R for Macroecology
Spatial models
Next week
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Any topics that we haven’t talked about?
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Group projects
SAR Models
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Augment standard OLS with an additional term to model
the spatial autocorrelation
We’ll focus on error SAR models, which focuses on
spatial pattern in the error part of the model
OLS
SARlag
SARerror
Y = βX + ε
Y = ρWY + βX + ε
Y = βX + λWu+ ε
Defining the spatial weights matrix, W, is crucial
Neighborhoods in R
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spdep
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dnearneigh()
knearneigh()
Coordinates (matrix or SpatialPoints)
dnearneigh(x, d1, d2, row.names = NULL, longlat = NULL)
Minimum and maximum distances
(in km if longlat = T)
Returns a list of vectors giving the neighbors for each point
Neighborhoods in R
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spdep
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dnearneigh()
knearneigh()
> x = c(1,3,2,5)
> y = c(3,2,4,4)
> n = dnearneigh(cbind(x,y),d1 = 0,d2 = 3)
> n
Neighbour list object:
Number of regions: 4
Number of nonzero links: 10
Percentage nonzero weights: 62.5
Average number of links: 2.5
> str(n)
List of 4
$ : int [1:2] 2 3
$ : int [1:3] 1 3 4
$ : int [1:3] 1 2 4
$ : int [1:2] 2 3
- attr(*, "class")= chr "nb"
- attr(*, "nbtype")= chr "distance”
...
Converting a neighborhood to weights
neighbors list
what to do with neighborless points
nb2listw(neighbours, style="W", zero.policy=NULL)
W = row standardized (rows sum to 1)
B = binary (0/1)
C = global standardized (all links sum to n)
U = C/n
S = variance stabilization (Tiefelsdorf et al. 1999)
Converting a neighborhood to weights
> nb2listw(n,style = "W")$weights
[[1]]
[1] 0.5 0.5
> nb2listw(n,style = "C")$weights
[[1]]
[1] 0.4 0.4
[[2]]
[1] 0.3333333 0.3333333 0.3333333
[[2]]
[1] 0.4 0.4 0.4
[[3]]
[1] 0.3333333 0.3333333 0.3333333
[[3]]
[1] 0.4 0.4 0.4
[[4]]
[1] 0.5 0.5
> nb2listw(n,style = "B")$weights
[[1]]
[1] 1 1
[[4]]
[1] 0.4 0.4>
> nb2listw(n,style = "S")$weights
[[1]]
[1] 0.4494897 0.4494897
[[2]]
[1] 1 1 1
[[2]]
[1] 0.3670068 0.3670068 0.3670068
[[3]]
[1] 1 1 1
[[3]]
[1] 0.3670068 0.3670068 0.3670068
[[4]]
[1] 1 1
[[4]]
[1] 0.4494897 0.4494897
Converting a neighborhood to weights
> nb2listw(n,style = "W")$weights
[[1]]
[1] 0.5 0.5
> nb2listw(n,style = "C")$weights
[[1]]
[1] 0.4 0.4
[[2]]
[1] 0.3333333
0.3333333
0.3333333
Emphasizes
weakly
[[2]]
[1] 0.4 0.4
0.4
Emphasizes
connected points
[[3]]
[1] 0.3333333 0.3333333 0.3333333
[[3]]
[1] 0.4 0.4 0.4
[[4]]
[1] 0.5 0.5
> nb2listw(n,style = "B")$weights
[[1]]
[1] 1 1
[[4]]
[1] 0.4 0.4
> nb2listw(n,style = "S")$weights
[[1]]
[1] 0.4494897 0.4494897
[[2]]
[1] 1 1 Emphasizes
1
[[2]]
[1] 0.3670068 0.3670068 0.3670068
[[3]]
[1] 1 1 1
[[3]]
[1] 0.3670068 0.3670068 0.3670068
[[4]]
[1] 1 1
strongly
connected points
strongly
connected points
Tries to balance
[[4]]
[1] 0.4494897 0.4494897
Lots of options – how to choose?
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Define the neighborhood
Define the spatial weights matrix
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Try things out!
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Look for stability in model estimates
Look for residual autocorrelation
Defining the neighborhood - d
#1. Small distance
n = dnearneigh(cbind(x,y),d1 = 0, d2 = 0.1)
w1 = nb2listw(n,zero.policy = T)
#2. Medium distance
n = dnearneigh(cbind(x,y),d1 = 0, d2 = 0.3)
w2 = nb2listw(n,zero.policy = T)
#2. Large distance
n = dnearneigh(cbind(x,y),d1 = 0, d2 = 0.5)
w3 = nb2listw(n,zero.policy = T)
par(mfrow = c(1,4))
plot(x,y,axes = F,xlab = "",ylab = "")
plot(w1,cbind(x,y))
plot(w2,cbind(x,y))
plot(w3,cbind(x,y))
Defining the neighborhood - K
#4. 2 neighbors
n = knn2nb(knearneigh(cbind(x,y),k=2,RANN = F))
w4 = nb2listw(n,zero.policy = T)
#5. 4 neighbors
n = knn2nb(knearneigh(cbind(x,y),k=4,RANN = F))
w5 = nb2listw(n,zero.policy = T)
#6. 8 neighbors
n = knn2nb(knearneigh(cbind(x,y),k=8,RANN = F))
w6 = nb2listw(n,zero.policy = T)
par(mfrow = c(1,4))
plot(x,y,axes = F,xlab = "",ylab = "")
plot(w4,cbind(x,y))
plot(w5,cbind(x,y))
plot(w6,cbind(x,y))
Neighborhoods on grids
x = rep(1:20,20)
y = rep(1:20,each = 20)
plot(x,y)
n = dnearneigh(cbind(x,y),d1=0,d2 = 1)
w = nb2listw(n)
plot(w,cbind(x,y))
n = dnearneigh(cbind(x,y),d1=0,d2 = sqrt(2))
w = nb2listw(n)
plot(w,cbind(x,y))
Rook’s case
Queen’s case
Data size
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SAR models can take a very long time to fit
2000 points is the maximum I have used
sample() is useful again
Fitting the SAR model
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errorsarlm()
just like lm()
what to do with neighborless points
errorsarlm(formula, listw, zero.policy=NULL)
The neighborhood weights
Try it out
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Build several SAR models with different W
Which one works best?
Spatial eigenvector maps
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Generate new predictors that represent the spatial
structure of the data
Three steps
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Calculate a pairwise distance matrix
Do a principal components analysis on this matrix
Select some of these PCA axes to add to an OLS model
Spatial eigenvector maps
Diniz-Filho and Bini 2005
Filter 1
Filter 2
Filter 3
Filter 4
Filter 10
Filter 30
Filter 20
Filter 40