Use RStudio to answer the questions below.

Load the New Haven dataset which contains US Census$-$derived demographic characteristics of each census block. Here, we are interested in measuring spatial autocorrelation in the proportion of residents of each census block who self$-$identify as white $($i$.$e$.,$ the variable blocks$P$\_$WHITE$).$ For this variable, you should:

$1.$ Create a lagged means plot for queen$$s case adjacency rules, and fit a linear regression between P$\_$WHITE $($independent variable$)$ and the lagged means $($dependent variable$).$ What does this regression indicate about spatial autocorrelation in the variable P$\_$WHITE?

$2.$ Calculate Moran$$s I using a k nearest neighbors rule for determining adjacency, for each of k $=1,2,3,...,10.$ Create a plot of k $($x axis$)$ vs$.$ Moran$$s I $($y axis$).$ What does your plot suggest about how the spatial autocorrelation of P$\_$WHITE changes with how neighboring census blocks are defined?

I started the script for Rstudio and at the end for answering Question $2$ is where I am having trouble.

Please see my script for both questions $($I cannot upload the Rstudio dataset called newhaven$).$

#Load libraries

library$($maptools$)$

library$($spdep$)$

library$($GISTools$)$

library$($pgirmess$)$

# Load data

data$($newhaven$)$

coords $<-$ coordinates$($blocks$)$ # get the centroids of blocks

# Question $1$: Lagged Means Plot and Linear Regression

coords $<-$ coordinates$($blocks$)$ # get the centroids of blocks

#Choropleth

par$($mfrow$=$c$(1,1))$

choropleth$($blocks$,$ blocks$PCTOWNHOME$)$

#Introducing the $\text{'}$nb$\text{'}$ objects and queen's case adjacency rule

$?$poly$2$nb

nb$.$queen $<-$ poly$2$nb$($blocks$)$

plot$($blocks$)$

plot$($nb$.$queen, coords, add$=$T$,$ pch$=16,$ col$=$"red"$)$

#rook's case adjacency rule

nb$.$rook $<-$ poly$2$nb$($blocks$,$ queen$=$F$)$

plot$($blocks$)$

plot$($nb$.$rook, coords, add$=$T$,$ pch$=16,$ col$=$"blue"$)$

#Plotting queen and rook side by side

par$($mfrow$=$c$(1,2))$

plot$($blocks$)$

plot$($nb$.$queen, coords, add$=$T$,$ pch$=16,$ col$=$"red"$)$

plot$($blocks$)$

plot$($nb$.$rook, coords, add$=$T$,$ pch$=16,$ col$=$"blue"$)$

#Creating a nb object from finding the k nearest polygons $($centroids$)$ as neighbors

IDs $<-$ row.names$($as$($blocks$,$ "data.frame"$))$

nb$.$k$1<-$ knn$2$nb$($knearneigh$($coords$,$ k$=1),$ row.names$=$IDs$)$

plot$($blocks$)$

plot$($nb$.$k$1,$ coords, add$=$T$,$ pch$=16,$ col$=$"red"$)$

#k$=2,3,4$ nearest neighbors

nb$.$k$2<-$ knn$2$nb$($knearneigh$($coords$,$ k$=2),$ row.names$=$IDs$)$

nb$.$k$3<-$ knn$2$nb$($knearneigh$($coords$,$ k$=3),$ row.names$=$IDs$)$

nb$.$k$4<-$ knn$2$nb$($knearneigh$($coords$,$ k$=4),$ row.names$=$IDs$)$

par$($mfrow$=$c$(2,2))$

plot$($blocks$)$

plot$($nb$.$k$1,$ coords, add$=$T$,$ pch$=16,$ col$=$"red"$)$

plot$($blocks$)$

plot$($nb$.$k$2,$ coords, add$=$T$,$ pch$=16,$ col$=$"purple"$)$

plot$($blocks$)$

plot$($nb$.$k$3,$ coords, add$=$T$,$ pch$=16,$ col$=$"green"$)$

plot$($blocks$)$

plot$($nb$.$k$4,$ coords, add$=$T$,$ pch$=16,$ col$=$"gold"$)$

#To determine neighbors based on distance itself, use the function dnearneigh to find neighbors between distance $$d$1$ and $$d$2$

nb$.$d$500<-$ dnearneigh$($coords$,$ d$1=0,$ d$2=500,$ row.names$=$IDs$)$

nb$.$d$1000<-$ dnearneigh$($coords$,$ d$1=0,$ d$2=1000,$ row.names$=$IDs$)$

nb$.$d$1500<-$ dnearneigh$($coords$,$ d$1=0,$ d$2=1500,$ row.names$=$IDs$)$

nb$.$d$2000<-$ dnearneigh$($coords$,$ d$1=0,$ d$2=2000,$ row.names$=$IDs$)$

plot$($blocks$)$

plot$($nb$.$d$500,$ coords, add$=$T$,$ pch$=16,$ col$=$"red"$)$

plot$($blocks$)$

plot$($nb$.$d$1000,$ coords, add$=$T$,$ pch$=16,$ col$=$"blue"$)$

plot$($blocks$)$

plot$($nb$.$d$1500,$ coords, add$=$T$,$ pch$=16,$ col$=$"green"$)$

plot$($blocks$)$

plot$($nb$.$d$2000,$ coords, add$=$T$,$ pch$=16,$ col$=$"gold"$)$

#Spatial autocorrelation

par$($mfrow$=$c$(1,1))$

choropleth$($blocks$,$ blocks$PCTOWNHOME$)$

#Lagged plot

# Calculate lagged means

lw$.$queen $<-$ nb$2$listw$($nb$.$queen, style $="$W$")$

lagged.means $<-$ lag.listw$($lw$.$queen, blocks$P$\_$WHITE$)$

plot$($lagged$.$means, blocks$P$\_$WHITE, xlab $=$ "Lagged Means", ylab $="$P$\_$WHITE"$)$

abline$($lm$($blocks$P$\_$WHITE ~ lagged.means$),$ col $=$ "red"$)$

#Question $2$

# Calculate Moran's I using k nearest neighbors rule

k $<-$ c$(1$:$10)$

estimates $<-$ rep$($NA$,10)$

for $($i in $1$:$10)\{$

lw$\_$k $<-$ knn$2$nb$($knearneigh$($coords$,$ k $=$ i$),$ row.names $=$ row.names$($blocks$))$ # create spatial weights with k nearest neighbors rule

estimates$[$i$]<-$ as$.$numeric$($moran$.$test$($blocks$P$\_$WHITE, lw$\_$k$)$$estimate$[1])$ # extract Moran's I

$\}$

# Develop a plot of k vs$.$ Moran's I

plot$($k$,$ estimates, type $="$b$",$ main $=$ "Moran's I vs$.$ k$",$ xlab $="$k$",$ ylab $=$ "Moran's I"$)$