pacman::p_load(sf, spdep, tmap, tidyverse)Exercise 2C: Local Measures of Spatial Autocorrelation
1. Getting Started
1.1 Install and launching R packages
The code chunk below uses p_load() of pacman package to check if sf, spdep, tmap, and tidyverse packages are installed into the R environment. If they are, then they will be launched into R.
1.2 Importing the OD data
We will be using two data sets for this exercise. Data were retrieved on 19th Nov 2023. They are :
- Hunan country boundary layer*. (data is in ESRI shapefile format) - Geospatial data
- Hunan_2012.csv*. (data is in csv file) - Attribute table
In this exercise, we are interested to examine the spatial pattern of GDPPC (a.k.a GPD per Capital) of Hunan Provice, People Republic of China.
1.2.1 Importing Hunan country boundary layer
The code chunk below uses st_read() of sf package to import the 1st data set 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
`/Users/smu/Rworkshop/jiawenoh/ISSS624/Hands-on_Ex/Hands-on_Ex02/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 841.2.2 Importing Hunan_2012.csv
Next, we will import the 2nd dataset (csv) into R. We will use read_csv() of readr package. The output is in R dataframe class.
hunan2012 <- read_csv("data/aspatial/Hunan_2012.csv")1.3 Performing relational join
After importing, we will update the attribute table of hunan’s Spatial Polygons Data Frame with the attribute fields of hunan2012 dataframe. We will performed a left_join() with the aid of dplyr package.
hunan <- left_join(hunan,hunan2012) %>%
select(1:4,7,15)We will be joining both tables by County. By doing the left_join, we will combined the 8 variables from hunan, with 29 variables from hunan2012 and uses select() to filter for the variables that we are interested in.
1.4 . Visualising Regional Development Indicator
After joining, we will do a quick visualization. We will be using the qtm() of tmap package to prepare a basemap and a choropleth map to see the distribution of GDPPC 2012.
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equal <- tm_shape(hunan) +
tm_fill("GDPPC",
n = 5,
style = "equal") +
tm_borders(alpha = 0.5) +
tm_layout(main.title = "Equal interval classification")
quantile <- tm_shape(hunan) +
tm_fill("GDPPC",
n = 5,
style = "quantile") +
tm_borders(alpha = 0.5) +
tm_layout(main.title = "Equal quantile classification")
tmap_arrange(equal,
quantile,
asp=1,
ncol=2)
2. Cluster and Outlier Analysis
In Exercise 2B, we covered the Global Indicators of Spatial Association.
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# incorporate global SA to do local
#computing contiguity spatial weights
wm_q <- poly2nb(hunan,
queen=TRUE)
#row-standardized weights matrix
rswm_q <- nb2listw(wm_q,
style="W",
zero.policy = TRUE)In this exercise, we will be covering the Local Indicators of Spatial Association or LISA to evaluate existence of clusters in the spatial arrangement of a given variable.
We will be applying local Moran’ I to detect cluster and/or outliers from GDP per capita 2012 of Hunan Province, PRC.
We will perform Local Moran’s I statistical testing using localmoran() of spdep. Given a set of Z.Li values, it computes Ii , and a listw object providding neighbor weighting information for the polygon associated with the Z.Ii values.
The code chunks below are used to compute local Moran’s I of GDPPC2012 at the county level.
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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
Note about
localmoran()
localmoran() returns a matrix of values whose columns are:
“Ii” : the local Moran’s I statistics
“E.Ii” : the expectation of local moran statistics under the randomisation hypothesis
“Var.Ii” : the variance of local moran statistic under the randomisation hypothesis
“Z.Ii” : the standard deviation of local moran statistic
“Pr()” : the p-value of local moran statistic
To see the content of the local Moran matrix, we can use printCoefmat() :
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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 0.4830Notably, we should append the local Moran’s I dataframe (i.e. localMI) onto hunan Spatial Polygon Data Frame before mapping the local Moran’s I map. To perform the tasks, we can use the code below:
Step 1: Append Local Moran’s I with SPDF
hunan.localMI <- cbind(hunan,localMI) %>%
rename(Pr.Ii = Pr.z....E.Ii..)Step 2: Map local Moran’s I values
Using choropleth mapping function of tmap package, we can plot the local Moran’s I values:
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tm_shape(hunan.localMI) +
tm_fill(col = "Ii",
style = "pretty",
palette = "RdBu",
title = "local moran statistics") +
tm_borders(alpha = 0.5)
Step 3: Map local Moran’s I p-values
As seen above, Ii contains both positive and negative values. Thus, it is useful for us to consider the p-values for each values.
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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)
Step 4 (Optional) : Map both local Moran’s I values and p-values
For effective interpretation, we can plot both I-values and corresponding p-values next to one another.
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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)
3. Creating LISA Cluster Map
We are able to observe significant locations color coded by type of spatial autocorrelation. To begin with, we will plot the Moran Scatterplot
Firstly, we will plot the Moran Scatterplot of GDPPC 2012 by using moran.plot() of spdep:
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nci <- moran.plot(hunan$GDPPC, rswm_q,
labels=as.character(hunan$County),
xlab="GDPPC 2012",
ylab="Spatially Lag GDPPC 2012")
From the result above, the plot is split into 4 quadrants. Quadrant 2 (top right) are the high-high locations. They belongs to area with high GDPPC and are surrounded by other areas with average level of GDPPC.
Next, we will use scale() to centers and scale the variable. Centering is done by subtracting the mean (omitting NAs) for the corresponding columns, and Scaling is done by dividing the (centered) varialbe by their standard deviations.
At the end, as.vector() is added to ensure data type is a vector.
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#add scale to the previous step
hunan$Z.GDPPC <- scale(hunan$GDPPC) %>%
as.vector
#plot moran scatterplot
nci2 <- moran.plot(hunan$Z.GDPPC, rswm_q,
labels=as.character(hunan$County),
xlab="z-GDPPC 2012",
ylab="Spatially Lag z-GDPPC 2012")
The code chunks below show the steps to prepare a LISA cluster map.
Step 1: Vectorize
quadrant <- vector(mode="numeric",length=nrow(localMI))Step 2: Derive Spatially lagged variable of interest and centers it around its mean
hunan$lag_GDPPC <- lag.listw(rswm_q, hunan$GDPPC)
DV <- hunan$lag_GDPPC - mean(hunan$lag_GDPPC) Step 3: Center the local Moran’s around the mean
LM_I <- localMI[,1] - mean(localMI[,1]) Step 4: Set statistical significance level for the local Moran
signif <- 0.05 Step 5: Define the quadrants
1 = low-low 2 = low - high 3 = high - low 4 = high - high
quadrant[DV <0 & LM_I>0] <- 1
quadrant[DV >0 & LM_I<0] <- 2
quadrant[DV <0 & LM_I<0] <- 3
quadrant[DV >0 & LM_I>0] <- 4 Step 6: Place non-significant Moran in Category 0
quadrant[localMI[,5]>signif] <- 0Final Output:
quadrant <- vector(mode="numeric",length=nrow(localMI))
hunan$lag_GDPPC <- lag.listw(rswm_q, hunan$GDPPC)
DV <- hunan$lag_GDPPC - mean(hunan$lag_GDPPC)
LM_I <- localMI[,1]
signif <- 0.05
quadrant[DV <0 & LM_I>0] <- 1
quadrant[DV >0 & LM_I<0] <- 2
quadrant[DV <0 & LM_I<0] <- 3
quadrant[DV >0 & LM_I>0] <- 4
quadrant[localMI[,5]>signif] <- 0With the preparation, we can now plot the LISA map by plotting both local Moran’s I values map and its corresponding p-values map next to one another.
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gdppc <- qtm(hunan, "GDPPC")
hunan.localMI$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)
#to plot I-value and P-value map
tmap_arrange(gdppc, LISAmap,
asp=1, ncol=2)
4. Hot Spot and Cold Spot Area Analysis
Apart from detecting cluster and outliers, localised spatial statistics can also be used to detect hot spot and/or cold spot areas. By definition, hot spot refers to region or value that is higher relative to its surroundings.
It consists of a three steps approach:
To begin, we need to define new set of neighbors. Instead of considering units through shared borders, Getis-Ord define neighbors based on distance.
There are two types of distance-based proximity matrix: - fixed distance weight matrix, and - adaptive distance weight matrix
1.1 Deriving the centriod
Before making our connectivity graph, we will need to associate each polygon. We will use a mapping function to get the longitude, latitude and thereafter, combine them together.
#get longitude
longitude <- map_dbl(hunan$geometry, ~st_centroid(.x)[[1]])
#get latitude
latitude <- map_dbl(hunan$geometry, ~st_centroid(.x)[[2]])
#combine
coords <- cbind(longitude, latitude)1.2 Determine the cut-off distance
We need to determine the upper list for distance band by :
- 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 usingknn2nb(). - 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().
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#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 As seen above, the max distance is 61.79km. We can infer that all units will have at least one neighbour.
1.3 Compute fixed distance weight matrix
We will compute the distance weight matrix by using dnearneigh():
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#compute distance weight matrix
wm_d62 <- dnearneigh(coords, 0, 62, longlat = TRUE)
wm_d62Neighbour 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.
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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 54401.4 Compute adaptive distance weight matrix
knn <- knn2nb(knearneigh(coords, k=8))
knnNeighbour list object:
Number of regions: 88
Number of nonzero links: 704
Percentage nonzero weights: 9.090909
Average number of links: 8
Non-symmetric neighbours listNext, nb2listw() is used to convert the nb object into spatial weights object.
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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 23014Gi Statistics using fixed distance
Notably, the output of localG() is a vector of G or Gstar values, with attributes gstari set to TRUE or FALSE. The Gi stats represents the z-score where greater values represent a greater intensity of clustering and the direction (positive or negative) indicates high or low clusters.
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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$GDPPC, listw = wm62_lw)
attr(,"class")
[1] "localG"Next, we will join the Gi values to their corresponding hunan sf data frame by using the code chunk below by converting the output vector (gi.fixed) into r matrix object by using as.matrix(). Then, cbind() to join hunan data and gi.fixed matrix to produce a new Spatial Polygon Data Frame called hunan.gi.
Lastly, it will be rename to gstat_fixed by using rename().
hunan.gi <- cbind(hunan, as.matrix(gi.fixed)) %>%
rename(gstat_fixed = as.matrix.gi.fixed.)fixed distance weights
Show the code
gdppc <- qtm(hunan, "GDPPC")
Gimap <-tm_shape(hunan.gi) +
tm_fill(col = "gstat_fixed",
style = "pretty",
palette="-RdBu",
title = "local Gi") +
tm_borders(alpha = 0.5)
tmap_arrange(gdppc, Gimap, asp=1, ncol=2)
adaptive distance weights
Show the code
# compute the Gi value
fips <- order(hunan$County)
gi.adaptive <- localG(hunan$GDPPC, knn_lw)
hunan.gi <- cbind(hunan, as.matrix(gi.adaptive)) %>%
rename(gstat_adaptive = as.matrix.gi.adaptive.)
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)