Hands-on Exercise 2: Spatial Weights and Application

Author

Muhamad Ameer Noor

Published

November 24, 2023

Modified

December 14, 2023

Weighing Space Illustration

1 Overview

Spatial analysis is a method used to understand the significance of spatial relationships between different objects. It’s like figuring out how different pieces on a chessboard influence each other’s moves. Spatial weights are concepts that help us measure and analyze how different locations or regions are related to each other based on their proximity, similarity, or interaction. Spatial weights are numerical values that represent the strength or intensity of the connection between two spatial units, such as points, polygons, or pixels. Applications of spatial weights include detecting patterns, clusters, outliers, hot spots, or cold spots in spatial data, and testing hypotheses about spatial processes or phenomena. summarized from: Getis, 2010

The data used for practice in this exercise includes a map outlining the boundaries of Hunan county, presented as a geospatial dataset in ESRI shapefile format, and a CSV file named “Hunan_2012.csv,” which includes specific local development indicators for Hunan in the year 2012.

This exercise will help to get familiar with importing geospatial data using functions from the sf package, reading CSV files with functions from the readr package, conducting relational joins through functions from the dplyr package, computing spatial weights calculating spatially lagged variables using functions from the spdep package.

2 Preparing the Library and Data

The following code chunk will import the required library:

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

The following panel will show how the data is imported and joined

the following code use st_read() from sf package to import Hunan shapefile into simple features Object

hunan <- st_read(dsn = "../data/geospatial", layer = "Hunan")
Reading layer `Hunan' from data source `C:\ameernoor\ISSS624\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
glimpse(hunan)
Rows: 88
Columns: 8
$ NAME_2     <chr> "Changde", "Changde", "Changde", "Changde", "Changde", "Cha…
$ ID_3       <int> 21098, 21100, 21101, 21102, 21103, 21104, 21109, 21110, 211…
$ NAME_3     <chr> "Anxiang", "Hanshou", "Jinshi", "Li", "Linli", "Shimen", "L…
$ ENGTYPE_3  <chr> "County", "County", "County City", "County", "County", "Cou…
$ Shape_Leng <dbl> 1.869074, 2.360691, 1.425620, 3.474325, 2.289506, 4.171918,…
$ Shape_Area <dbl> 0.10056190, 0.19978745, 0.05302413, 0.18908121, 0.11450357,…
$ County     <chr> "Anxiang", "Hanshou", "Jinshi", "Li", "Linli", "Shimen", "L…
$ geometry   <POLYGON [°]> POLYGON ((112.0625 29.75523..., POLYGON ((112.2288 …

the following code use read_csv() from readr package to import

hunan2012 <- read_csv("../data/aspatial/Hunan_2012.csv")
glimpse(hunan2012)
Rows: 88
Columns: 29
$ County      <chr> "Anhua", "Anren", "Anxiang", "Baojing", "Chaling", "Changn…
$ City        <chr> "Yiyang", "Chenzhou", "Changde", "Hunan West", "Zhuzhou", …
$ avg_wage    <dbl> 30544, 28058, 31935, 30843, 31251, 28518, 54540, 28597, 33…
$ deposite    <dbl> 10967.0, 4598.9, 5517.2, 2250.0, 8241.4, 10860.0, 24332.0,…
$ FAI         <dbl> 6831.7, 6386.1, 3541.0, 1005.4, 6508.4, 7920.0, 33624.0, 1…
$ Gov_Rev     <dbl> 456.72, 220.57, 243.64, 192.59, 620.19, 769.86, 5350.00, 1…
$ Gov_Exp     <dbl> 2703.0, 1454.7, 1779.5, 1379.1, 1947.0, 2631.6, 7885.5, 11…
$ GDP         <dbl> 13225.0, 4941.2, 12482.0, 4087.9, 11585.0, 19886.0, 88009.…
$ GDPPC       <dbl> 14567, 12761, 23667, 14563, 20078, 24418, 88656, 10132, 17…
$ GIO         <dbl> 9276.90, 4189.20, 5108.90, 3623.50, 9157.70, 37392.00, 513…
$ Loan        <dbl> 3954.90, 2555.30, 2806.90, 1253.70, 4287.40, 4242.80, 4053…
$ NIPCR       <dbl> 3528.3, 3271.8, 7693.7, 4191.3, 3887.7, 9528.0, 17070.0, 3…
$ Bed         <dbl> 2718, 970, 1931, 927, 1449, 3605, 3310, 582, 2170, 2179, 1…
$ Emp         <dbl> 494.310, 290.820, 336.390, 195.170, 330.290, 548.610, 670.…
$ EmpR        <dbl> 441.4, 255.4, 270.5, 145.6, 299.0, 415.1, 452.0, 127.6, 21…
$ EmpRT       <dbl> 338.0, 99.4, 205.9, 116.4, 154.0, 273.7, 219.4, 94.4, 174.…
$ Pri_Stu     <dbl> 54.175, 33.171, 19.584, 19.249, 33.906, 81.831, 59.151, 18…
$ Sec_Stu     <dbl> 32.830, 17.505, 17.819, 11.831, 20.548, 44.485, 39.685, 7.…
$ Household   <dbl> 290.4, 104.6, 148.1, 73.2, 148.7, 211.2, 300.3, 76.1, 139.…
$ Household_R <dbl> 234.5, 121.9, 135.4, 69.9, 139.4, 211.7, 248.4, 59.6, 110.…
$ NOIP        <dbl> 101, 34, 53, 18, 106, 115, 214, 17, 55, 70, 44, 84, 74, 17…
$ Pop_R       <dbl> 670.3, 243.2, 346.0, 184.1, 301.6, 448.2, 475.1, 189.6, 31…
$ RSCG        <dbl> 5760.60, 2386.40, 3957.90, 768.04, 4009.50, 5220.40, 22604…
$ Pop_T       <dbl> 910.8, 388.7, 528.3, 281.3, 578.4, 816.3, 998.6, 256.7, 45…
$ Agri        <dbl> 4942.253, 2357.764, 4524.410, 1118.561, 3793.550, 6430.782…
$ Service     <dbl> 5414.5, 3814.1, 14100.0, 541.8, 5444.0, 13074.6, 17726.6, …
$ Disp_Inc    <dbl> 12373, 16072, 16610, 13455, 20461, 20868, 183252, 12379, 1…
$ RORP        <dbl> 0.7359464, 0.6256753, 0.6549309, 0.6544614, 0.5214385, 0.5…
$ ROREmp      <dbl> 0.8929619, 0.8782065, 0.8041262, 0.7460163, 0.9052651, 0.7…

The following code use left_join() from dplyr package to merge the aspatial data to the geospatial data

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

this section will explore distribution of Gross Domestic Product Per Capita (GDPPC) 2012 in Hunan by creating base map and build choropleth map. qtm() from tmap package is used to build the map.

Code 
# Creating The Basemap
basemap <- tm_shape(hunan) +
  tm_polygons() +
  tm_text("NAME_3", size = 0.5) +
  tm_layout(main.title = "Basemap", main.title.position = "left")  # Add title

# Creating The Choropleth Map
gdppc <- qtm(hunan, "GDPPC") +
  tm_layout(main.title = "Choropleth Map", main.title.position = "left",
            legend.outside = TRUE, legend.outside.position = 'right')  # adjust the legend

# show the map
tmap_arrange(basemap, gdppc, asp=1, ncol=2, widths = c(0.4,0.6))

4 Computing Contiguity Spatial Weights

Contiguity Spatial Weights are used in spatial data analysis to understand how close or connected different geographic areas are to each other. Simply put, if two areas, like counties or neighborhoods, share a border, they’re considered “contiguous” or neighbors. This concept is important for understanding patterns like how a phenomenon in one area might affect neighboring areas. Two main criteria are used to define contiguity: ‘rook’ and ‘queen’. Rook contiguity means areas are neighbors if they share a common edge. Queen contiguity is a bit broader, including areas that share either a common edge or a corner. This is akin to the movements of rook and queen pieces in chess Summarized from: Anselin

Queen vs Rook Contiguity

Source: Research Gate

This section explore poly2nb() from spdep package to compute contiguity weight matrices. The function builds a neighbours list based on regions with contiguous boundaries. Using “queen” parameter that takes TRUE or FALSE as options, if it is set to TRUE, the function will return a list of first order neighbours using the Queen criteria.

wm_q <- poly2nb(hunan, queen=TRUE)
summary(wm_q)
Neighbour list object:
Number of regions: 88 
Number of nonzero links: 448 
Percentage nonzero weights: 5.785124 
Average number of links: 5.090909 
Link number distribution:

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

The output summarizes the spatial relationships in Hunan using Queen’s contiguity method. There are 88 regions, and the analysis reveals a total of 448 connections among them. The percentage of nonzero weights, indicating connected regions, is approximately 5.79%. On average, each region has around 5.09 links with other regions. The distribution of links shows that most regions have 4 or 5 connections, with the least connected regions being 30 and 65, each having only 1 link. The most connected region is labeled as 85, with 11 links.

to list all neighboring polygons of a unit, use wm_q as shown in the following code, where 1 represent the polygon Unit ID being shown, and the output shows the 5 negiboring polygon Unit ID

Code 
wm_q[[1]]
[1]  2  3  4 57 85

to retrieve the name of the county, use the following code

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

to retrieve county names of more than one polygons, use the following example that display the neigbor of Anxiang

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

additionally the GDDPC data of multiple countries can also be displayed using the following code

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

to display the complete weight matrix which represent the neigbors of each region, use the following code

Code 
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

Similar to the example of Queen method, Rook method can be executed by changing queen parameter to False

Code 
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

As expected from the stricter condition of Rook compared to Queen, the regions will have less neighbor on average

To create a connectivity graph, we first need to represent polygons as points. In our case, we’re working with polygons, so we’ll use polygon centroids as points for our graph. The common approach is to calculate these centroids using the sf package. To achieve this, we employ the st_centroid function on the geometry column of our spatial object (in this case, hunan). Since we require the coordinates in a separate data frame, we utilize a mapping function. This function applies st_centroid to each element of the geometry column and returns a vector of the same length. We specifically use the map_dbl variation from the purrr package. For latitude and longitude values, we extract them using double bracket notation, [[1]] for longitude and [[2]] for latitude. Finally, we combine these coordinates into a single object using cbind(), and we verify the formatting by checking the first few observations using head().

Code 
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

Next, the following code will be used to display and compare Queen and Rook contiguity neighbours maps

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

5 Computing Distance Based Neighbours

In this part, we will explore how to figure out which areas are close to each other using distances, by utilizing dnearneigh() function from the spdep package.

This function looks at points on a map and finds their neighbors based on how far apart they are. Range of distances can be set using bounds argument, with a lower limit d1= and an upper limit d2=. If the locations are given in regular coordinates (like x and y on a typical map) and latitude and longitude argument set to true (longlat=TRUE), the function measures distances in kilometers. It does this as if by figuring out how far it is on the Earth’s surface, using something called the WGS84 reference ellipsoid.

The WGS84 reference ellipsoid is a mathematical model that approximates the shape of the Earth. It’s not a perfect sphere but more like a slightly squashed ball, wider at the equator than at the poles. When measuring distances using this model, it considers the Earth’s curvature. This method provides a more accurate way to measure real distances on the Earth’s surface, especially over long distances where the Earth’s curvature becomes significant. It’s like tracing a line along the surface of an orange, rather than cutting straight through it.

The following part will explore how to find the right distance cut-off, fixed distance calculation, and adaptive distance calculation.

5.1 Determine the cut-off distance

To find the right distance for the analysis, execute the following steps:

  • Use knearneigh() to get a list of indices representing the k nearest neighbors for each point.
  • Convert this list into a neighbor list with knn2nb().
  • Find the lengths of these neighbor relationships with nbdists(). Remove any complex structure with unlist().
Code 
#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 shows that the maximum distance to the first nearest neighbor is 61.79 km. Use it as threshold to ensure each unit has at least one neighbor.

5.2 Computing fixed distance weight matrix

Based on the previous knowledge, create the distance weight matrix using the specified distance range (0 to 62 km).

Code 
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 “Average number of links: 3.681818” means that, on average, each location is linked to approximately 3.68 other locations within the specified distance range.

We can inspect the structure of the weight matrix using str() or combining table() and card() of spdep.

Code 
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

the output shows for who are the neighbors of each county (shown in unit ID list per row)

Code 
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
Code 
n_comp <- n.comp.nb(wm_d62)
table(n_comp$comp.id)

 1 
88

the table shows, for each county, how many neighbors it has.

Overlapping Visualization

The red lines represent 1st nearest neighbors, while the black lines are links within the 62 km cut-off distance.

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

Side by Side Visualization

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

5.3 Computing adaptive distance weight matrix

Using fixed distance, densely settled urban areas tend to have more neigbours compared to rural. Having many neighbours smoothes the neighbour relationship across more neighbours. Number of neighbors can be adapted by accepting asymmetric neighbours or imposing symmetry.

The following code chunk impose 6 neighbors in the argument, hence the average number of links is 6 as well.

Code 
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

Visualize the weight matrix.

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

6 Weights based on IDW

Another method to derive spatial weight matrix is based on Inversed Distance method (IDW).

Compute distance of areas using nbdists() of spdep.

Code 
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

Assign equal weights (style=“W”) to neighboring polygons. It’s calculated by assigning the fraction 1/(#ofneighbors) to each neighboring county then summing the weighted income values.

One downside of using this approach is that regions at the edges of the study area might rely on fewer neighboring regions. This could lead to either overestimating or underestimating the real spatial connections in the data.

For a stronger and more reliable choice, you can use “style=B.”

Code 
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

Be careful when setting zero.policy=TRUE because it lets you have lists of regions that are not neighbors. This can be risky because you might not notice if some neighbors are missing in your data. On the other hand, using zero.policy=FALSE would result in an error if there are missing neighbors.

Check the weight of the first polygon’s eight neighbors with the following code chunk.

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

Each neighbor gets a share of 0.125 from the total weight. This implies that when R calculates the average income of neighboring areas, it multiplies each neighbor’s income by 0.2 before adding them up.

We can apply a similar approach to create a distance weight matrix that is standardized by rows.

Code 
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
Code 
rswm_ids$weights[1]
[[1]]
[1] 0.01535405 0.03916350 0.01820896 0.02807922 0.01145113
Code 
summary(unlist(rswm_ids$weights))
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
0.008218 0.015088 0.018739 0.019614 0.022823 0.040338

7 Application of Spatial Weight Matrix

This part will explore how to create four types of spatial lagged variables as shown in the panel.

Code 
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

Recalling the GDPPC values obtained earlier for these five counties

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

Spatial Lag with Row-Standardized Weights method measures how much an observation at one location is influenced by observations at neighboring locations. The spatial lag is calculated as a weighted average, where the weights are standardized so that they add up to one for each location. This means that each location’s value is influenced equally by its neighbors, creating a balanced representation of neighboring influence. summarized from: Anselin

We can add the spatially lagged GDPPC values to the hunan sf data frame using the following code:

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

The table below shows the average neighboring income values for each region.

Code 
head(hunan)
Simple feature collection with 6 features and 7 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 110.4922 ymin: 28.61762 xmax: 112.3013 ymax: 30.12812
Geodetic CRS:  WGS 84
   NAME_2  ID_3  NAME_3   ENGTYPE_3  County GDPPC lag GDPPC
1 Changde 21098 Anxiang      County Anxiang 23667  24847.20
2 Changde 21100 Hanshou      County Hanshou 20981  22724.80
3 Changde 21101  Jinshi County City  Jinshi 34592  24143.25
4 Changde 21102      Li      County      Li 24473  27737.50
5 Changde 21103   Linli      County   Linli 25554  27270.25
6 Changde 21104  Shimen      County  Shimen 27137  21248.80
                        geometry
1 POLYGON ((112.0625 29.75523...
2 POLYGON ((112.2288 29.11684...
3 POLYGON ((111.8927 29.6013,...
4 POLYGON ((111.3731 29.94649...
5 POLYGON ((111.6324 29.76288...
6 POLYGON ((110.8825 30.11675...

Next, plot both GDPPC and spatially lagged GDPPC for comparison.

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

using row-standardized weights, the distribution of lagged GDPPC on the right shows how neighboring countries becomes more similar. note that some region which was originally much richer than it’s neighbors, becomes poorer than its neighbors while it’s neighbor becomes richer. this indicates caution when using the row-standardized weights

We can calculate spatial lag as a sum of neighboring values using binary weights. This involves going back to the neighbors list, applying a function to assign binary weights, and explicitly assigning these weights in the nb2listw function.

We start by assigning a value of 1 to each neighbor using lapply:

Code 
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, compute the lag variable from our weights and GDPPC.

Code 
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")

examine the result:

Code 
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

This method involves summing up the values of neighboring observations to calculate the spatial lag. Unlike the row-standardized method, this doesn’t involve any kind of averaging or standardization, so the total influence is simply the sum of the influences from each neighbor. This approach is particularly useful when dealing with binary data (like 0 or 1 values). summarized from: Anselin

append the lag_sum GDPPC field to the hunan sf data frame:

Code 
hunan <- left_join(hunan, lag.res)

plot both GDPPC and Spatial Lag Sum GDPPC for comparison.

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

Spatial window average uses row-standardized weights and includes the diagonal element. To achieve this in R, we need to add the diagonal element to the neighbors’ structure before assigning weights.

Add the diagonal element using include.self() from spdep

Code 
wm_qs <- include.self(wm_q)

obtain weights with nb2listw()

Code 
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

create the lag variable from our weight structure and GDPPC variable:

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

convert the lag variable listw object into a data.frame:

Code 
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")

append lag_window_avg GDPPC values to hunan:

Code 
hunan <- left_join(hunan, lag_wm_qs.res)

compare the values of lag GDPPC and Spatial window average by using kable()

Code 
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…

use qtm() to plot the lag_gdppc and w_ave_gdppc maps next to each other for quick comparison:

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

This concept extends the idea of spatial lag by including the observation itself in the average calculation. It’s like creating a window that includes the value at a specific location and its neighbors, and then computing the average of all these values. This method is useful when you want to take into account both the value at a specific point and the influence of its surroundings. summarized from: Anselin

Spatial window sum is similar to window average but without using row-standardized weights.

Let’s add the diagonal element to the neighbor list:

Code 
wm_qs <- include.self(wm_q)

Next, we assign binary weights:

Code 
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

Now, we can compute the lag variable with lag.listw():

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

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

Next, we convert the lag variable listw object into a data.frame:

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

Now, we append w_sum GDPPC values to hunan:

Code 
hunan <- left_join(hunan, w_sum_gdppc.res)

use kable() To compare the values of lag GDPPC and Spatial window average

Code 
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…

Finally, we’ll use qtm() to plot the lag_sum GDPPC and w_sum_gdppc maps next to each other for quick comparison:

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

This is similar to the spatial window average, but instead of averaging the values, it sums them up. This method calculates the total value by adding the value at a specific location to the sum of its neighboring values. It provides a more cumulative measure of spatial influence compared to the average. summarized from: Anselin

8 References