InClass_Ex01 : Spatial Weights and Applications

Author

Yogendra Shinde

Chapter # 3 : Spatial Weights and Applications - Hunan Province Case Study.

Published 25-Nov-2022

3.1 Overview.

In this hands-on exercise, we will learn how to compute spatial weights using R.

  • 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.

3.2 The Study Area and Data

Two data sets will be used in this hands-on exercise, they are :

  1. Hunan county boundary layer. This is a geospatial data set in ESRI shapefile format.

  2. Hunan_2012.csv: This csv file contains selected Hunan’s local development indicators in 2012.

3.2.1 Getting Started - The Winding Road of Hunan.

Before we get started, we need to ensure that spdep, sf, tmap and tidyverse packages of R are currently installed in our R.

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

3.3 Getting the Data Into R Environment.

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

3.3.1 Import shapefile into R environment.

hunan <- st_read(dsn = "Hunan_data/geospatial", 
                 layer = "Hunan")
Reading layer `Hunan' from data source 
  `C:\Yogendra345\ISSS624_A01\InClass_Ex01\Hunan_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

Here,let us first understand the complete information of the hunan dataframe - which is a POLYGON file by using head() command in dplyr package of R

head(hunan,n=7)
Simple feature collection with 7 features and 7 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 110.4922 ymin: 27.85304 xmax: 114.2544 ymax: 30.12812
Geodetic CRS:  WGS 84
    NAME_2  ID_3  NAME_3   ENGTYPE_3 Shape_Leng Shape_Area  County
1  Changde 21098 Anxiang      County   1.869074 0.10056190 Anxiang
2  Changde 21100 Hanshou      County   2.360691 0.19978745 Hanshou
3  Changde 21101  Jinshi County City   1.425620 0.05302413  Jinshi
4  Changde 21102      Li      County   3.474325 0.18908121      Li
5  Changde 21103   Linli      County   2.289506 0.11450357   Linli
6  Changde 21104  Shimen      County   4.171918 0.37194707  Shimen
7 Changsha 21109 Liuyang County City   4.060579 0.46016789 Liuyang
                        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,...

3.3.2 Import .csv file into R environment.

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

hunan2012 <- read_csv("Hunan_data/aspatial/Hunan_2012.csv")
Rows: 88 Columns: 29
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr  (2): County, City
dbl (27): avg_wage, deposite, FAI, Gov_Rev, Gov_Exp, GDP, GDPPC, GIO, Loan, ...

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
summary(hunan2012)
    County              City              avg_wage        deposite      
 Length:88          Length:88          Min.   :26832   Min.   :  564.1  
 Class :character   Class :character   1st Qu.:31480   1st Qu.: 4306.9  
 Mode  :character   Mode  :character   Median :33070   Median : 6677.3  
                                       Mean   :33465   Mean   : 7514.1  
                                       3rd Qu.:34247   3rd Qu.: 9502.6  
                                       Max.   :54540   Max.   :24332.0  
      FAI           Gov_Rev          Gov_Exp            GDP       
 Min.   : 1005   Min.   : 108.4   Min.   : 683.6   Min.   : 1490  
 1st Qu.: 3911   1st Qu.: 307.3   1st Qu.:1451.7   1st Qu.: 5844  
 Median : 6854   Median : 455.2   Median :2037.0   Median :10483  
 Mean   : 9021   Mean   : 646.7   Mean   :2155.9   Mean   :14931  
 3rd Qu.:10212   3rd Qu.: 637.3   3rd Qu.:2638.1   3rd Qu.:19131  
 Max.   :49234   Max.   :5350.0   Max.   :7885.5   Max.   :88009  
     GDPPC            GIO              Loan           NIPCR      
 Min.   : 8497   Min.   :   514   Min.   :  358   Min.   : 2895  
 1st Qu.:14566   1st Qu.:  4965   1st Qu.: 2242   1st Qu.: 3981  
 Median :20433   Median : 10698   Median : 3683   Median : 6119  
 Mean   :24405   Mean   : 19227   Mean   : 4687   Mean   : 7095  
 3rd Qu.:27224   3rd Qu.: 22223   3rd Qu.: 5082   3rd Qu.: 9768  
 Max.   :88656   Max.   :148976   Max.   :40534   Max.   :17070  
      Bed            Emp              EmpR           EmpRT      
 Min.   : 392   Min.   : 73.65   Min.   : 62.0   Min.   : 33.6  
 1st Qu.:1111   1st Qu.:231.56   1st Qu.:191.2   1st Qu.:105.5  
 Median :1642   Median :363.79   Median :284.2   Median :171.2  
 Mean   :1765   Mean   :388.71   Mean   :316.4   Mean   :190.4  
 3rd Qu.:2240   3rd Qu.:525.04   3rd Qu.:433.5   3rd Qu.:263.5  
 Max.   :6225   Max.   :919.62   Max.   :757.6   Max.   :451.6  
    Pri_Stu          Sec_Stu         Household       Household_R    
 Min.   :  5.91   Min.   : 3.201   Min.   : 27.15   Min.   : 30.40  
 1st Qu.: 24.09   1st Qu.:15.095   1st Qu.: 99.15   1st Qu.: 87.95  
 Median : 36.85   Median :24.834   Median :162.85   Median :152.85  
 Mean   : 43.08   Mean   :27.609   Mean   :175.04   Mean   :159.62  
 3rd Qu.: 58.26   3rd Qu.:36.248   3rd Qu.:245.32   3rd Qu.:218.22  
 Max.   :112.20   Max.   :68.853   Max.   :391.70   Max.   :369.80  
      NOIP           Pop_R            RSCG             Pop_T       
 Min.   : 10.0   Min.   : 57.7   Min.   :  354.5   Min.   :  92.3  
 1st Qu.: 47.0   1st Qu.:227.3   1st Qu.: 1757.2   1st Qu.: 333.1  
 Median : 80.5   Median :348.7   Median : 3437.8   Median : 572.5  
 Mean   :107.3   Mean   :369.3   Mean   : 4164.7   Mean   : 586.8  
 3rd Qu.:124.0   3rd Qu.:517.0   3rd Qu.: 5630.2   3rd Qu.: 792.0  
 Max.   :733.0   Max.   :834.1   Max.   :22604.0   Max.   :1285.5  
      Agri            Service           Disp_Inc           RORP       
 Min.   :  527.2   Min.   :    5.2   Min.   : 11954   Min.   :0.2357  
 1st Qu.: 2255.3   1st Qu.: 1576.1   1st Qu.: 14966   1st Qu.:0.5968  
 Median : 3700.7   Median : 5932.5   Median : 18542   Median :0.6517  
 Mean   : 4705.9   Mean   : 8678.9   Mean   : 26133   Mean   :0.6298  
 3rd Qu.: 6312.9   3rd Qu.:11727.5   3rd Qu.: 21774   3rd Qu.:0.6850  
 Max.   :18328.5   Max.   :53160.0   Max.   :183252   Max.   :0.7606  
     ROREmp      
 Min.   :0.4545  
 1st Qu.:0.7565  
 Median :0.8300  
 Mean   :0.8037  
 3rd Qu.:0.8653  
 Max.   :0.9179

3.3.3 Performing relational join.

The code chunk below will be used to update the attribute table of Hunan’s SpatialPolygonsDataFrame with the attribute fields of Hunan2012 dataframe. This is performed by using left_join() of dplyr package.

hunan <- left_join(hunan,hunan2012)%>%
  select(1:4, 7, 15)
Joining, by = "County"
head(hunan, n=10)
Simple feature collection with 10 features and 6 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 110.4922 ymin: 26.28322 xmax: 114.2544 ymax: 30.12812
Geodetic CRS:  WGS 84
     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...

3.4 Visualizing Regional Development Indicator.

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

basemap <- tm_shape(hunan) +
  tm_polygons() +
  tm_text("NAME_3", size=0.28)

gdppc <- qtm(hunan, "GDPPC")
tmap_arrange(basemap, gdppc, widths = 1, asp=0, ncol=2,outer.margins = 0.01)

3.5 Computing Contiguity Spatial Weights.

In this section, we will learn how to use poly2nb() of spdep package to compute contiguity weight matrices for the study area. This function builds a neighbours list based on regions with contiguous boundaries. If we look at the documentation, we will see that we can pass a “queen” argument that takes TRUE or FALSE as options. If we do not specify this argument ,the default is set to TRUE, that is, if we don’t specify queen = FALSE, this function will return a list of first order neighbours using the Queen’s criteria.

3.5.1 Computing (QUEEN) contiguity based neighbors.

The code chunk below is used to compute Queen contiguity weight matrix.

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

The summary report above shows that there are 88 area units in Hunan. The most connected area unit has 11 neighbors. There are two area units with only one heighbours.

For each polygon in our polygon object, wm_q lists all neighboring polygons. For example, to see the neighbors for the first polygon in the object, type:

wm_q[[1]]
[1]  2  3  4 57 85
wm_q[[85]]
 [1]  1  2  3  5  6 32 56 57 69 75 78

Polygon 1 has 5 neighbors and polygon 85 has 11 neighbors. The numbers represent the polygon IDs as stored in hunan SpatialPolygonsDataFrame class.

We can retrive the county name of Polygon ID=1 by using the code chunk below:

hunan$County[1]
[1] "Anxiang"

To reveal the county names of the five neighboring polygons, the code chunk will be used:

hunan$NAME_3[c(2,3,4,57,85)]
[1] "Hanshou" "Jinshi"  "Li"      "Nan"     "Taoyuan"

We can retrieve the GDPPC of these five counties by using the code chunk below.

nb1 <- wm_q[[1]]
nb1 <- hunan$GDPPC[nb1]
nb1
[1] 20981 34592 24473 21311 22879

The printed output above shows that the GDPPC of the five nearest neighbours based on Queen’s method are 20981, 34592, 24473, 21311 and 22879 respectively.

We can display the complete weight matrix 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

3.5.2 Creating (ROOK) contiguity based neighbors.

The code chunk below is used to compute Rook contiguity weight 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

3.5.3 Visualising contiguity weights.

A connectivity graph takes a point and displays a line to each neighboring point. We are working with polygons at the moment, so we will need to get points in order to make our connectivity graphs. The most typically method for this will be polygon centroids. We will calculate these in the sf package before moving onto the graphs. Getting Latitude and Longitude of Polygon Centroids.

We will need points to associate with each polygon before we can make our connectivity graph. It will be a little more complicated than just running st_centroid on the

sf object: us.bound.

We need the coordinates in a separate data frame for this to work. To do this we will use a mapping function. The mapping function applies a given function to each element of a vector and returns a vector of the same length. Our input vector will be the geometry column of us.bound. Our function will be st_centroid. We will be using map_dbl variation of map from the purrr package. For more documentation, check out map documentation.

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

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

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

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

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

coords <- cbind(longitude, latitude)

We 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

3.5.3.1 Plotting Queen contiguity based neighbors map.

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

3.5.3.2 Plotting Rook contiguity based neighbors map.

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")

3.6 Computing distance based neighbors.

In this section, we will learn how to derive distance-based weight matrices by using dnearneigh() of spdep() package.

The function identifies neighbors 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=TRUE, great circle distances in km will be calculated assuming the WGS84 reference ellipsoid.

3.6.1 Determine the cut-off distance

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

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

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

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

  • Remove the list structure of the returned object by using unlist().

#coords <- coordinates(hunan)
k1 <- knn2nb(knearneigh(coords))
k1dists <- unlist(nbdists(k1, coords, longlat = TRUE))
summary(k1dists)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  24.79   32.57   38.01   39.07   44.52   61.79

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

3.6.2 Computing fixed distance weight matrix.

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

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

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

==> It is ratio of Number of non-zero links and Number of Regions

==> 324 / 88 = 3.681

Next, we will use str() to display the content of wm_d62 weight matrix.

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 is to combine table() and card() of spdep.

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

 1 
88

3.6.2.1 Plotting fixed distance weight matrix.

Next, we will plot the distance weight matrix by 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 neighbors and the black lines show the links of neighbors within the cut-off distance of 62km.

Alternatively, we can plot both of them next to each other by using the code chunk below.

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")

3.6.3 Computing 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 neighbors and the less densely settled areas (usually the rural counties) tend to have lesser neighbors. Having many neighbors smoothes the neighbor relationship across more neighbors.

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

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, we can display the content of the matrix by 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"

3.6.3.1 Plotting distance based neighbors

We can plot the weight matrix using the code chunk below.

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

3.7 Weights based on IDW

In this section, you will learn how to derive a spatial weight matrix based on Inversed Distance method.

First, we will 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

3.7.1 Row-standardised weights matrix.

Next, we need to assign weights to each neighboring polygon. In our case, each neighboring polygon will be assigned equal weight (style=“W”). This is accomplished by assigning the fraction 1/(#ofneighbors) to each neighboring county then summing the weighted income values. While this is the most intuitive way to summaries the neighbors’ values it has one drawback in that polygons along the edges of the study area will base their lagged values on fewer polygons thus potentially over- or under-estimating the true nature of the spatial auto-correlation in the data. For this example, we’ll stick with the style=“W” option for simplicity’s sake but note that other more robust options are available, notably style=“B”

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 zero.policy=TRUE option allows for lists of non-neighbors. This should be used with caution since the user may not be aware of missing neighbors in their dataset however, a zero.policy of FALSE would return an error.

To see the weight of the first polygon’s eight neighbors type:

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 weight. This means that when R computes the average neighboring income values, each neighbor’s income will be multiplied by 0.2 before being tallied.

Using the same method, we can also derive a row standardized distance weight matrix by using 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$weights[1]
[[1]]
[1] 0.01535405 0.03916350 0.01820896 0.02807922 0.01145113
summary(unlist(rswm_ids$weights))
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
0.008218 0.015088 0.018739 0.019614 0.022823 0.040338

3.8 Application of Spatial Weight Matrix.

In this section, we will learn how to create four different spatial lagged variables, they are:

  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

3.8.1 Spatial lag with row-standardized weights.

Finally, we’ll compute the average neighbor GDPPC value for each polygon. These values are 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

Recalled in the previous section, we retrieved the GDPPC of these five countries by using the code chunk below.

nb1 <- wm_q[[1]]
nb1 <- hunan$GDPPC[nb1]
nb1
[1] 20981 34592 24473 21311 22879

We can append the spatially lag GDPPC values onto 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)
Joining, by = "NAME_3"

The following table shows the average neighboring income values (stored in the Inc.lag object) 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, we will 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=0.6, ncol=2)
Some legend labels were too wide. These labels have been resized to 0.62. Increase legend.width (argument of tm_layout) to make the legend wider and therefore the labels larger.

3.8.2 Spatial lag as a sum of neighboring values.

We can calculate spatial lag as a sum of neighboring values by assigning binary weights. This requires us to go back to our neighbors list, then apply a function that will assign binary weights, then we use glist = in the nb2listw function to explicitly assign these weights.

We start by applying a function that will assign a value of 1 per each neighbor. This is done with lapply, which we have been using to manipulate the neighbors structure throughout the past notebooks. Basically it applies a function across each value in the neighbors structure.

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, we can use lag.listw to compute a lag variable from our weight 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")
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

Next, we will append the lag_sum GDPPC field into hunan sf data frame by using the code chunk below.

hunan <- left_join(hunan, lag.res)
Joining, by = "NAME_3"

Now, We can 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=0.65, ncol=2)
Some legend labels were too wide. These labels have been resized to 0.64, 0.64, 0.64, 0.64. Increase legend.width (argument of tm_layout) to make the legend wider and therefore the labels larger.

3.8.3 Spatial window average.

The spatial window average uses row-standardized weights and includes the diagonal element. To do this in R, we need to go back to the neighbors structure and add the diagonal element before assigning weights. To begin we assign k6 to a new variable because we will directly alter its structure to add the diagonal elements.

wm_q1 <- wm_q

To add the diagonal element to the neighbor list, we just need to use include.self() from spdep.

include.self(wm_q1)
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 536 
Percentage nonzero weights: 6.921488 
Average number of links: 6.090909

Now we obtain weights with nb2listw()

wm_q1 <- nb2listw(wm_q1)
wm_q1
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
lag_w_avg_gpdpc <- lag.listw(wm_q1, 
                             hunan$GDPPC)
lag_w_avg_gpdpc
 [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

Next, we will convert the lag variable listw object into a data.frame by using as.data.frame().

lag.list.wm_q1 <- list(hunan$NAME_3, lag.listw(wm_q1, hunan$GDPPC))
lag_wm_q1.res <- as.data.frame(lag.list.wm_q1)
colnames(lag_wm_q1.res) <- c("NAME_3", "lag_window_avg GDPPC")

Note: The third command line on the code chunk above renames the field names of lag_wm_q1.res object into NAME_3 and lag_window_avg GDPPC respectively.

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

hunan <- left_join(hunan, lag_wm_q1.res)
Joining, by = "NAME_3"

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

gdppc <- qtm(hunan, "GDPPC")
w_avg_gdppc <- qtm(hunan, "lag_window_avg GDPPC")
tmap_arrange(gdppc, w_avg_gdppc, asp=0.65, ncol=2)

Note: For more effective comparison, it is advicible to use the core tmap mapping functions.

3.8.4 Spatial window sum.

The spatial window sum is the counter part of the window average, but without using row-standardized weights. To do this we assign binary weights to the neighbor structure that includes the diagonal element.

wm_q1 <- wm_q

To add the diagonal element to the neighbor list, we just need to use include.self() from spdep.

include.self(wm_q1)
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 536 
Percentage nonzero weights: 6.921488 
Average number of links: 6.090909 
wm_q1
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 448 
Percentage nonzero weights: 5.785124 
Average number of links: 5.090909 
b_weights <- lapply(wm_q1, function(x) 0*x + 1)
b_weights[1]
[[1]]
[1] 1 1 1 1 1

Again, we use nb2listw() and glist() to explicitly assign weight values.

b_weights2 <- nb2listw(wm_q1, 
                       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 our new weight structure, we can 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] 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

Next, we will convert the lag variable listw object into a data.frame by using as.data.frame().

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

Note: The second command line on the code chunk above renames the field names of w_sum_gdppc.res object into NAME_3 and w_sum GDPPC respectively.

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

hunan <- left_join(hunan, w_sum_gdppc.res)
Joining, by = "NAME_3"

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

gdppc <- qtm(hunan, "GDPPC")
w_sum_gdppc <- qtm(hunan, "w_sum GDPPC")
tmap_arrange(gdppc, w_sum_gdppc, asp=0.65, ncol=2)
Some legend labels were too wide. These labels have been resized to 0.64, 0.64, 0.64, 0.64. Increase legend.width (argument of tm_layout) to make the legend wider and therefore the labels larger.

Note: For more effective comparison, it is advisable to use the core tmap mapping functions.