R for Macroecology

Spatial models

Next week Any topics that we haven’t talked about?

Group projects

SAR Models 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 Y = βX + εSARlag Y = ρWY + βX + εSARerror Y = βX + λWu+ ε

Defining the spatial weights matrix, W, is crucial

Neighborhoods in R spdep

dnearneigh() knearneigh()

dnearneigh(x, d1, d2, row.names = NULL, longlat = NULL)

Coordinates (matrix or SpatialPoints)

Minimum and maximum distances(in km if longlat = T)

Returns a list of vectors giving the neighbors for each point

Neighborhoods in R spdep

dnearneigh() knearneigh()> x = c(1,3,2,5)> y = c(3,2,4,4)> n = dnearneigh(cbind(x,y),d1 = 0,d2 = 3)> nNeighbour 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

nb2listw(neighbours, style="W", zero.policy=NULL)

neighbors list what to do with neighborless points

W = row standardized (rows sum to 1)B = binary (0/1)C = global standardized (all links sum to n)U = C/nS = variance stabilization (Tiefelsdorf et al. 1999)

Converting a neighborhood to weights> nb2listw(n,style = "W")$weights[[1]][1] 0.5 0.5

[[2]][1] 0.3333333 0.3333333 0.3333333

[[3]][1] 0.3333333 0.3333333 0.3333333

[[4]][1] 0.5 0.5> nb2listw(n,style = "B")$weights[[1]][1] 1 1

[[2]][1] 1 1 1

[[3]][1] 1 1 1

[[4]][1] 1 1

> nb2listw(n,style = "C")$weights[[1]][1] 0.4 0.4

[[2]][1] 0.4 0.4 0.4

[[3]][1] 0.4 0.4 0.4

[[4]][1] 0.4 0.4> > nb2listw(n,style = "S")$weights[[1]][1] 0.4494897 0.4494897

[[2]][1] 0.3670068 0.3670068 0.3670068

[[3]][1] 0.3670068 0.3670068 0.3670068

[[4]][1] 0.4494897 0.4494897

Converting a neighborhood to weights> nb2listw(n,style = "W")$weights[[1]][1] 0.5 0.5

[[2]][1] 0.3333333 0.3333333 0.3333333

[[3]][1] 0.3333333 0.3333333 0.3333333

[[4]][1] 0.5 0.5> nb2listw(n,style = "B")$weights[[1]][1] 1 1

[[2]][1] 1 1 1

[[3]][1] 1 1 1

[[4]][1] 1 1

> nb2listw(n,style = "C")$weights[[1]][1] 0.4 0.4

[[2]][1] 0.4 0.4 0.4

[[3]][1] 0.4 0.4 0.4

[[4]][1] 0.4 0.4 > nb2listw(n,style = "S")$weights[[1]][1] 0.4494897 0.4494897

[[2]][1] 0.3670068 0.3670068 0.3670068

[[3]][1] 0.3670068 0.3670068 0.3670068

[[4]][1] 0.4494897 0.4494897

Emphasizes weakly

connected points

Emphasizes strongly

connected points

Emphasizes strongly

connected points

Tries to balance

Lots of options – how to choose? Define the neighborhood Define the spatial weights matrix

Try things out! Look for stability in model estimates Look for residual autocorrelation

Defining the neighborhood - d#1. Small distancen = dnearneigh(cbind(x,y),d1 = 0, d2 = 0.1)w1 = nb2listw(n,zero.policy = T)

#2. Medium distancen = dnearneigh(cbind(x,y),d1 = 0, d2 = 0.3)w2 = nb2listw(n,zero.policy = T)

#2. Large distancen = 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 neighborsn = knn2nb(knearneigh(cbind(x,y),k=2,RANN = F))w4 = nb2listw(n,zero.policy = T)

#5. 4 neighborsn = knn2nb(knearneigh(cbind(x,y),k=4,RANN = F))w5 = nb2listw(n,zero.policy = T)

#6. 8 neighborsn = 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 gridsx = 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 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 errorsarlm()

errorsarlm(formula, listw, zero.policy=NULL)

just like lm() what to do with neighborless points

The neighborhood weights

Try it out Build several SAR models with different W Which one works best?

Spatial eigenvector maps Generate new predictors that represent the

spatial structure of the data Three steps

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

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