ISSS415 Geospatial Analytics In-class Exercise #8: Second-order Analysis of Spatial Point Patterns
Taught by Dr. Kam Tin Seong

1.0 Overview

This is an exercise to gain experience using appropriate R functions to perform spatial point patter analysis. The case study aims to discover the spatial point processes of childecare centres in Singapore.

The research question we aim to answer: - are the childcare centre centres in Singapore randomly distributed throughout the country?

• if the answer is not, then the next logical question is where are the locations with higher concentration of childcare centres?

2.0 Installing Required Packages

packages = c('rgdal', 'maptools', 'spatstat','ggplot2','ggthemes','plotly')
for (p in packages){
if(!require(p, character.only = T)){
install.packages(p)
}
library(p,character.only = T)
}

3.0 Data Preparation

3.1 Reading in shapefiles

readORG() of rgdal package is used to import the three geospatial data in R. The output is a SpatialPolygonsDataframe object of the sp package.

childcare <- readOGR(dsn = "data", layer="CHILDCARE")
mpsz = readOGR(dsn = "data", layer="MP14_SUBZONE_WEB_PL")

## OGR data source with driver: ESRI Shapefile 
## Source: "D:\IS415-AY2019-20T3A\03-Hands-on Exercises\In-class_Ex08\In-class_Ex08(part2)\In-class_Ex08\data", layer: "CHILDCARE"
## with 1312 features
## It has 18 fields
## OGR data source with driver: ESRI Shapefile 
## Source: "D:\IS415-AY2019-20T3A\03-Hands-on Exercises\In-class_Ex08\In-class_Ex08(part2)\In-class_Ex08\data", layer: "MP14_SUBZONE_WEB_PL"
## with 323 features
## It has 15 fields

3.2 Converting the spatial point data frame into generic sp format

Spatstat package will be used to perform point pattern analysis. A new class of object ppp will created to use spatstat functions. However since there is no direct way to convert SpatialDataFrame into ppp, it is necessary to first convert the SpatialDataFrame into Spatial object. This can be achieved using as() function.

childcare_sp <- as(childcare, "SpatialPoints")

3.3 Converting the generic sp format into spatstat’s ppp format

as.ppp() function of spatstat is then used to convert the spatial data into spatstat’s ppp object format.

childcare_ppp <- as(childcare_sp, "ppp")
childcare_ppp

## Planar point pattern: 1312 points
## window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units

3.4 Checking for duplicated spatial point events

duplicated(childcare_ppp)

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The print out above shows that there are several duplication point events (i.e. TRUE). Duplicated points must be avoided when doing statistical test as the number of points used to compute the Poisson distribution will be different from the distribution of observed events, and this renders the test inaccurate.

The code chunk uses unique() to eliminate duplicate point events.

childcare_ppp_u <- unique(childcare_ppp)
duplicated(childcare_ppp_u) 

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Warning: In practice, it is not advisible to merely dropping the duplicated point events. We should study the context carefully. A better approach is to use the jittering approach discussed in previous hands-on exercise to address the issue.

The code chunk below implements the jittering approach.

childcare_ppp_jit <- rjitter(childcare_ppp, retry=TRUE, nsim=1, drop=TRUE)

duplicated(childcare_ppp_jit)

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3.5 Extracting study area

The code chunk below will be used to extract the target planning areas.

pg = mpsz[mpsz@data$PLN_AREA_N == "PUNGGOL",]
tm = mpsz[mpsz@data$PLN_AREA_N == "TAMPINES",]
ck = mpsz[mpsz@data$PLN_AREA_N == "CHOA CHU KANG",]
jw = mpsz[mpsz@data$PLN_AREA_N == "JURONG WEST",]

The four planning areas are plotted below:

par(mfrow=c(2,2))
plot(pg, main = "Ponggol")
plot(tm, main = "Tampines")
plot(ck, main = "Choa Chu Kang")
plot(jw, main = "Jurong West")

3.6 Converting the spatial point data frame to ppp objects

The SpatialPolygonsDataFrame layers are converted to generic spatialpolygons layers.

pg_sp = as(pg, "SpatialPolygons")
tm_sp = as(tm, "SpatialPolygons")
ck_sp = as(ck, "SpatialPolygons")
jw_sp = as(jw, "SpatialPolygons")

Next, we will convert these SpatialPolygons objects into owin objects that is required by spatstat.

pg_owin = as(pg_sp, "owin")
tm_owin = as(tm_sp, "owin")
ck_owin = as(ck_sp, "owin")
jw_owin = as(jw_sp, "owin")

Finally the ppp objects are created using the code chunk below, using the owin objects derived above to define the boundaries.

childcare_pg_ppp = childcare_ppp_jit[pg_owin]
childcare_tm_ppp = childcare_ppp_jit[tm_owin]
childcare_ck_ppp = childcare_ppp_jit[ck_owin]
childcare_jw_ppp = childcare_ppp_jit[jw_owin]

The four study areas are plotted below:

par(mfrow=c(2,2))
plot(childcare_pg_ppp, main="Punggol")
plot(childcare_tm_ppp, main="Tampines")
plot(childcare_ck_ppp, main="Choa Chu Kang")
plot(childcare_jw_ppp, main="Jurong West")

4.0 Analysing Spatial Point Process Using G-Function

The G function measures the distribution of the distances from an arbitrary event to its nearest event. In this section, you will learn how to compute G-function estimation by using Gest() of spatstat package. You will also learn how to perform monta carlo simulation test using envelope() of spatstat package.

4.1 Choa Chu Kang

4.1.1 Computing G-function estimate

G_CK = Gest(childcare_ck_ppp, correction = "border")
plot(G_CK)

4.1.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001 (the value 0.001 is dependent on the number of simulations performed).

Monte Carlo test with G-function

G_CK.csr <- envelope(childcare_ck_ppp, Gest, nsim = 999)
plot(G_CK.csr)

## Generating 999 simulations of CSR  ...
## 1, 2, 3, ......10.........20.........30.........40.........50.........60........
## .70.........80.........90.........100.........110.........120.........130......
## ...140.........150.........160.........170.........180.........190.........200....
## .....210.........220.........230.........240.........250.........260.........270..
## .......280.........290.........300.........310.........320.........330.........340
## .........350.........360.........370.........380.........390.........400........
## .410.........420.........430.........440.........450.........460.........470......
## ...480.........490.........500.........510.........520.........530.........540....
## .....550.........560.........570.........580.........590.........600.........610..
## .......620.........630.........640.........650.........660.........670.........680
## .........690.........700.........710.........720.........730.........740........
## .750.........760.........770.........780.........790.........800.........810......
## ...820.........830.........840.........850.........860.........870.........880....
## .....890.........900.........910.........920.........930.........940.........950..
## .......960.........970.........980.........990........ 999.
## 
## Done.

4.2 Tampines

4.2.1 Computing G-function estimation

G_tm = Gest(childcare_tm_ppp, correction = "best")
plot(G_tm)

4.2.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001 (the value 0.001 is dependent on the number of simulations performed)

The code chunk below is used to perform the hypothesis testing.

G_tm.csr <- envelope(childcare_tm_ppp, Gest, correction = "all", nsim = 999)
plot(G_tm.csr)

## Generating 999 simulations of CSR  ...
## 1, 2, 3, ......10.........20.........30.........40.........50.........60........
## .70.........80.........90.........100.........110.........120.........130......
## ...140.........150.........160.........170.........180.........190.........200....
## .....210.........220.........230.........240.........250.........260.........270..
## .......280.........290.........300.........310.........320.........330.........340
## .........350.........360.........370.........380.........390.........400........
## .410.........420.........430.........440.........450.........460.........470......
## ...480.........490.........500.........510.........520.........530.........540....
## .....550.........560.........570.........580.........590.........600.........610..
## .......620.........630.........640.........650.........660.........670.........680
## .........690.........700.........710.........720.........730.........740........
## .750.........760.........770.........780.........790.........800.........810......
## ...820.........830.........840.........850.........860.........870.........880....
## .....890.........900.........910.........920.........930.........940.........950..
## .......960.........970.........980.........990........ 999.
## 
## Done.

5.0 Analysing Spatial Point Process Using F-Function

The F function estimates the empty space function F(r) or its hazard rate h(r) from a point pattern in a window of arbitrary shape. In this section, you will learn how to compute F-function estimation by using Fest() of spatstat package. You will also learn how to perform monta carlo simulation test using envelope() of spatstat package.

5.1 Choa Chu Kang

5.1.1 Computing F-function estimation

The code chunk below is used to compute F-function using Fest().

F_CK = Fest(childcare_ck_ppp)
plot(F_CK)

5.1.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001 (the value 0.001 is dependent on the number of simulations performed).

Monte Carlo test with F-fucntion

F_CK.csr <- envelope(childcare_ck_ppp, Fest, nsim = 999)
plot(F_CK.csr)

## Generating 999 simulations of CSR  ...
## 1, 2, 3, ......10.........20.........30.........40.........50.........60........
## .70.........80.........90.........100.........110.........120.........130......
## ...140.........150.........160.........170.........180.........190.........200....
## .....210.........220.........230.........240.........250.........260.........270..
## .......280.........290.........300.........310.........320.........330.........340
## .........350.........360.........370.........380.........390.........400........
## .410.........420.........430.........440.........450.........460.........470......
## ...480.........490.........500.........510.........520.........530.........540....
## .....550.........560.........570.........580.........590.........600.........610..
## .......620.........630.........640.........650.........660.........670.........680
## .........690.........700.........710.........720.........730.........740........
## .750.........760.........770.........780.........790.........800.........810......
## ...820.........830.........840.........850.........860.........870.........880....
## .....890.........900.........910.........920.........930.........940.........950..
## .......960.........970.........980.........990........ 999.
## 
## Done.

5.2 Tampines

5.2.1 Computing F-function estimation

F_tm = Fest(childcare_tm_ppp, correction = "best")
plot(F_tm)

5.2.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001 (the value 0.001 is dependent on the number of simulations performed).

The code chunk below is used to perform the hypothesis testing.

F_tm.csr <- envelope(childcare_tm_ppp, Fest, correction = "all", nsim = 999)
plot(F_tm.csr)

## Generating 999 simulations of CSR  ...
## 1, 2, 3, ......10.........20.........30.........40.........50.........60........
## .70.........80.........90.........100.........110.........120.........130......
## ...140.........150.........160.........170.........180.........190.........200....
## .....210.........220.........230.........240.........250.........260.........270..
## .......280.........290.........300.........310.........320.........330.........340
## .........350.........360.........370.........380.........390.........400........
## .410.........420.........430.........440.........450.........460.........470......
## ...480.........490.........500.........510.........520.........530.........540....
## .....550.........560.........570.........580.........590.........600.........610..
## .......620.........630.........640.........650.........660.........670.........680
## .........690.........700.........710.........720.........730.........740........
## .750.........760.........770.........780.........790.........800.........810......
## ...820.........830.........840.........850.........860.........870.........880....
## .....890.........900.........910.........920.........930.........940.........950..
## .......960.........970.........980.........990........ 999.
## 
## Done.

It can be observed that below the distance of ~375m, the observed pattern resembles CSR pattern, while above that, the observed pattern resembles clustered pattern.

6.0 Analysing Spatial Point Process Using K-Function

K-function measures the number of events found up to a given distance of any particular event. In this section, you will learn how to compute K-function estimates by using Kest() of spatstat package. You will also learn how to perform monta carlo simulation test using envelope() of spatstat package.

6.1 Choa Chu Kang

6.1.1 Computing K-function estimate

K_ck = Kest(childcare_ck_ppp, correction = "Ripley")
plot(K_ck, . -r ~ r, ylab= "K(d)-r", xlab = "d(m)")

6.1.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

The code chunk below is used to perform the hypothesis testing.

K_ck.csr <- envelope(childcare_ck_ppp, Kest, nsim = 999, rank = 1, glocal=TRUE)
plot(K_ck.csr, . - r ~ r, xlab="d", ylab="K(d)-r")

## Generating 999 simulations of CSR  ...
## 1, 2, 3, ......10.........20.........30.........40.........50.........60........
## .70.........80.........90.........100.........110.........120.........130......
## ...140.........150.........160.........170.........180.........190.........200....
## .....210.........220.........230.........240.........250.........260.........270..
## .......280.........290.........300.........310.........320.........330.........340
## .........350.........360.........370.........380.........390.........400........
## .410.........420.........430.........440.........450.........460.........470......
## ...480.........490.........500.........510.........520.........530.........540....
## .....550.........560.........570.........580.........590.........600.........610..
## .......620.........630.........640.........650.........660.........670.........680
## .........690.........700.........710.........720.........730.........740........
## .750.........760.........770.........780.........790.........800.........810......
## ...820.........830.........840.........850.........860.........870.........880....
## .....890.........900.........910.........920.........930.........940.........950..
## .......960.........970.........980.........990........ 999.
## 
## Done.

6.2 Tampines

6.2.1 Computing K-function estimation

K_tm = Kest(childcare_tm_ppp, correction = "Ripley")
plot(K_tm, . -r ~ r, 
     ylab= "K(d)-r", xlab = "d(m)", 
     xlim=c(0,1000))

6.2.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

The code chunk below is used to perform the hypothesis testing.

K_tm.csr <- envelope(childcare_tm_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)
plot(K_tm.csr, . - r ~ r, 
     xlab="d", ylab="K(d)-r", xlim=c(0,500))

## Generating 99 simulations of CSR  ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
## 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
## 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98,  99.
## 
## Done.

It can be observed that beyond ~65m, there are signs of statistically significant clustering.

7.0 Analysing Spatial Point Process Using L-Function

In this section, you will learn how to compute L-function estimation by using Lest() of spatstat package. You will also learn how to perform monta carlo simulation test using envelope() of spatstat package.

7.1 Choa Chu Kang

7.1.1 Computing L Function estimation

L_ck = Lest(childcare_ck_ppp, correction = "Ripley")
plot(L_ck, . -r ~ r, 
     ylab= "L(d)-r", xlab = "d(m)")

7.1.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value if smaller than alpha value of 0.001.

The code chunk below is used to perform the hypothesis testing.

L_ck.csr <- envelope(childcare_ck_ppp, Lest, nsim = 999, rank = 1, glocal=TRUE)
plot(L_ck.csr, . - r ~ r, xlab="d", ylab="L(d)-r")

## Generating 999 simulations of CSR  ...
## 1, 2, 3, ......10.........20.........30.........40.........50.........60........
## .70.........80.........90.........100.........110.........120.........130......
## ...140.........150.........160.........170.........180.........190.........200....
## .....210.........220.........230.........240.........250.........260.........270..
## .......280.........290.........300.........310.........320.........330.........340
## .........350.........360.........370.........380.........390.........400........
## .410.........420.........430.........440.........450.........460.........470......
## ...480.........490.........500.........510.........520.........530.........540....
## .....550.........560.........570.........580.........590.........600.........610..
## .......620.........630.........640.........650.........660.........670.........680
## .........690.........700.........710.........720.........730.........740........
## .750.........760.........770.........780.........790.........800.........810......
## ...820.........830.........840.........850.........860.........870.........880....
## .....890.........900.........910.........920.........930.........940.........950..
## .......960.........970.........980.........990........ 999.
## 
## Done.

7.2 Tampines

7.2.1 Computing L-function estimate

L_tm = Lest(childcare_tm_ppp, correction = "Ripley")
plot(L_tm, . -r ~ r, 
     ylab= "L(d)-r", xlab = "d(m)", 
     xlim=c(0,1000))

7.2.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

The code chunk below will be used to perform the hypothesis testing.

L_tm.csr <- envelope(childcare_tm_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)
plot(L_tm.csr, . - r ~ r, 
     xlab="d", ylab="L(d)-r", xlim=c(0,500))

## Generating 99 simulations of CSR  ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
## 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
## 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98,  99.
## 
## Done.

7.3 Building an interative plot with ggplotly

The previous code chunks uses plot() to visualise the envelopes of the second-order summary statistics (such as L-function). The output is a static plot, therefore it can be difficult to make accurate guesstimates of the statistics and the corresponding distance, r.

The below code chunk converts the output into a dataframe, which can be used to generate a similar plot using appropriate aesthetic mappings from ggplot package. Finally, ggplotly() is used to convert the ggplot into an interactive plotly visualisation.

The codes were referenced from a R-blogger article. Further modifications were made to enhance the user experience by customizing the tooltips for greater clarity and intuition.

title <- "Pairwise Distance: L function"

Lcsr_df <- as.data.frame(L_tm.csr)

colour=c("#0D657D","#ee770d","#D3D3D3")
csr_plot <- ggplot(Lcsr_df, aes(r, obs-r))+
  # plot observed value
  geom_line(colour=c("#4d4d4d"))+
  geom_line(aes(r,theo-r), colour="red", linetype = "dashed")+
  # plot simulation envelopes
  geom_ribbon(aes(ymin=lo-r,ymax=hi-r),alpha=0.1, colour=c("#91bfdb")) +
  xlab("Distance r (km)") +
  ylab("L(r)-r") +
  geom_rug(data=Lcsr_df[Lcsr_df$obs > Lcsr_df$hi,], sides="b", colour=colour[1])  +
  geom_rug(data=Lcsr_df[Lcsr_df$obs < Lcsr_df$lo,], sides="b", colour=colour[2]) +
  geom_rug(data=Lcsr_df[Lcsr_df$obs >= Lcsr_df$lo & Lcsr_df$obs <= Lcsr_df$hi,], sides="b", color=colour[3]) +
  theme_tufte()+
  ggtitle(title)

text1<-"Significant clustering"
text2<-"Significant segregation"
text3<-"Not significant clustering/segregation"

# the below conditional statement is required to ensure that the labels (text1/2/3) are assigned to the correct traces
if (nrow(Lcsr_df[Lcsr_df$obs > Lcsr_df$hi,])==0){ 
  if (nrow(Lcsr_df[Lcsr_df$obs < Lcsr_df$lo,])==0){ 
    ggplotly(csr_plot, dynamicTicks=T) %>%
      style(text = text3, traces = 4) %>%
      rangeslider() 
  }else if (nrow(Lcsr_df[Lcsr_df$obs >= Lcsr_df$lo & Lcsr_df$obs <= Lcsr_df$hi,])==0){ 
    ggplotly(csr_plot, dynamicTicks=T) %>%
      style(text = text2, traces = 4) %>%
      rangeslider() 
  }else {
    ggplotly(csr_plot, dynamicTicks=T) %>%
      style(text = text2, traces = 4) %>%
      style(text = text3, traces = 5) %>%
      rangeslider() 
  }
} else if (nrow(Lcsr_df[Lcsr_df$obs < Lcsr_df$lo,])==0){
  if (nrow(Lcsr_df[Lcsr_df$obs >= Lcsr_df$lo & Lcsr_df$obs <= Lcsr_df$hi,])==0){
    ggplotly(csr_plot, dynamicTicks=T) %>%
      style(text = text1, traces = 4) %>%
      rangeslider() 
  } else{
    ggplotly(csr_plot, dynamicTicks=T) %>%
      style(text = text1, traces = 4) %>%
      style(text = text3, traces = 5) %>%
      rangeslider()
  }
} else{
  ggplotly(csr_plot, dynamicTicks=T) %>%
    style(text = text1, traces = 4) %>%
    style(text = text2, traces = 5) %>%
    style(text = text3, traces = 6) %>%
    rangeslider()
  }

The main features of the above visualisation:

• The observed and theoretical values of L(r)-r (including the upper and lower curves of the simulated envelope) can be found in the tooltip upon hovering the cursor over the geometry layers.
• The range slider below the plot enables users to pan and zoom in to a specific range of distance.
• The colored bands at the bottom of the line graph gives a clearer indication of significant or insignificant spatial segregation/ clustering at distance r. Dark green bands indicate significant clustering, orange indicate significant segregation, while grey indicates insignificant clustering/segregation. Tooltips were added to provide color legend information.