Hands-on Exercise 2A

Published

November 21, 2023

8 Spatial Weights and Applications

8.1 Overview

In this page, I show how I had completed the Hands-on Exercise 2A, on computing spatial weights.

The objectives are:

  • 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 spatial weights using appropriate functions of spdep package; and

  • Calculate spatially lagged variables using appropriate functions of spdep package.

8.2 Getting Started: Study Area and Data

The R packages used in this hands-on exercises are:

  • tmap for thematic mapping;

  • sf for importing, managing, and processing geospatial data;

  • tidyverse (i.e. readr, tidyr, dplyr) for performing data science tasks such as importing, tidying, and wrangling data;

  • knitr for embedding R code in different document formats (e.g., HTML) to facilitate dynamic report generation; and

  • spdep for analysing spatial dependence and spatial relationships in data.

They are loaded into the R environment using the following code:

pacman::p_load(sf, spdep, tmap, tidyverse, knitr)

Student Note: This allows for loading of multiple packages in one line of code.

8.3 Importing Data

The following data sets are used in this hands-on exercise:

  • Hunan’s County Boundary Layer. This is a geospatial data set in ESRI shapefile format.

  • Hunan’s Local Development Indicators 2012. This csv file contains data on selected Hunan’s local development indicators in 2012.

The data sets are placed under two sub-folders:

  • geospatial (County Boundary Layer), and

  • aspatial (Local Development Indicators 2012).

These two sub-folders are within the data folder of my Hands-on_Ex2 folder.

8.3.1 Importing shapefile

The st_read() (under sf package) is used to import the geospatial data set: hunan, a polygon feature layer in ESRI shapefile format.

Student Note: The geospatial objects are polygon features. There are a total of 88 features and 8 fields in hunan simple feature data frame. hunan is in wgs84 coordinate system.

hunan = st_read(dsn = "data/geospatial", 
                 layer = "Hunan")
Reading layer `Hunan' from data source 
  `C:\jmphosis\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

8.3.2 Importing csv file

The read_csv() (under readr package) is used to import the aspatial data set: hunan_2012, a csv file.

Student Note: The hunan_2012 tibble data frame contains 88 rows and 29 columns. There are two columns with character data - County and City.

hunan2012 = read_csv("data/aspatial/Hunan_2012.csv")

8.3.3 Performing Relational Join

The attribute table of the spatial polygons data frame, hunan, is updated using the attribute fields of the tibble data frame, hunan2012 using left_join() (under dplyr package).

Student Note: Without explicitly stating the “by” argument for left_join(), the two tables are joined by the ‘County’ columns.

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

hunan
Simple feature collection with 88 features and 6 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
1   Changde 21098   Anxiang      County   Anxiang 23667
2   Changde 21100   Hanshou      County   Hanshou 20981
3   Changde 21101    Jinshi County City    Jinshi 34592
4   Changde 21102        Li      County        Li 24473
5   Changde 21103     Linli      County     Linli 25554
6   Changde 21104    Shimen      County    Shimen 27137
7  Changsha 21109   Liuyang County City   Liuyang 63118
8  Changsha 21110 Ningxiang      County Ningxiang 62202
9  Changsha 21111 Wangcheng      County Wangcheng 70666
10 Chenzhou 21112     Anren      County     Anren 12761
                         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...

8.4 Visualising Regional Development Indicator

A basemap and a choropleth map are prepared usign qtm() (under tmap package) to visualise the 2012 Gross Domestic Product Per Capita (GDPPC).

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)

8.5 Computing Contiguity Spatial Weight

The poly2nb() (under spdep package) is used to compute contiguity weight matrices for the study area. This function builds a neighbours list based on regions with contiguous boundaries.

8.5.1 Computing (QUEEN) Contiguity Based Neighbours

The “queen” argument in the function is set to either TRUE (default) or FALSE. The TRUE option will return a list of first order neighbours using the Queen’s contiguity criteria.

Student Note: According to Queen’s criteria, two regions are considered neighbours if they share any part of their boundary (even if it is a single point). This results in a more inclusive definition of neighbour relationships.

Student Note: The summary report below shows that there are 88 area units in hunan. The most connected area unit (85) has 11 neighbours. The least connected area units (30 and 65) have only one neighbour each.

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

We can examine the details of a specific polygon and its neigbhours in the following manner:

  • Neighbours for a specific polygon. For example, for area unit 1, its neighbours are 2, 3, 4, 57 and 85.
wm_q[[1]]
[1]  2  3  4 57 85
  • County names. For example, area unit 1 is Anxiang, and it is surrounded by Hanshou, Jinshi, Li, Nan, and Taoyuan.

Student Note: Both “County” and “NAME_3” can be used to get the county names.

hunan$County[1]
[1] "Anxiang"
hunan$NAME_3[c(2,3,4,57,85)]
[1] "Hanshou" "Jinshi"  "Li"      "Nan"     "Taoyuan"
hunan$County[c(2,3,4,57,85)]
[1] "Hanshou" "Jinshi"  "Li"      "Nan"     "Taoyuan"
  • GDP Per Capita.
nb1 = wm_q[[1]]
hunan$GDPPC[nb1]
[1] 20981 34592 24473 21311 22879

The complete weight matrix, wm_q, can be displayed using str() (under utils package).

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

8.5.2 Creating (ROOK) Contiguity Based Neighbours

The “queen” argument in the function is set to either TRUE (default) or FALSE. The FALSE option will return a list of first order neighbours using the Rook’s continguity criteria instead.

Student Note: According to Rook’s criteria, two regions are considered neighbours if they share an entire edge (but not corners). This results in a stricter definition neighbour relationships.

Student Note: The summary report below shows that there are 88 area units in hunan. The most connected area unit (85) has 10 neighbours. The least connected area units (30 and 65) have only one neighbour each. This is the same outcome as setting “queen = TRUE”.

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

The complete weight matrix, wm_r, can be displayed using str() (under utils package).

Student Note: Due to the different contiguity criteria used, there are some differences in the outcomes. For example, for area unit 1, it has only four neighbours under the Rook’s criteria (3, 4, 57 and 85), whereas it has five neighbours under the Queen’s criteria (2, 3, 4, 57 and 85). This is expected given that the Rook’s criteria is stricter.

str(wm_r)
List of 88
 $ : int [1:4] 3 4 57 85
 $ : int [1:4] 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 [1:7] 16 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 [1:4] 10 17 22 83
 $ : int [1:6] 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 [1:5] 1 2 58 64 76
 $ : 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 [1:4] 10 11 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 [1:10] 1 2 3 5 6 32 56 69 75 78
 $ : 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 = FALSE)
 - attr(*, "type")= chr "rook"
 - attr(*, "sym")= logi TRUE

8.5.3 Visualising Contiguity Weights

A connectivity graph takes a point and displays a line to each neighboring point. The typical method used for polygons is to find the polygon centroids. These can be calculated using the sf package.

The mapping function, map_dbl() (under the purrr package) is utilised to apply a function, st_centroid() (under sf package), on each element of the geometry column, us.bound, returning a vector of a same length.

The longitude is then extracted by looking for the first value of each centroid, while the latitude is extracted by looking for the second value of each centroid. The cbind() is then used to put the two values together.

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

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

coords = cbind(longitude, latitude)

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

The neighbours maps using the Queen’s criteria and Rook’s criteria respectively are then plotted below.

Student Note: The par() (under graphics package) is used to plot the two maps side by side. The “main” argument is used to add titles for the two maps.

par(mfrow = c(1, 2))

plot(hunan$geometry, border="lightgrey", main = "Queen's Continguity Criteria")
plot(wm_q, coords, pch = 19, cex = 0.6, add = TRUE, col= "red")

plot(hunan$geometry, border="lightgrey", main = "Rook's Continguity Criteria")
plot(wm_r, coords, pch = 19, cex = 0.6, add = TRUE, col = "blue")

8.6 Computing Distance Based Neighbours

The dnearneigh() (under spdep package) is used to calculate distance-based weight matrices for the study area.

The function identifies neighbours of region points by Euclidean distance with a distance band with lower “d1” and upper “d2” bounds controlled by the bounds argument. If unprojected coordinates are used and either specified in the coordinates object x or with x as a two column matrix and “longlat” set as TRUE, great circle distances in km will be calculated assuming the WGS84 reference ellipsoid.

8.6.1 Determine the Cut-off Distance

The upper limit for the distance band is determined by using the following steps and functions under the spdep package:

  1. Return a matrix with the indices of points belonging to the set of the k nearest neighbours (knn) of each other by using knearneigh().

    Student Note: In the matrix, each row corresponds to a point, and the columns contain the indices of its knn.

  2. Convert the knn object into a neighbours list of class nb with a list of integer vectors containing neighbour region number ids by using knn2nb().

  3. Return the length of neighbour relationship edges by using nbdists(). This function returns in the units of the coordinates if the coordinates are projected, and in km otherwise.

  4. Remove the list structure of the returned object by using unlist() (under base package).

The summary report below shows that the largest first nearest neighbour distance is 61.79 km (i.e., max value). Thus, this is used as the upper threshold as it ensures that all units will have at least one neighbour.

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

8.6.2 Computing and Plotting Fixed Distance Weight Matrix

The dnearneigh() (under spdep package) is used to compute the distance weight matrix.

Student Note: knearneigh() computes knn, while dnearneigh() computes distance-based neighbours.

Question: What is the meaning of “Average number of links: 3.681818” shown above?

Answer: On average, each region is connected to approximately 3.68 neighbours.

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

The complete weight matrix, wm_d62, can be displayed using str() (under utils package).

str(wm_d62)
List of 88
 $ : int [1:5] 3 4 5 57 64
 $ : int [1:4] 57 58 78 85
 $ : int [1:4] 1 4 5 57
 $ : int [1:3] 1 3 5
 $ : int [1:4] 1 3 4 85
 $ : int 69
 $ : int [1:2] 67 84
 $ : int [1:4] 9 46 47 78
 $ : int [1:4] 8 46 68 84
 $ : int [1:4] 16 22 70 72
 $ : int [1:3] 14 17 72
 $ : int [1:5] 13 60 61 63 83
 $ : int [1:4] 12 15 60 83
 $ : int [1:2] 11 17
 $ : int 13
 $ : int [1:4] 10 17 22 83
 $ : int [1:3] 11 14 16
 $ : int [1:3] 20 22 63
 $ : int [1:5] 20 21 73 74 82
 $ : int [1:5] 18 19 21 22 82
 $ : int [1:6] 19 20 35 74 82 86
 $ : int [1:4] 10 16 18 20
 $ : int [1:3] 41 77 82
 $ : int [1:4] 25 28 31 54
 $ : int [1:4] 24 28 33 81
 $ : int [1:4] 27 33 42 81
 $ : int [1:2] 26 29
 $ : int [1:6] 24 25 33 49 52 54
 $ : int [1:2] 27 37
 $ : int 33
 $ : int [1:2] 24 36
 $ : int 50
 $ : int [1:5] 25 26 28 30 81
 $ : int [1:3] 36 45 80
 $ : int [1:6] 21 41 46 47 80 82
 $ : int [1:5] 31 34 45 56 80
 $ : int [1:2] 29 42
 $ : int [1:3] 44 77 79
 $ : int [1:4] 40 42 43 81
 $ : int [1:3] 39 45 79
 $ : int [1:5] 23 35 45 79 82
 $ : int [1:5] 26 37 39 43 81
 $ : int [1:3] 39 42 44
 $ : int [1:2] 38 43
 $ : int [1:6] 34 36 40 41 79 80
 $ : int [1:5] 8 9 35 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 [1:6] 32 48 51 52 54 55
 $ : int [1:4] 48 49 50 52
 $ : int [1:6] 28 48 49 50 51 54
 $ : int [1:2] 48 55
 $ : int [1:5] 24 28 49 50 52
 $ : int [1:4] 48 50 53 75
 $ : int 36
 $ : int [1:5] 1 2 3 58 64
 $ : int [1:5] 2 57 64 66 68
 $ : int [1:3] 60 87 88
 $ : int [1:4] 12 13 59 61
 $ : int [1:5] 12 60 62 63 87
 $ : int [1:4] 61 63 77 87
 $ : int [1:5] 12 18 61 62 83
 $ : int [1:4] 1 57 58 76
 $ : int 76
 $ : int [1:5] 58 67 68 76 84
 $ : int [1:2] 7 66
 $ : int [1:4] 9 58 66 84
 $ : int [1:2] 6 75
 $ : int [1:3] 10 72 73
 $ : int [1:2] 73 74
 $ : int [1:3] 10 11 70
 $ : int [1:4] 19 70 71 74
 $ : int [1:5] 19 21 71 73 86
 $ : int [1:2] 55 69
 $ : int [1:3] 64 65 66
 $ : int [1:3] 23 38 62
 $ : int [1:2] 2 8
 $ : int [1:4] 38 40 41 45
 $ : int [1:5] 34 35 36 45 47
 $ : int [1:5] 25 26 33 39 42
 $ : int [1:6] 19 20 21 23 35 41
 $ : int [1:4] 12 13 16 63
 $ : int [1:4] 7 9 66 68
 $ : int [1:2] 2 5
 $ : int [1:4] 21 46 47 74
 $ : 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 dnearneigh(x = coords, d1 = 0, d2 = 62, longlat = TRUE)
 - attr(*, "dnn")= num [1:2] 0 62
 - attr(*, "bounds")= chr [1:2] "GE" "LE"
 - attr(*, "nbtype")= chr "distance"
 - attr(*, "sym")= logi TRUE

Another way to display the structure of the weight matrix, wm_d62, is to combine table() (under base package) and card() (under spdep package).

table(hunan$County, card(wm_d62))
               
                1 2 3 4 5 6
  Anhua         1 0 0 0 0 0
  Anren         0 0 0 1 0 0
  Anxiang       0 0 0 0 1 0
  Baojing       0 0 0 0 1 0
  Chaling       0 0 1 0 0 0
  Changning     0 0 1 0 0 0
  Changsha      0 0 0 1 0 0
  Chengbu       0 1 0 0 0 0
  Chenxi        0 0 0 1 0 0
  Cili          0 1 0 0 0 0
  Dao           0 0 0 1 0 0
  Dongan        0 0 1 0 0 0
  Dongkou       0 0 0 1 0 0
  Fenghuang     0 0 0 1 0 0
  Guidong       0 0 1 0 0 0
  Guiyang       0 0 0 1 0 0
  Guzhang       0 0 0 0 0 1
  Hanshou       0 0 0 1 0 0
  Hengdong      0 0 0 0 1 0
  Hengnan       0 0 0 0 1 0
  Hengshan      0 0 0 0 0 1
  Hengyang      0 0 0 0 0 1
  Hongjiang     0 0 0 0 1 0
  Huarong       0 0 0 1 0 0
  Huayuan       0 0 0 1 0 0
  Huitong       0 0 0 1 0 0
  Jiahe         0 0 0 0 1 0
  Jianghua      0 0 1 0 0 0
  Jiangyong     0 1 0 0 0 0
  Jingzhou      0 1 0 0 0 0
  Jinshi        0 0 0 1 0 0
  Jishou        0 0 0 0 0 1
  Lanshan       0 0 0 1 0 0
  Leiyang       0 0 0 1 0 0
  Lengshuijiang 0 0 1 0 0 0
  Li            0 0 1 0 0 0
  Lianyuan      0 0 0 0 1 0
  Liling        0 1 0 0 0 0
  Linli         0 0 0 1 0 0
  Linwu         0 0 0 1 0 0
  Linxiang      1 0 0 0 0 0
  Liuyang       0 1 0 0 0 0
  Longhui       0 0 1 0 0 0
  Longshan      0 1 0 0 0 0
  Luxi          0 0 0 0 1 0
  Mayang        0 0 0 0 0 1
  Miluo         0 0 0 0 1 0
  Nan           0 0 0 0 1 0
  Ningxiang     0 0 0 1 0 0
  Ningyuan      0 0 0 0 1 0
  Pingjiang     0 1 0 0 0 0
  Qidong        0 0 1 0 0 0
  Qiyang        0 0 1 0 0 0
  Rucheng       0 1 0 0 0 0
  Sangzhi       0 1 0 0 0 0
  Shaodong      0 0 0 0 1 0
  Shaoshan      0 0 0 0 1 0
  Shaoyang      0 0 0 1 0 0
  Shimen        1 0 0 0 0 0
  Shuangfeng    0 0 0 0 0 1
  Shuangpai     0 0 0 1 0 0
  Suining       0 0 0 0 1 0
  Taojiang      0 1 0 0 0 0
  Taoyuan       0 1 0 0 0 0
  Tongdao       0 1 0 0 0 0
  Wangcheng     0 0 0 1 0 0
  Wugang        0 0 1 0 0 0
  Xiangtan      0 0 0 1 0 0
  Xiangxiang    0 0 0 0 1 0
  Xiangyin      0 0 0 1 0 0
  Xinhua        0 0 0 0 1 0
  Xinhuang      1 0 0 0 0 0
  Xinning       0 1 0 0 0 0
  Xinshao       0 0 0 0 0 1
  Xintian       0 0 0 0 1 0
  Xupu          0 1 0 0 0 0
  Yanling       0 0 1 0 0 0
  Yizhang       1 0 0 0 0 0
  Yongshun      0 0 0 1 0 0
  Yongxing      0 0 0 1 0 0
  You           0 0 0 1 0 0
  Yuanjiang     0 0 0 0 1 0
  Yuanling      1 0 0 0 0 0
  Yueyang       0 0 1 0 0 0
  Zhijiang      0 0 0 0 1 0
  Zhongfang     0 0 0 1 0 0
  Zhuzhou       0 0 0 0 1 0
  Zixing        0 0 1 0 0 0

The n.comp.nb() (under spdep package) is used to calculate the number of connected components in wm_d62, which is then accessed using “nc”. Using “comp.id”, the connected component(s) and their size(s) (i.e., number of spatial units belonging to each) are extracted.

Student Note: Since n_comp$nc = 1, it means that there is only one connected component in wm_d62. Having only one connected component suggests that the entire study area is spatially connected, and there are no distinct subgroups that are not linked to each other.

n_comp = n.comp.nb(wm_d62)
n_comp$nc
[1] 1
table(n_comp$comp.id)

 1 
88

The distance weight matrix is then plotted. The red lines show the links of the first nearest neighbours, while the black lines show the links of neighbours within cut-off distance of 62 km. Alternatively, they can be plotted side by side.

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

par(mfrow=c(1,2))
plot(hunan$geometry, border="lightgrey")
plot(k1, coords, add=TRUE, col="red", length=0.08, main="1st Nearest Neighbours")
plot(hunan$geometry, border="lightgrey")
plot(wm_d62, coords, add=TRUE, pch = 19, cex = 0.6, main="Distance Link (Cut-off = 62km)")

8.6.3 Computing and Plotting Adaptive Distance Weight Matrix

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 less neighbours. Having many neighbours smoothes the neighbour relationship across more neighbours.

It is possible to control the numbers of neighbours directly using knn, either accepting asymmetric neighbours or imposing symmetry as shown below.

Student Note: Fixing the k argument’s value in knearneigh() means that the number of neighbours for each region is exactly 6.

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
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"

The distance weight matrix is then plotted.

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

8.7 Weights Based on Inverse Distance Weighting (IDW)

The Inversed Distance Weighting (IDW) method is used to derive a different type of spatial weight matrix.

Student Note: IDW assigns higher weights to neighbours that are closer and lower weights to neighbours that are further away.

The nbdists() (under spdep package) is used to compute the distances between areas.

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

8.7.1 Row-standardised Weights Matrix

Weights are assigned to each neighbouring polygon. The nb21listw() (under spdep package) is used to convert a neighborhood object, wm_q, to a listw object, rswm_q (style=“W”) or rswm_ids (style=“B”). This allows row-standardised distance weight matrices to be created, whereby each row sums to 1.

The “style” argument influences the specific characteristics of the weights matrix.

The following example shows “style=”W””, each neighboring polygon is assigned equal weight. 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, its downside is 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.

Student Note:

  • “listw” stands for list of weights.

  • The “style” argument set to “W” specifies a binary spatial weight matrix, where the presence of a spatial relationship is indicated by 1, and absence by 0. All neighbouring units are considered equal in terms of their impact on the target unit, reflecting a uniform spatial relationship.

  • The “zero.policy” argument returns lists of non-neighbours when set to TRUE.

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

The weights of the first polygon’s eight neighbours are extracted below. It shows that each neighbour is assigned an equal weight of 0.2 of the total weight. This means that when computing the average neighbouring income values, each neighbour’s income will be multiplied by 0.2 before being tallied.

Student Note: In this example, each neighbour has an equal weight, indicating a form of uniform spatial relationship.

rswm_q$weights[1]
[[1]]
[1] 0.2 0.2 0.2 0.2 0.2

On the other hand, for “style=”B””, the spatial lag of a variable for a particular unit is the sum of that variable over all neighbouring units, with each neighbour contributing a binary indicator (1 or 0).

Student Note:

  • “glist=ids” means the list of inverse distances is used for the general list of weights argument.

  • The “style” argument set to “B” specifies a binary spatial lag matrix, where the presence of a spatial relationship is indicated by 1, and absence by 0. At the same time, the direction of the connection is considered for the calculations.

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

The weights of the first polygon’s five neighbours are extracted below. It shows that each neighbour is assigned varying weights. This means that when computing the average neighbouring income values, each neighbour’s income will be multiplied by their corresponding weight values before being tallied.

Student Note: In this example, each neighbour has an equal weight, indicating a form of uniform spatial relationship.

rswm_ids$weights[1]
[[1]]
[1] 0.01535405 0.03916350 0.01820896 0.02807922 0.01145113

Student Note: The summary below shows the range of weights using “style=”B”” and IDW. The summary statistics indicate that the weights are not uniform.

summary(unlist(rswm_ids$weights))
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
0.008218 0.015088 0.018739 0.019614 0.022823 0.040338

8.8 Application of Spatial Weight Matrix

Four different spatial lagged variables are created to compute the average neighbour GDP per capita for each polygon.

  1. Spatial lag with row-standardized weights;

  2. Spatial lag as a sum of neighbouring values;

  3. Spatial window average; and

  4. Spatial window sum.

8.8.1 Spatial Lag with Row-standardised Weights

Student Note: Spatial lag with row-standardized weights reflects average value of the variable in neighbouring units, with closer units having a higher influence. It is calculated using the weighted average of the values of a variable in neighbouring units, where the weights are row standardised.

Comparing the outcome below with the neighbouring GDP per capita values found in section 8.5.1 above, it is observed that each element GDPPC.lag is the average GDP per capita value of the neighbouring polygons. For example, for the first polygon, the average of 20981, 34592, 24473, 21311, and 22879 is indeed 24847.20.

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

The spatially lag GDP per capita values can be added to the hunan sf data frame.

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)

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...

The plots for the GDPPC and spatial lag GDPPC can then be plotted side by side.

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

8.8.2 Spatial Lag as a Sum of Neighbouring Values

Student Note: Spatial lag as a sum of neighbouring values represents the total values of the variable in the surrounding area. This is calculated by summing the values of a variable in neighbouring units, where each neighbouring units contributes equally, regardless of distance.

A value of 1 is assigned to each neighbour using lapply() (under base package).

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

The lag variable for using the weights and GPD per capita is then computed using lag.listw() (under spdep package).

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")

The following shows each county and the sum of the GDPPC of its neighbours. For example, for the first polygon, the total of 20981, 34592, 24473, 21311, and 22879 is indeed 124236.

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

Similarly, the spatially lag GDP per capita values can be added to the hunan sf data frame.

hunan = left_join(hunan, lag.res)

head(hunan)
Simple feature collection with 6 features and 8 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 lag_sum GDPPC
1 Changde 21098 Anxiang      County Anxiang 23667  24847.20        124236
2 Changde 21100 Hanshou      County Hanshou 20981  22724.80        113624
3 Changde 21101  Jinshi County City  Jinshi 34592  24143.25         96573
4 Changde 21102      Li      County      Li 24473  27737.50        110950
5 Changde 21103   Linli      County   Linli 25554  27270.25        109081
6 Changde 21104  Shimen      County  Shimen 27137  21248.80        106244
                        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...

The plots for the GDPPC and spatial lag sum GDPPC can then be plotted side by side.

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

8.8.3 Spatial Window Average

The spatial window average uses row-standardised weights and includes the diagonal element.

Student Note: Spatial window average provides a localised measure of central tendency by calculating the average value of a variable within a specified spatial window or neighbourhood. Unlike row-standardised spatial lag, this approach may consider a fixed-size spatial window, and all units within that window contribute equally to the average.

To add the diagonal element to the neighbour list, we need to use include.self() (under spdep package).

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

For example, polygon 1 now has six instead of five neighbours.

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

The weights are then obtained and assigned using nb2listw() (under spdep package).

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

The lag variable is then created from the weights 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

The lag variable listw object is then converted to a data frame and appended to hunan sf data frame using left_join() (under dplyr package).

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")

hunan = left_join(hunan, lag_wm_qs.res)
head(hunan)
Simple feature collection with 6 features and 9 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 lag_sum GDPPC
1 Changde 21098 Anxiang      County Anxiang 23667  24847.20        124236
2 Changde 21100 Hanshou      County Hanshou 20981  22724.80        113624
3 Changde 21101  Jinshi County City  Jinshi 34592  24143.25         96573
4 Changde 21102      Li      County      Li 24473  27737.50        110950
5 Changde 21103   Linli      County   Linli 25554  27270.25        109081
6 Changde 21104  Shimen      County  Shimen 27137  21248.80        106244
  lag_window_avg GDPPC                       geometry
1             24650.50 POLYGON ((112.0625 29.75523...
2             22434.17 POLYGON ((112.2288 29.11684...
3             26233.00 POLYGON ((111.8927 29.6013,...
4             27084.60 POLYGON ((111.3731 29.94649...
5             26927.00 POLYGON ((111.6324 29.76288...
6             22230.17 POLYGON ((110.8825 30.11675...

To compare the values of lag GDPPC and spatial window average, kable() (under knitr package) is used to prepare a table.

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…

The plots for the spatial lag GDPPC and spatial window average GDPPC can then be plotted side by side.

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

8.8.4 Spatial Window Sum

The spatial window sum is the counterpart of spatial window average, but without using row-standardised weights.

Student Note: Spatial window sum reflects the total accumulated value of the variable in the surrounding area, without normalisation by the number of neighbouring units. It is calculated by summing the values of a variable within a specified spatial window.

To add the diagonal element to the neighbour list, we need to use include.self() (under spdep package).

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

A value of 1 is assigned to each neighbour (including diagonal) using lapply() (under base package).

b_weights = lapply(wm_qs, function(x) 0*x + 1)

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

The lag variable for using the weights and GPD per capita is then computed using lag.listw() (under spdep package).

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

The lag variable listw object is then converted to a data frame and appended to hunan sf data frame using left_join() (under dplyr package).

w_sum_gdppc.res = as.data.frame(w_sum_gdppc)
colnames(w_sum_gdppc.res) = c("NAME_3", "w_sum GDPPC")
hunan = left_join(hunan, w_sum_gdppc.res)
head(hunan)
Simple feature collection with 6 features and 10 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 lag_sum GDPPC
1 Changde 21098 Anxiang      County Anxiang 23667  24847.20        124236
2 Changde 21100 Hanshou      County Hanshou 20981  22724.80        113624
3 Changde 21101  Jinshi County City  Jinshi 34592  24143.25         96573
4 Changde 21102      Li      County      Li 24473  27737.50        110950
5 Changde 21103   Linli      County   Linli 25554  27270.25        109081
6 Changde 21104  Shimen      County  Shimen 27137  21248.80        106244
  lag_window_avg GDPPC w_sum GDPPC                       geometry
1             24650.50      147903 POLYGON ((112.0625 29.75523...
2             22434.17      134605 POLYGON ((112.2288 29.11684...
3             26233.00      131165 POLYGON ((111.8927 29.6013,...
4             27084.60      135423 POLYGON ((111.3731 29.94649...
5             26927.00      134635 POLYGON ((111.6324 29.76288...
6             22230.17      133381 POLYGON ((110.8825 30.11675...

To compare the values of lag GDPPC and spatial window average, kable() (under knitr package) is used 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…

The plots for the spatial window average GDPPC and the spatial window sum GDPPC can then be plotted side by side.

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

~~~ End of Hands-on Exercise 2A ~~~