Matrices of influence

Note: Document prepared for Spatial Socioeconometric Modeling and materials build on Anselin (2020)

1 Matrices of influence

• Our last couple of weeks’ discussions dealt with objects, features and attributes

  • This week’s topic deal with what is probably the most controversial issue in spatial modeling

• Spatial modeling relies on the establishment of neighbors or connections

• These neighbors serve to model spatial dependence AKA as spatial autocorrelation

  • With these connections one can model the first law of geography: everything is related, but near things are more related than distant things (Tobler, 1930)

1.1 Approaches

1.1.1 Contiguity-Based Spatial Weights

• Contiguity means that two spatial units share a common border of non-zero length

  • Rook criterion defines neighbors by the existence of a common edge between two spatial units.

  • The queen criterion is somewhat more encompassing and defines neighbors as spatial units sharing a common edge or a common vertex.

  • Finally, the Bishop’s approach just relies on a common vertex (not available anymore, but conceptually works as follows)

Spatial weights matrix

• Distance-based spatial weights
  • These are employed with points
  • Two main forms: distance-based (radii-based) and number-based (kth-neighbor-based)
• Spatial weights are critically dependent on the quality of the spatial data source from which they are constructed.

2 Data management procedures

Once more, we would need to access data from repositories hosted in the internet

Specifically, we will rely on the following packages

install.packages("tigris")
install.packages("acs")
install.packages("spdep")
install.pacakges("tmaptools")
library(spdep)
library(tigris)
options(tigris_use_cache = TRUE)
library(acs)
library(stringr)
library(tmaptools)
library(plyr)

Let’s work with a map of the USA

2.1 Reading polygons at the country level

states <- states(cb = TRUE, class="sp")

## Warning in proj4string(obj): CRS object has comment, which is lost in output

"%ni%" <- Negate("%in%")
plot(states)

Geographic data obtained from Walker (2020)

Limit to the contiguous USA

states<-states[states$NAME %ni% c('Alaska','American Samoa','Commonwealth of the Northern Mariana Islands','Guam','Hawaii','United States Virgin Islands','Puerto Rico'),]
plot(states)

Geographic data obtained from Walker (2020)

Now let’s create our first weight matrices

m1f<-poly2nb(states, queen=TRUE)
m1f<- nb2listw(m1f,zero.policy=T)
coords<-coordinates(states)
str(m1f)

## List of 3
##  $ style     : chr "W"
##  $ neighbours:List of 49
##   ..$ : int [1:2] 15 16
##   ..$ : int [1:4] 12 23 26 32
##   ..$ : int [1:4] 12 32 33 39
##   ..$ : int [1:5] 7 18 36 44 45
##   ..$ : int [1:2] 16 43
##   ..$ : int 34
##   ..$ : int [1:2] 4 44
##   ..$ : int [1:2] 30 49
##   ..$ : int [1:6] 12 19 23 25 33 42
##   ..$ : int [1:2] 26 40
##   ..$ : int [1:5] 13 20 21 35 42
##   ..$ : int [1:6] 2 3 9 23 32 33
##   ..$ : int [1:4] 11 17 21 46
##   ..$ : int [1:3] 27 35 40
##   ..$ : int [1:4] 1 16 22 41
##   ..$ : int [1:5] 1 5 15 22 43
##   ..$ : int [1:6] 13 19 21 22 41 46
##   ..$ : int [1:6] 4 31 36 38 45 47
##   ..$ : int [1:8] 9 17 21 22 25 28 33 37
##   ..$ : int [1:6] 11 23 26 27 35 42
##   ..$ : int [1:6] 11 13 17 19 25 42
##   ..$ : int [1:8] 15 16 17 19 37 41 43 44
##   ..$ : int [1:6] 2 9 12 20 26 42
##   ..$ : int [1:4] 28 37 38 48
##   ..$ : int [1:4] 9 19 21 42
##   ..$ : int [1:6] 2 10 20 23 27 40
##   ..$ : int [1:5] 14 20 26 35 40
##   ..$ : int [1:5] 19 24 33 37 39
##   ..$ : int [1:3] 34 47 49
##   ..$ : int [1:3] 8 47 49
##   ..$ : int [1:3] 18 36 47
##   ..$ : int [1:3] 2 3 12
##   ..$ : int [1:6] 3 9 12 19 28 39
##   ..$ : int [1:3] 6 29 49
##   ..$ : int [1:5] 11 14 20 27 42
##   ..$ : int [1:3] 4 18 31
##   ..$ : int [1:7] 19 22 24 28 38 44 45
##   ..$ : int [1:5] 18 24 37 45 48
##   ..$ : int [1:4] 3 28 33 48
##   ..$ : int [1:4] 10 14 26 27
##   ..$ : int [1:4] 15 17 22 46
##   ..$ : int [1:7] 9 11 20 21 23 25 35
##   ..$ : int [1:4] 5 16 22 44
##   ..$ : int [1:6] 4 7 22 37 43 45
##   ..$ : int [1:5] 4 18 37 38 44
##   ..$ : int [1:3] 13 17 41
##   ..$ : int [1:5] 18 29 30 31 49
##   ..$ : int [1:3] 24 38 39
##   ..$ : int [1:5] 8 29 30 34 47
##   ..- attr(*, "class")= chr "nb"
##   ..- attr(*, "region.id")= chr [1:49] "0" "2" "3" "4" ...
##   ..- attr(*, "call")= language poly2nb(pl = states, queen = TRUE)
##   ..- attr(*, "type")= chr "queen"
##   ..- attr(*, "sym")= logi TRUE
##  $ weights   :List of 49
##   ..$ : num [1:2] 0.5 0.5
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:2] 0.5 0.5
##   ..$ : num 1
##   ..$ : num [1:2] 0.5 0.5
##   ..$ : num [1:2] 0.5 0.5
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:2] 0.5 0.5
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:8] 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:8] 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:7] 0.143 0.143 0.143 0.143 0.143 ...
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:7] 0.143 0.143 0.143 0.143 0.143 ...
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..- attr(*, "mode")= chr "binary"
##   ..- attr(*, "W")= logi TRUE
##   ..- attr(*, "comp")=List of 1
##   .. ..$ d: num [1:49] 2 4 4 5 2 1 2 2 6 2 ...
##  - attr(*, "class")= chr [1:2] "listw" "nb"
##  - attr(*, "region.id")= chr [1:49] "0" "2" "3" "4" ...
##  - attr(*, "call")= language nb2listw(neighbours = m1f, zero.policy = T)

Now we can plot:

plot(states, border="grey")
plot(m1f, coords, add=TRUE)

Geographic data obtained from Walker (2020)

To compare we need to repeat

m2f<-poly2nb(states, queen=FALSE)
m2f<- nb2listw(m2f,zero.policy=T)
coords<-coordinates(states)
str(m2f)

## List of 3
##  $ style     : chr "W"
##  $ neighbours:List of 49
##   ..$ : int [1:2] 15 16
##   ..$ : int [1:4] 12 23 26 32
##   ..$ : int [1:4] 12 32 33 39
##   ..$ : int [1:5] 7 18 36 44 45
##   ..$ : int [1:2] 16 43
##   ..$ : int 34
##   ..$ : int [1:2] 4 44
##   ..$ : int [1:2] 30 49
##   ..$ : int [1:6] 12 19 23 25 33 42
##   ..$ : int [1:2] 26 40
##   ..$ : int [1:4] 13 21 35 42
##   ..$ : int [1:6] 2 3 9 23 32 33
##   ..$ : int [1:4] 11 17 21 46
##   ..$ : int [1:3] 27 35 40
##   ..$ : int [1:4] 1 16 22 41
##   ..$ : int [1:5] 1 5 15 22 43
##   ..$ : int [1:6] 13 19 21 22 41 46
##   ..$ : int [1:6] 4 31 36 38 45 47
##   ..$ : int [1:8] 9 17 21 22 25 28 33 37
##   ..$ : int [1:5] 23 26 27 35 42
##   ..$ : int [1:6] 11 13 17 19 25 42
##   ..$ : int [1:8] 15 16 17 19 37 41 43 44
##   ..$ : int [1:6] 2 9 12 20 26 42
##   ..$ : int [1:4] 28 37 38 48
##   ..$ : int [1:4] 9 19 21 42
##   ..$ : int [1:6] 2 10 20 23 27 40
##   ..$ : int [1:5] 14 20 26 35 40
##   ..$ : int [1:5] 19 24 33 37 39
##   ..$ : int [1:3] 34 47 49
##   ..$ : int [1:3] 8 47 49
##   ..$ : int [1:3] 18 36 47
##   ..$ : int [1:3] 2 3 12
##   ..$ : int [1:6] 3 9 12 19 28 39
##   ..$ : int [1:3] 6 29 49
##   ..$ : int [1:4] 11 14 20 27
##   ..$ : int [1:3] 4 18 31
##   ..$ : int [1:7] 19 22 24 28 38 44 45
##   ..$ : int [1:5] 18 24 37 45 48
##   ..$ : int [1:4] 3 28 33 48
##   ..$ : int [1:4] 10 14 26 27
##   ..$ : int [1:4] 15 17 22 46
##   ..$ : int [1:6] 9 11 20 21 23 25
##   ..$ : int [1:4] 5 16 22 44
##   ..$ : int [1:6] 4 7 22 37 43 45
##   ..$ : int [1:5] 4 18 37 38 44
##   ..$ : int [1:3] 13 17 41
##   ..$ : int [1:5] 18 29 30 31 49
##   ..$ : int [1:3] 24 38 39
##   ..$ : int [1:5] 8 29 30 34 47
##   ..- attr(*, "class")= chr "nb"
##   ..- attr(*, "region.id")= chr [1:49] "0" "2" "3" "4" ...
##   ..- attr(*, "call")= language poly2nb(pl = states, queen = FALSE)
##   ..- attr(*, "type")= chr "rook"
##   ..- attr(*, "sym")= logi TRUE
##  $ weights   :List of 49
##   ..$ : num [1:2] 0.5 0.5
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:2] 0.5 0.5
##   ..$ : num 1
##   ..$ : num [1:2] 0.5 0.5
##   ..$ : num [1:2] 0.5 0.5
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:2] 0.5 0.5
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:8] 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:8] 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:7] 0.143 0.143 0.143 0.143 0.143 ...
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:4] 0.25 0.25 0.25 0.25
##   ..$ : num [1:6] 0.167 0.167 0.167 0.167 0.167 ...
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..$ : num [1:3] 0.333 0.333 0.333
##   ..$ : num [1:5] 0.2 0.2 0.2 0.2 0.2
##   ..- attr(*, "mode")= chr "binary"
##   ..- attr(*, "W")= logi TRUE
##   ..- attr(*, "comp")=List of 1
##   .. ..$ d: num [1:49] 2 4 4 5 2 1 2 2 6 2 ...
##  - attr(*, "class")= chr [1:2] "listw" "nb"
##  - attr(*, "region.id")= chr [1:49] "0" "2" "3" "4" ...
##  - attr(*, "call")= language nb2listw(neighbours = m2f, zero.policy = T)

plot(states, border="grey")
plot(m1f, coords, add=TRUE)
plot(m2f, coords, add=TRUE,  col="red")

Geographic data obtained from Walker (2020)

3 Point matrix weights

For this section we will access all colleges and universities in Philly

Distances in Philly, see González Canché (2019)

3.1 Row standardization with Radii

Standardization with Radii, see González Canché (2019)

3.2 Row standardization with kth

Standardization with kth, see González Canché (2019)

library(tmaptools)
a<-getwd()
url18 <- paste('http://nces.ed.gov/ipeds/datacenter/data/HD2018.zip')
download.file(url18, destfile = paste(a,"HD2018.zip",sep="/"))
d18<- read.csv(unz("HD2018.zip", "hd2018.csv"))

#keeping only five first digits of a zip code
d18$ZIP<- as.numeric(substr(d18$ZIP, 1, 5))
#Subsetting the file with institutions located in Philadelphia, these zip codes were retrieved from Zillow
d18 <-d18[d18$ZIP %in% c(19126, 19125,  19128,  19127,  19130,  19129,  19132,  19131,  19134,  19133,  19136,  19135,  19138,  19137,  19140,  19139,  19142,  19141,  19144,  19143,  19146,  19145,  19148,  19147,  19150,  19149,  19152,  19151,  19154,  19153,  19155,  19170,  19173,  19019,  19176,  19187,  19192,  19092,  19101,  19099,  19103,  19102,  19105,  19104,  19107,  19106,  19109,  19111,  19110,  19112,  19115,  19114,  19118,  19116,  19120,  19119,  19122,  19121,  19124,  19123),] 
#Looking at the dimensions of the subsetted dataset
dim(d18)

## [1] 37 73

table(d18$SECTOR)

## 
##  1  2  3  4  5  6  8  9 
##  1 17  4  1  3  5  1  5

d18<-d18[d18$SECTOR<3,c("UNITID", "INSTNM", "SECTOR", "LONGITUD", "LATITUDE")]
d18#Print Table 

#This will create the object coords as shown in equation (17)
coords = cbind(d18$LONGITUD, d18$LATITUDE)

Print distances:

dist <- spDists(coords, coords, longlat=T)
#Moving from KM to Miles
dist <- dist * 0.621371192237
dist #Prints distance table with actual data

##             [,1]      [,2]       [,3]       [,4]      [,5]     [,6]       [,7]
##  [1,]  0.0000000 10.012459  0.1712984  1.0006310 12.614185 6.306948  0.6350112
##  [2,] 10.0124592  0.000000 10.0012119  9.3062647 12.969018 5.237701  9.4395681
##  [3,]  0.1712984 10.001212  0.0000000  1.1159608 12.450388 6.219961  0.5718978
##  [4,]  1.0006310  9.306265  1.1159608  0.0000000 13.020336 6.038045  0.9614580
##  [5,] 12.6141852 12.969018 12.4503877 13.0203357  0.000000 8.970754 12.1660606
##  [6,]  6.3069480  5.237701  6.2199605  6.0380451  8.970754 0.000000  5.6724469
##  [7,]  0.6350112  9.439568  0.5718978  0.9614580 12.166061 5.672447  0.0000000
##  [8,]  0.4286572 10.295565  0.3296307  1.4273361 12.415880 6.420435  0.8567019
##  [9,]  0.7632002  9.684695  0.6094543  1.4124608 11.855834 5.704068  0.4690691
## [10,]  1.0907076  9.553368  1.2392903  0.3598743 13.356296 6.385156  1.2330556
## [11,]  5.2657047  4.806120  5.2380874  4.6705236 11.239352 2.343632  4.6691159
## [12,]  0.4377258 10.268306  0.3231432  1.4314644 12.374185 6.381259  0.8314790
## [13,]  4.5254724  5.690013  4.5542238  3.6950408 12.771047 4.048722  4.0410401
## [14,]  1.7697204  9.753059  1.9335857  1.0858638 14.106200 6.935817  1.9937067
## [15,]  4.6954162  6.391158  4.7722529  3.7275798 14.118122 5.430496  4.3597156
## [16,]  4.8623453  6.805905  4.9575557  3.8682887 14.718357 6.054051  4.5979672
## [17,]  2.4201742  8.261752  2.3069789  2.4816658 10.592531 3.967679  1.8101581
## [18,]  0.8217110 10.261436  0.6629135  1.7544007 11.964251 6.195292  0.9533728
##              [,8]       [,9]      [,10]     [,11]       [,12]     [,13]
##  [1,]  0.42865719  0.7632002  1.0907076  5.265705  0.43772580  4.525472
##  [2,] 10.29556481  9.6846953  9.5533676  4.806120 10.26830573  5.690013
##  [3,]  0.32963066  0.6094543  1.2392903  5.238087  0.32314315  4.554224
##  [4,]  1.42733611  1.4124608  0.3598743  4.670524  1.43146445  3.695041
##  [5,] 12.41588049 11.8558344 13.3562957 11.239352 12.37418545 12.771047
##  [6,]  6.42043470  5.7040683  6.3851561  2.343632  6.38125902  4.048722
##  [7,]  0.85670191  0.4690691  1.2330556  4.669116  0.83147901  4.041040
##  [8,]  0.00000000  0.7207223  1.5118626  5.519665  0.04374115  4.875763
##  [9,]  0.72072225  0.0000000  1.6500805  4.887682  0.67989128  4.395942
## [10,]  1.51186259  1.6500805  0.0000000  4.964794  1.52610354  3.904395
## [11,]  5.51966471  4.8876819  4.9647938  0.000000  5.49007527  1.757560
## [12,]  0.04374115  0.6798913  1.5261035  5.490075  0.00000000  4.857718
## [13,]  4.87576271  4.3959421  3.9043952  1.757560  4.85771823  0.000000
## [14,]  2.16059955  2.3992091  0.7618053  5.330715  2.18338801  4.063047
## [15,]  5.10124880  4.7904692  3.8178285  3.119028  5.09468693  1.383153
## [16,]  5.28181268  5.0481222  3.9017276  3.741531  5.28025566  2.006011
## [17,]  2.47085672  1.7501719  2.8362542  3.483868  2.43003711  3.563742
## [18,]  0.46961740  0.5812735  1.9015484  5.461316  0.43366595  4.967533
##            [,14]      [,15]      [,16]     [,17]      [,18]
##  [1,]  1.7697204  4.6954162  4.8623453  2.420174  0.8217110
##  [2,]  9.7530589  6.3911577  6.8059053  8.261752 10.2614357
##  [3,]  1.9335857  4.7722529  4.9575557  2.306979  0.6629135
##  [4,]  1.0858638  3.7275798  3.8682887  2.481666  1.7544007
##  [5,] 14.1061995 14.1181218 14.7183572 10.592531 11.9642507
##  [6,]  6.9358166  5.4304958  6.0540507  3.967679  6.1952919
##  [7,]  1.9937067  4.3597156  4.5979672  1.810158  0.9533728
##  [8,]  2.1605996  5.1012488  5.2818127  2.470857  0.4696174
##  [9,]  2.3992091  4.7904692  5.0481222  1.750172  0.5812735
## [10,]  0.7618053  3.8178285  3.9017276  2.836254  1.9015484
## [11,]  5.3307151  3.1190284  3.7415311  3.483868  5.4613164
## [12,]  2.1833880  5.0946869  5.2802557  2.430037  0.4336660
## [13,]  4.0630467  1.3831528  2.0060112  3.563742  4.9675333
## [14,]  0.0000000  3.7079946  3.6692842  3.554220  2.5911120
## [15,]  3.7079946  0.0000000  0.6237331  4.429434  5.3117886
## [16,]  3.6692842  0.6237331  0.0000000  4.872350  5.5376318
## [17,]  3.5542196  4.4294339  4.8723505  0.000000  2.2278248
## [18,]  2.5911120  5.3117886  5.5376318  2.227825  0.0000000

3.3 K-th neighbor function

test.nb_k<-knn2nb(knearneigh(coords,k=1))
test.listw1c<-nb2listw(test.nb_k)
# plot(states[states@data$NAME=="Pennsylvania",])
plot(test.nb_k, coords, lwd=.2, col="blue", cex = .5)

Four-year colleges in Philadelphia

3.4 Maximum distance

• Or critical threshold if using a distance based (more of this below)

critical.threshold <- max(unlist(nbdists(test.nb_k, coords, longlat = TRUE)))
critical.threshold

## [1] 14.43703

#First create a neighbors object reaching 1 mile (default is in KM)
test.nb<-dnearneigh(coords, 0, 1.60934, row.names = d18$UNITID, longlat = TRUE)
#Now create a list, similar to an edgelist in spatial form
test.listw<-nb2listw(test.nb, zero.policy=TRUE)
plot(test.nb, coords, lwd=.2, col="blue", cex = .5)

4 Distance functions

• In order to conform to Tobler’s first law of geography, a distance decay effect must be respected.

• The value of the function of distance needs to decrease with a growing distance

• Since the weights are inversely related to distance, large values for the latter will yield small values for the former, and vice versa.

• Recalculate back to 1-k neighbor test.nb_k and the critical.threshold along with all distances as follows

nb.dist.band <- dnearneigh(coords, 0, critical.threshold, longlat = TRUE)
distances <- nbdists(nb.dist.band,coords, longlat = TRUE)
distances[1]

## [[1]]
##  [1]  0.2756780  1.6103595 10.1500490  1.0219514  0.6898569  1.2282517
##  [7]  1.7553238  8.4743303  0.7044514  7.2830419  2.8480890  7.5565399
## [13]  7.8251862  3.8948929  1.3224157

Finally apply this function to obtain the inverse

invd1 <- lapply(distances, function(x) (1/x))
length(invd1)

## [1] 18

• If values are very small, we can divide by 10 or 100 (if not, by 1 :D)

invd1a <- lapply(distances, function(x) (1/(x/1)))
invd1a[1]

## [[1]]
##  [1] 3.62742056 0.62097934 0.09852169 0.97852013 1.44957603 0.81416538
##  [7] 0.56969546 0.11800342 1.41954436 0.13730527 0.35111263 0.13233570
## [13] 0.12779249 0.25674647 0.75619185

• And use these weights to create our matrix

invd.weights <- nb2listw(nb.dist.band, glist = invd1a, style = "B")

By specifying style = "B" we apply these weights, if we do not include that, we get row standardization

plot(invd.weights, coords, lwd=.2, col="blue", cex = .5)

Four-year colleges in Philadelphia

To see the weights, we can do the following

w3 = listw2mat(invd.weights)
w3

##          [,1]       [,2]       [,3]      [,4]       [,5]       [,6]      [,7]
## 1  0.00000000 0.00000000 3.62742056 0.6209793 0.00000000 0.09852169 0.9785201
## 2  0.00000000 0.00000000 0.00000000 0.0000000 0.00000000 0.11863435 0.0000000
## 3  3.62742056 0.00000000 0.00000000 0.5568038 0.00000000 0.09989954 1.0865073
## 4  0.62097934 0.00000000 0.55680378 0.0000000 0.00000000 0.10290933 0.6462802
## 5  0.00000000 0.00000000 0.00000000 0.0000000 0.00000000 0.06926633 0.0000000
## 6  0.09852169 0.11863435 0.09989954 0.1029093 0.06926633 0.00000000 0.1095420
## 7  0.97852013 0.00000000 1.08650735 0.6462802 0.00000000 0.10954200 0.0000000
## 8  1.44957603 0.00000000 1.88505281 0.4353363 0.00000000 0.09678024 0.7253062
## 9  0.81416538 0.00000000 1.01955330 0.4399210 0.00000000 0.10893474 1.3246900
## 10 0.56969546 0.00000000 0.50139277 1.7266341 0.00000000 0.09731496 0.5039279
## 11 0.11800342 0.12928748 0.11862559 0.1330410 0.00000000 0.26513175 0.1330811
## 12 1.41954436 0.00000000 1.92289761 0.4340808 0.00000000 0.09737439 0.7473083
## 13 0.13730527 0.10920382 0.13643844 0.1681636 0.00000000 0.15347342 0.1537652
## 14 0.35111263 0.00000000 0.32135694 0.5722368 0.00000000 0.08958876 0.3116663
## 15 0.13233570 0.09722357 0.13020500 0.1666956 0.00000000 0.11442255 0.1425256
## 16 0.12779249 0.09129883 0.12533822 0.1606321 0.00000000 0.10263726 0.1351404
## 17 0.25674647 0.07521059 0.26934412 0.2503847 0.00000000 0.15660822 0.3432690
## 18 0.75619185 0.00000000 0.93733371 0.3541786 0.00000000 0.10029732 0.6517610
##           [,8]      [,9]      [,10]     [,11]       [,12]     [,13]      [,14]
## 1   1.44957603 0.8141654 0.56969546 0.1180034  1.41954436 0.1373053 0.35111263
## 2   0.00000000 0.0000000 0.00000000 0.1292875  0.00000000 0.1092038 0.00000000
## 3   1.88505281 1.0195533 0.50139277 0.1186256  1.92289761 0.1364384 0.32135694
## 4   0.43533628 0.4399210 1.72663406 0.1330410  0.43408077 0.1681636 0.57223679
## 5   0.00000000 0.0000000 0.00000000 0.0000000  0.00000000 0.0000000 0.00000000
## 6   0.09678024 0.1089347 0.09731496 0.2651318  0.09737439 0.1534734 0.08958876
## 7   0.72530618 1.3246900 0.50392795 0.1330811  0.74730833 0.1537652 0.31166630
## 8   0.00000000 0.8621507 0.41099713 0.1125741 14.20564389 0.1274408 0.28759202
## 9   0.86215070 0.0000000 0.37657023 0.1271300  0.91392729 0.1413511 0.25899001
## 10  0.41099713 0.3765702 0.00000000 0.1251555  0.40716188 0.1591466 0.81565615
## 11  0.11257408 0.1271300 0.12515549 0.0000000  0.11318081 0.3535419 0.11656432
## 12 14.20564389 0.9139273 0.40716188 0.1131808  0.00000000 0.1279142 0.28459037
## 13  0.12744082 0.1413511 0.15914659 0.3535419  0.12791421 0.0000000 0.15293233
## 14  0.28759202 0.2589900 0.81565615 0.1165643  0.28459037 0.1529323 0.00000000
## 15  0.12180766 0.1297099 0.16275513 0.1992195  0.12196455 0.4492426 0.16757608
## 16  0.11764355 0.1230896 0.15925540 0.1660740  0.11767824 0.3097546 0.16934398
## 17  0.25148006 0.3550344 0.21908163 0.1783567  0.25570440 0.1743592 0.17482633
## 18  1.32314347 1.0689825 0.32677117 0.1137768  1.43283370 0.1250865 0.23980870
##         [,15]      [,16]      [,17]     [,18]
## 1  0.13233570 0.12779249 0.25674647 0.7561919
## 2  0.09722357 0.09129883 0.07521059 0.0000000
## 3  0.13020500 0.12533822 0.26934412 0.9373337
## 4  0.16669561 0.16063206 0.25038472 0.3541786
## 5  0.00000000 0.00000000 0.00000000 0.0000000
## 6  0.11442255 0.10263726 0.15660822 0.1002973
## 7  0.14252563 0.13514042 0.34326902 0.6517610
## 8  0.12180766 0.11764355 0.25148006 1.3231435
## 9  0.12970988 0.12308957 0.35503439 1.0689825
## 10 0.16275513 0.15925540 0.21908163 0.3267712
## 11 0.19921947 0.16607404 0.17835669 0.1137768
## 12 0.12196455 0.11767824 0.25570440 1.4328337
## 13 0.44924262 0.30975459 0.17435920 0.1250865
## 14 0.16757608 0.16934398 0.17482633 0.2398087
## 15 0.00000000 0.99621325 0.14028230 0.1169797
## 16 0.99621325 0.00000000 0.12753007 0.1122088
## 17 0.14028230 0.12753007 0.00000000 0.2789139
## 18 0.11697965 0.11220883 0.27891385 0.0000000

4.1 With k-neighbors

We do not need to get critical threshold

test.nb_k<-knn2nb(knearneigh(coords,k=3))
k.distances <- nbdists(test.nb_k, coords, longlat = TRUE)
invd2a <- lapply(k.distances, function(x) (1/(x/1)))
invd2a[1]

## [[1]]
## [1] 3.627421 1.449576 1.419544

invd.weights.knn <- nb2listw(test.nb_k, glist = invd2a,style = "B")
plot(invd.weights.knn, coords, lwd=.2, col="blue", cex = .5)

Four-year colleges in Philadelphia with weights

Once more, to see the weights, we can do the following

w4 = listw2mat(invd.weights.knn)
w4

##         [,1] [,2]     [,3]      [,4] [,5]       [,6]      [,7]      [,8]
## 1  0.0000000    0 3.627421 0.0000000    0 0.00000000 0.0000000  1.449576
## 2  0.0000000    0 0.000000 0.0000000    0 0.11863435 0.0000000  0.000000
## 3  3.6274206    0 0.000000 0.0000000    0 0.00000000 0.0000000  1.885053
## 4  0.6209793    0 0.000000 0.0000000    0 0.00000000 0.6462802  0.000000
## 5  0.0000000    0 0.000000 0.0000000    0 0.06926633 0.0000000  0.000000
## 6  0.0000000    0 0.000000 0.0000000    0 0.00000000 0.0000000  0.000000
## 7  0.9785201    0 1.086507 0.0000000    0 0.00000000 0.0000000  0.000000
## 8  1.4495760    0 1.885053 0.0000000    0 0.00000000 0.0000000  0.000000
## 9  0.0000000    0 0.000000 0.0000000    0 0.00000000 1.3246900  0.000000
## 10 0.5696955    0 0.000000 1.7266341    0 0.00000000 0.0000000  0.000000
## 11 0.0000000    0 0.000000 0.0000000    0 0.26513175 0.0000000  0.000000
## 12 0.0000000    0 1.922898 0.0000000    0 0.00000000 0.0000000 14.205644
## 13 0.0000000    0 0.000000 0.0000000    0 0.00000000 0.0000000  0.000000
## 14 0.3511126    0 0.000000 0.5722368    0 0.00000000 0.0000000  0.000000
## 15 0.0000000    0 0.000000 0.0000000    0 0.00000000 0.0000000  0.000000
## 16 0.0000000    0 0.000000 0.0000000    0 0.00000000 0.0000000  0.000000
## 17 0.0000000    0 0.000000 0.0000000    0 0.00000000 0.3432690  0.000000
## 18 0.0000000    0 0.000000 0.0000000    0 0.00000000 0.0000000  1.323143
##         [,9]     [,10]     [,11]      [,12]     [,13]     [,14]     [,15]
## 1  0.0000000 0.0000000 0.0000000  1.4195444 0.0000000 0.0000000 0.0000000
## 2  0.0000000 0.0000000 0.1292875  0.0000000 0.1092038 0.0000000 0.0000000
## 3  0.0000000 0.0000000 0.0000000  1.9228976 0.0000000 0.0000000 0.0000000
## 4  0.0000000 1.7266341 0.0000000  0.0000000 0.0000000 0.0000000 0.0000000
## 5  0.0000000 0.0000000 0.0000000  0.0000000 0.0000000 0.0000000 0.0000000
## 6  0.0000000 0.0000000 0.2651318  0.0000000 0.1534734 0.0000000 0.0000000
## 7  1.3246900 0.0000000 0.0000000  0.0000000 0.0000000 0.0000000 0.0000000
## 8  0.0000000 0.0000000 0.0000000 14.2056439 0.0000000 0.0000000 0.0000000
## 9  0.0000000 0.0000000 0.0000000  0.9139273 0.0000000 0.0000000 0.0000000
## 10 0.0000000 0.0000000 0.0000000  0.0000000 0.0000000 0.8156561 0.0000000
## 11 0.0000000 0.0000000 0.0000000  0.0000000 0.3535419 0.0000000 0.1992195
## 12 0.0000000 0.0000000 0.0000000  0.0000000 0.0000000 0.0000000 0.0000000
## 13 0.0000000 0.0000000 0.3535419  0.0000000 0.0000000 0.0000000 0.4492426
## 14 0.0000000 0.8156561 0.0000000  0.0000000 0.0000000 0.0000000 0.0000000
## 15 0.0000000 0.0000000 0.1992195  0.0000000 0.4492426 0.0000000 0.0000000
## 16 0.0000000 0.0000000 0.0000000  0.0000000 0.3097546 0.1693440 0.9962132
## 17 0.3550344 0.0000000 0.0000000  0.0000000 0.0000000 0.0000000 0.0000000
## 18 1.0689825 0.0000000 0.0000000  1.4328337 0.0000000 0.0000000 0.0000000
##        [,16]      [,17]      [,18]
## 1  0.0000000 0.00000000 0.00000000
## 2  0.0000000 0.00000000 0.00000000
## 3  0.0000000 0.00000000 0.00000000
## 4  0.0000000 0.00000000 0.00000000
## 5  0.0000000 0.05866126 0.05193565
## 6  0.0000000 0.15660822 0.00000000
## 7  0.0000000 0.00000000 0.00000000
## 8  0.0000000 0.00000000 0.00000000
## 9  0.0000000 0.00000000 1.06898254
## 10 0.0000000 0.00000000 0.00000000
## 11 0.0000000 0.00000000 0.00000000
## 12 0.0000000 0.00000000 1.43283370
## 13 0.3097546 0.00000000 0.00000000
## 14 0.0000000 0.00000000 0.00000000
## 15 0.9962132 0.00000000 0.00000000
## 16 0.0000000 0.00000000 0.00000000
## 17 0.0000000 0.00000000 0.27891385
## 18 0.0000000 0.00000000 0.00000000

References

Anselin, Luc. 2020. Spatial Data Science. University of Chicago Center for Spatial Data Science. https://spatialanalysis.github.io/tutorials/.

González Canché, Manuel S. 2019. “Geographical, Statistical, and Qualitative Network Analysis: A Multifaceted Method-Bridging Tool to Reveal and Model Meaningful Structures in Education Research.” In Higher Education: Handbook of Theory and Research: Volume 34, edited by Michael B. Paulsen and Laura W. Perna, 535–634. Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-03457-3_12.

Walker, Kyle. 2020. Tigris: Load Census Tiger/Line Shapefiles. https://CRAN.R-project.org/package=tigris.