2a: Spatial Weights and Applications

Author

Magdalene Chan

Published

November 18, 2023

Modified

November 25, 2023

Spatial weights (Wij) are a way to define spatial neighbourhoods.

Getting Started

The code chunk below uses p_load() of pacman package to check if the required packages have been installed on the computer. If they are, the packages will be launched.

  • sf package is used for importing, managing, and processing geospatial data.
  • tmap package is used for thematic mapping.
  • spdep package is used to create spatial weights matrix objects.
pacman::p_load(sf, spdep, tmap, tidyverse, knitr)

The data sets used are:

  • Hunan county boundary layer: a geospatial data set in ESRI shapefile format.
  • Hunan_2012.csv: csv file that contains selected Hunan’s local development indicators in 2012.

Importing Data

Import shapefile into R

The code chunk below uses the st_read() function of sf package to import Hunan county boundary shapefile into R as a simple feature data frame called hunan.

hunan <- st_read(dsn = "data/geospatial", 
                 layer = "Hunan")
Reading layer `Hunan' from data source 
  `C:\magdalenecjw\ISSS624 Geospatial\Hands_on_Exercise\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

There are a total of 88 polygon features and 7 fields in hunan simple feature data frame. hunan is in wgs84 GCS.

Import aspatial data into R

The code chunk below uses the read_csv() function of readr package to import Hunan_2012.csv file into R and save it as a R dataframe called hunan2012.

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

Performing relational join

left_join() of dplyr is used to join the geographical data and attribute table using County as the common identifier.

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

Visualising Regional Development Indicator

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=0.5)

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

The choropleth map generated using qtm() is based on equal intervals. All except nine regions have a GDPPC of either “0 to 20,000” or “20,000 to 40,000” – the distribution of GDPPC appears to be right-skewed.

Computing Contiguity Spatial Weights

Contiguity means that two spatial units share a common border of non-zero length. This can be further divided into rook or queen criterion of contiguity, in analogy to the moves allowed for the such-named pieces on a chess board.

The rook criterion defines neighbors by the existence of a common edge between two spatial units, while the queen criterion defines neighbors as spatial units sharing a common edge or a common vertex.

Hence, the number of neighbors according to the queen criterion will always be larger than or equal to the rook criterion.

Compute contiguity based neighbours based on queen criterion

poly2nb() of spdep package computes contiguity weights matrices for the study area by building a neighbours list based on regions with contiguous boundaries. The code chunk below computes the queen contiguity weights matrix.

In poly2nb(), the queen argument takes TRUE (default) or FALSE as options.

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

Based on the summary report above, there are 88 area units in Hunan. The most connected area unit has 11 neighbours while there are two area units with only one neighbour each.

For each polygon in the polygon object, wm_q lists all neighboring polygons. The code chunk below can be used to see the neighbors for the individual polygons in the object.

1wm_q[[1]]
2hunan$County[1]
3hunan$NAME_3[c(2,3,4,57,85)]
4hunan$GDPPC[wm_q[[1]]]
1
Show the neighbors for the first polygon. Each number shown represents one polygon ID stored in hunan SpatialPolygonsDataFrame class.
2
Retrieve the county name of Polygon ID=1.
3
Retrieve the county names of the five neighboring polygons.
4
Retrieve the GDPPC of the five neighboring counties. 
[1]  2  3  4 57 85
[1] "Anxiang"
[1] "Hanshou" "Jinshi"  "Li"      "Nan"     "Taoyuan"
[1] 20981 34592 24473 21311 22879

The complete weights matrix can be displayed by using str().

str(wm_q)
List of 88
 $ : int [1:5] 2 3 4 57 85
 $ : int [1:5] 1 57 58 78 85
 $ : int [1:4] 1 4 5 85
 $ : int [1:4] 1 3 5 6
 $ : int [1:4] 3 4 6 85
 $ : int [1:5] 4 5 69 75 85
 $ : int [1:4] 67 71 74 84
 $ : int [1:7] 9 46 47 56 78 80 86
 $ : int [1:6] 8 66 68 78 84 86
 $ : int [1:8] 16 17 19 20 22 70 72 73
 $ : int [1:3] 14 17 72
 $ : int [1:5] 13 60 61 63 83
 $ : int [1:4] 12 15 60 83
 $ : int [1:3] 11 15 17
 $ : int [1:4] 13 14 17 83
 $ : int [1:5] 10 17 22 72 83
 $ : int [1:7] 10 11 14 15 16 72 83
 $ : int [1:5] 20 22 23 77 83
 $ : int [1:6] 10 20 21 73 74 86
 $ : int [1:7] 10 18 19 21 22 23 82
 $ : int [1:5] 19 20 35 82 86
 $ : int [1:5] 10 16 18 20 83
 $ : int [1:7] 18 20 38 41 77 79 82
 $ : int [1:5] 25 28 31 32 54
 $ : int [1:5] 24 28 31 33 81
 $ : int [1:4] 27 33 42 81
 $ : int [1:3] 26 29 42
 $ : int [1:5] 24 25 33 49 54
 $ : int [1:3] 27 37 42
 $ : int 33
 $ : int [1:8] 24 25 32 36 39 40 56 81
 $ : int [1:8] 24 31 50 54 55 56 75 85
 $ : int [1:5] 25 26 28 30 81
 $ : int [1:3] 36 45 80
 $ : int [1:6] 21 41 47 80 82 86
 $ : int [1:6] 31 34 40 45 56 80
 $ : int [1:4] 29 42 43 44
 $ : int [1:4] 23 44 77 79
 $ : int [1:5] 31 40 42 43 81
 $ : int [1:6] 31 36 39 43 45 79
 $ : int [1:6] 23 35 45 79 80 82
 $ : int [1:7] 26 27 29 37 39 43 81
 $ : int [1:6] 37 39 40 42 44 79
 $ : int [1:4] 37 38 43 79
 $ : int [1:6] 34 36 40 41 79 80
 $ : int [1:3] 8 47 86
 $ : int [1:5] 8 35 46 80 86
 $ : int [1:5] 50 51 52 53 55
 $ : int [1:4] 28 51 52 54
 $ : int [1:5] 32 48 52 54 55
 $ : int [1:3] 48 49 52
 $ : int [1:5] 48 49 50 51 54
 $ : int [1:3] 48 55 75
 $ : int [1:6] 24 28 32 49 50 52
 $ : int [1:5] 32 48 50 53 75
 $ : int [1:7] 8 31 32 36 78 80 85
 $ : int [1:6] 1 2 58 64 76 85
 $ : int [1:5] 2 57 68 76 78
 $ : int [1:4] 60 61 87 88
 $ : int [1:4] 12 13 59 61
 $ : int [1:7] 12 59 60 62 63 77 87
 $ : int [1:3] 61 77 87
 $ : int [1:4] 12 61 77 83
 $ : int [1:2] 57 76
 $ : int 76
 $ : int [1:5] 9 67 68 76 84
 $ : int [1:4] 7 66 76 84
 $ : int [1:5] 9 58 66 76 78
 $ : int [1:3] 6 75 85
 $ : int [1:3] 10 72 73
 $ : int [1:3] 7 73 74
 $ : int [1:5] 10 11 16 17 70
 $ : int [1:5] 10 19 70 71 74
 $ : int [1:6] 7 19 71 73 84 86
 $ : int [1:6] 6 32 53 55 69 85
 $ : int [1:7] 57 58 64 65 66 67 68
 $ : int [1:7] 18 23 38 61 62 63 83
 $ : int [1:7] 2 8 9 56 58 68 85
 $ : int [1:7] 23 38 40 41 43 44 45
 $ : int [1:8] 8 34 35 36 41 45 47 56
 $ : int [1:6] 25 26 31 33 39 42
 $ : int [1:5] 20 21 23 35 41
 $ : int [1:9] 12 13 15 16 17 18 22 63 77
 $ : int [1:6] 7 9 66 67 74 86
 $ : int [1:11] 1 2 3 5 6 32 56 57 69 75 ...
 $ : int [1:9] 8 9 19 21 35 46 47 74 84
 $ : int [1:4] 59 61 62 88
 $ : int [1:2] 59 87
 - attr(*, "class")= chr "nb"
 - attr(*, "region.id")= chr [1:88] "1" "2" "3" "4" ...
 - attr(*, "call")= language poly2nb(pl = hunan, queen = TRUE)
 - attr(*, "type")= chr "queen"
 - attr(*, "sym")= logi TRUE
Warning

The output may cut across several pages. To print out the report, it is advised to save the trees.

Compute contiguity based neighbours based on rook criterion

The code chunk below computes the rook contiguity weights matrix.

wm_r <- poly2nb(hunan, queen=FALSE)
summary(wm_r)
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 440 
Percentage nonzero weights: 5.681818 
Average number of links: 5 
Link number distribution:

 1  2  3  4  5  6  7  8  9 10 
 2  2 12 20 21 14 11  3  2  1 
2 least connected regions:
30 65 with 1 link
1 most connected region:
85 with 10 links

Based on the summary report above, there are 88 area units in Hunan. The most connected area unit has 10 neighbours while there are two area units with only one neighbour each.

Visualising contiguity weights

Connectivity graphs can be used to visualise contiguity weights. It takes a point and displays a line to each neighboring point. However, as the hunan simple feature data frame contains polygon geometry, points are needed in order to create the connectivity graphs. The most typical method to do so is to use polygon centroids, which can be calculated using the sf packages.

Getting Latitude and Longitude of Polygon Centroids

st_centroid() can be used to obtain the polygon centroids. However, the coordinates need to be in a separate data frame in order to create the connectivity graphs. This can be done using a mapping function (map() from the purrr package), which applies a given function to each element of a vector and returns a vector of the same length.

As st_centroid() returns a dbl data type, map_dbl() variation of the map() function from the purrr package should be used. Other variations of this function include map_lgl() for logical vectors, map_int() for integer vectors and map_chr() for string vectors.

To compute the longitude values, map st_centroid() over the geometry column of hunan and access the longitude value through double bracket notation [[]] and 1. This returns only the longitude, which is the first value in each centroid.

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

To compute the latitude values, map st_centroid() over the geometry column of hunan and access the latitude value through double bracket notation [[]] and 2. This returns only the latitude, which is the second value in each centroid.

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

cbind() can be used to put longitude and latitude into the same object.

coords <- cbind(longitude, latitude)

Using head(), check the first few observations to see if things are formatted correctly.

head(coords)
     longitude latitude
[1,]  112.1531 29.44362
[2,]  112.0372 28.86489
[3,]  111.8917 29.47107
[4,]  111.7031 29.74499
[5,]  111.6138 29.49258
[6,]  111.0341 29.79863

Once the coordinates of the polygon centroids are obtained, connectivity graphs can be plotted to visualise contiguity weights.

plot(hunan$geometry, border="lightgrey")
plot(wm_q, coords, pch = 19, cex = 0.6, add = TRUE, col= "red")

plot(hunan$geometry, border="lightgrey")
plot(wm_r, coords, pch = 19, cex = 0.6, add = TRUE, col = "red")

par(mfrow=c(1,2))
plot(hunan$geometry, border="lightgrey")
plot(wm_q, coords, pch = 19, cex = 0.6, add = TRUE, col= "red", main="Queen Contiguity")
plot(hunan$geometry, border="lightgrey")
plot(wm_r, coords, pch = 19, cex = 0.6, add = TRUE, col = "red", main="Rook Contiguity")

Computing distance based neighbours

Distance-based weights matrices can be derived using dnearneigh() of spdep package. This function identifies neighbours of region points by Euclidean distance with a distance band defined by lower (d1) and upper (d2) bounds, controlled by the bounds argument. Regions falling within this distance range are considered neighbors.

If coordinates are unprojected (i.e. not transformed to a projected coordinate system) and in latitude and longitude format i.e. longlat=TRUE, the function calculates distances using the great circle distance formula i.e. the shortest distance between two points on the surface of a sphere, assuming the WGS84 reference ellipsoid.

Determine the cut-off distance

First, determine the upper limit for distance band using the steps below:

#coords <- coordinates(hunan)
1k1 <- knn2nb(knearneigh(coords))
2k1dists <- unlist(nbdists(k1, coords, longlat = TRUE))
summary(k1dists)
1
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 returned knn object into a neighbours list of class nb with a list of integer vectors containing neighbour region number ids by using knn2nb().
2
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, and in km if otherwise. Remove the list structure of the returned object by using unlist(). 
   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 67.50 km, so using this as the upper threshold (rounded up to the next integer) gives certainty that all units will have at least one neighbour.

Computing fixed distance weights matrix

Next, compute the distance weights matrix using dnearneigh() as shown in the code chunk below.

wm_d62 <- dnearneigh(coords, 0, 68, longlat = TRUE)
wm_d62
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 416 
Percentage nonzero weights: 5.371901 
Average number of links: 4.727273

The summary report above shows that the average number of links is 5.14. This means that on average, each point in the hunan dataset has approximately 5.14 neighboring points within the specified distance range.

Next, either of the following two methods can be used to display the content of wm_d62 weights matrix.

str(wm_d62)
List of 88
 $ : int [1:7] 2 3 4 5 57 58 64
 $ : int [1:5] 1 57 58 78 85
 $ : int [1:4] 1 4 5 57
 $ : int [1:4] 1 3 5 6
 $ : int [1:6] 1 3 4 6 69 85
 $ : int [1:3] 4 5 69
 $ : int [1:3] 67 71 84
 $ : int [1:6] 9 46 47 78 80 86
 $ : int [1:7] 8 46 66 68 78 84 86
 $ : int [1:6] 16 19 22 70 72 73
 $ : int [1:3] 14 17 72
 $ : int [1:6] 13 15 60 61 63 83
 $ : int [1:5] 12 15 60 61 83
 $ : int [1:2] 11 17
 $ : int [1:3] 12 13 83
 $ : int [1:5] 10 17 22 72 83
 $ : int [1:4] 11 14 16 72
 $ : int [1:5] 20 22 23 63 77
 $ : int [1:6] 10 20 21 73 74 82
 $ : int [1:5] 18 19 21 22 82
 $ : int [1:7] 19 20 35 47 74 82 86
 $ : int [1:5] 10 16 18 20 83
 $ : int [1:5] 18 41 77 79 82
 $ : int [1:5] 25 28 31 52 54
 $ : int [1:5] 24 28 31 33 81
 $ : int [1:4] 27 33 42 81
 $ : int [1:3] 26 29 42
 $ : int [1:6] 24 25 33 49 52 54
 $ : int [1:2] 27 37
 $ : int 33
 $ : int [1:4] 24 25 36 40
 $ : int 50
 $ : int [1:5] 25 26 28 30 81
 $ : int [1:6] 36 40 41 45 56 80
 $ : int [1:7] 21 41 46 47 80 82 86
 $ : int [1:6] 31 34 40 45 56 80
 $ : int [1:4] 29 42 43 44
 $ : int [1:4] 43 44 77 79
 $ : int [1:4] 40 42 43 81
 $ : int [1:7] 31 34 36 39 43 45 79
 $ : int [1:7] 23 34 35 45 79 80 82
 $ : int [1:6] 26 27 37 39 43 81
 $ : int [1:7] 37 38 39 40 42 44 79
 $ : int [1:4] 37 38 43 79
 $ : int [1:6] 34 36 40 41 79 80
 $ : int [1:5] 8 9 35 47 86
 $ : int [1:6] 8 21 35 46 80 86
 $ : int [1:5] 50 51 52 53 55
 $ : int [1:4] 28 51 52 54
 $ : int [1:6] 32 48 51 52 54 55
 $ : int [1:5] 48 49 50 52 54
 $ : int [1:7] 24 28 48 49 50 51 54
 $ : int [1:2] 48 55
 $ : int [1:6] 24 28 49 50 51 52
 $ : int [1:4] 48 50 53 75
 $ : int [1:2] 34 36
 $ : int [1:6] 1 2 3 58 64 68
 $ : int [1:7] 1 2 57 64 66 68 78
 $ : int [1:3] 60 87 88
 $ : int [1:6] 12 13 59 61 63 87
 $ : int [1:6] 12 13 60 62 63 87
 $ : int [1:4] 61 63 77 87
 $ : int [1:6] 12 18 60 61 62 83
 $ : int [1:4] 1 57 58 76
 $ : int 76
 $ : int [1:6] 9 58 67 68 76 84
 $ : int [1:2] 7 66
 $ : int [1:6] 9 57 58 66 78 84
 $ : int [1:3] 5 6 75
 $ : int [1:3] 10 72 73
 $ : int [1:4] 7 73 74 86
 $ : int [1:5] 10 11 16 17 70
 $ : int [1:5] 10 19 70 71 74
 $ : int [1:5] 19 21 71 73 86
 $ : int [1:2] 55 69
 $ : int [1:3] 64 65 66
 $ : int [1:4] 18 23 38 62
 $ : int [1:5] 2 8 9 58 68
 $ : int [1:7] 23 38 40 41 43 44 45
 $ : int [1:7] 8 34 35 36 41 45 47
 $ : int [1:5] 25 26 33 39 42
 $ : int [1:6] 19 20 21 23 35 41
 $ : int [1:6] 12 13 15 16 22 63
 $ : int [1:4] 7 9 66 68
 $ : int [1:2] 2 5
 $ : int [1:8] 8 9 21 35 46 47 71 74
 $ : int [1:5] 59 60 61 62 88
 $ : int [1:2] 59 87
 - attr(*, "class")= chr "nb"
 - attr(*, "region.id")= chr [1:88] "1" "2" "3" "4" ...
 - attr(*, "call")= language dnearneigh(x = coords, d1 = 0, d2 = 68, longlat = TRUE)
 - attr(*, "dnn")= num [1:2] 0 68
 - attr(*, "bounds")= chr [1:2] "GE" "LE"
 - attr(*, "nbtype")= chr "distance"
 - attr(*, "sym")= logi TRUE
table(hunan$County, card(wm_d62))
               
                1 2 3 4 5 6 7 8
  Anhua         0 1 0 0 0 0 0 0
  Anren         0 0 0 0 0 1 0 0
  Anxiang       0 0 0 0 0 0 1 0
  Baojing       0 0 0 0 1 0 0 0
  Chaling       0 0 1 0 0 0 0 0
  Changning     0 0 0 0 1 0 0 0
  Changsha      0 0 0 1 0 0 0 0
  Chengbu       0 0 0 1 0 0 0 0
  Chenxi        0 0 0 0 1 0 0 0
  Cili          0 0 1 0 0 0 0 0
  Dao           0 0 0 0 1 0 0 0
  Dongan        0 0 0 1 0 0 0 0
  Dongkou       0 0 0 1 0 0 0 0
  Fenghuang     0 0 0 1 0 0 0 0
  Guidong       0 0 1 0 0 0 0 0
  Guiyang       0 0 0 0 0 1 0 0
  Guzhang       0 0 0 0 0 1 0 0
  Hanshou       0 0 0 0 1 0 0 0
  Hengdong      0 0 0 0 0 1 0 0
  Hengnan       0 0 0 0 1 0 0 0
  Hengshan      0 0 0 0 0 0 1 0
  Hengyang      0 0 0 0 0 1 0 0
  Hongjiang     0 0 0 0 1 0 0 0
  Huarong       0 0 0 1 0 0 0 0
  Huayuan       0 0 0 0 1 0 0 0
  Huitong       0 0 0 1 0 0 0 0
  Jiahe         0 0 0 0 0 1 0 0
  Jianghua      0 0 1 0 0 0 0 0
  Jiangyong     0 1 0 0 0 0 0 0
  Jingzhou      0 0 1 0 0 0 0 0
  Jinshi        0 0 0 1 0 0 0 0
  Jishou        0 0 0 0 0 0 1 0
  Lanshan       0 0 0 0 0 1 0 0
  Leiyang       0 0 0 0 1 0 0 0
  Lengshuijiang 0 0 0 0 0 1 0 0
  Li            0 0 0 1 0 0 0 0
  Lianyuan      0 0 0 0 0 0 1 0
  Liling        0 0 0 1 0 0 0 0
  Linli         0 0 0 0 0 1 0 0
  Linwu         0 0 0 0 1 0 0 0
  Linxiang      1 0 0 0 0 0 0 0
  Liuyang       0 0 1 0 0 0 0 0
  Longhui       0 0 0 0 0 0 1 0
  Longshan      0 1 0 0 0 0 0 0
  Luxi          0 0 0 0 0 1 0 0
  Mayang        0 0 0 0 0 1 0 0
  Miluo         0 0 0 0 0 1 0 0
  Nan           0 0 0 0 0 1 0 0
  Ningxiang     0 0 0 0 0 1 0 0
  Ningyuan      0 0 0 0 0 1 0 0
  Pingjiang     0 1 0 0 0 0 0 0
  Qidong        0 0 0 0 1 0 0 0
  Qiyang        0 0 0 1 0 0 0 0
  Rucheng       0 1 0 0 0 0 0 0
  Sangzhi       0 1 0 0 0 0 0 0
  Shaodong      0 0 0 0 0 0 1 0
  Shaoshan      0 0 0 0 1 0 0 0
  Shaoyang      0 0 0 0 0 0 1 0
  Shimen        0 0 1 0 0 0 0 0
  Shuangfeng    0 0 0 0 0 0 1 0
  Shuangpai     0 0 0 1 0 0 0 0
  Suining       0 0 0 0 0 1 0 0
  Taojiang      0 0 0 0 1 0 0 0
  Taoyuan       0 1 0 0 0 0 0 0
  Tongdao       0 1 0 0 0 0 0 0
  Wangcheng     0 0 0 0 0 0 1 0
  Wugang        0 0 0 0 0 0 1 0
  Xiangtan      0 0 0 0 0 0 0 1
  Xiangxiang    0 0 0 0 0 1 0 0
  Xiangyin      0 0 0 0 0 1 0 0
  Xinhua        0 0 0 0 0 1 0 0
  Xinhuang      1 0 0 0 0 0 0 0
  Xinning       0 0 0 1 0 0 0 0
  Xinshao       0 0 0 0 0 1 0 0
  Xintian       0 0 0 0 0 1 0 0
  Xupu          0 0 0 1 0 0 0 0
  Yanling       0 0 0 0 1 0 0 0
  Yizhang       0 0 1 0 0 0 0 0
  Yongshun      0 0 0 1 0 0 0 0
  Yongxing      0 0 0 0 1 0 0 0
  You           0 0 0 0 1 0 0 0
  Yuanjiang     0 0 0 0 0 0 1 0
  Yuanling      1 0 0 0 0 0 0 0
  Yueyang       0 0 1 0 0 0 0 0
  Zhijiang      0 0 0 0 1 0 0 0
  Zhongfang     0 0 0 0 1 0 0 0
  Zhuzhou       0 0 0 0 1 0 0 0
  Zixing        0 0 0 1 0 0 0 0
n_comp <- n.comp.nb(wm_d62)
table(n_comp$comp.id)

 1 
88

The fixed distance weights matrix can then be plotted using the code chunk below.

plot(hunan$geometry, border="lightgrey")
plot(wm_d62, coords, add=TRUE)
plot(k1, coords, add=TRUE, col="red", length=0.08)

The red lines show the links of 1st nearest neighbours and the black lines show the links of neighbours within the cut-off distance of 62km. Alternatively, the red and the black lines could be plotted in two separate graphs next to each other.

par(mfrow=c(1,2))
plot(hunan$geometry, border="lightgrey", main="1st nearest neighbours")
plot(k1, coords, add=TRUE, col="red", length=0.08)
plot(hunan$geometry, border="lightgrey", main="Distance link")
plot(wm_d62, coords, add=TRUE, pch = 19, cex = 0.6)

Computing adaptive distance weights matrix

A characteristic of fixed distance weights 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.

As compared to just determining the cut-off distance when computing the fixed distance weights matrix, an additional argument k is specified for adaptive distance weights matrix.

knn6 <- knn2nb(knearneigh(coords, k=6))
knn6
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 528 
Percentage nonzero weights: 6.818182 
Average number of links: 6 
Non-symmetric neighbours list

Similarly, display the content of the matrix using str().

str(knn6)
List of 88
 $ : int [1:6] 2 3 4 5 57 64
 $ : int [1:6] 1 3 57 58 78 85
 $ : int [1:6] 1 2 4 5 57 85
 $ : int [1:6] 1 3 5 6 69 85
 $ : int [1:6] 1 3 4 6 69 85
 $ : int [1:6] 3 4 5 69 75 85
 $ : int [1:6] 9 66 67 71 74 84
 $ : int [1:6] 9 46 47 78 80 86
 $ : int [1:6] 8 46 66 68 84 86
 $ : int [1:6] 16 19 22 70 72 73
 $ : int [1:6] 10 14 16 17 70 72
 $ : int [1:6] 13 15 60 61 63 83
 $ : int [1:6] 12 15 60 61 63 83
 $ : int [1:6] 11 15 16 17 72 83
 $ : int [1:6] 12 13 14 17 60 83
 $ : int [1:6] 10 11 17 22 72 83
 $ : int [1:6] 10 11 14 16 72 83
 $ : int [1:6] 20 22 23 63 77 83
 $ : int [1:6] 10 20 21 73 74 82
 $ : int [1:6] 18 19 21 22 23 82
 $ : int [1:6] 19 20 35 74 82 86
 $ : int [1:6] 10 16 18 19 20 83
 $ : int [1:6] 18 20 41 77 79 82
 $ : int [1:6] 25 28 31 52 54 81
 $ : int [1:6] 24 28 31 33 54 81
 $ : int [1:6] 25 27 29 33 42 81
 $ : int [1:6] 26 29 30 37 42 81
 $ : int [1:6] 24 25 33 49 52 54
 $ : int [1:6] 26 27 37 42 43 81
 $ : int [1:6] 26 27 28 33 49 81
 $ : int [1:6] 24 25 36 39 40 54
 $ : int [1:6] 24 31 50 54 55 56
 $ : int [1:6] 25 26 28 30 49 81
 $ : int [1:6] 36 40 41 45 56 80
 $ : int [1:6] 21 41 46 47 80 82
 $ : int [1:6] 31 34 40 45 56 80
 $ : int [1:6] 26 27 29 42 43 44
 $ : int [1:6] 23 43 44 62 77 79
 $ : int [1:6] 25 40 42 43 44 81
 $ : int [1:6] 31 36 39 43 45 79
 $ : int [1:6] 23 35 45 79 80 82
 $ : int [1:6] 26 27 37 39 43 81
 $ : int [1:6] 37 39 40 42 44 79
 $ : int [1:6] 37 38 39 42 43 79
 $ : int [1:6] 34 36 40 41 79 80
 $ : int [1:6] 8 9 35 47 78 86
 $ : int [1:6] 8 21 35 46 80 86
 $ : int [1:6] 49 50 51 52 53 55
 $ : int [1:6] 28 33 48 51 52 54
 $ : int [1:6] 32 48 51 52 54 55
 $ : int [1:6] 28 48 49 50 52 54
 $ : int [1:6] 28 48 49 50 51 54
 $ : int [1:6] 48 50 51 52 55 75
 $ : int [1:6] 24 28 49 50 51 52
 $ : int [1:6] 32 48 50 52 53 75
 $ : int [1:6] 32 34 36 78 80 85
 $ : int [1:6] 1 2 3 58 64 68
 $ : int [1:6] 2 57 64 66 68 78
 $ : int [1:6] 12 13 60 61 87 88
 $ : int [1:6] 12 13 59 61 63 87
 $ : int [1:6] 12 13 60 62 63 87
 $ : int [1:6] 12 38 61 63 77 87
 $ : int [1:6] 12 18 60 61 62 83
 $ : int [1:6] 1 3 57 58 68 76
 $ : int [1:6] 58 64 66 67 68 76
 $ : int [1:6] 9 58 67 68 76 84
 $ : int [1:6] 7 65 66 68 76 84
 $ : int [1:6] 9 57 58 66 78 84
 $ : int [1:6] 4 5 6 32 75 85
 $ : int [1:6] 10 16 19 22 72 73
 $ : int [1:6] 7 19 73 74 84 86
 $ : int [1:6] 10 11 14 16 17 70
 $ : int [1:6] 10 19 21 70 71 74
 $ : int [1:6] 19 21 71 73 84 86
 $ : int [1:6] 6 32 50 53 55 69
 $ : int [1:6] 58 64 65 66 67 68
 $ : int [1:6] 18 23 38 61 62 63
 $ : int [1:6] 2 8 9 46 58 68
 $ : int [1:6] 38 40 41 43 44 45
 $ : int [1:6] 34 35 36 41 45 47
 $ : int [1:6] 25 26 28 33 39 42
 $ : int [1:6] 19 20 21 23 35 41
 $ : int [1:6] 12 13 15 16 22 63
 $ : int [1:6] 7 9 66 68 71 74
 $ : int [1:6] 2 3 4 5 56 69
 $ : int [1:6] 8 9 21 46 47 74
 $ : int [1:6] 59 60 61 62 63 88
 $ : int [1:6] 59 60 61 62 63 87
 - attr(*, "region.id")= chr [1:88] "1" "2" "3" "4" ...
 - attr(*, "call")= language knearneigh(x = coords, k = 6)
 - attr(*, "sym")= logi FALSE
 - attr(*, "type")= chr "knn"
 - attr(*, "knn-k")= num 6
 - attr(*, "class")= chr "nb"

Based on the output above, each county has exactly six neighbours.

The weights matrix is then plotted using the code chunk below.

plot(hunan$geometry, border="lightgrey")
plot(knn6, coords, pch = 19, cex = 0.6, add = TRUE, col = "red")

Computing weights based on IDW (Inversed Distance Method)

First, compute the distances between areas by using nbdists() of spdep.

dist <- nbdists(wm_q, coords, longlat = TRUE)
ids <- lapply(dist, function(x) 1/(x))
ids
[[1]]
[1] 0.01535405 0.03916350 0.01820896 0.02807922 0.01145113

[[2]]
[1] 0.01535405 0.01764308 0.01925924 0.02323898 0.01719350

[[3]]
[1] 0.03916350 0.02822040 0.03695795 0.01395765

[[4]]
[1] 0.01820896 0.02822040 0.03414741 0.01539065

[[5]]
[1] 0.03695795 0.03414741 0.01524598 0.01618354

[[6]]
[1] 0.015390649 0.015245977 0.021748129 0.011883901 0.009810297

[[7]]
[1] 0.01708612 0.01473997 0.01150924 0.01872915

[[8]]
[1] 0.02022144 0.03453056 0.02529256 0.01036340 0.02284457 0.01500600 0.01515314

[[9]]
[1] 0.02022144 0.01574888 0.02109502 0.01508028 0.02902705 0.01502980

[[10]]
[1] 0.02281552 0.01387777 0.01538326 0.01346650 0.02100510 0.02631658 0.01874863
[8] 0.01500046

[[11]]
[1] 0.01882869 0.02243492 0.02247473

[[12]]
[1] 0.02779227 0.02419652 0.02333385 0.02986130 0.02335429

[[13]]
[1] 0.02779227 0.02650020 0.02670323 0.01714243

[[14]]
[1] 0.01882869 0.01233868 0.02098555

[[15]]
[1] 0.02650020 0.01233868 0.01096284 0.01562226

[[16]]
[1] 0.02281552 0.02466962 0.02765018 0.01476814 0.01671430

[[17]]
[1] 0.01387777 0.02243492 0.02098555 0.01096284 0.02466962 0.01593341 0.01437996

[[18]]
[1] 0.02039779 0.02032767 0.01481665 0.01473691 0.01459380

[[19]]
[1] 0.01538326 0.01926323 0.02668415 0.02140253 0.01613589 0.01412874

[[20]]
[1] 0.01346650 0.02039779 0.01926323 0.01723025 0.02153130 0.01469240 0.02327034

[[21]]
[1] 0.02668415 0.01723025 0.01766299 0.02644986 0.02163800

[[22]]
[1] 0.02100510 0.02765018 0.02032767 0.02153130 0.01489296

[[23]]
[1] 0.01481665 0.01469240 0.01401432 0.02246233 0.01880425 0.01530458 0.01849605

[[24]]
[1] 0.02354598 0.01837201 0.02607264 0.01220154 0.02514180

[[25]]
[1] 0.02354598 0.02188032 0.01577283 0.01949232 0.02947957

[[26]]
[1] 0.02155798 0.01745522 0.02212108 0.02220532

[[27]]
[1] 0.02155798 0.02490625 0.01562326

[[28]]
[1] 0.01837201 0.02188032 0.02229549 0.03076171 0.02039506

[[29]]
[1] 0.02490625 0.01686587 0.01395022

[[30]]
[1] 0.02090587

[[31]]
[1] 0.02607264 0.01577283 0.01219005 0.01724850 0.01229012 0.01609781 0.01139438
[8] 0.01150130

[[32]]
[1] 0.01220154 0.01219005 0.01712515 0.01340413 0.01280928 0.01198216 0.01053374
[8] 0.01065655

[[33]]
[1] 0.01949232 0.01745522 0.02229549 0.02090587 0.01979045

[[34]]
[1] 0.03113041 0.03589551 0.02882915

[[35]]
[1] 0.01766299 0.02185795 0.02616766 0.02111721 0.02108253 0.01509020

[[36]]
[1] 0.01724850 0.03113041 0.01571707 0.01860991 0.02073549 0.01680129

[[37]]
[1] 0.01686587 0.02234793 0.01510990 0.01550676

[[38]]
[1] 0.01401432 0.02407426 0.02276151 0.01719415

[[39]]
[1] 0.01229012 0.02172543 0.01711924 0.02629732 0.01896385

[[40]]
[1] 0.01609781 0.01571707 0.02172543 0.01506473 0.01987922 0.01894207

[[41]]
[1] 0.02246233 0.02185795 0.02205991 0.01912542 0.01601083 0.01742892

[[42]]
[1] 0.02212108 0.01562326 0.01395022 0.02234793 0.01711924 0.01836831 0.01683518

[[43]]
[1] 0.01510990 0.02629732 0.01506473 0.01836831 0.03112027 0.01530782

[[44]]
[1] 0.01550676 0.02407426 0.03112027 0.01486508

[[45]]
[1] 0.03589551 0.01860991 0.01987922 0.02205991 0.02107101 0.01982700

[[46]]
[1] 0.03453056 0.04033752 0.02689769

[[47]]
[1] 0.02529256 0.02616766 0.04033752 0.01949145 0.02181458

[[48]]
[1] 0.02313819 0.03370576 0.02289485 0.01630057 0.01818085

[[49]]
[1] 0.03076171 0.02138091 0.02394529 0.01990000

[[50]]
[1] 0.01712515 0.02313819 0.02551427 0.02051530 0.02187179

[[51]]
[1] 0.03370576 0.02138091 0.02873854

[[52]]
[1] 0.02289485 0.02394529 0.02551427 0.02873854 0.03516672

[[53]]
[1] 0.01630057 0.01979945 0.01253977

[[54]]
[1] 0.02514180 0.02039506 0.01340413 0.01990000 0.02051530 0.03516672

[[55]]
[1] 0.01280928 0.01818085 0.02187179 0.01979945 0.01882298

[[56]]
[1] 0.01036340 0.01139438 0.01198216 0.02073549 0.01214479 0.01362855 0.01341697

[[57]]
[1] 0.028079221 0.017643082 0.031423501 0.029114131 0.013520292 0.009903702

[[58]]
[1] 0.01925924 0.03142350 0.02722997 0.01434859 0.01567192

[[59]]
[1] 0.01696711 0.01265572 0.01667105 0.01785036

[[60]]
[1] 0.02419652 0.02670323 0.01696711 0.02343040

[[61]]
[1] 0.02333385 0.01265572 0.02343040 0.02514093 0.02790764 0.01219751 0.02362452

[[62]]
[1] 0.02514093 0.02002219 0.02110260

[[63]]
[1] 0.02986130 0.02790764 0.01407043 0.01805987

[[64]]
[1] 0.02911413 0.01689892

[[65]]
[1] 0.02471705

[[66]]
[1] 0.01574888 0.01726461 0.03068853 0.01954805 0.01810569

[[67]]
[1] 0.01708612 0.01726461 0.01349843 0.01361172

[[68]]
[1] 0.02109502 0.02722997 0.03068853 0.01406357 0.01546511

[[69]]
[1] 0.02174813 0.01645838 0.01419926

[[70]]
[1] 0.02631658 0.01963168 0.02278487

[[71]]
[1] 0.01473997 0.01838483 0.03197403

[[72]]
[1] 0.01874863 0.02247473 0.01476814 0.01593341 0.01963168

[[73]]
[1] 0.01500046 0.02140253 0.02278487 0.01838483 0.01652709

[[74]]
[1] 0.01150924 0.01613589 0.03197403 0.01652709 0.01342099 0.02864567

[[75]]
[1] 0.011883901 0.010533736 0.012539774 0.018822977 0.016458383 0.008217581

[[76]]
[1] 0.01352029 0.01434859 0.01689892 0.02471705 0.01954805 0.01349843 0.01406357

[[77]]
[1] 0.014736909 0.018804247 0.022761507 0.012197506 0.020022195 0.014070428
[7] 0.008440896

[[78]]
[1] 0.02323898 0.02284457 0.01508028 0.01214479 0.01567192 0.01546511 0.01140779

[[79]]
[1] 0.01530458 0.01719415 0.01894207 0.01912542 0.01530782 0.01486508 0.02107101

[[80]]
[1] 0.01500600 0.02882915 0.02111721 0.01680129 0.01601083 0.01982700 0.01949145
[8] 0.01362855

[[81]]
[1] 0.02947957 0.02220532 0.01150130 0.01979045 0.01896385 0.01683518

[[82]]
[1] 0.02327034 0.02644986 0.01849605 0.02108253 0.01742892

[[83]]
[1] 0.023354289 0.017142433 0.015622258 0.016714303 0.014379961 0.014593799
[7] 0.014892965 0.018059871 0.008440896

[[84]]
[1] 0.01872915 0.02902705 0.01810569 0.01361172 0.01342099 0.01297994

[[85]]
 [1] 0.011451133 0.017193502 0.013957649 0.016183544 0.009810297 0.010656545
 [7] 0.013416965 0.009903702 0.014199260 0.008217581 0.011407794

[[86]]
[1] 0.01515314 0.01502980 0.01412874 0.02163800 0.01509020 0.02689769 0.02181458
[8] 0.02864567 0.01297994

[[87]]
[1] 0.01667105 0.02362452 0.02110260 0.02058034

[[88]]
[1] 0.01785036 0.02058034

Row-standardised weights matrix

Next, assign equal weights to each neighboring polygon (style=“W”). This is accomplished by assigning the fraction 1/(#ofneighbors) to each neighboring county then summing the weighted income values. While this is the most intuitive way to summarise 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. Other more robust options are available to correct such drawbacks, notably style=“B”.

The style can take values “W”, “B”, “C”, “U”, “minmax” and “S”. B is the basic binary coding, W is row standardised (sums over all links to n), C is globally standardised (sums over all links to n), U is equal to C divided by the number of neighbours (sums over all links to unity), while S is the variance-stabilizing coding scheme proposed by Tiefelsdorf et al. 1999, p. 167-168 (sums over all links to n).

The zero.policy=TRUE option allows for lists of non-neighbors. When set, 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.

zero.policy=TRUE should be used with caution since users may not be aware of missing neighbors in their dataset. Using zero.policy=FALSE at first instance may be more advised as it returns an error if there are empty neighbour sets.

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

To see the weights of the first polygon’s eight neighbors type, use the following code chunk:

rswm_q$weights[10]
[[1]]
[1] 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125

Each neighbor is assigned a 0.125 of the total weights. This means that when R computes the average neighboring income values, each neighbor’s income will be multiplied by 0.125 before being tallied. A row standardised distance weights matrix can be derived using the same method in the code chunk below.

rswm_ids <- nb2listw(wm_q, glist=ids, style="B", zero.policy=TRUE)
rswm_ids
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: B 
Weights constants summary:
   n   nn       S0        S1     S2
B 88 7744 8.786867 0.3776535 3.8137
rswm_ids$weights[1]
[[1]]
[1] 0.01535405 0.03916350 0.01820896 0.02807922 0.01145113

Application of Spatial Weights Matrix

This section will focus on the creation of four different spatial lagged variables:

  • spatial lag with row-standardized weights,
  • spatial lag as a sum of neighbouring values,
  • spatial window average, and
  • spatial window sum.

Spatial lag with row-standardized weights

Compute the average neighbor GDPPC value for each polygon, which is also often referred to as spatially lagged values.

GDPPC.lag <- lag.listw(rswm_q, hunan$GDPPC)
GDPPC.lag
 [1] 24847.20 22724.80 24143.25 27737.50 27270.25 21248.80 43747.00 33582.71
 [9] 45651.17 32027.62 32671.00 20810.00 25711.50 30672.33 33457.75 31689.20
[17] 20269.00 23901.60 25126.17 21903.43 22718.60 25918.80 20307.00 20023.80
[25] 16576.80 18667.00 14394.67 19848.80 15516.33 20518.00 17572.00 15200.12
[33] 18413.80 14419.33 24094.50 22019.83 12923.50 14756.00 13869.80 12296.67
[41] 15775.17 14382.86 11566.33 13199.50 23412.00 39541.00 36186.60 16559.60
[49] 20772.50 19471.20 19827.33 15466.80 12925.67 18577.17 14943.00 24913.00
[57] 25093.00 24428.80 17003.00 21143.75 20435.00 17131.33 24569.75 23835.50
[65] 26360.00 47383.40 55157.75 37058.00 21546.67 23348.67 42323.67 28938.60
[73] 25880.80 47345.67 18711.33 29087.29 20748.29 35933.71 15439.71 29787.50
[81] 18145.00 21617.00 29203.89 41363.67 22259.09 44939.56 16902.00 16930.00

This computation can be verified by comparing to the results ran in the previous section. Running the code chunk hunan$GDPPC[wm_q[[1]]] gave the GDPPC of the five neighbouring counties for Polygon ID=1 gave the output of: [1] 20981 34592 24473 21311 22879 The average of these five neighbouring counties is 24847.20 which corresponds to the first output to the above code chunk.

Append the spatially lagged GDPPC values to hunan sf data frame by using the code chunk below:

lag.list <- list(hunan$NAME_3, lag.listw(rswm_q, hunan$GDPPC))
lag.res <- as.data.frame(lag.list)
colnames(lag.res) <- c("NAME_3", "lag GDPPC")
hunan <- left_join(hunan,lag.res)

The following table shows the average neighboring income values in the column lag GDPPC for each county.

head(hunan)
Simple feature collection with 6 features and 7 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 110.4922 ymin: 28.61762 xmax: 112.3013 ymax: 30.12812
Geodetic CRS:  WGS 84
   NAME_2  ID_3  NAME_3   ENGTYPE_3  County GDPPC lag GDPPC
1 Changde 21098 Anxiang      County Anxiang 23667  24847.20
2 Changde 21100 Hanshou      County Hanshou 20981  22724.80
3 Changde 21101  Jinshi County City  Jinshi 34592  24143.25
4 Changde 21102      Li      County      Li 24473  27737.50
5 Changde 21103   Linli      County   Linli 25554  27270.25
6 Changde 21104  Shimen      County  Shimen 27137  21248.80
                        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...

Next, plot both the GDPPC and Spatial Lag GDPPC for comparison using the code chunk below.

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

Spatial lag as a sum of neighboring values

Another way to compute spatial lag is as a sum of neighboring values by assigning binary weights: from the neighbors list, apply a function that will assign binary weights, then use the glist argument in the nb2listw() function to explicitly assign these weights.

Start by applying a function (lapply()) that will assign a value of 1 per each neighbor as shown in the code chunk below.

b_weights <- lapply(wm_q, function(x) 0*x + 1)
b_weights2 <- nb2listw(wm_q, 
                       glist = b_weights, 
                       style = "B")
b_weights2
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: B 
Weights constants summary:
   n   nn  S0  S1    S2
B 88 7744 448 896 10224

With the proper weights assigned, use lag.listw() to compute a lag variable from the weights and GDPPC.

lag_sum <- list(hunan$NAME_3, lag.listw(b_weights2, hunan$GDPPC))
lag.res <- as.data.frame(lag_sum)
colnames(lag.res) <- c("NAME_3", "lag_sum GDPPC")

Next, examine the results using the code chunk below.

lag_sum
[[1]]
 [1] "Anxiang"       "Hanshou"       "Jinshi"        "Li"           
 [5] "Linli"         "Shimen"        "Liuyang"       "Ningxiang"    
 [9] "Wangcheng"     "Anren"         "Guidong"       "Jiahe"        
[13] "Linwu"         "Rucheng"       "Yizhang"       "Yongxing"     
[17] "Zixing"        "Changning"     "Hengdong"      "Hengnan"      
[21] "Hengshan"      "Leiyang"       "Qidong"        "Chenxi"       
[25] "Zhongfang"     "Huitong"       "Jingzhou"      "Mayang"       
[29] "Tongdao"       "Xinhuang"      "Xupu"          "Yuanling"     
[33] "Zhijiang"      "Lengshuijiang" "Shuangfeng"    "Xinhua"       
[37] "Chengbu"       "Dongan"        "Dongkou"       "Longhui"      
[41] "Shaodong"      "Suining"       "Wugang"        "Xinning"      
[45] "Xinshao"       "Shaoshan"      "Xiangxiang"    "Baojing"      
[49] "Fenghuang"     "Guzhang"       "Huayuan"       "Jishou"       
[53] "Longshan"      "Luxi"          "Yongshun"      "Anhua"        
[57] "Nan"           "Yuanjiang"     "Jianghua"      "Lanshan"      
[61] "Ningyuan"      "Shuangpai"     "Xintian"       "Huarong"      
[65] "Linxiang"      "Miluo"         "Pingjiang"     "Xiangyin"     
[69] "Cili"          "Chaling"       "Liling"        "Yanling"      
[73] "You"           "Zhuzhou"       "Sangzhi"       "Yueyang"      
[77] "Qiyang"        "Taojiang"      "Shaoyang"      "Lianyuan"     
[81] "Hongjiang"     "Hengyang"      "Guiyang"       "Changsha"     
[85] "Taoyuan"       "Xiangtan"      "Dao"           "Jiangyong"    

[[2]]
 [1] 124236 113624  96573 110950 109081 106244 174988 235079 273907 256221
[11]  98013 104050 102846  92017 133831 158446 141883 119508 150757 153324
[21] 113593 129594 142149 100119  82884  74668  43184  99244  46549  20518
[31] 140576 121601  92069  43258 144567 132119  51694  59024  69349  73780
[41]  94651 100680  69398  52798 140472 118623 180933  82798  83090  97356
[51]  59482  77334  38777 111463  74715 174391 150558 122144  68012  84575
[61] 143045  51394  98279  47671  26360 236917 220631 185290  64640  70046
[71] 126971 144693 129404 284074 112268 203611 145238 251536 108078 238300
[81] 108870 108085 262835 248182 244850 404456  67608  33860

Again, comparing this computation to the GDPPC of the five neighbouring counties for Polygon ID=1: [1] 20981 34592 24473 21311 22879 The sum of these five neighbouring counties is 124236 which corresponds to the first output to the above code chunk.

Append the lag_sum GDPPC field into hunan sf data frame by using the code chunk below.

hunan <- left_join(hunan, lag.res)

Now, plot both the GDPPC and Spatial Lag Sum GDPPC for comparison using the code chunk below.

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

Spatial window average

The spatial window average uses row-standardized weights and includes the diagonal element. To do this in R, add the diagonal element before assigning weights in the neighbors structure. The function include.self() from spdep can be used to add the diagonal element to the neighbour list.

wm_qs <- include.self(wm_q)
wm_qs
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 536 
Percentage nonzero weights: 6.921488 
Average number of links: 6.090909

Notice that the Number of nonzero links, Percentage nonzero weights and Average number of links are 536, 6.921488 and 6.090909 respectively as compared to wm_q of 448, 5.785124 and 5.090909.

The neighbour list of area[1] can be accessed using the code chunk below.

wm_qs[[1]]
[1]  1  2  3  4 57 85

Now [1] has six neighbours instead of five. Next, obtain weights with nb2listw() as shown in the code chunk below.

wm_qs <- nb2listw(wm_qs)
wm_qs
Characteristics of weights list object:
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 536 
Percentage nonzero weights: 6.921488 
Average number of links: 6.090909 

Weights style: W 
Weights constants summary:
   n   nn S0       S1       S2
W 88 7744 88 30.90265 357.5308

Again, use nb2listw() and glist() to explicitly assign weights values. Lastly, create the lag variable from the weights structure and GDPPC variable.

lag_w_avg_gpdpc <- lag.listw(wm_qs, hunan$GDPPC)
lag_w_avg_gpdpc
 [1] 24650.50 22434.17 26233.00 27084.60 26927.00 22230.17 47621.20 37160.12
 [9] 49224.71 29886.89 26627.50 22690.17 25366.40 25825.75 30329.00 32682.83
[17] 25948.62 23987.67 25463.14 21904.38 23127.50 25949.83 20018.75 19524.17
[25] 18955.00 17800.40 15883.00 18831.33 14832.50 17965.00 17159.89 16199.44
[33] 18764.50 26878.75 23188.86 20788.14 12365.20 15985.00 13764.83 11907.43
[41] 17128.14 14593.62 11644.29 12706.00 21712.29 43548.25 35049.00 16226.83
[49] 19294.40 18156.00 19954.75 18145.17 12132.75 18419.29 14050.83 23619.75
[57] 24552.71 24733.67 16762.60 20932.60 19467.75 18334.00 22541.00 26028.00
[65] 29128.50 46569.00 47576.60 36545.50 20838.50 22531.00 42115.50 27619.00
[73] 27611.33 44523.29 18127.43 28746.38 20734.50 33880.62 14716.38 28516.22
[81] 18086.14 21244.50 29568.80 48119.71 22310.75 43151.60 17133.40 17009.33

Convert the lag variable listw object into a data frame using as.data.frame(). The third command line renames the field names of lag_wm_q1.res object into NAME_3 and lag_window_avg GDPPC respectively.

lag.list.wm_qs <- list(hunan$NAME_3, lag.listw(wm_qs, hunan$GDPPC))
lag_wm_qs.res <- as.data.frame(lag.list.wm_qs)
colnames(lag_wm_qs.res) <- c("NAME_3", "lag_window_avg GDPPC")

Next, the code chunk below will be used to append lag_window_avg GDPPC values into hunan sf data frame using left_join() of dplyr package.

hunan <- left_join(hunan, lag_wm_qs.res)

To compare the values of lag GDPPC and the spatial window average (lag_window_avg GDPPC), kable() of Knitr package is used to prepare a table as shown in the code chunk below.

hunan %>%
  select("County", "lag GDPPC", "lag_window_avg GDPPC") %>%
  kable()
County lag GDPPC lag_window_avg GDPPC geometry
Anxiang 24847.20 24650.50 POLYGON ((112.0625 29.75523…
Hanshou 22724.80 22434.17 POLYGON ((112.2288 29.11684…
Jinshi 24143.25 26233.00 POLYGON ((111.8927 29.6013,…
Li 27737.50 27084.60 POLYGON ((111.3731 29.94649…
Linli 27270.25 26927.00 POLYGON ((111.6324 29.76288…
Shimen 21248.80 22230.17 POLYGON ((110.8825 30.11675…
Liuyang 43747.00 47621.20 POLYGON ((113.9905 28.5682,…
Ningxiang 33582.71 37160.12 POLYGON ((112.7181 28.38299…
Wangcheng 45651.17 49224.71 POLYGON ((112.7914 28.52688…
Anren 32027.62 29886.89 POLYGON ((113.1757 26.82734…
Guidong 32671.00 26627.50 POLYGON ((114.1799 26.20117…
Jiahe 20810.00 22690.17 POLYGON ((112.4425 25.74358…
Linwu 25711.50 25366.40 POLYGON ((112.5914 25.55143…
Rucheng 30672.33 25825.75 POLYGON ((113.6759 25.87578…
Yizhang 33457.75 30329.00 POLYGON ((113.2621 25.68394…
Yongxing 31689.20 32682.83 POLYGON ((113.3169 26.41843…
Zixing 20269.00 25948.62 POLYGON ((113.7311 26.16259…
Changning 23901.60 23987.67 POLYGON ((112.6144 26.60198…
Hengdong 25126.17 25463.14 POLYGON ((113.1056 27.21007…
Hengnan 21903.43 21904.38 POLYGON ((112.7599 26.98149…
Hengshan 22718.60 23127.50 POLYGON ((112.607 27.4689, …
Leiyang 25918.80 25949.83 POLYGON ((112.9996 26.69276…
Qidong 20307.00 20018.75 POLYGON ((111.7818 27.0383,…
Chenxi 20023.80 19524.17 POLYGON ((110.2624 28.21778…
Zhongfang 16576.80 18955.00 POLYGON ((109.9431 27.72858…
Huitong 18667.00 17800.40 POLYGON ((109.9419 27.10512…
Jingzhou 14394.67 15883.00 POLYGON ((109.8186 26.75842…
Mayang 19848.80 18831.33 POLYGON ((109.795 27.98008,…
Tongdao 15516.33 14832.50 POLYGON ((109.9294 26.46561…
Xinhuang 20518.00 17965.00 POLYGON ((109.227 27.43733,…
Xupu 17572.00 17159.89 POLYGON ((110.7189 28.30485…
Yuanling 15200.12 16199.44 POLYGON ((110.9652 28.99895…
Zhijiang 18413.80 18764.50 POLYGON ((109.8818 27.60661…
Lengshuijiang 14419.33 26878.75 POLYGON ((111.5307 27.81472…
Shuangfeng 24094.50 23188.86 POLYGON ((112.263 27.70421,…
Xinhua 22019.83 20788.14 POLYGON ((111.3345 28.19642…
Chengbu 12923.50 12365.20 POLYGON ((110.4455 26.69317…
Dongan 14756.00 15985.00 POLYGON ((111.4531 26.86812…
Dongkou 13869.80 13764.83 POLYGON ((110.6622 27.37305…
Longhui 12296.67 11907.43 POLYGON ((110.985 27.65983,…
Shaodong 15775.17 17128.14 POLYGON ((111.9054 27.40254…
Suining 14382.86 14593.62 POLYGON ((110.389 27.10006,…
Wugang 11566.33 11644.29 POLYGON ((110.9878 27.03345…
Xinning 13199.50 12706.00 POLYGON ((111.0736 26.84627…
Xinshao 23412.00 21712.29 POLYGON ((111.6013 27.58275…
Shaoshan 39541.00 43548.25 POLYGON ((112.5391 27.97742…
Xiangxiang 36186.60 35049.00 POLYGON ((112.4549 28.05783…
Baojing 16559.60 16226.83 POLYGON ((109.7015 28.82844…
Fenghuang 20772.50 19294.40 POLYGON ((109.5239 28.19206…
Guzhang 19471.20 18156.00 POLYGON ((109.8968 28.74034…
Huayuan 19827.33 19954.75 POLYGON ((109.5647 28.61712…
Jishou 15466.80 18145.17 POLYGON ((109.8375 28.4696,…
Longshan 12925.67 12132.75 POLYGON ((109.6337 29.62521…
Luxi 18577.17 18419.29 POLYGON ((110.1067 28.41835…
Yongshun 14943.00 14050.83 POLYGON ((110.0003 29.29499…
Anhua 24913.00 23619.75 POLYGON ((111.6034 28.63716…
Nan 25093.00 24552.71 POLYGON ((112.3232 29.46074…
Yuanjiang 24428.80 24733.67 POLYGON ((112.4391 29.1791,…
Jianghua 17003.00 16762.60 POLYGON ((111.6461 25.29661…
Lanshan 21143.75 20932.60 POLYGON ((112.2286 25.61123…
Ningyuan 20435.00 19467.75 POLYGON ((112.0715 26.09892…
Shuangpai 17131.33 18334.00 POLYGON ((111.8864 26.11957…
Xintian 24569.75 22541.00 POLYGON ((112.2578 26.0796,…
Huarong 23835.50 26028.00 POLYGON ((112.9242 29.69134…
Linxiang 26360.00 29128.50 POLYGON ((113.5502 29.67418…
Miluo 47383.40 46569.00 POLYGON ((112.9902 29.02139…
Pingjiang 55157.75 47576.60 POLYGON ((113.8436 29.06152…
Xiangyin 37058.00 36545.50 POLYGON ((112.9173 28.98264…
Cili 21546.67 20838.50 POLYGON ((110.8822 29.69017…
Chaling 23348.67 22531.00 POLYGON ((113.7666 27.10573…
Liling 42323.67 42115.50 POLYGON ((113.5673 27.94346…
Yanling 28938.60 27619.00 POLYGON ((113.9292 26.6154,…
You 25880.80 27611.33 POLYGON ((113.5879 27.41324…
Zhuzhou 47345.67 44523.29 POLYGON ((113.2493 28.02411…
Sangzhi 18711.33 18127.43 POLYGON ((110.556 29.40543,…
Yueyang 29087.29 28746.38 POLYGON ((113.343 29.61064,…
Qiyang 20748.29 20734.50 POLYGON ((111.5563 26.81318…
Taojiang 35933.71 33880.62 POLYGON ((112.0508 28.67265…
Shaoyang 15439.71 14716.38 POLYGON ((111.5013 27.30207…
Lianyuan 29787.50 28516.22 POLYGON ((111.6789 28.02946…
Hongjiang 18145.00 18086.14 POLYGON ((110.1441 27.47513…
Hengyang 21617.00 21244.50 POLYGON ((112.7144 26.98613…
Guiyang 29203.89 29568.80 POLYGON ((113.0811 26.04963…
Changsha 41363.67 48119.71 POLYGON ((112.9421 28.03722…
Taoyuan 22259.09 22310.75 POLYGON ((112.0612 29.32855…
Xiangtan 44939.56 43151.60 POLYGON ((113.0426 27.8942,…
Dao 16902.00 17133.40 POLYGON ((111.498 25.81679,…
Jiangyong 16930.00 17009.33 POLYGON ((111.3659 25.39472…

Lastly, qtm() of tmap package is used to plot the lag_gdppc and w_ave_gdppc maps next to each other for quick comparison.

For a more effective comparison, use core tmap mapping functions.

w_avg_gdppc <- qtm(hunan, "lag_window_avg GDPPC")
tmap_arrange(lag_gdppc, w_avg_gdppc, asp=1, ncol=2)

Spatial window sum

The spatial window sum is the counterpart of spatial window average without using row-standardized weights. To add the diagonal element to the neighbour list, use include.self() from spdep.

wm_qs <- include.self(wm_q)
wm_qs
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 536 
Percentage nonzero weights: 6.921488 
Average number of links: 6.090909

Next, assign binary weights to the neighbour structure that includes the diagonal element.

b_weights <- lapply(wm_qs, function(x) 0*x + 1)
b_weights[1]
[[1]]
[1] 1 1 1 1 1 1

Now [1] has six neighbours instead of five. Again, use nb2listw() and glist() to explicitly assign weights values.

b_weights2 <- nb2listw(wm_qs, 
                       glist = b_weights, 
                       style = "B")
b_weights2
Characteristics of weights list object:
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 536 
Percentage nonzero weights: 6.921488 
Average number of links: 6.090909 

Weights style: B 
Weights constants summary:
   n   nn  S0   S1    S2
B 88 7744 536 1072 14160

With the new weights structure, compute the lag variable with lag.listw().

w_sum_gdppc <- list(hunan$NAME_3, lag.listw(b_weights2, hunan$GDPPC))
w_sum_gdppc
[[1]]
 [1] "Anxiang"       "Hanshou"       "Jinshi"        "Li"           
 [5] "Linli"         "Shimen"        "Liuyang"       "Ningxiang"    
 [9] "Wangcheng"     "Anren"         "Guidong"       "Jiahe"        
[13] "Linwu"         "Rucheng"       "Yizhang"       "Yongxing"     
[17] "Zixing"        "Changning"     "Hengdong"      "Hengnan"      
[21] "Hengshan"      "Leiyang"       "Qidong"        "Chenxi"       
[25] "Zhongfang"     "Huitong"       "Jingzhou"      "Mayang"       
[29] "Tongdao"       "Xinhuang"      "Xupu"          "Yuanling"     
[33] "Zhijiang"      "Lengshuijiang" "Shuangfeng"    "Xinhua"       
[37] "Chengbu"       "Dongan"        "Dongkou"       "Longhui"      
[41] "Shaodong"      "Suining"       "Wugang"        "Xinning"      
[45] "Xinshao"       "Shaoshan"      "Xiangxiang"    "Baojing"      
[49] "Fenghuang"     "Guzhang"       "Huayuan"       "Jishou"       
[53] "Longshan"      "Luxi"          "Yongshun"      "Anhua"        
[57] "Nan"           "Yuanjiang"     "Jianghua"      "Lanshan"      
[61] "Ningyuan"      "Shuangpai"     "Xintian"       "Huarong"      
[65] "Linxiang"      "Miluo"         "Pingjiang"     "Xiangyin"     
[69] "Cili"          "Chaling"       "Liling"        "Yanling"      
[73] "You"           "Zhuzhou"       "Sangzhi"       "Yueyang"      
[77] "Qiyang"        "Taojiang"      "Shaoyang"      "Lianyuan"     
[81] "Hongjiang"     "Hengyang"      "Guiyang"       "Changsha"     
[85] "Taoyuan"       "Xiangtan"      "Dao"           "Jiangyong"    

[[2]]
 [1] 147903 134605 131165 135423 134635 133381 238106 297281 344573 268982
[11] 106510 136141 126832 103303 151645 196097 207589 143926 178242 175235
[21] 138765 155699 160150 117145 113730  89002  63532 112988  59330  35930
[31] 154439 145795 112587 107515 162322 145517  61826  79925  82589  83352
[41] 119897 116749  81510  63530 151986 174193 210294  97361  96472 108936
[51]  79819 108871  48531 128935  84305 188958 171869 148402  83813 104663
[61] 155742  73336 112705  78084  58257 279414 237883 219273  83354  90124
[71] 168462 165714 165668 311663 126892 229971 165876 271045 117731 256646
[81] 126603 127467 295688 336838 267729 431516  85667  51028

Next, convert the lag variable listw object into a data frame. The second command line in the code chunk below renames the field names of w_sum_gdppc.res object into NAME_3 and w_sum GDPPC respectively.

w_sum_gdppc.res <- as.data.frame(w_sum_gdppc)
colnames(w_sum_gdppc.res) <- c("NAME_3", "w_sum GDPPC")

Next, the code chunk below will be used to append w_sum GDPPC values into hunan sf data frame using left_join() of dplyr.

hunan <- left_join(hunan, w_sum_gdppc.res)

To compare the values of lag GDPPC and the spatial window sum, use kable() of Knitr to prepare a table.

hunan %>%
  select("County", "lag_sum GDPPC", "w_sum GDPPC") %>%
  kable()
County lag_sum GDPPC w_sum GDPPC geometry
Anxiang 124236 147903 POLYGON ((112.0625 29.75523…
Hanshou 113624 134605 POLYGON ((112.2288 29.11684…
Jinshi 96573 131165 POLYGON ((111.8927 29.6013,…
Li 110950 135423 POLYGON ((111.3731 29.94649…
Linli 109081 134635 POLYGON ((111.6324 29.76288…
Shimen 106244 133381 POLYGON ((110.8825 30.11675…
Liuyang 174988 238106 POLYGON ((113.9905 28.5682,…
Ningxiang 235079 297281 POLYGON ((112.7181 28.38299…
Wangcheng 273907 344573 POLYGON ((112.7914 28.52688…
Anren 256221 268982 POLYGON ((113.1757 26.82734…
Guidong 98013 106510 POLYGON ((114.1799 26.20117…
Jiahe 104050 136141 POLYGON ((112.4425 25.74358…
Linwu 102846 126832 POLYGON ((112.5914 25.55143…
Rucheng 92017 103303 POLYGON ((113.6759 25.87578…
Yizhang 133831 151645 POLYGON ((113.2621 25.68394…
Yongxing 158446 196097 POLYGON ((113.3169 26.41843…
Zixing 141883 207589 POLYGON ((113.7311 26.16259…
Changning 119508 143926 POLYGON ((112.6144 26.60198…
Hengdong 150757 178242 POLYGON ((113.1056 27.21007…
Hengnan 153324 175235 POLYGON ((112.7599 26.98149…
Hengshan 113593 138765 POLYGON ((112.607 27.4689, …
Leiyang 129594 155699 POLYGON ((112.9996 26.69276…
Qidong 142149 160150 POLYGON ((111.7818 27.0383,…
Chenxi 100119 117145 POLYGON ((110.2624 28.21778…
Zhongfang 82884 113730 POLYGON ((109.9431 27.72858…
Huitong 74668 89002 POLYGON ((109.9419 27.10512…
Jingzhou 43184 63532 POLYGON ((109.8186 26.75842…
Mayang 99244 112988 POLYGON ((109.795 27.98008,…
Tongdao 46549 59330 POLYGON ((109.9294 26.46561…
Xinhuang 20518 35930 POLYGON ((109.227 27.43733,…
Xupu 140576 154439 POLYGON ((110.7189 28.30485…
Yuanling 121601 145795 POLYGON ((110.9652 28.99895…
Zhijiang 92069 112587 POLYGON ((109.8818 27.60661…
Lengshuijiang 43258 107515 POLYGON ((111.5307 27.81472…
Shuangfeng 144567 162322 POLYGON ((112.263 27.70421,…
Xinhua 132119 145517 POLYGON ((111.3345 28.19642…
Chengbu 51694 61826 POLYGON ((110.4455 26.69317…
Dongan 59024 79925 POLYGON ((111.4531 26.86812…
Dongkou 69349 82589 POLYGON ((110.6622 27.37305…
Longhui 73780 83352 POLYGON ((110.985 27.65983,…
Shaodong 94651 119897 POLYGON ((111.9054 27.40254…
Suining 100680 116749 POLYGON ((110.389 27.10006,…
Wugang 69398 81510 POLYGON ((110.9878 27.03345…
Xinning 52798 63530 POLYGON ((111.0736 26.84627…
Xinshao 140472 151986 POLYGON ((111.6013 27.58275…
Shaoshan 118623 174193 POLYGON ((112.5391 27.97742…
Xiangxiang 180933 210294 POLYGON ((112.4549 28.05783…
Baojing 82798 97361 POLYGON ((109.7015 28.82844…
Fenghuang 83090 96472 POLYGON ((109.5239 28.19206…
Guzhang 97356 108936 POLYGON ((109.8968 28.74034…
Huayuan 59482 79819 POLYGON ((109.5647 28.61712…
Jishou 77334 108871 POLYGON ((109.8375 28.4696,…
Longshan 38777 48531 POLYGON ((109.6337 29.62521…
Luxi 111463 128935 POLYGON ((110.1067 28.41835…
Yongshun 74715 84305 POLYGON ((110.0003 29.29499…
Anhua 174391 188958 POLYGON ((111.6034 28.63716…
Nan 150558 171869 POLYGON ((112.3232 29.46074…
Yuanjiang 122144 148402 POLYGON ((112.4391 29.1791,…
Jianghua 68012 83813 POLYGON ((111.6461 25.29661…
Lanshan 84575 104663 POLYGON ((112.2286 25.61123…
Ningyuan 143045 155742 POLYGON ((112.0715 26.09892…
Shuangpai 51394 73336 POLYGON ((111.8864 26.11957…
Xintian 98279 112705 POLYGON ((112.2578 26.0796,…
Huarong 47671 78084 POLYGON ((112.9242 29.69134…
Linxiang 26360 58257 POLYGON ((113.5502 29.67418…
Miluo 236917 279414 POLYGON ((112.9902 29.02139…
Pingjiang 220631 237883 POLYGON ((113.8436 29.06152…
Xiangyin 185290 219273 POLYGON ((112.9173 28.98264…
Cili 64640 83354 POLYGON ((110.8822 29.69017…
Chaling 70046 90124 POLYGON ((113.7666 27.10573…
Liling 126971 168462 POLYGON ((113.5673 27.94346…
Yanling 144693 165714 POLYGON ((113.9292 26.6154,…
You 129404 165668 POLYGON ((113.5879 27.41324…
Zhuzhou 284074 311663 POLYGON ((113.2493 28.02411…
Sangzhi 112268 126892 POLYGON ((110.556 29.40543,…
Yueyang 203611 229971 POLYGON ((113.343 29.61064,…
Qiyang 145238 165876 POLYGON ((111.5563 26.81318…
Taojiang 251536 271045 POLYGON ((112.0508 28.67265…
Shaoyang 108078 117731 POLYGON ((111.5013 27.30207…
Lianyuan 238300 256646 POLYGON ((111.6789 28.02946…
Hongjiang 108870 126603 POLYGON ((110.1441 27.47513…
Hengyang 108085 127467 POLYGON ((112.7144 26.98613…
Guiyang 262835 295688 POLYGON ((113.0811 26.04963…
Changsha 248182 336838 POLYGON ((112.9421 28.03722…
Taoyuan 244850 267729 POLYGON ((112.0612 29.32855…
Xiangtan 404456 431516 POLYGON ((113.0426 27.8942,…
Dao 67608 85667 POLYGON ((111.498 25.81679,…
Jiangyong 33860 51028 POLYGON ((111.3659 25.39472…

Lastly, qtm() of tmap package is used to plot the lag_sum GDPPC and w_sum_gdppc maps next to each other for quick comparison.

w_sum_gdppc <- qtm(hunan, "w_sum GDPPC")
tmap_arrange(lag_sum_gdppc, w_sum_gdppc, asp=1, ncol=2)