Autokorelasi SpasialStatistika Spasial | Praktikum ke-4

rrahmaanisa@apps.ipb.ac.id

Autokorelasi temporal

set.seed(0)

d <- sample(100, 10)

d

a <- d[-length(d)]

b <- d[-1]

plot(a, b, xlab='t', ylab='t-1')

cor(a, b)

d <- sort(d)

d

a <- d[-length(d)]

b <- d[-1]

plot(a, b, xlab='t', ylab='t-1')

acf(d)

AutokorelasiSpasial

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Contoh Data Spasial

library(raster)

p <- shapefile(system.file("external/lux.shp", package="raster"))

p <- p[p$NAME_1=="Diekirch", ]

p$value <- c(10, 6, 4, 11, 6)

data.frame(p)

par(mai=c(0,0,0,0))

plot(p, col=2:7)

xy <- coordinates(p)

points(xy, cex=6, pch=20, col='white')

text(p, 'ID_2', cex=1.5)

1. Choose a neighborhood criterion

• Which areas are linked?

2. Assign weights to the areas that are linked

• Create a spatial weights matrix

3. Run statistical test to examine spatial autocorrelation

Steps in determining the extent of spatial autocorrelation in your data :

Step 1: Choose a neighborhood criterion

Spatial weights matrices

• Neighborhoods can be defined in a number of ways• Contiguity (common boundary)

• What is a “shared” boundary?

• Distance (distance band, K-nearest neighbors)• How many “neighbors” to include, what distance do we use?

• General weights (social distance, distance decay)

Contiguity based neighbors• Areas sharing any boundary point (QUEEN) are taken as neighbors, using the poly2nb function,

which accepts a SpatialPolygonsDataFrame

> library(spdep)

> w<-poly2nb(p)

• If contiguity is defined as areas sharing more than one boundary point (ROOK), the queen= argument is set to FALSE

> w.rook<-poly2nb(p, queen=FALSE)

> coords<-coordinates(p)

> plot(p)

> plot(w, coords, add=T)

Queen contiguity Rook contiguity

Distance based neighborsk nearest neighbors• Can also choose the k nearest points as neighbors

> coords<-coordinates(p)

> IDs<-row.names(as(p, "data.frame"))

> p_kn1<-knn2nb(knearneigh(coords, k=1), row.names=IDs)

> p_kn2<-knn2nb(knearneigh(coords, k=2), row.names=IDs)

> p_kn4<-knn2nb(knearneigh(coords, k=4), row.names=IDs)

> plot(p)

> plot(p_kn2, coords, add=T)

k=1

k=2

k=3

k=1 k=2 k=4

Distance based neighbors : Specified distance

• Can also assign neighbors based on a specified distance

> dist<-unlist(nbdists(p_kn1, coords))

> summary(dist)

Min. 1st Qu. Median Mean 3rd Qu. Max.

0.07316 0.07316 0.14159 0.11832 0.14159 0.16213

> max_k1<-max(dist)

> p_kd1<-dnearneigh(coords, d1=0, d2=0.75*max_k1, row.names=IDs)

> p_kd2<-dnearneigh(coords, d1=0, d2=1*max_k1, row.names=IDs)

> p_kd3<-dnearneigh(coords, d1=0, d2=1.5*max_k1, row.names=IDs)

OR by raw distance> p_ran1<-dnearneigh(coords, d1=0, d2=0.16123, row.names=IDs)

dist=1*max_k1

dist=1.5*max_k1

Distance=0.75*max_k1 Distance=1*max_k1 Distance=1.5*max_k1

Step 2: Assign weights to the areas that are linkedCreating spatial weights matrices using neighborhood lists

Spatial weights matrices

• Once our list of neighbors has been created, we assign spatial weights to each relationship• Can be binary or variable

• Even when the values are binary 0/1, the issue of what to do with no-neighbor observations arises

• Binary weighting will, for a target feature, assign a value of 1 to neighboring features and 0 to all other features• Used with fixed distance, k nearest neighbors, and contiguity

Row-standardized weights matrix

> p_nbq_w<- nb2listw(w)

> p_nbq_w

• Row standardization is used to create proportional weights in cases where features have an unequal number of neighbors• Divide each neighbor weight for a

feature by the sum of all neighbor weights• Obs i has 3 neighbors, each has a weight

of 1/3

• Obs j has 2 neighbors, each has a weight of 1/2

• Use is you want comparable spatial parameters across different data sets with different connectivity structures

Binary weights

> w_nbq_wb<-nb2listw(w, style="B")

> w_nbq_wb

• Row-standardised weights increase the influence of links from observations with few neighbours

• Binary weights vary the influence of observations• Those with many neighbours are up-

weighted compared to those with few

Binary vs. row-standardized

• A binary weights matrix looks like:

• A row-standardized matrix it looks like:

0 1 0 0

0 0 1 1

1 1 0 0

0 1 1 1

0 1 0 0

0 0 .5 .5

.5 .5 0 0

0 .33 .33 .33

Style Options

Code Description

B basic binary coding

W row standardised (sums over all links to n)

C globally standardised (sums over all links to n)

U equal to C divided by the number of neighbours (sums over all links to unity)

S the variance-stabilizing coding scheme

Regions with no neighbors

• If you ever get the following error:

Error in nb2listw(filename): Empty neighbor sets found

• You have some regions that have NO neighbors

> p_nbq_w<-nb2listw(p_nbq, zero.policy=T)

Step 3: Examine spatial autocorrelationUsing spatial weights matrices, run statistical tests of spatial autocorrelation

Spatial autocorrelation

• Test for the presence of spatial autocorrelation• Global

• Moran’s I

• Geary’s C

• Local (LISA – Local Indicators of Spatial Autocorrelation)• Local Moran’s I and Getis Gi*

• We’ll just focus on the “industry standard” – Moran’s I

Autokorelasi Spasial

library(spdep)

w <- poly2nb(p, row.names=p$Id)

class(w)

summary(w)

str(w)

plot(p, col='gray', border='blue', lwd=2)

plot(w, xy, col='red', lwd=2, add=TRUE)

wm <- nb2mat(w, style='B')

wm

Autokorelasi Spasial

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Menghitung Indeks Moran (1)

#compute Moran's Index

n <- length(p)

y <- p$value

ybar <- mean(y)

x

#####1st method

dy <- y - ybar

g <- expand.grid(dy, dy)

yiyj <- g[,1] * g[,2]

#####2nd method

yi <- rep(dy, each=n)

yj <- rep(dy)

yiyj <- yi * yj

pm <- matrix(yiyj, ncol=n)

pmw <- pm * wm

pmw

spmw <- sum(pmw)

spmw

smw <- sum(wm)

sw <- spmw / smw

vr <- n / sum(dy^2)

MI <- vr * sw

MI

EI <- -1/(n-1)

EI

Menghitung Indeks Moran (2)

#compute moran's using spdep functionww <- nb2listw(w, style='B')ww

moran(p$value, ww, n=length(ww$neighbours), S0=Szero(ww))

#Note thatSzero(ww)# is the same aspmw

Moran’s I in R

> moran.test(p$value, listw=ww, randomisation=FALSE, alternative=‘less’)

“two.sided” → HA: I ≠ I0

“greater” → HA: I > I0

Diagram Pencar Moran

n <- length(p)

ms <- cbind(id=rep(1:n, each=n), y=rep(y, each=n), value=as.vector(wm * y))

ms <- ms[ms[,3] > 0, ]

ams <- aggregate(ms[,2:3], list(ms[,1]), FUN=mean)

ams <- ams[,-1]

colnames(ams) <- c('y', 'spatially lagged y')

head(ams)

plot(ams)

reg <- lm(ams[,2] ~ ams[,1])

abline(reg, lwd=2)

abline(h=mean(ams[,2]), lt=2)

abline(v=ybar, lt=2)

coefficients(reg)[2]

rwm <- mat2listw(wm, style='W')

# Checking if rows add up to 1

mat <- listw2mat(rwm)

apply(mat, 1, sum)[1:15]

moran.plot(y, rwm)

Moran Scatter Plot

rwm <- mat2listw(wm, style='W')

# Checking if rows add up to 1

mat <- listw2mat(rwm)

apply(mat, 1, sum)

moran.plot(y, rwm)

Moran Scatter Plot

Latihan

• Input data berikut:

kemiskinan<-read.csv("http://bit.ly/dataKemiskinan",sep=',',header=T)

• Input data bobot berikut:

bobot<-read.csv("http://bit.ly/bobot_kemiskinan",sep=',',header=F)

Hitunglah indeks moran untuk data kemiskinan tsb!

Latihan

• Mengubah data bobot ke dalam bentuk matriks

bot<-as.matrix(bobot)

• Menghitung indeks moran global

w=mat2listw(bot)

moran(kemiskinan$Y, listw=w, n=112, S0=Szero(w))

Latihan

• Menghitung indeks moran lokal

localmoran(kemiskinan$Y, w)

• Membuat Moran scatter plot

moran.plot(kemiskinan$Y, mat2listw(bot,style='W'), labels=kemiskinan$Nama.Kabupaten)