sums over all links to n)
      • B: basic binary coding
      • C: globally standardised (sums over all links to n)
      • U: equal to C divided by the number of neighbours (sums over all links to unity)
      • S: variance-stabilizing coding scheme (sums over all links to n)
      • minmax: divides the weights by min of the max row sums and max column sums of the input weights; similar to C/U

  • The input of nb2listw() must be an object of class nb. The syntax of the function has two major arguments, namely style and zero.poly.

    rswm_q <- nb2listw(wm_q, 
                   style="W", 
                   zero.policy = TRUE) 
rswm_q
    Characteristics of weights list object:
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 448 
Percentage nonzero weights: 5.785124 
Average number of links: 5.090909 

Weights style: W 
Weights constants summary:
   n   nn S0       S1       S2
W 88 7744 88 37.86334 365.9147

  • zero.policy=TRUE: allows for lists of non-neighbors. This should be used with caution since the user may not be aware of missing neighbors in their dataset however, a zero.policy of FALSE would return an error. - If zero policy is set to TRUE, weights vectors of zero length are inserted for regions without neighbour in the neighbours list. These will in turn generate lag values of zero, equivalent to the sum of products of the zero row t(rep(0, length=length(neighbours))) %*% x, for arbitrary numerical vector x of length length(neighbours). The spatially lagged value of x for the zero-neighbour region will then be zero, which may (or may not) be a sensible choice.

4.3 Global Spatial Autocorrelation: Moran’s I

  • Describe how features differ from the values in the study area as a whole

  • Moran I (Z value) is:

    • positive (I>0): Clustered, observations tend to be similar;

    • negative(I<0): Dispersed, observations tend to be dissimilar;

    • approximately zero: observations are arranged randomly over space.

H0: Observed spatial patterns of values is equally likely as any other spatial pattern i.e. data is randomly disbursed, no spatial pattern H1: Data is more spatially clustered than expected by chance alone.

Moran’s I statistical testing using moran.test() of spdep:

moran.test(hunan$GDPPC, 
           listw=rswm_q, 
           zero.policy = TRUE, 
           na.action=na.omit)

    Moran I test under randomisation

data:  hunan$GDPPC  
weights: rswm_q    

Moran I statistic standard deviate = 4.7351, p-value = 1.095e-06
alternative hypothesis: greater
sample estimates:
Moran I statistic       Expectation          Variance 
      0.300749970      -0.011494253       0.004348351

Permutation test for Moran’s I statistic by using moran.mc() of spdep. A total of 1000 simulation will be performed.

set.seed(1234)
bperm= moran.mc(hunan$GDPPC, 
                listw=rswm_q, 
                nsim=999, 
                zero.policy = TRUE, 
                na.action=na.omit)
bperm

    Monte-Carlo simulation of Moran I

data:  hunan$GDPPC 
weights: rswm_q  
number of simulations + 1: 1000 

statistic = 0.30075, observed rank = 1000, p-value = 0.001
alternative hypothesis: greater

p-value <0.05, reject null hypothesis. Positive Moran’s I suggest variable is spatially clustered and tend to be similar.

Plot the distribution of the statistical values as histrogram to examine the simulated Moran’s I test statistics in greater detail: hist() and abline() of R Graphics are used.

mean(bperm$res[1:999])
[1] -0.01504572
var(bperm$res[1:999])
[1] 0.004371574
summary(bperm$res[1:999])
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
-0.18339 -0.06168 -0.02125 -0.01505  0.02611  0.27593 
hist(bperm$res, 
     freq=TRUE, 
     breaks=20, 
     xlab="Simulated Moran's I")
abline(v=0, 
       col="#e0218a") 

plot2 <- bperm$res
mu <- mean(plot2)

ggplot(data=data.frame(plot2),
       aes(x=plot2)
  ) +
  geom_histogram(
    bins=30,
    fill="#69b3a2",
    color="black",
    size=0.2
  ) +
  geom_vline(
    xintercept = mu,
    color="purple"
  )

4.4 Global Spatial Autocorrelation: Geary’s

In this section, you will learn how to perform Geary’s c statistics testing by using appropriate functions of spdep package.

  • Describes how features differ from their immediate neighbours.

  • Geary c (Z value) is:

    • Large c value (>1) : Dispersed, observations tend to be dissimilar;
    • Small c value (<1) : Clustered, observations tend to be similar;
    • c = 1: observations are arranged randomly over space.

The code chunk below performs Geary’s C test for spatial autocorrelation by using geary.test() of spdep.

geary.test(hunan$GDPPC, listw=rswm_q)

    Geary C test under randomisation

data:  hunan$GDPPC 
weights: rswm_q 

Geary C statistic standard deviate = 3.6108, p-value = 0.0001526
alternative hypothesis: Expectation greater than statistic
sample estimates:
Geary C statistic       Expectation          Variance 
        0.6907223         1.0000000         0.0073364

p-value <0.05, reject null hypothesis. Conclude that Geary’s C statistic of 0.69 suggest variable is not randomly arranged, and is spatially clusters. Observations tend to be similar.

Performs permutation test for Geary’s C statistic by using geary.mc() of spdep.

set.seed(1234)
bperm=geary.mc(hunan$GDPPC, 
               listw=rswm_q, 
               nsim=999)
bperm

    Monte-Carlo simulation of Geary C

data:  hunan$GDPPC 
weights: rswm_q 
number of simulations + 1: 1000 

statistic = 0.69072, observed rank = 1, p-value = 0.001
alternative hypothesis: greater

Plot a histogram to reveal the distribution of the simulated values by using the code chunk below.

mean(bperm$res[1:999])
[1] 1.004402
var(bperm$res[1:999])
[1] 0.007436493
summary(bperm$res[1:999])
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.7142  0.9502  1.0052  1.0044  1.0595  1.2722 
hist(bperm$res, freq=TRUE, breaks=20, xlab="Simulated Geary c")
abline(v=1, col="red")

5. Spatial Correlogram

  • Spatial correlograms useful examine patterns of spatial autocorrelation in your data or model residuals.
  • Show how correlated are pairs of spatial observations when you increase the distance (lag) between them - they are plots of some index of autocorrelation (Moran’s I or Geary’s c) against distance.
  • Although correlograms are not as fundamental as variograms (a keystone concept of geostatistics), they are very useful as an exploratory and descriptive tool. For this purpose they actually provide richer information than variograms.
  • sp.correlogram() of spdep package: computes a 6-lag spatial correlogram of GDPPC.
  • The global spatial autocorrelation used in Moran’s I.
  • The plot() of base Graph is then used to plot the output.
MI_corr <- sp.correlogram(wm_q, 
                          hunan$GDPPC, 
                          order=6, 
                          method="I", 
                          style="W")
plot(MI_corr)

  • Plotting the output might not allow us to provide complete interpretation, as not all autocorrelation values are statistically significant.

  • Important for us to examine the full analysis report by printing out the analysis results as in the code chunk below.

print(MI_corr)
Spatial correlogram for hunan$GDPPC 
method: Moran's I
         estimate expectation   variance standard deviate Pr(I) two sided    
1 (88)  0.3007500  -0.0114943  0.0043484           4.7351       2.189e-06 ***
2 (88)  0.2060084  -0.0114943  0.0020962           4.7505       2.029e-06 ***
3 (88)  0.0668273  -0.0114943  0.0014602           2.0496        0.040400 *  
4 (88)  0.0299470  -0.0114943  0.0011717           1.2107        0.226015    
5 (88) -0.1530471  -0.0114943  0.0012440          -4.0134       5.984e-05 ***
6 (88) -0.1187070  -0.0114943  0.0016791          -2.6164        0.008886 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
  • sp.correlogram() of spdep package: used to compute a 6-lag spatial correlogram of GDPPC.
  • The global spatial autocorrelation used in Geary’s C.
  • The plot() of base Graph is then used to plot the output.
GC_corr <- sp.correlogram(wm_q, 
                          hunan$GDPPC, 
                          order=6, 
                          method="C", 
                          style="W")
plot(GC_corr)

Similar to the previous step, we will print out the analysis report by using the code chunk below.

print(GC_corr)
Spatial correlogram for hunan$GDPPC 
method: Geary's C
        estimate expectation  variance standard deviate Pr(I) two sided    
1 (88) 0.6907223   1.0000000 0.0073364          -3.6108       0.0003052 ***
2 (88) 0.7630197   1.0000000 0.0049126          -3.3811       0.0007220 ***
3 (88) 0.9397299   1.0000000 0.0049005          -0.8610       0.3892612    
4 (88) 1.0098462   1.0000000 0.0039631           0.1564       0.8757128    
5 (88) 1.2008204   1.0000000 0.0035568           3.3673       0.0007592 ***
6 (88) 1.0773386   1.0000000 0.0058042           1.0151       0.3100407    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

6. Cluster and Outlier Analysis

  • Local Indicators of Spatial Association (LISA): statistics that evaluate the existence of clusters in the spatial arrangement of a given variable.

  • Eg if we are studying cancer rates among census tracts in a given city local clusters in the rates mean that there are areas that have higher or lower rates than is to be expected by chance alone; that is, the values occurring are above or below those of a random distribution in space.

In this section, we learn how to apply appropriate Local Indicators for Spatial Association (LISA), especially local Moran’s I to detect cluster and/or outlier from GDP per capita 2012 of Hunan Province, PRC.

Computing Local Moran’s I

  • localmoran() function of spdep computes Ii values, given a set of zi values and a listw object providing neighbour weighting information for the polygon associated with the zi values
  • The code chunks below are used to compute local Moran’s I of GDPPC2012 at the county level.
fips <- order(hunan$County)
localMI <- localmoran(hunan$GDPPC, rswm_q)
head(localMI)
            Ii          E.Ii       Var.Ii        Z.Ii Pr(z != E(Ii))
1 -0.001468468 -2.815006e-05 4.723841e-04 -0.06626904      0.9471636
2  0.025878173 -6.061953e-04 1.016664e-02  0.26266425      0.7928094
3 -0.011987646 -5.366648e-03 1.133362e-01 -0.01966705      0.9843090
4  0.001022468 -2.404783e-07 5.105969e-06  0.45259801      0.6508382
5  0.014814881 -6.829362e-05 1.449949e-03  0.39085814      0.6959021
6 -0.038793829 -3.860263e-04 6.475559e-03 -0.47728835      0.6331568
  • Ii: the local Moran’s I statistics
  • E.Ii: the expectation of local moran statistic under the randomisation hypothesis
  • Var.Ii: the variance of local moran statistic under the randomisation hypothesis
  • Z.Ii:the standard deviate of local moran statistic
  • Pr(): the p-value of local moran statistic

The code chunk below list the content of the local Moran matrix derived by using printCoefmat().

printCoefmat(data.frame(
  localMI[fips,], 
  row.names=hunan$County[fips]),
  check.names=FALSE)
                       Ii        E.Ii      Var.Ii        Z.Ii Pr.z....E.Ii..
Anhua         -2.2493e-02 -5.0048e-03  5.8235e-02 -7.2467e-02         0.9422
Anren         -3.9932e-01 -7.0111e-03  7.0348e-02 -1.4791e+00         0.1391
Anxiang       -1.4685e-03 -2.8150e-05  4.7238e-04 -6.6269e-02         0.9472
Baojing        3.4737e-01 -5.0089e-03  8.3636e-02  1.2185e+00         0.2230
Chaling        2.0559e-02 -9.6812e-04  2.7711e-02  1.2932e-01         0.8971
Changning     -2.9868e-05 -9.0010e-09  1.5105e-07 -7.6828e-02         0.9388
Changsha       4.9022e+00 -2.1348e-01  2.3194e+00  3.3590e+00         0.0008
Chengbu        7.3725e-01 -1.0534e-02  2.2132e-01  1.5895e+00         0.1119
Chenxi         1.4544e-01 -2.8156e-03  4.7116e-02  6.8299e-01         0.4946
Cili           7.3176e-02 -1.6747e-03  4.7902e-02  3.4200e-01         0.7324
Dao            2.1420e-01 -2.0824e-03  4.4123e-02  1.0297e+00         0.3032
Dongan         1.5210e-01 -6.3485e-04  1.3471e-02  1.3159e+00         0.1882
Dongkou        5.2918e-01 -6.4461e-03  1.0748e-01  1.6338e+00         0.1023
Fenghuang      1.8013e-01 -6.2832e-03  1.3257e-01  5.1198e-01         0.6087
Guidong       -5.9160e-01 -1.3086e-02  3.7003e-01 -9.5104e-01         0.3416
Guiyang        1.8240e-01 -3.6908e-03  3.2610e-02  1.0305e+00         0.3028
Guzhang        2.8466e-01 -8.5054e-03  1.4152e-01  7.7931e-01         0.4358
Hanshou        2.5878e-02 -6.0620e-04  1.0167e-02  2.6266e-01         0.7928
Hengdong       9.9964e-03 -4.9063e-04  6.7742e-03  1.2742e-01         0.8986
Hengnan        2.8064e-02 -3.2160e-04  3.7597e-03  4.6294e-01         0.6434
Hengshan      -5.8201e-03 -3.0437e-05  5.1076e-04 -2.5618e-01         0.7978
Hengyang       6.2997e-02 -1.3046e-03  2.1865e-02  4.3486e-01         0.6637
Hongjiang      1.8790e-01 -2.3019e-03  3.1725e-02  1.0678e+00         0.2856
Huarong       -1.5389e-02 -1.8667e-03  8.1030e-02 -4.7503e-02         0.9621
Huayuan        8.3772e-02 -8.5569e-04  2.4495e-02  5.4072e-01         0.5887
Huitong        2.5997e-01 -5.2447e-03  1.1077e-01  7.9685e-01         0.4255
Jiahe         -1.2431e-01 -3.0550e-03  5.1111e-02 -5.3633e-01         0.5917
Jianghua       2.8651e-01 -3.8280e-03  8.0968e-02  1.0204e+00         0.3076
Jiangyong      2.4337e-01 -2.7082e-03  1.1746e-01  7.1800e-01         0.4728
Jingzhou       1.8270e-01 -8.5106e-04  2.4363e-02  1.1759e+00         0.2396
Jinshi        -1.1988e-02 -5.3666e-03  1.1334e-01 -1.9667e-02         0.9843
Jishou        -2.8680e-01 -2.6305e-03  4.4028e-02 -1.3543e+00         0.1756
Lanshan        6.3334e-02 -9.6365e-04  2.0441e-02  4.4972e-01         0.6529
Leiyang        1.1581e-02 -1.4948e-04  2.5082e-03  2.3422e-01         0.8148
Lengshuijiang -1.7903e+00 -8.2129e-02  2.1598e+00 -1.1623e+00         0.2451
Li             1.0225e-03 -2.4048e-07  5.1060e-06  4.5260e-01         0.6508
Lianyuan      -1.4672e-01 -1.8983e-03  1.9145e-02 -1.0467e+00         0.2952
Liling         1.3774e+00 -1.5097e-02  4.2601e-01  2.1335e+00         0.0329
Linli          1.4815e-02 -6.8294e-05  1.4499e-03  3.9086e-01         0.6959
Linwu         -2.4621e-03 -9.0703e-06  1.9258e-04 -1.7676e-01         0.8597
Linxiang       6.5904e-02 -2.9028e-03  2.5470e-01  1.3634e-01         0.8916
Liuyang        3.3688e+00 -7.7502e-02  1.5180e+00  2.7972e+00         0.0052
Longhui        8.0801e-01 -1.1377e-02  1.5538e-01  2.0787e+00         0.0376
Longshan       7.5663e-01 -1.1100e-02  3.1449e-01  1.3690e+00         0.1710
Luxi           1.8177e-01 -2.4855e-03  3.4249e-02  9.9561e-01         0.3194
Mayang         2.1852e-01 -5.8773e-03  9.8049e-02  7.1663e-01         0.4736
Miluo          1.8704e+00 -1.6927e-02  2.7925e-01  3.5715e+00         0.0004
Nan           -9.5789e-03 -4.9497e-04  6.8341e-03 -1.0988e-01         0.9125
Ningxiang      1.5607e+00 -7.3878e-02  8.0012e-01  1.8274e+00         0.0676
Ningyuan       2.0910e-01 -7.0884e-03  8.2306e-02  7.5356e-01         0.4511
Pingjiang     -9.8964e-01 -2.6457e-03  5.6027e-02 -4.1698e+00         0.0000
Qidong         1.1806e-01 -2.1207e-03  2.4747e-02  7.6396e-01         0.4449
Qiyang         6.1966e-02 -7.3374e-04  8.5743e-03  6.7712e-01         0.4983
Rucheng       -3.6992e-01 -8.8999e-03  2.5272e-01 -7.1814e-01         0.4727
Sangzhi        2.5053e-01 -4.9470e-03  6.8000e-02  9.7972e-01         0.3272
Shaodong      -3.2659e-02 -3.6592e-05  5.0546e-04 -1.4510e+00         0.1468
Shaoshan       2.1223e+00 -5.0227e-02  1.3668e+00  1.8583e+00         0.0631
Shaoyang       5.9499e-01 -1.1253e-02  1.3012e-01  1.6807e+00         0.0928
Shimen        -3.8794e-02 -3.8603e-04  6.4756e-03 -4.7729e-01         0.6332
Shuangfeng     9.2835e-03 -2.2867e-03  3.1516e-02  6.5174e-02         0.9480
Shuangpai      8.0591e-02 -3.1366e-04  8.9838e-03  8.5358e-01         0.3933
Suining        3.7585e-01 -3.5933e-03  4.1870e-02  1.8544e+00         0.0637
Taojiang      -2.5394e-01 -1.2395e-03  1.4477e-02 -2.1002e+00         0.0357
Taoyuan        1.4729e-02 -1.2039e-04  8.5103e-04  5.0903e-01         0.6107
Tongdao        4.6482e-01 -6.9870e-03  1.9879e-01  1.0582e+00         0.2900
Wangcheng      4.4220e+00 -1.1067e-01  1.3596e+00  3.8873e+00         0.0001
Wugang         7.1003e-01 -7.8144e-03  1.0710e-01  2.1935e+00         0.0283
Xiangtan       2.4530e-01 -3.6457e-04  3.2319e-03  4.3213e+00         0.0000
Xiangxiang     2.6271e-01 -1.2703e-03  2.1290e-02  1.8092e+00         0.0704
Xiangyin       5.4525e-01 -4.7442e-03  7.9236e-02  1.9539e+00         0.0507
Xinhua         1.1810e-01 -6.2649e-03  8.6001e-02  4.2409e-01         0.6715
Xinhuang       1.5725e-01 -4.1820e-03  3.6648e-01  2.6667e-01         0.7897
Xinning        6.8928e-01 -9.6674e-03  2.0328e-01  1.5502e+00         0.1211
Xinshao        5.7578e-02 -8.5932e-03  1.1769e-01  1.9289e-01         0.8470
Xintian       -7.4050e-03 -5.1493e-03  1.0877e-01 -6.8395e-03         0.9945
Xupu           3.2406e-01 -5.7468e-03  5.7735e-02  1.3726e+00         0.1699
Yanling       -6.9021e-02 -5.9211e-04  9.9306e-03 -6.8667e-01         0.4923
Yizhang       -2.6844e-01 -2.2463e-03  4.7588e-02 -1.2202e+00         0.2224
Yongshun       6.3064e-01 -1.1350e-02  1.8830e-01  1.4795e+00         0.1390
Yongxing       4.3411e-01 -9.0735e-03  1.5088e-01  1.1409e+00         0.2539
You            7.8750e-02 -7.2728e-03  1.2116e-01  2.4714e-01         0.8048
Yuanjiang      2.0004e-04 -1.7760e-04  2.9798e-03  6.9181e-03         0.9945
Yuanling       8.7298e-03 -2.2981e-06  2.3221e-05  1.8121e+00         0.0700
Yueyang        4.1189e-02 -1.9768e-04  2.3113e-03  8.6085e-01         0.3893
Zhijiang       1.0476e-01 -7.8123e-04  1.3100e-02  9.2214e-01         0.3565
Zhongfang     -2.2685e-01 -2.1455e-03  3.5927e-02 -1.1855e+00         0.2358
Zhuzhou        3.2864e-01 -5.2432e-04  7.2391e-03  3.8688e+00         0.0001
Zixing        -7.6849e-01 -8.8210e-02  9.4057e-01 -7.0144e-01         0.4830

Before mapping the local Moran’s I map, it is wise to append the local Moran’s I dataframe (i.e. localMI) onto hunan SpatialPolygonDataFrame.

hunan.localMI <- cbind(hunan,localMI) %>%
  rename(Pr.Ii = Pr.z....E.Ii..)
hunan.localMI
Simple feature collection with 88 features and 11 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 108.7831 ymin: 24.6342 xmax: 114.2544 ymax: 30.12812
Geodetic CRS:  WGS 84
First 10 features:
     NAME_2  ID_3    NAME_3   ENGTYPE_3    County GDPPC           Ii
1   Changde 21098   Anxiang      County   Anxiang 23667 -0.001468468
2   Changde 21100   Hanshou      County   Hanshou 20981  0.025878173
3   Changde 21101    Jinshi County City    Jinshi 34592 -0.011987646
4   Changde 21102        Li      County        Li 24473  0.001022468
5   Changde 21103     Linli      County     Linli 25554  0.014814881
6   Changde 21104    Shimen      County    Shimen 27137 -0.038793829
7  Changsha 21109   Liuyang County City   Liuyang 63118  3.368821673
8  Changsha 21110 Ningxiang      County Ningxiang 62202  1.560689600
9  Changsha 21111 Wangcheng      County Wangcheng 70666  4.421958618
10 Chenzhou 21112     Anren      County     Anren 12761 -0.399322576
            E.Ii       Var.Ii        Z.Ii        Pr.Ii
1  -2.815006e-05 4.723841e-04 -0.06626904 0.9471636332
2  -6.061953e-04 1.016664e-02  0.26266425 0.7928093714
3  -5.366648e-03 1.133362e-01 -0.01966705 0.9843089778
4  -2.404783e-07 5.105969e-06  0.45259801 0.6508382339
5  -6.829362e-05 1.449949e-03  0.39085814 0.6959020959
6  -3.860263e-04 6.475559e-03 -0.47728835 0.6331568039
7  -7.750185e-02 1.518028e+00  2.79715225 0.0051555232
8  -7.387766e-02 8.001247e-01  1.82735933 0.0676457604
9  -1.106694e-01 1.359593e+00  3.88727819 0.0001013746
10 -7.011066e-03 7.034768e-02 -1.47912938 0.1391057404
                         geometry
1  POLYGON ((112.0625 29.75523...
2  POLYGON ((112.2288 29.11684...
3  POLYGON ((111.8927 29.6013,...
4  POLYGON ((111.3731 29.94649...
5  POLYGON ((111.6324 29.76288...
6  POLYGON ((110.8825 30.11675...
7  POLYGON ((113.9905 28.5682,...
8  POLYGON ((112.7181 28.38299...
9  POLYGON ((112.7914 28.52688...
10 POLYGON ((113.1757 26.82734...

Plot the local Moran’s I values by using choropleth mapping functions of tmap package.

tm_shape(hunan.localMI) +
  tm_fill(col = "Ii", 
          style = "pretty",
          palette = "RdBu",
          title = "local moran statistics") +
  tm_borders(alpha = 0.5)

The choropleth shows there is evidence for both positive and negative Ii values. However, it is useful to consider the p-values for each of these values, as consider above.

The code chunks below produce a choropleth map of Moran’s I p-values by using functions of tmap package.

tm_shape(hunan.localMI) +
  tm_fill(col = "Pr.Ii", 
          breaks=c(-Inf, 0.001, 0.01, 0.05, 0.1, Inf),
          palette="-Blues", 
          title = "local Moran's I p-values") +
  tm_borders(alpha = 0.5)

Plot both the local Moran’s I values map and its corresponding p-values map next to each other for easier comparison.

localMI.map <- tm_shape(hunan.localMI) +
  tm_fill(col = "Ii", 
          style = "pretty", 
          title = "local moran statistics") +
  tm_borders(alpha = 0.5)

pvalue.map <- tm_shape(hunan.localMI) +
  tm_fill(col = "Pr.Ii", 
          breaks=c(-Inf, 0.001, 0.01, 0.05, 0.1, Inf),
          palette="-Blues", 
          title = "local Moran's I p-values") +
  tm_borders(alpha = 0.5)

tmap_arrange(localMI.map, pvalue.map, asp=1, ncol=2)

7. Creating a LISA Cluster Map

The LISA Cluster Map shows the significant locations color coded by type of spatial autocorrelation. The first step before we can generate the LISA cluster map is to plot the Moran scatterplot.

Plotting Moran scatterplot

The Moran scatterplot is an illustration of the relationship between the values of the chosen attribute at each location and the average value of the same attribute at neighboring locations.

The code chunk below plots the Moran scatterplot of GDPPC 2012 by using moran.plot() of spdep.

nci <- moran.plot(hunan$GDPPC, rswm_q,
                  labels=as.character(hunan$County), #seems like no difference if as.character is removed
                  xlab="GDPPC 2012", 
                  ylab="Spatially Lag GDPPC 2012")

Notice that the plot is split in 4 quadrants. The top right corner belongs to areas that have high GDPPC and are surrounded by other areas that have the average level of GDPPC.

Note

This is high-high locations in the lesson slide. HH Autocorrelation: Positive Cluster: “I’m high and my neighbours are high.”

Plotting Moran scatterplot with standardised variable

First, use scale() to center and scale the variable. Here centering is done by subtracting the mean (omitting NAs) the corresponding columns, and scaling is done by dividing the (centered) variable by their standard deviations.

hunan$Z.GDPPC <- scale(hunan$GDPPC) %>% 
  as.vector

The as.vector() added to the end is to make sure that the data type we get out of this is a vector, that map neatly into out dataframe.

Now, we are ready to plot the Moran scatterplot again by using the code chunk below.

nci2 <- moran.plot(hunan$Z.GDPPC, rswm_q,
                   labels=as.character(hunan$County),
                   xlab="z-GDPPC 2012", 
                   ylab="Spatially Lag z-GDPPC 2012")

Preparing LISA map classes

The code chunks below show the steps to prepare a LISA cluster map.

quadrant <- vector(mode="numeric",length=nrow(localMI))

Next, derive the spatially lagged variable of interest (i.e. GDPPC) and centers the spatially lagged variable around its mean.

hunan$lag_GDPPC <- lag.listw(rswm_q, hunan$GDPPC)
DV <- hunan$lag_GDPPC - mean(hunan$lag_GDPPC)

This is follow by centering the local Moran’s around the mean.

LM_I <- localMI[,1] - mean(localMI[,1])

Next, we will set a statistical significance level for the local Moran.

signif <- 0.05

These four command lines define the low-low (1), low-high (2), high-low (3) and high-high (4) categories.

quadrant[DV <0 & LM_I>0] <- 1
quadrant[DV >0 & LM_I<0] <- 2
quadrant[DV <0 & LM_I<0] <- 3  
quadrant[DV >0 & LM_I>0] <- 4

Lastly, place non-significant Moran in the category 0.

quadrant[localMI[,5]>signif] <- 0

In fact, we can combined all the steps into one single code chunk as shown below:

quadrant <- vector(mode="numeric",length=nrow(localMI))
hunan$lag_GDPPC <- lag.listw(rswm_q, hunan$GDPPC)
DV <- hunan$lag_GDPPC - mean(hunan$lag_GDPPC)     
LM_I <- localMI[,1]   
signif <- 0.05       
quadrant[DV <0 & LM_I>0] <- 1
quadrant[DV >0 & LM_I<0] <- 2
quadrant[DV <0 & LM_I<0] <- 3  
quadrant[DV >0 & LM_I>0] <- 4    
quadrant[localMI[,5]>signif] <- 0

Plotting LISA map

Now, we can build the LISA map by using the code chunks below.

hunan.localMI$quadrant <- quadrant
colors <- c("#ffffff", "#2c7bb6", "#abd9e9", "#fdae61", "#d7191c")
clusters <- c("insignificant", "low-low", "low-high", "high-low", "high-high")

tm_shape(hunan.localMI) +
  tm_fill(col = "quadrant", 
          style = "cat", 
          palette = colors[c(sort(unique(quadrant)))+1], 
          labels = clusters[c(sort(unique(quadrant)))+1],
          popup.vars = c("")) +
  tm_view(set.zoom.limits = c(11,17)) +
  tm_borders(alpha=0.5)

Plot both the local Moran’s I values map and its corresponding p-values map next to each other for easier comparison.

The code chunk below will be used to create such visualisation.

gdppc <- qtm(hunan, "GDPPC")

hunan.localMI$quadrant <- quadrant
colors <- c("#ffffff", "#2c7bb6", "#abd9e9", "#fdae61", "#d7191c")
clusters <- c("insignificant", "low-low", "low-high", "high-low", "high-high")

LISAmap <- tm_shape(hunan.localMI) +
  tm_fill(col = "quadrant", 
          style = "cat", 
          palette = colors[c(sort(unique(quadrant)))+1], 
          labels = clusters[c(sort(unique(quadrant)))+1],
          popup.vars = c("")) +
  tm_view(set.zoom.limits = c(11,17)) +
  tm_borders(alpha=0.5)

tmap_arrange(gdppc, LISAmap, 
             asp=1, ncol=2)

We can also include the local Moran’s I map and p-value map as shown below for easy comparison.

Hot Spot and Cold Spot Area Analysis

Localised spatial statistics can be also used to detect hot spot and/or cold spot areas.

Getis and Ord’s G-Statistics

  • Used to to detect spatial anomalies is the Getis and Ord’s G-statistics .

  • Looks at neighbours within a defined proximity to identify where either high or low values clutser spatially.

  • Here, statistically significant hot-spots are recognised as areas of high values where other areas within a neighbourhood range also share high values too.

The analysis consists of three steps:

  1. Deriving spatial weight matrix
  2. Computing Gi statistics
  3. Mapping Gi statistics

1. Deriving distance-based weight matrix

First, we need to define a new set of neighbours. While the spatial autocorrelation considered units which shared borders, for Getis-Ord we are defining neighbours based on distance.

There are two type of distance-based proximity matrix, they are:

  • fixed distance weight matrix; and
  • adaptive distance weight matrix.

We will need points to associate with each polygon before we can make our connectivity graph. It will be a little more complicated than just running st_centroid() on the sf object: us.bound. We need the coordinates in a separate data frame for this to work. To do this we will use a mapping function. The mapping function applies a given function to each element of a vector and returns a vector of the same length. Our input vector will be the geometry column of us.bound. Our function will be st_centroid(). We will be using map_dbl variation of map from the purrr package. For more documentation, check out map documentation

To get our longitude values we map the st_centroid() function over the geometry column of us.bound and access the longitude value through double bracket notation [[]] and 1. This allows us to get only the longitude, which is the first value in each centroid.

longitude <- map_dbl(hunan$geometry, ~st_centroid(.x)[[1]])

We do the same for latitude with one key difference. We access the second value per each centroid with [[2]].

latitude <- map_dbl(hunan$geometry, ~st_centroid(.x)[[2]])

Now that we have latitude and longitude, we use cbind to put longitude and latitude into the same object.

coords <- cbind(longitude, latitude)

Firstly, we need to determine the upper limit for distance band by using the steps below:

  • Return a matrix with the indices of points belonging to the set of the k nearest neighbours of each other by using knearneigh() of spdep.
  • Convert the knn object returned by knearneigh() into a neighbours list of class nb with a list of integer vectors containing neighbour region number ids by using knn2nb().
  • Return the length of neighbour relationship edges by using nbdists() of spdep. The function returns in the units of the coordinates if the coordinates are projected, in km otherwise.
  • Remove the list structure of the returned object by using unlist().
#coords <- coordinates(hunan)
k1 <- knn2nb(knearneigh(coords))
k1dists <- unlist(nbdists(k1, coords, longlat = TRUE))
summary(k1dists)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  24.79   32.57   38.01   39.07   44.52   61.79

The summary report shows that the largest first nearest neighbour distance is 61.79 km, so using this as the upper threshold gives certainty that all units will have at least one neighbour.

Now, we will compute the distance weight matrix by using dnearneigh() as shown in the code chunk below.

wm_d62 <- dnearneigh(coords, 0, 62, longlat = TRUE)
wm_d62
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 324 
Percentage nonzero weights: 4.183884 
Average number of links: 3.681818

Next, nb2listw() is used to convert the nb object into spatial weights object.

The output spatial weights object is called wm62_lw.

wm62_lw <- nb2listw(wm_d62, style = 'B')
summary(wm62_lw)
Characteristics of weights list object:
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 324 
Percentage nonzero weights: 4.183884 
Average number of links: 3.681818 
Link number distribution:

 1  2  3  4  5  6 
 6 15 14 26 20  7 
6 least connected regions:
6 15 30 32 56 65 with 1 link
7 most connected regions:
21 28 35 45 50 52 82 with 6 links

Weights style: B 
Weights constants summary:
   n   nn  S0  S1   S2
B 88 7744 324 648 5440

One of the characteristics of fixed distance weight matrix is that more densely settled areas (usually the urban areas) tend to have more neighbours and the less densely settled areas (usually the rural counties) tend to have lesser neighbours. Having many neighbours smoothes the neighbour relationship across more neighbours.

It is possible to control the numbers of neighbours directly using k-nearest neighbours, either accepting asymmetric neighbours or imposing symmetry as shown in the code chunk below.

knn <- knn2nb(knearneigh(coords, k=8))
knn
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 704 
Percentage nonzero weights: 9.090909 
Average number of links: 8 
Non-symmetric neighbours list

Next, nb2listw() is used to convert the nb object into spatial weights object.

knn_lw <- nb2listw(knn, style = 'B')
summary(knn_lw)
Characteristics of weights list object:
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 704 
Percentage nonzero weights: 9.090909 
Average number of links: 8 
Non-symmetric neighbours list
Link number distribution:

 8 
88 
88 least connected regions:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 with 8 links
88 most connected regions:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 with 8 links

Weights style: B 
Weights constants summary:
   n   nn  S0   S1    S2
B 88 7744 704 1300 23014

2. Computing Gi statistics

fips <- order(hunan$County)
gi.fixed <- localG(hunan$GDPPC, wm62_lw)
gi.fixed
 [1]  0.436075843 -0.265505650 -0.073033665  0.413017033  0.273070579
 [6] -0.377510776  2.863898821  2.794350420  5.216125401  0.228236603
[11]  0.951035346 -0.536334231  0.176761556  1.195564020 -0.033020610
[16]  1.378081093 -0.585756761 -0.419680565  0.258805141  0.012056111
[21] -0.145716531 -0.027158687 -0.318615290 -0.748946051 -0.961700582
[26] -0.796851342 -1.033949773 -0.460979158 -0.885240161 -0.266671512
[31] -0.886168613 -0.855476971 -0.922143185 -1.162328599  0.735582222
[36] -0.003358489 -0.967459309 -1.259299080 -1.452256513 -1.540671121
[41] -1.395011407 -1.681505286 -1.314110709 -0.767944457 -0.192889342
[46]  2.720804542  1.809191360 -1.218469473 -0.511984469 -0.834546363
[51] -0.908179070 -1.541081516 -1.192199867 -1.075080164 -1.631075961
[56] -0.743472246  0.418842387  0.832943753 -0.710289083 -0.449718820
[61] -0.493238743 -1.083386776  0.042979051  0.008596093  0.136337469
[66]  2.203411744  2.690329952  4.453703219 -0.340842743 -0.129318589
[71]  0.737806634 -1.246912658  0.666667559  1.088613505 -0.985792573
[76]  1.233609606 -0.487196415  1.626174042 -1.060416797  0.425361422
[81] -0.837897118 -0.314565243  0.371456331  4.424392623 -0.109566928
[86]  1.364597995 -1.029658605 -0.718000620
attr(,"internals")
               Gi      E(Gi)        V(Gi)        Z(Gi) Pr(z != E(Gi))
 [1,] 0.064192949 0.05747126 2.375922e-04  0.436075843   6.627817e-01
 [2,] 0.042300020 0.04597701 1.917951e-04 -0.265505650   7.906200e-01
 [3,] 0.044961480 0.04597701 1.933486e-04 -0.073033665   9.417793e-01
 [4,] 0.039475779 0.03448276 1.461473e-04  0.413017033   6.795941e-01
 [5,] 0.049767939 0.04597701 1.927263e-04  0.273070579   7.847990e-01
 [6,] 0.008825335 0.01149425 4.998177e-05 -0.377510776   7.057941e-01
 [7,] 0.050807266 0.02298851 9.435398e-05  2.863898821   4.184617e-03
 [8,] 0.083966739 0.04597701 1.848292e-04  2.794350420   5.200409e-03
 [9,] 0.115751554 0.04597701 1.789361e-04  5.216125401   1.827045e-07
[10,] 0.049115587 0.04597701 1.891013e-04  0.228236603   8.194623e-01
[11,] 0.045819180 0.03448276 1.420884e-04  0.951035346   3.415864e-01
[12,] 0.049183846 0.05747126 2.387633e-04 -0.536334231   5.917276e-01
[13,] 0.048429181 0.04597701 1.924532e-04  0.176761556   8.596957e-01
[14,] 0.034733752 0.02298851 9.651140e-05  1.195564020   2.318667e-01
[15,] 0.011262043 0.01149425 4.945294e-05 -0.033020610   9.736582e-01
[16,] 0.065131196 0.04597701 1.931870e-04  1.378081093   1.681783e-01
[17,] 0.027587075 0.03448276 1.385862e-04 -0.585756761   5.580390e-01
[18,] 0.029409313 0.03448276 1.461397e-04 -0.419680565   6.747188e-01
[19,] 0.061466754 0.05747126 2.383385e-04  0.258805141   7.957856e-01
[20,] 0.057656917 0.05747126 2.371303e-04  0.012056111   9.903808e-01
[21,] 0.066518379 0.06896552 2.820326e-04 -0.145716531   8.841452e-01
[22,] 0.045599896 0.04597701 1.928108e-04 -0.027158687   9.783332e-01
[23,] 0.030646753 0.03448276 1.449523e-04 -0.318615290   7.500183e-01
[24,] 0.035635552 0.04597701 1.906613e-04 -0.748946051   4.538897e-01
[25,] 0.032606647 0.04597701 1.932888e-04 -0.961700582   3.362000e-01
[26,] 0.035001352 0.04597701 1.897172e-04 -0.796851342   4.255374e-01
[27,] 0.012746354 0.02298851 9.812587e-05 -1.033949773   3.011596e-01
[28,] 0.061287917 0.06896552 2.773884e-04 -0.460979158   6.448136e-01
[29,] 0.014277403 0.02298851 9.683314e-05 -0.885240161   3.760271e-01
[30,] 0.009622875 0.01149425 4.924586e-05 -0.266671512   7.897221e-01
[31,] 0.014258398 0.02298851 9.705244e-05 -0.886168613   3.755267e-01
[32,] 0.005453443 0.01149425 4.986245e-05 -0.855476971   3.922871e-01
[33,] 0.043283712 0.05747126 2.367109e-04 -0.922143185   3.564539e-01
[34,] 0.020763514 0.03448276 1.393165e-04 -1.162328599   2.451020e-01
[35,] 0.081261843 0.06896552 2.794398e-04  0.735582222   4.619850e-01
[36,] 0.057419907 0.05747126 2.338437e-04 -0.003358489   9.973203e-01
[37,] 0.013497133 0.02298851 9.624821e-05 -0.967459309   3.333145e-01
[38,] 0.019289310 0.03448276 1.455643e-04 -1.259299080   2.079223e-01
[39,] 0.025996272 0.04597701 1.892938e-04 -1.452256513   1.464303e-01
[40,] 0.016092694 0.03448276 1.424776e-04 -1.540671121   1.233968e-01
[41,] 0.035952614 0.05747126 2.379439e-04 -1.395011407   1.630124e-01
[42,] 0.031690963 0.05747126 2.350604e-04 -1.681505286   9.266481e-02
[43,] 0.018750079 0.03448276 1.433314e-04 -1.314110709   1.888090e-01
[44,] 0.015449080 0.02298851 9.638666e-05 -0.767944457   4.425202e-01
[45,] 0.065760689 0.06896552 2.760533e-04 -0.192889342   8.470456e-01
[46,] 0.098966900 0.05747126 2.326002e-04  2.720804542   6.512325e-03
[47,] 0.085415780 0.05747126 2.385746e-04  1.809191360   7.042128e-02
[48,] 0.038816536 0.05747126 2.343951e-04 -1.218469473   2.230456e-01
[49,] 0.038931873 0.04597701 1.893501e-04 -0.511984469   6.086619e-01
[50,] 0.055098610 0.06896552 2.760948e-04 -0.834546363   4.039732e-01
[51,] 0.033405005 0.04597701 1.916312e-04 -0.908179070   3.637836e-01
[52,] 0.043040784 0.06896552 2.829941e-04 -1.541081516   1.232969e-01
[53,] 0.011297699 0.02298851 9.615920e-05 -1.192199867   2.331829e-01
[54,] 0.040968457 0.05747126 2.356318e-04 -1.075080164   2.823388e-01
[55,] 0.023629663 0.04597701 1.877170e-04 -1.631075961   1.028743e-01
[56,] 0.006281129 0.01149425 4.916619e-05 -0.743472246   4.571958e-01
[57,] 0.063918654 0.05747126 2.369553e-04  0.418842387   6.753313e-01
[58,] 0.070325003 0.05747126 2.381374e-04  0.832943753   4.048765e-01
[59,] 0.025947288 0.03448276 1.444058e-04 -0.710289083   4.775249e-01
[60,] 0.039752578 0.04597701 1.915656e-04 -0.449718820   6.529132e-01
[61,] 0.049934283 0.05747126 2.334965e-04 -0.493238743   6.218439e-01
[62,] 0.030964195 0.04597701 1.920248e-04 -1.083386776   2.786368e-01
[63,] 0.058129184 0.05747126 2.343319e-04  0.042979051   9.657182e-01
[64,] 0.046096514 0.04597701 1.932637e-04  0.008596093   9.931414e-01
[65,] 0.012459080 0.01149425 5.008051e-05  0.136337469   8.915545e-01
[66,] 0.091447733 0.05747126 2.377744e-04  2.203411744   2.756574e-02
[67,] 0.049575872 0.02298851 9.766513e-05  2.690329952   7.138140e-03
[68,] 0.107907212 0.04597701 1.933581e-04  4.453703219   8.440175e-06
[69,] 0.019616151 0.02298851 9.789454e-05 -0.340842743   7.332220e-01
[70,] 0.032923393 0.03448276 1.454032e-04 -0.129318589   8.971056e-01
[71,] 0.030317663 0.02298851 9.867859e-05  0.737806634   4.606320e-01
[72,] 0.019437582 0.03448276 1.455870e-04 -1.246912658   2.124295e-01
[73,] 0.055245460 0.04597701 1.932838e-04  0.666667559   5.049845e-01
[74,] 0.074278054 0.05747126 2.383538e-04  1.088613505   2.763244e-01
[75,] 0.013269580 0.02298851 9.719982e-05 -0.985792573   3.242349e-01
[76,] 0.049407829 0.03448276 1.463785e-04  1.233609606   2.173484e-01
[77,] 0.028605749 0.03448276 1.455139e-04 -0.487196415   6.261191e-01
[78,] 0.039087662 0.02298851 9.801040e-05  1.626174042   1.039126e-01
[79,] 0.031447120 0.04597701 1.877464e-04 -1.060416797   2.889550e-01
[80,] 0.064005294 0.05747126 2.359641e-04  0.425361422   6.705732e-01
[81,] 0.044606529 0.05747126 2.357330e-04 -0.837897118   4.020885e-01
[82,] 0.063700493 0.06896552 2.801427e-04 -0.314565243   7.530918e-01
[83,] 0.051142205 0.04597701 1.933560e-04  0.371456331   7.102977e-01
[84,] 0.102121112 0.04597701 1.610278e-04  4.424392623   9.671399e-06
[85,] 0.021901462 0.02298851 9.843172e-05 -0.109566928   9.127528e-01
[86,] 0.064931813 0.04597701 1.929430e-04  1.364597995   1.723794e-01
[87,] 0.031747344 0.04597701 1.909867e-04 -1.029658605   3.031703e-01
[88,] 0.015893319 0.02298851 9.765131e-05 -0.718000620   4.727569e-01
attr(,"cluster")
 [1] Low  Low  High High High High High High High Low  Low  High Low  Low  Low 
[16] High High High High Low  High High Low  Low  High Low  Low  Low  Low  Low 
[31] Low  Low  Low  High Low  Low  Low  Low  Low  Low  High Low  Low  Low  Low 
[46] High High Low  Low  Low  Low  High Low  Low  Low  Low  Low  High Low  Low 
[61] Low  Low  Low  High High High Low  High Low  Low  High Low  High High Low 
[76] High Low  Low  Low  Low  Low  Low  High High Low  High Low  Low 
Levels: Low High
attr(,"gstari")
[1] FALSE
attr(,"call")
localG(x = hunan$GDPPC, listw = wm62_lw)
attr(,"class")
[1] "localG"

The output of localG() is a vector of G or Gstar values, with attributes “gstari” set to TRUE or FALSE, “call” set to the function call, and class “localG”.

The Gi statistics is represented as a Z-score. Greater values represent a greater intensity of clustering and the direction (positive or negative) indicates high or low clusters.

Next, we will join the Gi values to their corresponding hunan sf data frame by using the code chunk below.

hunan.gi <- cbind(hunan, as.matrix(gi.fixed)) %>%
  rename(gstat_fixed = as.matrix.gi.fixed.)

hunan.gi
Simple feature collection with 88 features and 9 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 108.7831 ymin: 24.6342 xmax: 114.2544 ymax: 30.12812
Geodetic CRS:  WGS 84
First 10 features:
     NAME_2  ID_3    NAME_3   ENGTYPE_3    County GDPPC      Z.GDPPC lag_GDPPC
1   Changde 21098   Anxiang      County   Anxiang 23667 -0.049205949  24847.20
2   Changde 21100   Hanshou      County   Hanshou 20981 -0.228341158  22724.80
3   Changde 21101    Jinshi County City    Jinshi 34592  0.679406172  24143.25
4   Changde 21102        Li      County        Li 24473  0.004547952  27737.50
5   Changde 21103     Linli      County     Linli 25554  0.076642204  27270.25
6   Changde 21104    Shimen      County    Shimen 27137  0.182215933  21248.80
7  Changsha 21109   Liuyang County City   Liuyang 63118  2.581867439  43747.00
8  Changsha 21110 Ningxiang      County Ningxiang 62202  2.520777398  33582.71
9  Changsha 21111 Wangcheng      County Wangcheng 70666  3.085260051  45651.17
10 Chenzhou 21112     Anren      County     Anren 12761 -0.776550918  32027.62
   gstat_fixed                       geometry
1   0.43607584 POLYGON ((112.0625 29.75523...
2  -0.26550565 POLYGON ((112.2288 29.11684...
3  -0.07303367 POLYGON ((111.8927 29.6013,...
4   0.41301703 POLYGON ((111.3731 29.94649...
5   0.27307058 POLYGON ((111.6324 29.76288...
6  -0.37751078 POLYGON ((110.8825 30.11675...
7   2.86389882 POLYGON ((113.9905 28.5682,...
8   2.79435042 POLYGON ((112.7181 28.38299...
9   5.21612540 POLYGON ((112.7914 28.52688...
10  0.22823660 POLYGON ((113.1757 26.82734...

Code chunk above performs three tasks:

  1. as.matrix(): to convert the output vector (i.e. gi.fixed) into r matrix object by using .
  2. cbind(): to join hunan@data and gi.fixed matrix to produce a new SpatialPolygonDataFrame called hunan.gi.
  3. rename(): rename the field name of the gi values to gstat_fixed by using

The code chunk below shows the functions used to map the Gi values derived using fixed distance weight matrix.

gdppc <- qtm(hunan, "GDPPC")

Gimap <-tm_shape(hunan.gi) +
  tm_fill(col = "gstat_fixed", 
          style = "pretty",
          palette="-RdBu",
          title = "local Gi") +
  tm_borders(alpha = 0.5)

tmap_arrange(gdppc, Gimap, asp=1, ncol=2)

The code chunk below are used to compute the Gi values for GDPPC2012 by using an adaptive distance weight matrix (i.e knb_lw).

fips <- order(hunan$County)
gi.adaptive <- localG(hunan$GDPPC, knn_lw)
hunan.gi <- cbind(hunan, as.matrix(gi.adaptive)) %>%
  rename(gstat_adaptive = as.matrix.gi.adaptive.)

It is time for us to visualise the locations of hot spot and cold spot areas. The choropleth mapping functions of tmap package will be used to map the Gi values.

The code chunk below shows the functions used to map the Gi values derived using fixed distance weight matrix.

gdppc<- qtm(hunan, "GDPPC")

Gimap <- tm_shape(hunan.gi) + 
  tm_fill(col = "gstat_adaptive", 
          style = "pretty", 
          palette="-RdBu", 
          title = "local Gi") + 
  tm_borders(alpha = 0.5)

tmap_arrange(gdppc, 
             Gimap, 
             asp=1, 
             ncol=2)