Hands-On Exercise 4

Today’s Adventure: Spatial Point Patterns Analysis! With the spatstat package, let’s discover the spatial point processes and learn how to perform spatial point patterns analysis 😄

1.0 Overview

We’ve learned data wrangling and thematic mapping - and next on our list is spatial points analysis! Spatial Point Pattern Analysis (SPPA for short) is the evaluation of a pattern/distribution of a set of points on a surface. Said points could refer to:

• where certain events such as crime, traffic accident and disease onset took place
• where business services (fast food outlets, cafe chains) or facilities (childcare centres) are located

2.0 Setup

2.1 Packages Used

The R packages we’ll be introducing today are:

• spatstat: used for point pattern analysis
• raster: reads, writes, manipulates, analyses and models gridded spatial data (i.e. raster)
• maptools: a set of tools for manipulating geographic data

In addition, we’ll be using the packages from our last lesson:

• sf: used for importing, managing, and processing geospatial data
• tmap: used for creating thematic maps, such as choropleth and bubble maps

packages = c('maptools', 'sf', 'raster','spatstat', 'tmap')
for (p in packages){
  if(!require(p, character.only = T)){
    install.packages(p)
  }
  library(p,character.only = T)
}

2.2 Data Used

The datasets used for this exercise are:

• CHILDCARE, a point feature data in geojson format from data.gov.sg, providing both location and attribute information of childcare centres
• MP14_SUBZONE_WEB_PL, a polygon feature data in ESRI shapefile format from data.gov.sg, providing information of URA 2014 Master Plan Planning Subzone boundary data
• CostalOutline, a polygon feature data in ESRI shapefile format from sla.gov.sg, showing the national boundary of Singapore

Note: Our aspatial data file does not contain any coordinates values, but its PA and SZ fields can be used as unique identifiers to geocode to MP14_SUBZONE_WEB_PL shapefile!

2.3 Importing Data

We’ve imported data in our previous hands-on exercises, so whatever we’ve written here is par for course!

2.3.1 Importing Spatial Data

childcare_sf <- st_read("data/child-care-services-geojson.geojson") %>%
  st_transform(crs = 3414)

Reading layer `child-care-services-geojson' from data source 
  `C:\IS415\IS415_msty\_posts\2021-09-05-hands-on-exercise-4\data\child-care-services-geojson.geojson' 
  using driver `GeoJSON'
Simple feature collection with 1545 features and 2 fields
Geometry type: POINT
Dimension:     XYZ
Bounding box:  xmin: 103.6824 ymin: 1.248403 xmax: 103.9897 ymax: 1.462134
z_range:       zmin: 0 zmax: 0
Geodetic CRS:  WGS 84

sg_sf <- st_read(dsn = "data", layer="CostalOutline")

Reading layer `CostalOutline' from data source 
  `C:\IS415\IS415_msty\_posts\2021-09-05-hands-on-exercise-4\data' 
  using driver `ESRI Shapefile'
Simple feature collection with 60 features and 4 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 2663.926 ymin: 16357.98 xmax: 56047.79 ymax: 50244.03
Projected CRS: SVY21

mpsz_sf <- st_read(dsn = "data", 
                layer = "MP14_SUBZONE_WEB_PL")

Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `C:\IS415\IS415_msty\_posts\2021-09-05-hands-on-exercise-4\data' 
  using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21

DIY Section 1: Using the appropriate sf function you learned in Hands-on Exercise 2, retrieve the referencing system information of these geospatial data.

In our Hands-On Exercise 02, we learned how to retrieve the referencing system information, and even how to change it! Hint: check out section 6.2.1 from my Hands-On Ex02! And if you need a helping hand, click “show code” below:

Show code

st_crs(childcare_sf)

Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

Show code

st_crs(mpsz_sf)

Coordinate Reference System:
  User input: SVY21 
  wkt:
PROJCRS["SVY21",
    BASEGEOGCRS["SVY21[WGS84]",
        DATUM["World Geodetic System 1984",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]],
            ID["EPSG",6326]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["Degree",0.0174532925199433]]],
    CONVERSION["unnamed",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["(E)",east,
            ORDER[1],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]],
        AXIS["(N)",north,
            ORDER[2],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]]]

Show code

st_crs(sg_sf)

Coordinate Reference System:
  User input: SVY21 
  wkt:
PROJCRS["SVY21",
    BASEGEOGCRS["SVY21[WGS84]",
        DATUM["World Geodetic System 1984",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]],
            ID["EPSG",6326]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["Degree",0.0174532925199433]]],
    CONVERSION["unnamed",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["(E)",east,
            ORDER[1],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]],
        AXIS["(N)",north,
            ORDER[2],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]]]

DIY Section 2: Using the method you learned in Lesson 2, assign the correct crs to mpsz_sf and sg_sf simple feature data frames.

Hmmm… it seems that the cr information isn’t appropriate. childcare_sf is in WGS84, while the other two are in SVY21. Let’s remedy that:

mpsz_sf <- st_transform(mpsz_sf, crs= 3414)
sg_sf <- st_transform(sg_sf, crs= 3414)
st_crs(mpsz_sf)

Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

st_crs(sg_sf)

Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

DIY Section 3: Using the mapping methods you learned in Hands-on Exercise 3, prepare a map as shown below.

Now that we’ve checked the referencing system, we should get a preliminary look of the data by plotting a point map:

tm_shape(sg_sf) +
  tm_polygons() +
tm_shape(mpsz_sf) +
  tm_polygons() +
tm_shape(childcare_sf)+
  tm_dots()

2.3.2 Importing Geospatial Data

childcare_sf <- st_read("data/child-care-services-geojson.geojson") %>%
  st_transform(crs = 3414)

Reading layer `child-care-services-geojson' from data source 
  `C:\IS415\IS415_msty\_posts\2021-09-05-hands-on-exercise-4\data\child-care-services-geojson.geojson' 
  using driver `GeoJSON'
Simple feature collection with 1545 features and 2 fields
Geometry type: POINT
Dimension:     XYZ
Bounding box:  xmin: 103.6824 ymin: 1.248403 xmax: 103.9897 ymax: 1.462134
z_range:       zmin: 0 zmax: 0
Geodetic CRS:  WGS 84

sg_sf <- st_read(dsn = "data", layer="CostalOutline")

Reading layer `CostalOutline' from data source 
  `C:\IS415\IS415_msty\_posts\2021-09-05-hands-on-exercise-4\data' 
  using driver `ESRI Shapefile'
Simple feature collection with 60 features and 4 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 2663.926 ymin: 16357.98 xmax: 56047.79 ymax: 50244.03
Projected CRS: SVY21

mpsz_sf <- st_read(dsn = "data", 
                layer = "MP14_SUBZONE_WEB_PL")

Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `C:\IS415\IS415_msty\_posts\2021-09-05-hands-on-exercise-4\data' 
  using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21

3.0 Data Wrangling

3.1 Converting sf data frames to sp’s Spatial* class

childcare <- as_Spatial(childcare_sf)
mpsz <- as_Spatial(mpsz_sf)
sg <- as_Spatial(sg_sf)

DIY Section: Using appropriate function, display the information of these three Spatial* classes.

childcare

class       : SpatialPointsDataFrame 
features    : 1545 
extent      : 11203.01, 45404.24, 25667.6, 49300.88  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 2
names       :    Name,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Description 
min values  :   kml_1, <center><table><tr><th colspan='2' align='center'><em>Attributes</em></th></tr><tr bgcolor="#E3E3F3"> <th>ADDRESSBLOCKHOUSENUMBER</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSBUILDINGNAME</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSPOSTALCODE</th> <td>018989</td> </tr><tr bgcolor=""> <th>ADDRESSSTREETNAME</th> <td>1, MARINA BOULEVARD, #B1 - 01, ONE MARINA BOULEVARD, SINGAPORE 018989</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSTYPE</th> <td></td> </tr><tr bgcolor=""> <th>DESCRIPTION</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>HYPERLINK</th> <td></td> </tr><tr bgcolor=""> <th>LANDXADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor="#E3E3F3"> <th>LANDYADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor=""> <th>NAME</th> <td>THE LITTLE SKOOL-HOUSE INTERNATIONAL PTE. LTD.</td> </tr><tr bgcolor="#E3E3F3"> <th>PHOTOURL</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSFLOORNUMBER</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>INC_CRC</th> <td>08F73931F4A691F4</td> </tr><tr bgcolor=""> <th>FMEL_UPD_D</th> <td>20200826094036</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSUNITNUMBER</th> <td></td> </tr></table></center> 
max values  : kml_999,                  <center><table><tr><th colspan='2' align='center'><em>Attributes</em></th></tr><tr bgcolor="#E3E3F3"> <th>ADDRESSBLOCKHOUSENUMBER</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSBUILDINGNAME</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSPOSTALCODE</th> <td>829646</td> </tr><tr bgcolor=""> <th>ADDRESSSTREETNAME</th> <td>200, PONGGOL SEVENTEENTH AVENUE, SINGAPORE 829646</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSTYPE</th> <td></td> </tr><tr bgcolor=""> <th>DESCRIPTION</th> <td>Child Care Services</td> </tr><tr bgcolor="#E3E3F3"> <th>HYPERLINK</th> <td></td> </tr><tr bgcolor=""> <th>LANDXADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor="#E3E3F3"> <th>LANDYADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor=""> <th>NAME</th> <td>RAFFLES KIDZ @ PUNGGOL PTE LTD</td> </tr><tr bgcolor="#E3E3F3"> <th>PHOTOURL</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSFLOORNUMBER</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>INC_CRC</th> <td>379D017BF244B0FA</td> </tr><tr bgcolor=""> <th>FMEL_UPD_D</th> <td>20200826094036</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSUNITNUMBER</th> <td></td> </tr></table></center> 

mpsz

class       : SpatialPolygonsDataFrame 
features    : 323 
extent      : 2667.538, 56396.44, 15748.72, 50256.33  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +datum=WGS84 +units=m +no_defs 
variables   : 15
names       : OBJECTID, SUBZONE_NO, SUBZONE_N, SUBZONE_C, CA_IND, PLN_AREA_N, PLN_AREA_C,       REGION_N, REGION_C,          INC_CRC, FMEL_UPD_D,     X_ADDR,     Y_ADDR,    SHAPE_Leng,    SHAPE_Area 
min values  :        1,          1, ADMIRALTY,    AMSZ01,      N, ANG MO KIO,         AM, CENTRAL REGION,       CR, 00F5E30B5C9B7AD8,      16409,  5092.8949,  19579.069, 871.554887798, 39437.9352703 
max values  :      323,         17,    YUNNAN,    YSSZ09,      Y,     YISHUN,         YS,    WEST REGION,       WR, FFCCF172717C2EAF,      16409, 50424.7923, 49552.7904, 68083.9364708,  69748298.792 

sg

class       : SpatialPolygonsDataFrame 
features    : 60 
extent      : 2663.926, 56047.79, 16357.98, 50244.03  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +datum=WGS84 +units=m +no_defs 
variables   : 4
names       : GDO_GID, MSLINK, MAPID,              COSTAL_NAM 
min values  :       1,      1,     0,             ISLAND LINK 
max values  :      60,     67,     0, SINGAPORE - MAIN ISLAND 

3.2 Converting the Spatial* class into generic sp format

spatstat requires the analytical data in ppp object form, but since there is no way to directly convert a Spatial* classes into ppp object, we’ll need to convert the Spatial classes* into a generic Spatial object first, then convert the generic sp object into ppp object form,

childcare_sp <- as(childcare, "SpatialPoints")
sg_sp <- as(sg, "SpatialPolygons")

childcare_sp

class       : SpatialPoints 
features    : 1545 
extent      : 11203.01, 45404.24, 25667.6, 49300.88  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 

sg_sp

class       : SpatialPolygons 
features    : 60 
extent      : 2663.926, 56047.79, 16357.98, 50244.03  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +datum=WGS84 +units=m +no_defs 

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

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

Planar point pattern: 1545 points
window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units

To examine the difference, let’s plot out childcare_ppp and check its summary statisstics:

plot(childcare_ppp)

summary(childcare_ppp)

Planar point pattern:  1545 points
Average intensity 1.91145e-06 points per square unit

*Pattern contains duplicated points*

Coordinates are given to 3 decimal places
i.e. rounded to the nearest multiple of 0.001 units

Window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
                    (34200 x 23630 units)
Window area = 808287000 square units

Notice the warning message about duplicates - it’s an issue of significance in spatial point patterns analysis. The statistical methodology used for spatial point patterns processes is based largely on the assumption that process are simple, that is, that the points cannot be coincidental. As such, we’ll need to remove said duplicates - as seen in the following section.

3.4 Handling Duplicated Points

To check for duplication in a ppp object, we’ll use duplicated():

any(duplicated(childcare_ppp))

[1] TRUE

Meanwhile, to count the number of co-indicence points, we’ll use multiplicity():

multiplicity(childcare_ppp)

And to get the number of locations that have more than 1 point (duplicated) event:

sum(multiplicity(childcare_ppp) > 1)

[1] 128

128 events! Let’s look at that in context of a map:

tmap_mode('view')
tm_shape(childcare) +
  tm_dots(alpha=0.4, 
          size=0.05)

tmap_mode('plot')

3.5 Overcoming the duplicates problem

There are 3 main ways of overcoming the issue of duplicates: - 1. Delete the duplicates: results in loss of useful point events - 2. Use jittering approach: Add a small perturbation to the duplicate points so that they do not occupy the exact same space - 3. Make each point unique and then attach the duplicates of the points to the patterns as marks, as attributes of the points: requires analytical techniques to take account of said marks

3.5.1 Jittering

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

DIY: Using the method you learned in previous section, check if any dusplicated point in this geospatial data.

any(duplicated(childcare_ppp_jit))

[1] FALSE

3.6 Introducing the owin object

3.6.1 Creating a owin object

Usually, when analysing spatial point patterns, we’ll confine our analysis within a certain geographical area - such as the Singapore boundary. In spatstat, an object called owin is specially designed to represent this polygonal region.

sg_owin <- as(sg_sp, "owin")

plot(sg_owin)

summary(sg_owin)

Window: polygonal boundary
60 separate polygons (no holes)
            vertices        area relative.area
polygon 1         38 1.56140e+04      2.09e-05
polygon 2        735 4.69093e+06      6.27e-03
polygon 3         49 1.66986e+04      2.23e-05
polygon 4         76 3.12332e+05      4.17e-04
polygon 5       5141 6.36179e+08      8.50e-01
polygon 6         42 5.58317e+04      7.46e-05
polygon 7         67 1.31354e+06      1.75e-03
polygon 8         15 4.46420e+03      5.96e-06
polygon 9         14 5.46674e+03      7.30e-06
polygon 10        37 5.26194e+03      7.03e-06
polygon 11        53 3.44003e+04      4.59e-05
polygon 12        74 5.82234e+04      7.78e-05
polygon 13        69 5.63134e+04      7.52e-05
polygon 14       143 1.45139e+05      1.94e-04
polygon 15       165 3.38736e+05      4.52e-04
polygon 16       130 9.40465e+04      1.26e-04
polygon 17        19 1.80977e+03      2.42e-06
polygon 18        16 2.01046e+03      2.69e-06
polygon 19        93 4.30642e+05      5.75e-04
polygon 20        90 4.15092e+05      5.54e-04
polygon 21       721 1.92795e+06      2.57e-03
polygon 22       330 1.11896e+06      1.49e-03
polygon 23       115 9.28394e+05      1.24e-03
polygon 24        37 1.01705e+04      1.36e-05
polygon 25        25 1.66227e+04      2.22e-05
polygon 26        10 2.14507e+03      2.86e-06
polygon 27       190 2.02489e+05      2.70e-04
polygon 28       175 9.25904e+05      1.24e-03
polygon 29      1993 9.99217e+06      1.33e-02
polygon 30        38 2.42492e+04      3.24e-05
polygon 31        24 6.35239e+03      8.48e-06
polygon 32        53 6.35791e+05      8.49e-04
polygon 33        41 1.60161e+04      2.14e-05
polygon 34        22 2.54368e+03      3.40e-06
polygon 35        30 1.08382e+04      1.45e-05
polygon 36       327 2.16921e+06      2.90e-03
polygon 37       111 6.62927e+05      8.85e-04
polygon 38        90 1.15991e+05      1.55e-04
polygon 39        98 6.26829e+04      8.37e-05
polygon 40       415 3.25384e+06      4.35e-03
polygon 41       222 1.51142e+06      2.02e-03
polygon 42       107 6.33039e+05      8.45e-04
polygon 43         7 2.48299e+03      3.32e-06
polygon 44        17 3.28303e+04      4.38e-05
polygon 45        26 8.34758e+03      1.11e-05
polygon 46       177 4.67446e+05      6.24e-04
polygon 47        16 3.19460e+03      4.27e-06
polygon 48        15 4.87296e+03      6.51e-06
polygon 49        66 1.61841e+04      2.16e-05
polygon 50       149 5.63430e+06      7.53e-03
polygon 51       609 2.62570e+07      3.51e-02
polygon 52         8 7.82256e+03      1.04e-05
polygon 53       976 2.33447e+07      3.12e-02
polygon 54        55 8.25379e+04      1.10e-04
polygon 55       976 2.33447e+07      3.12e-02
polygon 56        61 3.33449e+05      4.45e-04
polygon 57         6 1.68410e+04      2.25e-05
polygon 58         4 9.45963e+03      1.26e-05
polygon 59        46 6.99702e+05      9.35e-04
polygon 60        13 7.00873e+04      9.36e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 748741000 square units
Fraction of frame area: 0.414

3.6.2 Combining point events object and owin object

Let’s extract the childcare events that are located within the Singapore boundary:

childcareSG_ppp = childcare_ppp[sg_owin]

summary(childcareSG_ppp)

Planar point pattern:  1545 points
Average intensity 2.063463e-06 points per square unit

*Pattern contains duplicated points*

Coordinates are given to 3 decimal places
i.e. rounded to the nearest multiple of 0.001 units

Window: polygonal boundary
60 separate polygons (no holes)
            vertices        area relative.area
polygon 1         38 1.56140e+04      2.09e-05
polygon 2        735 4.69093e+06      6.27e-03
polygon 3         49 1.66986e+04      2.23e-05
polygon 4         76 3.12332e+05      4.17e-04
polygon 5       5141 6.36179e+08      8.50e-01
polygon 6         42 5.58317e+04      7.46e-05
polygon 7         67 1.31354e+06      1.75e-03
polygon 8         15 4.46420e+03      5.96e-06
polygon 9         14 5.46674e+03      7.30e-06
polygon 10        37 5.26194e+03      7.03e-06
polygon 11        53 3.44003e+04      4.59e-05
polygon 12        74 5.82234e+04      7.78e-05
polygon 13        69 5.63134e+04      7.52e-05
polygon 14       143 1.45139e+05      1.94e-04
polygon 15       165 3.38736e+05      4.52e-04
polygon 16       130 9.40465e+04      1.26e-04
polygon 17        19 1.80977e+03      2.42e-06
polygon 18        16 2.01046e+03      2.69e-06
polygon 19        93 4.30642e+05      5.75e-04
polygon 20        90 4.15092e+05      5.54e-04
polygon 21       721 1.92795e+06      2.57e-03
polygon 22       330 1.11896e+06      1.49e-03
polygon 23       115 9.28394e+05      1.24e-03
polygon 24        37 1.01705e+04      1.36e-05
polygon 25        25 1.66227e+04      2.22e-05
polygon 26        10 2.14507e+03      2.86e-06
polygon 27       190 2.02489e+05      2.70e-04
polygon 28       175 9.25904e+05      1.24e-03
polygon 29      1993 9.99217e+06      1.33e-02
polygon 30        38 2.42492e+04      3.24e-05
polygon 31        24 6.35239e+03      8.48e-06
polygon 32        53 6.35791e+05      8.49e-04
polygon 33        41 1.60161e+04      2.14e-05
polygon 34        22 2.54368e+03      3.40e-06
polygon 35        30 1.08382e+04      1.45e-05
polygon 36       327 2.16921e+06      2.90e-03
polygon 37       111 6.62927e+05      8.85e-04
polygon 38        90 1.15991e+05      1.55e-04
polygon 39        98 6.26829e+04      8.37e-05
polygon 40       415 3.25384e+06      4.35e-03
polygon 41       222 1.51142e+06      2.02e-03
polygon 42       107 6.33039e+05      8.45e-04
polygon 43         7 2.48299e+03      3.32e-06
polygon 44        17 3.28303e+04      4.38e-05
polygon 45        26 8.34758e+03      1.11e-05
polygon 46       177 4.67446e+05      6.24e-04
polygon 47        16 3.19460e+03      4.27e-06
polygon 48        15 4.87296e+03      6.51e-06
polygon 49        66 1.61841e+04      2.16e-05
polygon 50       149 5.63430e+06      7.53e-03
polygon 51       609 2.62570e+07      3.51e-02
polygon 52         8 7.82256e+03      1.04e-05
polygon 53       976 2.33447e+07      3.12e-02
polygon 54        55 8.25379e+04      1.10e-04
polygon 55       976 2.33447e+07      3.12e-02
polygon 56        61 3.33449e+05      4.45e-04
polygon 57         6 1.68410e+04      2.25e-05
polygon 58         4 9.45963e+03      1.26e-05
polygon 59        46 6.99702e+05      9.35e-04
polygon 60        13 7.00873e+04      9.36e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 748741000 square units
Fraction of frame area: 0.414

DIY: Using the method you learned in previous exercise, plot the newly derived childcareSG_ppp.

plot(childcareSG_ppp)

4.0 First Order Spatial Point Patterns Analysis

Now, let’s get into the meat of the matter: performing spatial point analysis! In this section, we’ll be doing first-order SPPA:

• deriving kernel density estimation (KDE) layer for visualising and exploring the intensity of point processes
• performing Confirmatory Spatial Point Patterns Analysis by using Nearest Neighbour statistics

4.1 Kernel Density Estimation

Before we start, you might be asking: just what is Kerndel Density Estimation? Well, in KDE, we apply a function (also known as a “kernel”) to each data point, which then averages the location of that point with respect to the location of other data points based on the bandwidth of the kernel.

4.1.1 Computing kernel density estimation using automatic bandwidth selection method

kde_childcareSG_bw <- density(childcareSG_ppp,
                              sigma=bw.diggle,
                              edge=TRUE,
                            kernel="gaussian") 

plot(kde_childcareSG_bw)

We’ve run into a snag! The density values of the output range from 0 to 0.000035: because the default unit of measurement of svy21 is in meters, the density values computed is in “number of points per square meter”. What we take away from this is that this scale is too small for your viewers to comprehend - so let’s see what we can do about it.

Fun fact! You can retrieve the bandwidth used to compute the kde layer with this:

bw <- bw.diggle(childcareSG_ppp)
bw

   sigma 
298.4095 

4.1.2 Rescaling KDE values with rescale()

Let’s convert the unit of measurement from meters to kilometers:

childcareSG_ppp.km <- rescale(childcareSG_ppp, 1000, "km")

kde_childcareSG.bw <- density(childcareSG_ppp.km, sigma=bw.diggle, edge=TRUE, kernel="gaussian")
plot(kde_childcareSG.bw)

4.1.3 Working with different automatic badwidth methods

Beside bw.diggle(), there are three other spatstat functions can be used to determine the bandwidth, they are: bw.CvL(), bw.scott(), and bw.ppl(). Let’s take a look at how they differ in terms of bandwidth:

 bw.CvL(childcareSG_ppp.km)

   sigma 
4.543278 

bw.scott(childcareSG_ppp.km)

 sigma.x  sigma.y 
2.224898 1.450966 

bw.ppl(childcareSG_ppp.km)

    sigma 
0.3897114 

bw.diggle(childcareSG_ppp.km)

    sigma 
0.2984095 

While there’s debate on which of these methods are the best, a study suggests to use bw.ppl() in patterns comprised primarily of tight clusters, and bw.diggle() to detect a single tight cluster in the midst of random noise. If we were to compare bw.diggle() and bw.ppl():

kde_childcareSG.ppl <- density(childcareSG_ppp.km, 
                               sigma=bw.ppl, 
                               edge=TRUE,
                               kernel="gaussian")
par(mfrow=c(1,2))
plot(kde_childcareSG.bw, main = "bw.diggle")
plot(kde_childcareSG.ppl, main = "bw.ppl")

4.1.4 Working with different kernel methods

By default, the kernel method used in density.ppp() is gaussian. But there are three other options, namely: Epanechnikov, Quartic and Dics. Let’s take a look at what they’re like:

par(mfrow=c(2,2))
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="gaussian"), 
     main="Gaussian")
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="epanechnikov"), 
     main="Epanechnikov")
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="quartic"), 
     main="Quartic")
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="disc"), 
     main="Disc")

4.1.5 Fixed and Adaptive KDE - Fixed Bandwidth

Let’s compute a KDE layer by defining a bandwidth of 600 meters. We’ll use a sigma value of 0.6 in this case, as the unit of measurement of our childcareSG_ppp.km object is in kilometers, hence the 600m is 0.6km.

kde_childcareSG_600 <- density(childcareSG_ppp.km, sigma=0.6, edge=TRUE, kernel="gaussian")
plot(kde_childcareSG_600)

A downside of fixed bandwidth is that this method is very sensitive to highly skewed distributions - so what do you do if your data is highly skewed? No fear, adaptive bandwidth is here!

4.1.6 Fixed and Adaptive KDE - Adaptive Bandwidth

kde_childcareSG_adaptive <- adaptive.density(childcareSG_ppp.km, method="kernel")
plot(kde_childcareSG_adaptive)

Let’s compare the fixed and adaptive kernel density estimation outputs:

par(mfrow=c(1,2))
plot(kde_childcareSG.bw, main = "Fixed bandwidth")
plot(kde_childcareSG_adaptive, main = "Adaptive bandwidth")

4.1.7 Converting KDE output into grid object into raster

Converting KDE output into grid object:

gridded_kde_childcareSG_bw <- as.SpatialGridDataFrame.im(kde_childcareSG.bw)
spplot(gridded_kde_childcareSG_bw)

Converting gridded output into raster with raster():

kde_childcareSG_bw_raster <- raster(gridded_kde_childcareSG_bw)

Let us take a look at the properties of kde_childcareSG_bw_raster RasterLayer:

kde_childcareSG_bw_raster

class      : RasterLayer 
dimensions : 128, 128, 16384  (nrow, ncol, ncell)
resolution : 0.4170614, 0.2647348  (x, y)
extent     : 2.663926, 56.04779, 16.35798, 50.24403  (xmin, xmax, ymin, ymax)
crs        : NA 
source     : memory
names      : v 
values     : -8.476185e-15, 28.51831  (min, max)

Note that the crs property is NA - let’s do something about that. To assign projection systems:

projection(kde_childcareSG_bw_raster) <- CRS("+init=EPSG:3414")
kde_childcareSG_bw_raster

class      : RasterLayer 
dimensions : 128, 128, 16384  (nrow, ncol, ncell)
resolution : 0.4170614, 0.2647348  (x, y)
extent     : 2.663926, 56.04779, 16.35798, 50.24403  (xmin, xmax, ymin, ymax)
crs        : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +units=m +no_defs 
source     : memory
names      : v 
values     : -8.476185e-15, 28.51831  (min, max)

Lastly, to visualise this output:

tm_shape(kde_childcareSG_bw_raster) + 
  tm_raster("v") +
  tm_layout(legend.position = c("right", "bottom"), frame = FALSE)

4.1.8 Comparing Spatial Point Patterns using KDE

Let’s compare the KDE of childcare at Punggol, Tampines, Chua Chu Kang and Jurong West planning areas.

4.1.8.1 Extracting study area

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",]

4.1.8.2 Plotting target planning areas

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

4.1.8.3 Converting the spatial point data frame into generic sp format

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

4.1.8.4 Creating owin object

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

4.1.8.5 Combining childcare points and the study area

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]

We’ll use rescale() to transform the unit of measurement from meters to kilometers:

childcare_pg_ppp.km = rescale(childcare_pg_ppp, 1000, "km")
childcare_tm_ppp.km = rescale(childcare_tm_ppp, 1000, "km")
childcare_ck_ppp.km = rescale(childcare_ck_ppp, 1000, "km")
childcare_jw_ppp.km = rescale(childcare_jw_ppp, 1000, "km")

Lastly, plotting it out:

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

4.1.8.6 Computing KDE

par(mfrow=c(2,2))
plot(density(childcare_pg_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Punggol")
plot(density(childcare_tm_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Tempines")
plot(density(childcare_ck_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Choa Chu Kang")
plot(density(childcare_jw_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="JUrong West")

4.1.8.6 Computing Fixed Bandwidth KDE

With 250m as the bandwidth:

par(mfrow=c(2,2))
plot(density(childcare_ck_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Chou Chu Kang")
plot(density(childcare_jw_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="JUrong West")
plot(density(childcare_pg_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Punggol")
plot(density(childcare_tm_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Tampines")

4.2 Nearest Neighbours Analysis

In this section, we’ll be performing the Clark-Evans test of aggregation for SPPA, using the clarkevans.test()

For our reference, the test hypotheses are: - Ho = The distribution of childcare services are randomly distributed. - H1= The distribution of childcare services are not randomly distributed. - The 95% confident interval will be used.

4.2.1 Testing spatial point patterns using Clark and Evans Test

clarkevans.test(childcareSG_ppp,
                correction="none",
                clipregion="sg_owin",
                alternative=c("clustered"),
                nsim=99)

    Clark-Evans test
    No edge correction
    Monte Carlo test based on 99 simulations of CSR with fixed n

data:  childcareSG_ppp
R = 0.54756, p-value = 0.01
alternative hypothesis: clustered (R < 1)

4.2.2 Clark-Evans Test: Choa Chu Kang planning area

clarkevans.test(childcare_ck_ppp,
                correction="none",
                clipregion=NULL,
                alternative=c("two.sided"),
                nsim=999)

    Clark-Evans test
    No edge correction
    Monte Carlo test based on 999 simulations of CSR with fixed n

data:  childcare_ck_ppp
R = 0.97844, p-value = 0.238
alternative hypothesis: two-sided

4.2.3 Clark-Evans Test: Tampines planning area

clarkevans.test(childcare_tm_ppp,
                correction="none",
                clipregion=NULL,
                alternative=c("two.sided"),
                nsim=999)

    Clark-Evans test
    No edge correction
    Monte Carlo test based on 999 simulations of CSR with fixed n

data:  childcare_tm_ppp
R = 0.79367, p-value = 0.002
alternative hypothesis: two-sided

5.0 Second Order Spatial Point Patterns Analysis

The G function measures the distribution of the distances from an arbitrary event to its nearest event. We’ll learn how to compute the G-function estimation with our handy Gest() function, as well as perform the Monte Carlo Simulation test with envelope().

5.1 Analysing Spatial Point Process Using G-Function

5.1.1 Choa Chu Kang planning area

5.1.1.1 Computing G-function estimation

G_CK = Gest(childcare_ck_ppp, correction = "border")
plot(G_CK, xlim=c(0,500))

5.1.1.2 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.

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

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.

plot(G_CK.csr)

5.1.2 Tampines planning area

5.1.2.1 Computing G-function estimation

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

5.1.2.2 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.

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

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.

plot(G_tm.csr)

5.2 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. Here, we’ll be using Fest() in place of Gest().

5.2.1 Choa Chu Kang planning area

5.2.1.1 Computing F-function estimation

F_CK = Fest(childcare_ck_ppp)
plot(F_CK)

5.2.1.2 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.

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

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.

plot(F_CK.csr)

5.2.2 Tampines planning area

5.2.2.1 Computing F-function estimation

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

5.2.2.2 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.

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

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.

plot(F_tm.csr)

5.3 Analysing Spatial Point Process Using K-Function

K-function measures the number of events found up to a given distance of any particular event.

5.3.1 Choa Chu Kang planning area

5.3.1.1 Computing K-function estimation

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

5.3.1.2 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.

K_ck.csr <- envelope(childcare_ck_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)

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.

plot(K_ck.csr, . - r ~ r, xlab="d", ylab="K(d)-r")

5.3.2 Tampines planning area

5.3.2.1 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))

5.3.2.2 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.

K_tm.csr <- envelope(childcare_tm_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)

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.

plot(K_tm.csr, . - r ~ r, 
     xlab="d", ylab="K(d)-r", xlim=c(0,500))

5.4 Analysing Spatial Point Process Using L-Function

5.4.1 Choa Chu Kang planning area

5.4.1.1 Computing L-function estimation

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

5.4.1.2 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 if smaller than alpha value of 0.001.

L_ck.csr <- envelope(childcare_ck_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)

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.

plot(L_ck.csr, . - r ~ r, xlab="d", ylab="L(d)-r")

5.4.2 Tampines planning area

5.4.2.1 Computing L-function estimation

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

5.4.2.2 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 if p-value is smaller than alpha value of 0.001.

L_tm.csr <- envelope(childcare_tm_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)

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.

plot(L_tm.csr, . - r ~ r, 
     xlab="d", ylab="L(d)-r", xlim=c(0,500))