2: Applied Spatial Interaction Models - Case study of Singapore public bus commuter flows

Author

Magdalene Chan

Published

December 11, 2023

Modified

December 17, 2023

Objectives

Traditionally, commuter surveys are used by transport operators and urban managers to uncover insights related to urban mobility challenges. However, commuter surveys are costly, time-consuming and often yield outdated information due to their extended completion timelines. As urban infrastructures become increasingly digitized e.g. with widespread adoption of GPS in vehicles and transport cards, digital data sets offer a framework for tracking movement patterns over space and time, promising a new way to understand commuter behavior.

Despite increasing amounts of open data available, there has not been significant practice research carried out to show how these disparate data sources can be integrated, analysed, and modelled to support policy making decisions. There is also a general lack of practical research to show how geospatial data science and analysis (GDSA) can be used to support decision-making.

This study aims to demonstrate the potential value of GDSA to integrate publicly available data from multiple sources. Specifically, it seeks to build spatial interaction models to determine factors affecting urban mobility patterns of public bus commuter flows.

Tasks

The following tasks will be undertaken in this exercise:

Geospatial Data Science

Derive an analytical hexagon data of 375m (this distance is the perpendicular distance between the hexagon centre and its edges) to represent the Traffic Analysis Zone (TAZ).
Construct an Origin-Destination (O-D) matrix of commuter flows at the analytics hexagon level for Weekdays Morning Peak: 6am-9am (inclusive) by integrating Passenger Volume by Origin Destination Bus Stops and Bus Stop Location from LTA DataMall.
Display the O-D flows of the passenger trips by using appropriate geovisualisation methods (not more than 5 maps).
Describe the spatial patterns revealed by the geovisualisation (not more than 100 words per visual).
Assemble at least three propulsive and three attractiveness variables by using aspatial and geospatial data from publicly available sources.
Compute a distance matrix by using the analytical hexagon data derived earlier.

Spatial Interaction Modelling

Calibrate spatial interactive models to determine factors affecting urban commuting flows at the selected time interval.
Present the modelling results by using appropriate geovisualisation and graphical visualisation method (not more than 5 visuals).
With reference to the Spatial Interaction Model output tables, maps and data visualisation prepared, describe the modelling results (not more than 100 words per visual).

Getting Started

The code chunk below uses p_load() of pacman package to check if the required packages have been installed on the computer. If they are, the packages will be launched. The following packages will be used:

sf package is used for importing, managing, and processing geospatial data.
sp package is used for processing geospatial data.
spdep package is used for computing spatial weights.
tmap package is used for thematic mapping.
tidyverse package is used for aspatial data wrangling.
knitr package is used for dynamic report generation in R.
stplanr package is used for plotting desire lines on maps.
httr package is used for working with HTTP.
performance package is used for computing statistical metrics such as RMSE.

pacman::p_load(sf, sp, spdep, tmap, tidyverse, knitr, stplanr, httr, performance)

Data Sets used

The open data sets used are:

Bus Stop Location (Last updated Jul 2023) from LTADataMall retrieved on 18 Nov 2023.
Passenger Volume by Origin Destination Bus Stops for August 2023 from LTADataMall retrieved on 18 Nov 2023.

The derived data sets used are:

Geospatial data sets of the locations of businesses and financial services specially compiled for ISSS624 AY2023-2024 Nov Sem.
Geocoded HDB Property Information data set based on Sep 2021 data.
Geocoded General Information of schools data set (Accurate as at 24 Mar 2021) from data.gov.sg retrieved on 8 Dec 2023.
Train Station Location (Last updated Feb 2023) from LTADataMall retrieved on 13 Dec 2023.

Derive Traffic Analysis Zone

Traffic Analysis Zones (TAZ) are used in travel demand modelling to depict the spatial layout of trip origins, destinations, population, employment, and other influential factors shaping travel demand. Urban areas are divided into zones, simplifying trips from one zone to another, despite actual travel between points.

There are at least two ways to define TAZ – in Singapore, planning subzones have been marked out by URA for urban planning purposes. However, the use of such planning subzones would divide the study area into irregular-sized polygons. An alternative would be to create spatial grids that divide the study area into equal-sized, regular polygons. This study will make use of hexagon-spatial grids as the TAZ for analysis.

Import Geospatial data: MPSZ

st_read() function of sf package is used to import MPSZ-2019 shapefile into R as a simple feature data frame called mpsz. As MPSZ-2019 uses svy21 projected coordinate system, the crs argument is set to 3414.

mpsz <- st_read(dsn = "data/geospatial", layer = "MPSZ-2019") %>%
  st_transform(crs = 3414)

Reading layer `MPSZ-2019' from data source 
  `C:\magdalenecjw\ISSS624 Geospatial\Take_Home_Exercise\Ex2\data\geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 332 features and 6 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 103.6057 ymin: 1.158699 xmax: 104.0885 ymax: 1.470775
Geodetic CRS:  WGS 84

Create Spatial Grids

For this study, a hexagon layer of 375m (where 375m is the perpendicular distance between the hexagon centre and its edges) will be created to represent the TAZ.

In the code chunk below, st_make_grid() of sf package is used to create the hexagon grid TAZ using the svy21 projected coordinate system. The cellsize argument refers to the cell size in the units that the crs of the spatial data is using and can be defined as the distance between opposite edges. Since the data is set in svy21 projected coordinate system, which uses metres as the unit, the value is set as c(750,750) to create a hexagon layer of 375m. The resulting data frame is then converted into a sf data frame and an index is added for each hexagon grid.

However, this hexagon layer created covers the entire plot area, including non-land areas. As this study is only interested in the land areas of Singapore, the mpsz sf data frame is also used to create a border of the land areas. This border is then used to clip the hexagon grid layer using st_intersection() of sf package, resulting in the output hex_grid_bounded sf data frame.

Code

# Create hexagon grid layer
hex_grid <- st_make_grid(mpsz, cellsize = c(750, 750), 
                         crs = 3413, what = "polygons", square = FALSE) %>%
  st_sf() %>%
  # Apply as.factor() since index will be used as the identifier to link to other data sets
  mutate(index = as.factor(row_number()))

# Create border of Singapore's land area
mpsz_border <- mpsz %>%
  summarise()

# Clip the hexagon grid layer
hex_grid_bounded <- st_intersection(hex_grid, mpsz_border)

To ensure that the hex_grid_bounded sf data frame is created correctly, it will be plotted on a map for visual inspection using tmap functions such as tm_shape() and tm_polygons().

Code

tmap_mode("plot")

tm_shape(hex_grid_bounded) +
  tm_polygons()

While the map above shows the TAZ for the land areas correctly, some zones are cut off by the borders and hence, incomplete. This should be rectified such that all zones have a complete hexagon shape.

The code chunk below checks if the hex_grid sf data frame intersects any polygons in the hex_grid_bounded sf data frame using st_intersects(). The grid index of these intersecting hexagons are then retrieved and filtered out from the hex_grid sf data frame to create a new hex_grid_bounded2 sf data frame.

Code

# Check if hex grid intersects any polygons using st_intersects
# Returns a list of intersecting hexagons
intersection_list = hex_grid$index[lengths(st_intersects(hex_grid, hex_grid_bounded)) > 0]

# Filter for the intersecting hexagons
hex_grid_bounded2 = hex_grid %>%
  filter(index %in% intersection_list)

To ensure that the hex_grid_bounded2 sf data frame is created correctly, it will be plotted on a map for visual inspection using tmap functions.

Code

tm_shape(hex_grid_bounded2) +
  tm_polygons()

The map above now shows the complete analytical hexagon data of 375m (perpendicular distance between the centre of hexagon and its edges) that represents the TAZ.

As not all TAZ have bus stops, the map could be further refined to reflect the Bus Stop Density within each TAZ.

Import Geospatial Data: Bus Stop Locations

The code chunk below uses the st_read() function of sf package to import BusStop shapefile into R as a simple feature data frame called BusStop. As BusStop uses svy21 projected coordinate system, the crs argument is set to 3414.

BusStop <- st_read(dsn = "data/geospatial", 
                layer = "BusStop") %>%
  st_transform(crs=3414)

Reading layer `BusStop' from data source 
  `C:\magdalenecjw\ISSS624 Geospatial\Take_Home_Exercise\Ex2\data\geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 5161 features and 3 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 3970.122 ymin: 26482.1 xmax: 48284.56 ymax: 52983.82
Projected CRS: SVY21

# Examine the structure of the data frame
str(BusStop)

Classes 'sf' and 'data.frame':  5161 obs. of  4 variables:
 $ BUS_STOP_N: chr  "22069" "32071" "44331" "96081" ...
 $ BUS_ROOF_N: chr  "B06" "B23" "B01" "B05" ...
 $ LOC_DESC  : chr  "OPP CEVA LOGISTICS" "AFT TRACK 13" "BLK 239" "GRACE INDEPENDENT CH" ...
 $ geometry  :sfc_POINT of length 5161; first list element:  'XY' num  13576 32884
 - attr(*, "sf_column")= chr "geometry"
 - attr(*, "agr")= Factor w/ 3 levels "constant","aggregate",..: NA NA NA
  ..- attr(*, "names")= chr [1:3] "BUS_STOP_N" "BUS_ROOF_N" "LOC_DESC"

There are a total of 5161 features in the BusStop shapefile. Notably, BUS_STOP_N is listed as a character variable. As this variable will be used as the identifier to link to the aspatial data, it should be transformed to a factor so that R treats it as a grouping variable.

Code

# Apply as.factor() to the column
BusStop$BUS_STOP_N <- as.factor(BusStop$BUS_STOP_N)

# Re-examine the structure of the data frame
str(BusStop)

Classes 'sf' and 'data.frame':  5161 obs. of  4 variables:
 $ BUS_STOP_N: Factor w/ 5145 levels "01012","01013",..: 1008 1724 2118 4972 431 3787 1160 2941 1610 4983 ...
 $ BUS_ROOF_N: chr  "B06" "B23" "B01" "B05" ...
 $ LOC_DESC  : chr  "OPP CEVA LOGISTICS" "AFT TRACK 13" "BLK 239" "GRACE INDEPENDENT CH" ...
 $ geometry  :sfc_POINT of length 5161; first list element:  'XY' num  13576 32884
 - attr(*, "sf_column")= chr "geometry"
 - attr(*, "agr")= Factor w/ 3 levels "constant","aggregate",..: NA NA NA
  ..- attr(*, "names")= chr [1:3] "BUS_STOP_N" "BUS_ROOF_N" "LOC_DESC"

Based on the output above, BUS_STOP_N is now a factor of 5145 levels. However, there are a total of 5161 observations, suggesting the presence of duplicate records. The code chunk below generates a list of duplicated bus stop codes using group_by() and filter() functions of dplyr package and displays them using datatable() function of DT package.

Code

duplicate <- BusStop %>%
  group_by(BUS_STOP_N) %>%
  filter(n() > 1) %>%
  arrange(BUS_STOP_N, LOC_DESC)

DT::datatable(duplicate)

The code chunk below uses the distinct() function of dplyr package to keep only the unique rows based on the BUS_STOP_N while preserving all other fields (through the argument .keep_all = TRUE). By default, distinct() keeps the first occurrence of each unique combination of values in the specified columns. Once duplicates are removed, the resultant sf data frame is re-examined for its structure.

Code

BusStop <- BusStop %>%
  distinct(BUS_STOP_N, .keep_all = TRUE)

# Re-examine the structure of the data frame
str(BusStop)

Classes 'sf' and 'data.frame':  5145 obs. of  4 variables:
 $ BUS_STOP_N: Factor w/ 5145 levels "01012","01013",..: 1008 1724 2118 4972 431 3787 1160 2941 1610 4983 ...
 $ BUS_ROOF_N: chr  "B06" "B23" "B01" "B05" ...
 $ LOC_DESC  : chr  "OPP CEVA LOGISTICS" "AFT TRACK 13" "BLK 239" "GRACE INDEPENDENT CH" ...
 $ geometry  :sfc_POINT of length 5145; first list element:  'XY' num  13576 32884
 - attr(*, "sf_column")= chr "geometry"
 - attr(*, "agr")= Factor w/ 3 levels "constant","aggregate",..: NA NA NA
  ..- attr(*, "names")= chr [1:3] "BUS_STOP_N" "BUS_ROOF_N" "LOC_DESC"

There are now a total of 5145 observations, aligned with the number of factor levels.

Compute Bus Stop Density

The code chunk below uses st_intersects() of sf package to return lists of bus stops that lie inside each TAZ and lengths() to count the number of bus stops in each list. This is then appended back to hex_grid_bounded2 sf data frame in a new column called busstop_count.

Code

hex_grid_bounded2$busstop_count <- lengths(st_intersects(hex_grid_bounded2, BusStop))

The map is then updated to fill by colour based on the bus stop density. This is the base map that will be used for future geovisualisation in this study.

Code

tm_shape(hex_grid_bounded2) +
  tm_fill(col = "busstop_count",
          palette = "Blues",
          style = "cont", 
          title = "Bus Stop Density") + 
  tm_borders(col = "grey")

Construct O-D matrix of Commuter Flow

The O-D matrix is a description of movement in a certain area and is used to assess the demand for transportation. In an O-D matrix, each cell is an intersection of a trip from an origin to a destination, and a higher number of trips implies a more in-demand bus route.

Import Passenger Volume by Origin-Destination Bus Stops

The code chunk below uses the read_csv() function of readr package (imported with the tidyverse package) to import the csv files into R and glimpse() is used to examine the data frame.

odbus <- read_csv("data/aspatial/origin_destination_bus_202308.csv")

# Examine the data frame
glimpse(odbus)

Rows: 5,709,512
Columns: 7
$ YEAR_MONTH          <chr> "2023-08", "2023-08", "2023-08", "2023-08", "2023-…
$ DAY_TYPE            <chr> "WEEKDAY", "WEEKENDS/HOLIDAY", "WEEKENDS/HOLIDAY",…
$ TIME_PER_HOUR       <dbl> 16, 16, 14, 14, 17, 17, 17, 17, 7, 17, 14, 10, 10,…
$ PT_TYPE             <chr> "BUS", "BUS", "BUS", "BUS", "BUS", "BUS", "BUS", "…
$ ORIGIN_PT_CODE      <chr> "04168", "04168", "80119", "80119", "44069", "4406…
$ DESTINATION_PT_CODE <chr> "10051", "10051", "90079", "90079", "17229", "1722…
$ TOTAL_TRIPS         <dbl> 7, 2, 3, 10, 5, 4, 3, 22, 3, 3, 7, 1, 3, 1, 3, 1, …

Based on the data frame structure seen above, ORIGIN_PT_CODE and DESTINATION_PT_CODE are listed as character variables. These variables are equivalent to BUS_STOP_N of BusStop sf data frame and should be transformed to factors so that R treats them as grouping variables.

Code

# Columns to convert to factors
columns_to_convert <- c("ORIGIN_PT_CODE", "DESTINATION_PT_CODE")

# Apply as.factor() to the adjusted columns
odbus[columns_to_convert] <- lapply(odbus[columns_to_convert], as.factor)

# Re-examine the data frame
glimpse(odbus)

Rows: 5,709,512
Columns: 7
$ YEAR_MONTH          <chr> "2023-08", "2023-08", "2023-08", "2023-08", "2023-…
$ DAY_TYPE            <chr> "WEEKDAY", "WEEKENDS/HOLIDAY", "WEEKENDS/HOLIDAY",…
$ TIME_PER_HOUR       <dbl> 16, 16, 14, 14, 17, 17, 17, 17, 7, 17, 14, 10, 10,…
$ PT_TYPE             <chr> "BUS", "BUS", "BUS", "BUS", "BUS", "BUS", "BUS", "…
$ ORIGIN_PT_CODE      <fct> 04168, 04168, 80119, 80119, 44069, 44069, 20281, 2…
$ DESTINATION_PT_CODE <fct> 10051, 10051, 90079, 90079, 17229, 17229, 20141, 2…
$ TOTAL_TRIPS         <dbl> 7, 2, 3, 10, 5, 4, 3, 22, 3, 3, 7, 1, 3, 1, 3, 1, …

Based on the output above, ORIGIN_PT_CODE and DESTINATION_PT_CODE are now factors.

Extract Commuting Flow data

The code chunk below extracts commuting flows for the target time period of Weekdays morning peak period (defined as between 6am to 9am (inclusive)).

Code

# Commute commuting flow for target time period
od_wkday_morn <- odbus %>%
  filter(DAY_TYPE == "WEEKDAY" & TIME_PER_HOUR >= 6 & TIME_PER_HOUR <=9) %>%
  group_by(ORIGIN_PT_CODE, DESTINATION_PT_CODE) %>%
  summarise(TRIPS = sum(TOTAL_TRIPS)) %>%
  ungroup()

The extracted commuting flow data is displayed for inspection using head() function shown in the code chunk below.

Code

head(od_wkday_morn, 10)

# A tibble: 10 × 3
   ORIGIN_PT_CODE DESTINATION_PT_CODE TRIPS
   <fct>          <fct>               <dbl>
 1 01012          01112                 276
 2 01012          01113                 143
 3 01012          01121                  66
 4 01012          01211                 134
 5 01012          01311                 256
 6 01012          07371                  22
 7 01012          60011                  26
 8 01012          60021                  20
 9 01012          60031                  21
10 01012          60159                  26

Geospatial Data Wrangling

The extracted commuting flows above are in aspatial format and will need to be converted into geospatial data.

The code chunk below populates the hexagon grid index (i.e. index) of hex_grid_bounded2 sf data frame into BusStop sf data frame. st_intersection() is used to perform point and polygon overlay and the output will be in point sf object. select() of dplyr package is then use to retain only BUS_STOP_N and index in the BusStop_hex sf data frame.

Code

# Identify hexagon grid index for each bus stop
BusStop_hex <- st_intersection(BusStop, hex_grid_bounded2) %>%
  select(BUS_STOP_N, index) %>%
  st_drop_geometry()

glimpse(BusStop_hex)

Rows: 5,140
Columns: 2
$ BUS_STOP_N <fct> 25059, 25751, 26379, 26299, 25761, 26399, 25719, 25711, 263…
$ index      <fct> 118, 146, 174, 175, 200, 201, 201, 201, 201, 202, 202, 203,…

From the print result above, there are now a total of 5140 observations. Five bus stops have been excluded in the resultant data frame as they are outside of Singapore’s boundary.

Next, append the hexagon grid index from BusStop_hex data frame to od_wkday_morn data frame for both ORIGIN_PT_CODE and DESTINATION_PT_CODE fields.

Code

# Join hexagon grid index for ORIGIN_PT_CODE
od_data <- left_join(od_wkday_morn , BusStop_hex,
            by = c("ORIGIN_PT_CODE" = "BUS_STOP_N")) %>%
  rename("ORIGIN_hex" = "index")

# Join hexagon grid index for DESTINATION_PT_CODE
od_data <- left_join(od_data , BusStop_hex,
            by = c("DESTINATION_PT_CODE" = "BUS_STOP_N")) %>%
  rename("DESTIN_hex" = "index") %>%
  drop_na() %>%
  group_by(ORIGIN_hex, DESTIN_hex) %>%
  summarise(TOTAL_TRIPS = sum(TRIPS))

The resultant data frame is checked for duplicates, if any.

Code

duplicate <- od_data %>%
  group_by_all() %>%
  filter(n()>1) %>%
  ungroup()

DT::datatable(duplicate)

The output above shows that there are no duplicates in the data frame.

Visualisation of O-D flows

The constructed O-D matrix can now be visualised using desire lines, which are rays connecting a site to associated location points.

Remove intra-zonal flows

The code chunk below will be used to remove intra-zonal flows that will not be plotted.

Code

od_plot <- od_data[od_data$ORIGIN_hex!=od_data$DESTIN_hex,]

Create desire lines

In this code chunk below, od2line() of stplanr package is used to create the desire lines.

Code

flowLine <- od2line(flow = od_plot, 
                    zones = hex_grid_bounded2,
                    zone_code = "index")

Visualise desire lines

In the code chunk below, tmap functions are used to visualise the resulting desire lines. To aid in a clearer and less cluttered visualization, only desire lines with at least 5000 trips are shown.

Code

tm_shape(hex_grid_bounded2) +
  tm_fill(col = "busstop_count",
          palette = "Blues",
          style = "cont",
          title = "Bus Stop Density") +
  tm_borders(col = "grey") +
flowLine %>%  
  filter(TOTAL_TRIPS >= 5000) %>%
tm_shape() +
  tm_lines(lwd = "TOTAL_TRIPS",
           style = "fixed",
           scale = c(1,2,3,4,5,7,9),
           n = 6, 
           alpha = 0.7,
           title.lwd = "Total Trips") + 
  tm_layout(main.title = "Desire Lines with at least 5000 trips \nbetween Traffic Analysis Zones for Weekday Morning Peak Period",
            main.title.position = "center",
            main.title.size = 1,
            frame = TRUE)

Insights

It is possible to identify centroids where people are travelling to or from during weekday morning peak period. Most centroids are in residential areas (e.g. Tampines, Jurong East, Punggol), though some are in the Central region. Notably, there are several long-distance bus routes linking the East to the North, and the Central region to the North-Western part of Singapore. The most in-demand bus route on the map appears to be between Woodlands Checkpoint and Kranji, but neither TAZ has the highest bus stop count.

It is also possible to zoom in to individual regions of Singapore to observe commuting trends. In the code chunk below, a left join is performed for hex_grid_bounded2 and mpsz sf data frames using st_join() and the left = TRUE argument. When plotting the desire lines, a filter can now be applied on the REGION_N column to zoom in to a specific region of Singapore e.g. West region.

Code

hex_grid_mpsz <- st_join(hex_grid_bounded2, mpsz, left = TRUE)

tmap_mode("view")

hex_grid_mpsz %>%
  filter(REGION_N == "WEST REGION") %>%  # Filter for West region
tm_shape() +
  tm_fill(col = "busstop_count",
          palette = "Blues",
          style = "cont",
          title = "Bus Stop Density",
          popup.vars = c("SUBZONE_N")) + 
  tm_view(set.zoom.limits = c(11,14)) + 
  tm_borders(col = "grey") +
flowLine %>%  
  filter(TOTAL_TRIPS >= 5000) %>%
tm_shape() +
  tm_lines(lwd = "TOTAL_TRIPS",
           style = "fixed",
           scale = c(1,2,3,4,5,7,9),
           n = 6, 
           alpha = 0.7,
           popup.vars = c("TOTAL_TRIPS"))

Code

tmap_mode("plot")

Insights

When zoomed into a specific region such as the West Region, it can be observed that bus stop density does not appear to directly correlate with the number of desire lines or the line thickness. For example, while hexagon grid index 772 has the highest bus stop density in the map above, its neighbouring grid 826 has more high volume flows that start from / end at that TAZ – of which, one of the high volume flows has as many as 89k trips being made per month.

As such, there is a need to further understand what drives such trends in commuting flows.

Assemble Propulsive and Attractiveness variable

Spatial interaction represents the flow of people, material, or information between locations in geographical space. As such, Spatial Interaction Models (SIM) can be used to understand more about what propels commuters from an origin zone and what attracts commuters to a destination zone.

Before such models can be built, there is a need to assemble propulsive and attractiveness variables. The following propulsive and attractiveness variables will be used:

Population Density: The population density in an area can significantly impact movement patterns as higher population densities in an area can act as a propulsive force, pushing people to travel from that area to other destinations for work. However, as there is no census data publicly available on a TAZ level, the number of HDB housing units will be used as a proxy for the population density – the more housing units there are within a TAZ, it can be implied that there is likely going to be a higher population within that zone.
Employment Opportunities Density: The density of employment opportunities in an area can significantly impact movement patterns as fewer employment opportunities in an area can act as a propulsive force, attracting people to travel from that area to other destinations for work. Conversely, more employment opportunities in an area can act as an attractiveness force, attracting people to travel from other origins to that area for work. However, as there is no employment data publicly available on a TAZ level, the number of businesses registered under a particular address will be used as a proxy for the employment opportunities density – more businesses registered within a TAZ suggests more employment opportunities within that zone.
School Density: The density of education institutions in an area can significantly impact movement patterns as fewer schools in an area can act as a propulsive force, attracting people to travel from that area to other destinations for education. Conversely, more education opportunities in an area can act as an attractiveness force, attracting people to travel from other origins to that area for school.
Financial Services Density: Close proximity of financial services such as banks, investment firms, or financial advisors to the workplace can contribute to the overall attractiveness of a destination for employment by offering convenience, business support, and signaling economic strength, particularly for industries where financial services are integral.
Public Transportation Nodes Density: Accessibility to public transportation can serve as an origin propulsive variable since areas with better access tend to generate more movement since they provide easier means to travel to various destinations. At the same time, accessibility to public transportation can also serve as a destination attractiveness variable since better access also makes an area more attractive to travel to for work. While not perfectly correlated, in this study, the density of public transportation nodes will be taken as an indicator of accessibility to public transportation.

Population Density

Import Aspatial Data: HDB

The code chunk below uses read_csv() function of readr package to import the specially prepared hdb csv data. The output R object is a tibble data frame called hdb. glimpse() is used to examine the structure of the tibble data frame.

hdb <- read_csv("data/aspatial/hdb.csv")

glimpse(hdb)

Rows: 12,442
Columns: 37
$ ...1                  <dbl> 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14…
$ blk_no                <chr> "1", "1", "1", "1", "1", "1", "1", "1", "1", "1"…
$ street                <chr> "BEACH RD", "BEDOK STH AVE 1", "CANTONMENT RD", …
$ max_floor_lvl         <dbl> 16, 14, 2, 15, 4, 25, 12, 14, 12, 2, 15, 15, 13,…
$ year_completed        <dbl> 1970, 1975, 2010, 1982, 1975, 1982, 1975, 1977, …
$ residential           <chr> "Y", "Y", "N", "Y", "Y", "Y", "Y", "Y", "Y", "N"…
$ commercial            <chr> "Y", "N", "Y", "N", "Y", "N", "N", "N", "Y", "Y"…
$ market_hawker         <chr> "N", "N", "N", "N", "N", "N", "N", "N", "N", "N"…
$ miscellaneous         <chr> "N", "Y", "N", "N", "N", "N", "Y", "Y", "N", "N"…
$ multistorey_carpark   <chr> "N", "N", "N", "N", "N", "N", "N", "N", "N", "N"…
$ precinct_pavilion     <chr> "N", "N", "N", "N", "N", "N", "N", "N", "N", "N"…
$ bldg_contract_town    <chr> "KWN", "BD", "CT", "BD", "PRC", "BM", "QT", "GL"…
$ total_dwelling_units  <dbl> 142, 206, 0, 102, 55, 96, 125, 247, 95, 0, 220, …
$ `1room_sold`          <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ `2room_sold`          <dbl> 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ `3room_sold`          <dbl> 138, 204, 0, 0, 54, 0, 118, 0, 62, 0, 216, 214, …
$ `4room_sold`          <dbl> 1, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ `5room_sold`          <dbl> 2, 2, 0, 92, 1, 96, 7, 0, 33, 0, 4, 5, 0, 4, 0, …
$ exec_sold             <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ multigen_sold         <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ studio_apartment_sold <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ `1room_rental`        <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 319, 0, 0, 0…
$ `2room_rental`        <dbl> 0, 0, 0, 0, 0, 0, 0, 247, 0, 0, 0, 0, 0, 0, 56, …
$ `3room_rental`        <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 1,…
$ other_room_rental     <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0,…
$ lat                   <dbl> 1.295097, 1.320852, 1.275488, 1.327969, 1.388610…
$ lng                   <dbl> 103.8541, 103.9337, 103.8414, 103.9227, 103.9881…
$ building              <chr> "RAFFLES HOTEL", "NIL", "PINNACLE @ DUXTON", "PI…
$ addr                  <chr> "1 BEACH ROAD RAFFLES HOTEL SINGAPORE 189673", "…
$ postal                <chr> "189673", "460001", "080001", "461001", "500001"…
$ SUBZONE_NO            <dbl> 2, 6, 3, 3, 1, 9, 10, 5, 3, 5, 1, 5, 2, 2, 1, 7,…
$ SUBZONE_N             <chr> "CITY HALL", "BEDOK SOUTH", "CHINATOWN", "KEMBAN…
$ SUBZONE_C             <chr> "DTSZ02", "BDSZ06", "OTSZ03", "BDSZ03", "CHSZ01"…
$ PLN_AREA_N            <chr> "DOWNTOWN CORE", "BEDOK", "OUTRAM", "BEDOK", "CH…
$ PLN_AREA_C            <chr> "DT", "BD", "OT", "BD", "CH", "BM", "QT", "GL", …
$ REGION_N              <chr> "CENTRAL REGION", "EAST REGION", "CENTRAL REGION…
$ REGION_C              <chr> "CR", "ER", "CR", "ER", "ER", "CR", "CR", "CR", …

The hdb tibble data frame consists of 12,442 rows and 37 columns. Each row of data shows one address (with postal code) and its corresponding number of dwelling units at that address. Based on the column names, there appears to be different building types i.e. “residential”, “commercial” , “market_hawker” and “miscellaneous”. For the purpose of computing a proxy for population density, filter() of dplyr package will be used to extract residential units.

Code

hdb_residential <- hdb %>%
  filter(residential == "Y")

# Examine the data
head(hdb_residential, 10)

# A tibble: 10 × 37
    ...1 blk_no street   max_f…¹ year_…² resid…³ comme…⁴ marke…⁵ misce…⁶ multi…⁷
   <dbl> <chr>  <chr>      <dbl>   <dbl> <chr>   <chr>   <chr>   <chr>   <chr>  
 1     0 1      BEACH RD      16    1970 Y       Y       N       N       N      
 2     1 1      BEDOK S…      14    1975 Y       N       N       Y       N      
 3     3 1      CHAI CH…      15    1982 Y       N       N       N       N      
 4     4 1      CHANGI …       4    1975 Y       Y       N       N       N      
 5     5 1      DELTA A…      25    1982 Y       N       N       N       N      
 6     6 1      DOVER RD      12    1975 Y       N       N       Y       N      
 7     7 1      EUNOS C…      14    1977 Y       N       N       Y       N      
 8     8 1      EVERTON…      12    1980 Y       Y       N       N       N      
 9    10 1      GHIM MO…      15    1975 Y       N       N       Y       N      
10    11 1      HAIG RD       15    1976 Y       N       N       Y       N      
# … with 27 more variables: precinct_pavilion <chr>, bldg_contract_town <chr>,
#   total_dwelling_units <dbl>, `1room_sold` <dbl>, `2room_sold` <dbl>,
#   `3room_sold` <dbl>, `4room_sold` <dbl>, `5room_sold` <dbl>,
#   exec_sold <dbl>, multigen_sold <dbl>, studio_apartment_sold <dbl>,
#   `1room_rental` <dbl>, `2room_rental` <dbl>, `3room_rental` <dbl>,
#   other_room_rental <dbl>, lat <dbl>, lng <dbl>, building <chr>, addr <chr>,
#   postal <chr>, SUBZONE_NO <dbl>, SUBZONE_N <chr>, SUBZONE_C <chr>, …

Among the residential units, there appears to be some outliers such as Raffles Hotel being indicated as a residential dwelling. Given this, the data will be checked for buildings which contain a “hotel” in its name using the grepl() function.

Code

hotels <- hdb_residential %>%
  filter(grepl("HOTEL", building, ignore.case = TRUE))

kable(hotels)

…1	blk_no	street	max_floor_lvl	year_completed	residential	commercial	market_hawker	miscellaneous	multistorey_carpark	precinct_pavilion	bldg_contract_town	total_dwelling_units	1room_sold	2room_sold	3room_sold	4room_sold	5room_sold	exec_sold	multigen_sold	studio_apartment_sold	1room_rental	2room_rental	3room_rental	other_room_rental	lat	lng	building	addr	postal	SUBZONE_NO	SUBZONE_N	SUBZONE_C	PLN_AREA_N	PLN_AREA_C	REGION_N	REGION_C
0	1	BEACH RD	16	1970	Y	Y	N	N	N	N	KWN	142	0	1	138	1	2	0	0	0	0	0	0	0	1.295097	103.8541	RAFFLES HOTEL	1 BEACH ROAD RAFFLES HOTEL SINGAPORE 189673	189673	2	CITY HALL	DTSZ02	DOWNTOWN CORE	DT	CENTRAL REGION	CR
4580	3	BEACH RD	16	1970	Y	Y	N	N	N	N	KWN	138	0	1	134	0	3	0	0	0	0	0	0	0	1.294801	103.8545	RAFFLES HOTEL SINGAPORE	3 BEACH ROAD RAFFLES HOTEL SINGAPORE SINGAPORE 189674	189674	2	CITY HALL	DTSZ02	DOWNTOWN CORE	DT	CENTRAL REGION	CR

The output above shows two listings which appear to be hotels. Upon checking, there appears to be an error with the geocoding due to similarities in address names. “Block 1 Beach Road” is indeed a HDB block while Raffles Hotel has a similar address name of “1 Beach Road”. However, both of these addresses have different postal codes. This could be due to an issue with the geocoding process as the original data set did not provide postal codes, leading to a confusion when there are similar addresses.

Given that 1 Beach Rd and 2 Beach Rd faced issues with geocoding, the data will be filtered for other similar addresses to ensure the geocoding has been done correctly.

Code

beach_rd <- hdb_residential %>%
  filter(grepl("BEACH RD", street, ignore.case = TRUE))

kable(beach_rd)

…1	blk_no	street	max_floor_lvl	year_completed	residential	commercial	market_hawker	miscellaneous	multistorey_carpark	precinct_pavilion	bldg_contract_town	total_dwelling_units	2room_sold	3room_sold	4room_sold	5room_sold	1room_rental	2room_rental	lat	lng	building	addr	postal	SUBZONE_NO	SUBZONE_N	SUBZONE_C	PLN_AREA_N	PLN_AREA_C	REGION_N	REGION_C
0	1	BEACH RD	16	1970	Y	Y	N	N	N	N	KWN	142	1	138	1	2	0	0	1.295097	103.8541	RAFFLES HOTEL	1 BEACH ROAD RAFFLES HOTEL SINGAPORE 189673	189673	2	CITY HALL	DTSZ02	DOWNTOWN CORE	DT	CENTRAL REGION	CR
1660	15	BEACH RD	20	1974	Y	Y	N	N	N	N	KWN	76	0	0	76	0	0	0	1.295796	103.8555	NIL	15 BEACH ROAD	NIL	1	BUGIS	DTSZ01	DOWNTOWN CORE	DT	CENTRAL REGION	CR
2079	17	BEACH RD	20	1974	Y	Y	N	N	N	N	KWN	76	0	0	76	0	0	0	1.303689	103.8636	GOLDEN BEACH VISTA	17 BEACH ROAD GOLDEN BEACH VISTA SINGAPORE 190017	190017	9	CRAWFORD	KLSZ09	KALLANG	KL	CENTRAL REGION	CR
2567	2	BEACH RD	16	1970	Y	Y	N	N	N	N	KWN	139	1	136	0	2	0	0	1.390462	103.9753	CHANGI BEACH CLUB	2 ANDOVER ROAD CHANGI BEACH CLUB SINGAPORE 509984	509984	1	CHANGI POINT	CHSZ01	CHANGI	CH	EAST REGION	ER
4580	3	BEACH RD	16	1970	Y	Y	N	N	N	N	KWN	138	1	134	0	3	0	0	1.294801	103.8545	RAFFLES HOTEL SINGAPORE	3 BEACH ROAD RAFFLES HOTEL SINGAPORE SINGAPORE 189674	189674	2	CITY HALL	DTSZ02	DOWNTOWN CORE	DT	CENTRAL REGION	CR
6028	4	BEACH RD	16	1968	Y	N	N	Y	N	N	KWN	336	0	0	0	0	336	0	1.304716	103.8652	NIL	4 BEACH ROAD SINGAPORE 190004	190004	9	CRAWFORD	KLSZ09	KALLANG	KL	CENTRAL REGION	CR
7743	5	BEACH RD	16	1968	Y	N	N	Y	N	N	KWN	336	0	0	0	0	336	0	1.298092	103.8569	BEACH ROAD CONSERVATION AREA	5 TAN QUEE LAN STREET BEACH ROAD CONSERVATION AREA SINGAPORE 188094	188094	1	BUGIS	DTSZ01	DOWNTOWN CORE	DT	CENTRAL REGION	CR
8956	6	BEACH RD	16	1968	Y	Y	N	N	N	N	KWN	198	45	1	28	0	57	67	1.303992	103.8644	BEACH ROAD GARDENS	6 BEACH ROAD BEACH ROAD GARDENS SINGAPORE 190006	190006	9	CRAWFORD	KLSZ09	KALLANG	KL	CENTRAL REGION	CR

According to postal code conventions, the postal codes for these HDB dwellings should be in the format of 1900XX, where XX is replaced by the block number. However, apart from 1 and 3 Beach Road identified earlier, 2, 5 and 15 Beach Road also do not have the correct postal codes, which suggests that the coordinates for these addresses are inaccurate.

The data will be modified using the mutate() and ifelse() functions of dplyr package. The correct coordinates values will be searched for using the OneMap service. The final output is then saved out as a new tibble data frame hdb_residential2.

Code

hdb_residential2 <- hdb_residential %>%
  mutate(postal = ifelse(blk_no == 1 & street == "BEACH RD", 190001, postal)) %>%
  mutate(lat = ifelse(blk_no == 1 & street == "BEACH RD", 1.3036714, lat)) %>%
  mutate(lng = ifelse(blk_no == 1 & street == "BEACH RD", 103.8644787, lng)) %>%
  mutate(postal = ifelse(blk_no == 2 & street == "BEACH RD", 190002, postal)) %>%
  mutate(lat = ifelse(blk_no == 2 & street == "BEACH RD", 1.3040331, lat)) %>%
  mutate(lng = ifelse(blk_no == 2 & street == "BEACH RD", 103.8649285, lng)) %>%
  mutate(postal = ifelse(blk_no == 3 & street == "BEACH RD", 190003, postal)) %>%
  mutate(lat = ifelse(blk_no == 3 & street == "BEACH RD", 1.3041872, lat)) %>%
  mutate(lng = ifelse(blk_no == 3 & street == "BEACH RD", 103.8651934, lng)) %>%
  mutate(postal = ifelse(blk_no == 5 & street == "BEACH RD", 190005, postal)) %>%
  mutate(lat = ifelse(blk_no == 5 & street == "BEACH RD", 1.3043463, lat)) %>%
  mutate(lng = ifelse(blk_no == 5 & street == "BEACH RD", 103.8648158, lng)) %>%
  mutate(postal = ifelse(blk_no == 15 & street == "BEACH RD", 190015, postal)) %>%
  mutate(lat = ifelse(blk_no == 15 & street == "BEACH RD", 1.3034254, lat)) %>%
  mutate(lng = ifelse(blk_no == 15 & street == "BEACH RD", 103.8631535, lng))

The resultant data frame is checked for duplicates, if any.

Code

duplicate <- hdb_residential2 %>%
  group_by_all() %>%
  filter(n()>1) %>%
  ungroup()

DT::datatable(duplicate)

The output above shows that there are no duplicate rows in the data set.

Convert Aspatial data to Geospatial data

To convert the aspatial data to geospatial data, the latitude (lat) and longitude (lng) columns will be used. These columns are in decimal degree format, indicating that the data is in wgs84 geographic coordinate system.

The code chunk below converts hdb_residential2 data frame into a sf data frame using st_as_sf() of sf package. The coords argument requires the column name of the x-coordinates first followed by the column name of the y-coordinates. Lastly, as the resultant data frame has many columns, only the required columns will be selected using select() function of dplyr package.

Code

hdb_residential_sf <- st_as_sf(hdb_residential2, 
                   coords = c("lng", "lat"),
                   crs=4326) %>%
  st_transform(crs = 3414) %>%
  select(postal, total_dwelling_units, geometry)

# Examine the structure of the data frame
str(hdb_residential_sf)

sf [10,181 × 3] (S3: sf/tbl_df/tbl/data.frame)
 $ postal              : chr [1:10181] "190001" "460001" "461001" "500001" ...
 $ total_dwelling_units: num [1:10181] 142 206 102 55 96 125 247 95 220 219 ...
 $ geometry            :sfc_POINT of length 10181; first list element:  'XY' num [1:2] 31468 31779
 - attr(*, "sf_column")= chr "geometry"
 - attr(*, "agr")= Factor w/ 3 levels "constant","aggregate",..: NA NA
  ..- attr(*, "names")= chr [1:2] "postal" "total_dwelling_units"

The table above shows the structure of hdb_residential_sf. A new column called geometry has been added into the data frame while the lng and lat columns have been dropped. To ensure that the hdb_residential_sf sf data frame is projected and converted correctly, it will be plotted on a map for visual inspection.

Code

# boundary map
tm_shape(hex_grid_bounded2) +
  tm_fill(col = "busstop_count",
          palette = "Blues",
          style = "cont",
          title = "Bus Stop Density") +
  tm_borders(col = "grey") +
# plot housing locations
tm_shape(hdb_residential_sf) +
  tm_dots() + 
  tm_layout(main.title = "Location of HDB Residential Units",
            main.title.position = "center",
            main.title.size = 1,
            frame = TRUE)

All the HDB Residential Units are encompassed in TAZ with bus stops, which aligns with the urban planning for Singapore where all HDB estates would have access to public transportation.

Performing in-polygon count

In the code chunk below, st_join() and st_intersects() is used to join hex_grid_bounded2 and hdb_residential_sf. The left = TRUE argument ensures that all elements from the hex_grid_bounded2 are retained in the result. Once the hexagon grid indexes are obtained for each point feature, the geometry can be dropped using st_drop_geometry(). group_by() function of dplyr package is then used to group the data by hexagon grid index before adding up the total count of dwelling units within each TAZ. Lastly, is.na() is used to check for missing counts and replaces them with 0 using the mutate() and ifelse() functions, ensuring a clean data set without missing values for further analysis and visualization.

Code

housing_count <- st_join(hex_grid_bounded2, hdb_residential_sf, 
                     join = st_intersects, left = TRUE) %>%
  st_drop_geometry() %>%
  group_by(index) %>%
  summarise(housing_count = sum(total_dwelling_units)) %>%
  ungroup() %>%
  mutate(housing_count = ifelse(is.na(housing_count), 0, housing_count))

A new hex_grid_bounded3 sf data frame is created using left_join() on hex_grid_bounded2 and housing_count. A summary of this new column is generated using summary().

Code

hex_grid_bounded3 <- left_join(hex_grid_bounded2, housing_count,
                               by = c("index" = "index"))

summary(hex_grid_bounded3$housing_count)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
    0.0     0.0     0.0   560.8     0.0  8324.0

The new variable is also plotted on a map for visual inspection.

Code

tm_shape(hex_grid_bounded3) +
  tm_fill(col = "housing_count",
          palette = "Blues",
          style = "cont", 
          title = "Housing Density") + 
  tm_borders(col = "grey")

Based on the summary stats and the choropleth map above, it can be observed that there is a large number of zero values in the field housing_count. If log() is going to be used to transform this field, an additional step is required to ensure that all zero values are replaced with a negligible offset that is between the values of 0 and 1 (but not either value).

Employment Opportunities Density

Import Geospatial data: Business

The code chunk below uses the st_read() function of sf package to import the specially prepared Business shapefile into R as a simple feature data frame called biz. As Business uses svy21 projected coordinate system, the crs argument is set to 3414.

Code

biz <- st_read(dsn = "data/geospatial", layer = "Business") %>%
  st_transform(crs = 3414)

Reading layer `Business' from data source 
  `C:\magdalenecjw\ISSS624 Geospatial\Take_Home_Exercise\Ex2\data\geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 6550 features and 3 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 3669.148 ymin: 25408.41 xmax: 47034.83 ymax: 50148.54
Projected CRS: SVY21 / Singapore TM

To ensure that the biz sf data frame is projected and converted correctly, it will be plotted on a map for visual inspection.

Code

# boundary map
tm_shape(hex_grid_bounded3) +
  tm_fill(col = "busstop_count",
          palette = "Blues",
          style = "cont",
          title = "Bus Stop Density") +
  tm_borders(col = "grey") +
# plot biz locations
tm_shape(biz) +
  tm_dots() + 
  tm_layout(main.title = "Location of Businesses",
            main.title.position = "center",
            main.title.size = 1,
            frame = TRUE)

The locations of businesses appear to be most densely concentrated in the central and west regions, which aligns with known business zones i.e. CBD and Jurong Industrial Area.

Perform point-in-polygon count

The code chunk below uses st_intersects() to return a list of businesses that lie inside each TAZ and lengths() to count the number of businesses in each list. This is then appended to the hex_grid_bounded3 sf data frame in a new column called biz_count and a summary of this new column is generated.

Code

hex_grid_bounded3$biz_count <- lengths(st_intersects(hex_grid_bounded3, biz))

summary(hex_grid_bounded3$biz_count)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  0.000   0.000   0.000   3.368   1.000 126.000

The new variable is also plotted on a map for visual inspection.

Code

tm_shape(hex_grid_bounded3) +
  tm_fill(col = "biz_count",
          palette = "Blues",
          style = "cont", 
          title = "Business Density") + 
  tm_borders(col = "grey")

Based on the summary stats and the choropleth map above, it can be observed that the distribution of biz_count is extremely skewed with a large number of zero values. If log() is going to be used to transform this field, an additional step is required to ensure that all zero values are replaced with a negligible offset that is between the values of 0 and 1 (but not either value).

Schools Density

Geocode Aspatial data: Schools

Address geocoding is the process of taking an aspatial description of a location, such as an address or postcode, and returning geographic coordinates, frequently latitude/longitude pair, to identify a location on the Earth’s surface.

Singapore Land Authority (SLA) supports an online geocoding service called OneMap API. Within which, the Search API service can be used to look up the address data or 6-digit postal code for an entered value, and returns both latitude, longitude and x,y coordinates of the searched location.

The code chunks below will perform geocoding using SLA OneMap API. The input data Generalinformationofschools is given in csv file format and read into R Studio environment using read_csv() function of readr package. A collection of http call functions of httr package of R will then be used to pass the individual records to the geocoding server at OneMap.

After completion of the geocoding process, two tibble data frames will be created: found and not_found. found contains all records that are geocoded correctly and not_found contains records that fail to be geocoded. The found data table will then be joined with the initial csv data table by using a unique identifier (i.e. postal code) common to both data tables and saved out as a csv file called schools.

Code

url <- "https://www.onemap.gov.sg/api/common/elastic/search"

csv <- read_csv("data/aspatial/Generalinformationofschools.csv")
postcodes <- csv$postal_code

found <- data.frame()
not_found <- data.frame()

for (postcode in postcodes) {
  query <- list('searchVal'=postcode, 'returnGeom'='Y', 'getAddrDetails'='Y', 'pageNum'='1')
  res <- GET(url, query=query)
  
  if ((content(res)$found)!=0){
    found <- rbind(found, data.frame(content(res))[4:13])
  } else {
    not_found = data.frame(postcode)
  }
}

merged <- merge(csv, found, by.x = "postal_code", by.y = "results.POSTAL", all = T)

write_csv(merged, "data/aspatial/schools.csv")
write_csv(not_found, "data/aspatial/not_found.csv")

Next, the csv file for the school(s) where the geocoding could not be completed is manually updated outside of the R environment before it is reloaded as a schools tibble data frame. A pipe operation is done to rename the coordinates columns and select only required columns.

Code

schools <- read_csv("data/aspatial/schools.csv")

schools <- schools %>%
  rename("latitude" = "results.LATITUDE",
         "longitude" = "results.LONGITUDE") %>%
  select(postal_code, school_name, latitude, longitude)

Conversion of Aspatial data to Geospatial data

Next, convert the aspatial data frame into a sf tibble data frame called schools_sf using st_as_sf() of sf package. The data frame is then converted into the svy21 projected coordinate system. The coordinates in the coords argument are specified in the order of longitude followed by latitude.

Code

schools_sf <- st_as_sf(schools, coords = c("longitude", "latitude"), 
                       crs = st_crs(4326)) %>%
  st_transform(crs = 3414)

To ensure that the schools_sf data frame is projected and converted correctly, it will be plotted on a map for visual inspection.

Code

tm_shape(hex_grid_bounded3) + 
  tm_fill(col = "busstop_count",
          palette = "Blues",
          style = "cont",
          title = "Bus Stop Density") +
  tm_borders(col = "grey") +
tm_shape(schools_sf) + 
  tm_dots() + 
  tm_layout(main.title = "Location of Schools",
            main.title.position = "center",
            main.title.size = 1,
            frame = TRUE)

All schools plotted are either encompassed in TAZ with bus stops, or are at the edge of a neighbouring TAZ, which aligns with the urban planning for Singapore where all schools are in close proximity to a bus stop.

Perform point-in-polygon count

The code chunk below uses st_intersects() to return a list of schools that lie inside each TAZ and lengths() to count the number of schools in each list. This is then appended back to hex_grid_bounded3 sf data frame in a new column called school_count. Lastly, a summary of this new column is generated.

Code

hex_grid_bounded3$school_count <- lengths(st_intersects(hex_grid_bounded3, schools_sf))

summary(hex_grid_bounded3$school_count)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.0000  0.0000  0.0000  0.1799  0.0000  5.0000

The new column is also plotted on a map for visual inspection.

Code

tm_shape(hex_grid_bounded3) +
  tm_fill(col = "school_count",
          palette = "Blues",
          style = "cont", 
          title = "School Density") + 
  tm_borders(col = "grey")

Based on the summary stats and the choropleth map above, it can be observed that there is a large number of zero values in the field school_count. If log() is going to be used to transform this field, an additional step is required to ensure that all zero values are replaced with a negligible offset that is between the values of 0 and 1 (but not either value).

Financial Services Density

Import Geospatial data: Train Stations

The code chunk below uses the st_read() function of sf package to import FinServ shapefile into R as a simple feature data frame called FinServ. As FinServ uses svy21 projected coordinate system, the crs argument is set to 3414.

Code

FinServ <- st_read(dsn = "data/geospatial", layer = "FinServ") %>%
  st_transform(crs = 3414)

Reading layer `FinServ' from data source 
  `C:\magdalenecjw\ISSS624 Geospatial\Take_Home_Exercise\Ex2\data\geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 3320 features and 3 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 4881.527 ymin: 25171.88 xmax: 46526.16 ymax: 49338.02
Projected CRS: SVY21 / Singapore TM

The FinServ sf data frame will now be plotted on a map for visual inspection.

Code

tm_shape(hex_grid_bounded3) + 
  tm_fill(col = "busstop_count",
          palette = "Blues",
          style = "cont",
          title = "Bus Stop Density") +
  tm_borders(col = "grey") +
tm_shape(FinServ) + 
  tm_dots() + 
  tm_layout(main.title = "Location of Financial Services",
            main.title.position = "center",
            main.title.size = 1,
            frame = TRUE)

The location of most financial services overlaps with TAZ with bus stops, though there are several financial services in the south region that are not served by bus stops.

Perform point-in-polygon count

The code chunk below uses st_intersects() to return a list of financial service centres that lie inside each TAZ and lengths() to count the number of financial service centres in each list. This is then appended back to hex_grid_bounded3 sf data frame in a new column called fin_count. Lastly, a summary of this new column is generated.

Code

hex_grid_bounded3$fin_count <- lengths(st_intersects(hex_grid_bounded3, FinServ))

summary(hex_grid_bounded3$fin_count)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  0.000   0.000   0.000   1.707   0.000 176.000

The new column is also plotted on a map for visual inspection.

Code

tm_shape(hex_grid_bounded3) +
  tm_fill(col = "fin_count",
          palette = "Blues",
          style = "cont", 
          title = "Financial Services Density") + 
  tm_borders(col = "grey")

Based on the summary stats and the choropleth map above, the distribution of fin_count is extremely skewed with a large number of zero values. If log() is going to be used to transform this field, an additional step is required to ensure that all zero values are replaced with a negligible offset that is between the values of 0 and 1 (but not either value).

Public Transportation Density

Bus stop counts has previously been computed. Another critical component of Singapore’s public transportation is the MRT / LRT system.

Import Geospatial data: Train Stations

The code chunk below uses the st_read() function of sf package to import RapidTransitSystemStation shapefile into R as a simple feature data frame called mrt_lrt. As RapidTransitSystemStation uses svy21 projected coordinate system, the crs argument is set to 3414.

Code

mrt_lrt <- st_read(dsn = "data/geospatial", layer = "RapidTransitSystemStation") %>%
  st_transform(crs = 3414)

Reading layer `RapidTransitSystemStation' from data source 
  `C:\magdalenecjw\ISSS624 Geospatial\Take_Home_Exercise\Ex2\data\geospatial' 
  using driver `ESRI Shapefile'

Warning in CPL_read_ogr(dsn, layer, query, as.character(options), quiet, :
GDAL Message 1: Non closed ring detected. To avoid accepting it, set the
OGR_GEOMETRY_ACCEPT_UNCLOSED_RING configuration option to NO

Simple feature collection with 220 features and 4 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 6068.209 ymin: 27478.44 xmax: 45377.5 ymax: 47913.58
Projected CRS: SVY21

In the output above, there is a warning that a non closed ring is detected in the geometry of the sf data frame. The code chunk below uses st_is_valid() to check for the error.

Code

st_is_valid(mrt_lrt)

  [1]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
 [13]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
 [25]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
 [37]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
 [49]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
 [61]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
 [73]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
 [85]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
 [97]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[109]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[121]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[133]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[145]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[157]  TRUE  TRUE FALSE  TRUE  TRUE  TRUE    NA  TRUE  TRUE  TRUE  TRUE  TRUE
[169]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[181]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[193]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[205]  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
[217]  TRUE FALSE  TRUE  TRUE

From the output above, it can be observed that there are two types of issues:

There are two invalid geometries (i.e. FALSE outcomes).
There is one geometry with an NA outcome.

The specific stations can be identified and addressed using the following code chunks.

Invalid geometries
NA geometries

Code

# Identify problematic geometries
problematic_geoms <- mrt_lrt[which(!st_is_valid(mrt_lrt)), ]

kable(problematic_geoms)

	TYP_CD	STN_NAM	TYP_CD_DES	STN_NAM_DE	geometry
159	0	NA	MRT	HARBOURFRONT MRT STATION	POLYGON ((26736.44 27495.44…
218	0	NA	MRT	UPPER THOMSON MRT STATION	POLYGON ((27808.12 37518.2,…

Based on the output above, the invalid geometries are for Harbourfront MRT Station and Upper Thomson MRT Station. These two invalid geometries will be fixed using the st_make_valid() function from the sf package.

Code

# Identify NA geometries
na_geoms <- mrt_lrt[is.na(st_is_valid(mrt_lrt)), ]

kable(na_geoms)

	TYP_CD	STN_NAM	TYP_CD_DES	STN_NAM_DE	geometry
163	0	NA	MRT	BOCC	POLYGON ((42614.11 35130.09…

Based on the output above, the NA geometry is for BOCC, which refers to the Bus Operations Control Centre. Given that this is not a public transportation node that is accessible to the general public, this feature will be dropped from the final mrt_lrt sf data frame.

The code chunk below fixes the mrt_lrt data frame by first filtering out the 163th feature, which corresponds to the BOCC station with the NA geometry, then applying st_make_valid() to resolve the two invalid geometries. The geometries are verified again after fixing.

Code

mrt_lrt_fixed <- mrt_lrt[-163, ] %>%
  st_make_valid()

# Verify that the geometries are valid after fixing
st_is_valid(mrt_lrt_fixed)

  [1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
 [16] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
 [31] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
 [46] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
 [61] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
 [76] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
 [91] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
[106] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
[121] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
[136] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
[151] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
[166] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
[181] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
[196] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
[211] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

The output above now shows that all the geometries are valid. The mrt_lrt_fixed sf data frame will now be plotted on a map for visual inspection.

Code

tm_shape(hex_grid_bounded3) + 
  tm_fill(col = "busstop_count",
          palette = "Blues",
          style = "cont", 
          title = "Bus Stop Density") +
  tm_borders(col = "grey") +
tm_shape(mrt_lrt_fixed) + 
  tm_polygons(col = "brown") + 
  tm_layout(main.title = "Location of MRT / LRT Stations",
            main.title.position = "center",
            main.title.size = 1,
            frame = TRUE)

Most MRT / LRT Stations are in a TAZ that contains at least one bus stop. However, some MRT / LRT Stations appear to cut across multiple TAZ.

Perform point-in-polygon count

As the geometries for the MRT / LRT stations are polygons that may cut across multiple TAZ, there is a need to convert them into points before Perform point-in-polygon count. A simple way could be to use st_centroid() to get the centroid of each polygon. Using centroids requires fewer computational resources than computing based on all points along polygon edges, especially with complex polygons featuring many vertices. Additionally, centroids act as singular points summarizing the entire shape, providing a simplified representation in contrast to edges.

Code

mrt_lrt_points <- st_centroid(mrt_lrt_fixed)

The code chunk below then uses st_intersects() to return a list of MRT / LRT stations that lie inside each TAZ and lengths() to count the number of stations in each list. This is then appended back to hex_grid_bounded2 sf data frame in a new column called mrtlrt_count. Lastly, a summary of this new column is generated.

Code

hex_grid_bounded3$mrtlrt_count <- lengths(st_intersects(hex_grid_bounded3, mrt_lrt_points))

summary(hex_grid_bounded3$mrtlrt_count)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.0000  0.0000  0.0000  0.1126  0.0000  4.0000

The new column is also plotted on a map for visual inspection.

Code

tm_shape(hex_grid_bounded3) +
  tm_fill(col = "mrtlrt_count",
          palette = "Blues",
          style = "cont", 
          title = "MRT/LRT Station Count") + 
  tm_borders(col = "grey")

Based on the summary stats and the choropleth map above, it can be observed that there is a large number of zero values in the field mrtlrt_count. If log() is going to be used to transform this field, an additional step is required to ensure that all zero values are replaced with a negligible offset that is between the values of 0 and 1 (but not either value).

Assemble the variables

The propulsive and attractiveness variables will be compiled into a tibble data frame called propulsive to be used for the Spatial Interaction Model later. A prefix of “o_” will be added to the column names to identify them as origin variables.

Code

propulsive <- hex_grid_bounded3 %>%
  st_drop_geometry() %>%
  select(index, housing_count, biz_count, 
         school_count, busstop_count, mrtlrt_count)
  

origin <- names(propulsive) %>%
  modify_at(-1, ~ paste0("o_", .))  # Add prefix to all except index

# Assign modified names back to the data frame
names(propulsive) <- origin

Prepare destination variables

The attractiveness variables will be compiled into a tibble data frame called attractiveness to be used for the Spatial Interaction Model later. A prefix of “d_” will be added to the column names to identify them as origin variables.

Code

attractiveness <- hex_grid_bounded3 %>%
  st_drop_geometry() %>%
  select(index, biz_count, school_count, 
         fin_count, busstop_count, mrtlrt_count)

destin <- names(attractiveness) %>%
  modify_at(-1, ~ paste0("d_", .))  # Add prefix to all except index

# Assign modified names back to the data frame
names(attractiveness) <- destin

Compute Distance Matrix

In spatial interaction, a distance matrix is a table that shows the distance between pairs of locations. A location’s distance from itself, which is shown in the main diagonal of a distance matrix table, is 0.

There are at least two ways to compute the required distance matrix – one based on sf and the other based on sp. However, past experience has shown that computing the distance matrix using sf function takes relatively longer than sp method especially when the data set is large. In view of this, sp method will be used in the code chunks below.

Converting sf data frame to SpatialPolygonsDataFrame

First as.Spatial() is used to convert the hex_grid_bounded2 sf tibble data frame to SpatialPolygonsDataFrame of sp object as shown in the code chunk below.

Code

hex_grid_sp <- as(hex_grid_bounded2, "Spatial")
hex_grid_sp

class       : SpatialPolygonsDataFrame 
features    : 1945 
extent      : 2292.538, 57042.54, 15315.71, 50606.24  (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       : index, busstop_count 
min values  :  1001,             0 
max values  :   997,            20

The code chunk below is used to measure the distance between the central points of each pair of spatial shapes using the function spDists() of sp package. This method is widely used due to its computational simplicity, providing a reasonably accurate indication of spatial connections between the shapes. As mentioned above, calculating distances between centroids demands less computational resources than calculations between all points along polygon edges, especially for intricate polygons with many vertices. Centroids also act as single point representations that provide an overview of the shape. While edges offer intricate shape details, the generalized perspective of centroids could be valuable when precise edge shapes are less critical.

Code

dist <- spDists(hex_grid_sp, longlat = FALSE)

# Examine the resultant matrix
head(dist, n=c(10, 10))

          [,1]     [,2]     [,3]     [,4]     [,5]     [,6]     [,7]     [,8]
 [1,]    0.000 1299.038 2598.076 3897.114 5196.152  750.000 1984.313 3269.174
 [2,] 1299.038    0.000 1299.038 2598.076 3897.114  750.000  750.000 1984.313
 [3,] 2598.076 1299.038    0.000 1299.038 2598.076 1984.313  750.000  750.000
 [4,] 3897.114 2598.076 1299.038    0.000 1299.038 3269.174 1984.313  750.000
 [5,] 5196.152 3897.114 2598.076 1299.038    0.000 4562.072 3269.174 1984.313
 [6,]  750.000  750.000 1984.313 3269.174 4562.072    0.000 1299.038 2598.076
 [7,] 1984.313  750.000  750.000 1984.313 3269.174 1299.038    0.000 1299.038
 [8,] 3269.174 1984.313  750.000  750.000 1984.313 2598.076 1299.038    0.000
 [9,] 4562.072 3269.174 1984.313  750.000  750.000 3897.114 2598.076 1299.038
[10,] 5857.687 4562.072 3269.174 1984.313  750.000 5196.152 3897.114 2598.076
          [,9]    [,10]
 [1,] 4562.072 5857.687
 [2,] 3269.174 4562.072
 [3,] 1984.313 3269.174
 [4,]  750.000 1984.313
 [5,]  750.000  750.000
 [6,] 3897.114 5196.152
 [7,] 2598.076 3897.114
 [8,] 1299.038 2598.076
 [9,]    0.000 1299.038
[10,] 1299.038    0.000

As seen in the output above, the result is a matrix object class and the column and row headers are not labeled with the hexagon grid index representing the TAZ.

Label column and row headers of distance matrix

To add the column and row headers, first create a list sorted according to the the distance matrix by TAZ.

Code

hex_names <- hex_grid_bounded2$index

Next, attach the hexagon grid index to row and column for distance matrix matching.

Code

colnames(dist) <- paste0(hex_names)
rownames(dist) <- paste0(hex_names)

Pivot distance value by hexagon grid index

Next, the distance matrix is pivoted into a long table by using the melt() function of reshape2 package and the row and column TAZ codes as show in the code chunk below.

Code

distPair <- reshape2::melt(dist) %>%
  rename(dist = value)

head(distPair, 10)

   Var1 Var2     dist
1    33   33    0.000
2    34   33 1299.038
3    35   33 2598.076
4    36   33 3897.114
5    37   33 5196.152
6    60   33  750.000
7    61   33 1984.313
8    62   33 3269.174
9    63   33 4562.072
10   64   33 5857.687

Notice that the within zone distance (intra-zonal distance) is 0.

Update intra-zonal distances

A constant value will be appended to the data frame to replace the intra-zonal distance of 0. To do so, first find the minimum non-zero value of the distance by using summary().

Code

distPair %>%
  filter(dist > 0) %>%
  summary()

      Var1           Var2           dist      
 Min.   :  33   Min.   :  33   Min.   :  750  
 1st Qu.: 966   1st Qu.: 966   1st Qu.:10894  
 Median :1667   Median :1667   Median :17477  
 Mean   :1752   Mean   :1752   Mean   :18932  
 3rd Qu.:2420   3rd Qu.:2420   3rd Qu.:25401  
 Max.   :3937   Max.   :3937   Max.   :57862

Given that the minimum distance is 750m, any values smaller than 750m can be used to represent intra-zonal distance. As such, in the code chunk below, a value of 375m (half of 750m) will be appended to intra-zonal distance to replace the current value of 0.

Code

distPair$dist <- ifelse(distPair$dist == 0,
                        375, distPair$dist)

# Examine data frame
summary(distPair)

      Var1           Var2           dist      
 Min.   :  33   Min.   :  33   Min.   :  375  
 1st Qu.: 966   1st Qu.: 966   1st Qu.:10894  
 Median :1667   Median :1667   Median :17477  
 Mean   :1752   Mean   :1752   Mean   :18923  
 3rd Qu.:2420   3rd Qu.:2420   3rd Qu.:25401  
 Max.   :3937   Max.   :3937   Max.   :57862

The minimum value is now 375m.

Lastly, the code chunk below is used to rename the origin and destination fields and to convert these two fields into factor data type.

Code

distPair <- distPair %>%
  rename(orig = Var1,
         dest = Var2) %>%
  mutate(across(c(orig, dest), as.factor))

Calibrate Spatial Interaction Model using Poisson Regression method

Spatial Interaction Models (SIM) are mathematical models for estimating flows between spatial entities.

There are four main types of traditional SIM: Unconstrained, Production-constrained, Attraction-constrained and Doubly-constrained.

All four SIM will be done for comparison.

Ordinary least square (OLS), log-normal, Poisson and negative binomial (NB) regression methods have been used extensively to calibrate OD flow models by processing flow data as different types of dependent variables.

The Poisson regression method is used for modelling when the dependent variable represents count outcomes that are discrete and non-negative, such as the number of occurrences of an event within a fixed unit of time or space.

Separating intra-flow from passenger volume

The code chunk below is used to add three new fields into the od_data dataframe.

Firstly, a new column FlowNoIntra is created to differentiate intra-zone trips from inter-zone trips based on the comparison of origin and destination zones. The ifelse() function is used to check if the values in columns ORIGIN_hex and DESTIN_hex are equal; if so, it is an indication that such trips start and end in the same zone, hence, the FlowNoIntra column is set to a value of 0. Otherwise, it holds the value from the TOTAL_TRIPS column.

Next, an offset column is created using another ifelse() statement where for intra-zone trips, the offset column is set to a negligible offset value of 0.000001. Otherwise, for inter-zone trips, it is set to 1.

Code

od_data$FlowNoIntra <- ifelse(
  od_data$ORIGIN_hex == od_data$DESTIN_hex, 0, od_data$TOTAL_TRIPS)
od_data$offset <- ifelse(
  od_data$ORIGIN_hex == od_data$DESTIN_hex, 0.000001, 1)

Next, inter-zonal flow will be selected from od_data and save into a new output data.frame called inter_zonal_flow by using the code chunk below.

Code

od_data <- od_data %>%
  filter(FlowNoIntra > 0)

Combining passenger volume data with distance value

Before joining od_data and distPair, convert the data value type of ORIGIN_hex and DESTIN_hex fields of od_data dataframe into factor data type.

Code

od_data$ORIGIN_hex <- as.factor(od_data$ORIGIN_hex)
od_data$DESTIN_hex <- as.factor(od_data$DESTIN_hex)

Next, left_join() of dplyr will be used to join od_data data frame and distPair data frame to give an output called flow_data.

Code

flow_data <- od_data %>%
  left_join (distPair,
             by = c("ORIGIN_hex" = "orig",
                    "DESTIN_hex" = "dest"))

Prepare origin attributes

Next, left_join() is used to join flow_data to the origin attributes prepared earlier (propulsive tibble data frame) using the hexagon grid index as a common identifier.

Code

flow_data <- flow_data %>%
  left_join(propulsive, by = c("ORIGIN_hex" = "index"))

Prepare destination attributes

Lastly, left_join() is used to join flow_data to the destination attributes prepared earlier (attractiveness tibble data frame) using the hexagon grid index as a common identifier.

Code

flow_data <- flow_data %>%
  left_join(attractiveness, by = c("DESTIN_hex" = "index"))

Visualising the dependent variable

The code chunk below plots the distribution of the dependent variable (i.e. TOTAL_TRIPS) as a histogram.

Code

ggplot(data = flow_data,
       aes(x = TOTAL_TRIPS)) +
  geom_histogram() + 
  labs(title = "Distribution of Trips across TAZ",
       x = "Total Trips",
       y = "Counts") +
  theme(plot.title = element_text(hjust = 0.5))

The distribution shown is highly skewed and does not resemble a normal distribution (bell-shaped curve).

The code chunk below visualises the relationship between the dependent variable and a key independent variable (i.e. dist) of the SIM. Specifying the argument method = lm within the geom_smooth() function fits a linear regression line to the data.

Code

ggplot(data = flow_data,
       aes(x = dist,
           y = TOTAL_TRIPS)) +
  geom_point() +
  geom_smooth(method = lm) + 
  labs(title = "Relationship between Distance and Total Trips", 
       x = "Distance", 
       y = "Total Trips") +
  theme(plot.title = element_text(hjust = 0.5))

The relationship between these two variables appears to be non-linear.

The scatterplot is re-plotted after applying a log transformation on both variables.

Code

ggplot(data = flow_data,
       aes(x = log(dist),
           y = log(TOTAL_TRIPS))) +
  geom_point() +
  geom_smooth(method = lm) + 
  labs(title = "Relationship between Log Distance and Log Total Trips", 
       x = "Log Distance", 
       y = "Log Total Trips") +
  theme(plot.title = element_text(hjust = 0.5))

While the relationship is still non-linear, it is less skewed than before.

Check for variables with zero values

Since Poisson Regression is based on log and log 0 is undefined, it is important to ensure that no 0 values in the explanatory variables before running the model. In the code chunk below, summary() is used to compute the summary statistics of all variables in the flow_data data frame.

Code

summary(flow_data)

   ORIGIN_hex      DESTIN_hex     TOTAL_TRIPS       FlowNoIntra     
 2634   :  285   2010   :  364   Min.   :    1.0   Min.   :    1.0  
 1849   :  284   1982   :  355   1st Qu.:    7.0   1st Qu.:    7.0  
 2010   :  283   2009   :  342   Median :   37.0   Median :   37.0  
 2038   :  282   1984   :  317   Mean   :  378.3   Mean   :  378.3  
 1987   :  281   2634   :  299   3rd Qu.:  177.0   3rd Qu.:  177.0  
 1990   :  280   1849   :  298   Max.   :89347.0   Max.   :89347.0  
 (Other):63356   (Other):63076                                      
     offset       dist       o_housing_count  o_biz_count      o_school_count  
 Min.   :1   Min.   :  750   Min.   :   0    Min.   :  0.000   Min.   :0.0000  
 1st Qu.:1   1st Qu.: 3000   1st Qu.:   0    1st Qu.:  0.000   1st Qu.:0.0000  
 Median :1   Median : 5408   Median : 976    Median :  1.000   Median :0.0000  
 Mean   :1   Mean   : 6280   Mean   :1848    Mean   :  6.571   Mean   :0.5865  
 3rd Qu.:1   3rd Qu.: 8842   3rd Qu.:3399    3rd Qu.:  6.000   3rd Qu.:1.0000  
 Max.   :1   Max.   :25200   Max.   :8324    Max.   :126.000   Max.   :5.0000  
                                                                               
 o_busstop_count  o_mrtlrt_count    d_biz_count      d_school_count  
 Min.   : 1.000   Min.   :0.0000   Min.   :  0.000   Min.   :0.0000  
 1st Qu.: 5.000   1st Qu.:0.0000   1st Qu.:  0.000   1st Qu.:0.0000  
 Median : 8.000   Median :0.0000   Median :  1.000   Median :0.0000  
 Mean   : 7.825   Mean   :0.4422   Mean   :  7.355   Mean   :0.5578  
 3rd Qu.:10.000   3rd Qu.:1.0000   3rd Qu.:  7.000   3rd Qu.:1.0000  
 Max.   :20.000   Max.   :4.0000   Max.   :126.000   Max.   :5.0000  
                                                                     
  d_fin_count      d_busstop_count  d_mrtlrt_count  
 Min.   :  0.000   Min.   : 1.000   Min.   :0.0000  
 1st Qu.:  0.000   1st Qu.: 5.000   1st Qu.:0.0000  
 Median :  3.000   Median : 8.000   Median :0.0000  
 Mean   :  8.999   Mean   : 7.766   Mean   :0.4669  
 3rd Qu.:  9.000   3rd Qu.:10.000   3rd Qu.:1.0000  
 Max.   :176.000   Max.   :20.000   Max.   :4.0000

The print report above reveals that all the count variables except o_busstop_count and d_busstop_count consist of 0 values, which should be replaced to a negligible value of 0.99.

In the code chunk below, vars(ends_with("_count")) selects all columns with names ending in “_count”, while ifelse() function checks if the values in these columns are equal to 0. If so, it replaces them with 0.99; otherwise, it leaves them unchanged.

Code

flow_data <- flow_data %>%
  mutate_at(vars(ends_with("_count")), ~ ifelse(. == 0, 0.99, .))

Run summary() again to check that the replacement has been done correctly.

Code

summary(flow_data)

   ORIGIN_hex      DESTIN_hex     TOTAL_TRIPS       FlowNoIntra     
 2634   :  285   2010   :  364   Min.   :    1.0   Min.   :    1.0  
 1849   :  284   1982   :  355   1st Qu.:    7.0   1st Qu.:    7.0  
 2010   :  283   2009   :  342   Median :   37.0   Median :   37.0  
 2038   :  282   1984   :  317   Mean   :  378.3   Mean   :  378.3  
 1987   :  281   2634   :  299   3rd Qu.:  177.0   3rd Qu.:  177.0  
 1990   :  280   1849   :  298   Max.   :89347.0   Max.   :89347.0  
 (Other):63356   (Other):63076                                      
     offset       dist       o_housing_count    o_biz_count     
 Min.   :1   Min.   :  750   Min.   :   0.99   Min.   :  0.990  
 1st Qu.:1   1st Qu.: 3000   1st Qu.:   0.99   1st Qu.:  0.990  
 Median :1   Median : 5408   Median : 976.00   Median :  1.000  
 Mean   :1   Mean   : 6280   Mean   :1848.20   Mean   :  6.987  
 3rd Qu.:1   3rd Qu.: 8842   3rd Qu.:3399.00   3rd Qu.:  6.000  
 Max.   :1   Max.   :25200   Max.   :8324.00   Max.   :126.000  
                                                                
 o_school_count  o_busstop_count  o_mrtlrt_count   d_biz_count     
 Min.   :0.990   Min.   : 1.000   Min.   :0.990   Min.   :  0.990  
 1st Qu.:0.990   1st Qu.: 5.000   1st Qu.:0.990   1st Qu.:  0.990  
 Median :0.990   Median : 8.000   Median :0.990   Median :  1.000  
 Mean   :1.183   Mean   : 7.825   Mean   :1.103   Mean   :  7.749  
 3rd Qu.:1.000   3rd Qu.:10.000   3rd Qu.:1.000   3rd Qu.:  7.000  
 Max.   :5.000   Max.   :20.000   Max.   :4.000   Max.   :126.000  
                                                                   
 d_school_count   d_fin_count      d_busstop_count  d_mrtlrt_count 
 Min.   :0.990   Min.   :  0.990   Min.   : 1.000   Min.   :0.990  
 1st Qu.:0.990   1st Qu.:  0.990   1st Qu.: 5.000   1st Qu.:0.990  
 Median :0.990   Median :  3.000   Median : 8.000   Median :0.990  
 Mean   :1.173   Mean   :  9.268   Mean   : 7.766   Mean   :1.122  
 3rd Qu.:1.000   3rd Qu.:  9.000   3rd Qu.:10.000   3rd Qu.:1.000  
 Max.   :5.000   Max.   :176.000   Max.   :20.000   Max.   :4.000

All 0 values in the count columns have now been replaced by 0.99.

Apply log() transformation to explanatory variables

As Poisson Regression is based on log, log() will be applied to all the explanatory variables before calibrating the various SIM.

Code

flow_data_log <- flow_data %>%
  mutate_at(vars(ends_with("_count")), log) %>%
  mutate(dist = log(dist))

summary(flow_data_log)

   ORIGIN_hex      DESTIN_hex     TOTAL_TRIPS       FlowNoIntra     
 2634   :  285   2010   :  364   Min.   :    1.0   Min.   :    1.0  
 1849   :  284   1982   :  355   1st Qu.:    7.0   1st Qu.:    7.0  
 2010   :  283   2009   :  342   Median :   37.0   Median :   37.0  
 2038   :  282   1984   :  317   Mean   :  378.3   Mean   :  378.3  
 1987   :  281   2634   :  299   3rd Qu.:  177.0   3rd Qu.:  177.0  
 1990   :  280   1849   :  298   Max.   :89347.0   Max.   :89347.0  
 (Other):63356   (Other):63076                                      
     offset       dist        o_housing_count     o_biz_count      
 Min.   :1   Min.   : 6.620   Min.   :-0.01005   Min.   :-0.01005  
 1st Qu.:1   1st Qu.: 8.006   1st Qu.:-0.01005   1st Qu.:-0.01005  
 Median :1   Median : 8.596   Median : 6.88346   Median : 0.00000  
 Mean   :1   Mean   : 8.484   Mean   : 4.87339   Mean   : 0.91389  
 3rd Qu.:1   3rd Qu.: 9.087   3rd Qu.: 8.13124   3rd Qu.: 1.79176  
 Max.   :1   Max.   :10.135   Max.   : 9.02690   Max.   : 4.83628  
                                                                   
 o_school_count     o_busstop_count o_mrtlrt_count      d_biz_count      
 Min.   :-0.01005   Min.   :0.000   Min.   :-0.01005   Min.   :-0.01005  
 1st Qu.:-0.01005   1st Qu.:1.609   1st Qu.:-0.01005   1st Qu.:-0.01005  
 Median :-0.01005   Median :2.079   Median :-0.01005   Median : 0.00000  
 Mean   : 0.11104   Mean   :1.934   Mean   : 0.05941   Mean   : 0.99195  
 3rd Qu.: 0.00000   3rd Qu.:2.303   3rd Qu.: 0.00000   3rd Qu.: 1.94591  
 Max.   : 1.60944   Max.   :2.996   Max.   : 1.38629   Max.   : 4.83628  
                                                                         
 d_school_count      d_fin_count       d_busstop_count d_mrtlrt_count    
 Min.   :-0.01005   Min.   :-0.01005   Min.   :0.000   Min.   :-0.01005  
 1st Qu.:-0.01005   1st Qu.:-0.01005   1st Qu.:1.609   1st Qu.:-0.01005  
 Median :-0.01005   Median : 1.09861   Median :2.079   Median :-0.01005  
 Mean   : 0.10488   Mean   : 1.29848   Mean   :1.920   Mean   : 0.07033  
 3rd Qu.: 0.00000   3rd Qu.: 2.19722   3rd Qu.:2.303   3rd Qu.: 0.00000  
 Max.   : 1.60944   Max.   : 5.17048   Max.   :2.996   Max.   : 1.38629

Unconstrained Spatial Interaction Model

Next, calibrate an unconstrained SIM by using glm() of Base Stats. The explanatory variables are all the origin and destination variables created earlier and distance between origin and destination (i.e. dist).

The code chunk used to calibrate the model is shown below:

Code

# Generate propulsive variables names
origin_var <- propulsive %>%
  select(-(index)) %>%
  names()

# Generate attractiveness variables names
destin_var <- attractiveness %>%
  select(-(index)) %>%
  names()

# Generate the formula dynamically
formula_string <- paste("TOTAL_TRIPS ~", paste(origin_var, collapse = " + "), 
                        "+", paste(destin_var, collapse = " + "), "+ dist")

# Convert the string to a formula
formula <- as.formula(formula_string)

uncSIM <- glm(formula,
              family = poisson(link = "log"),
              data = flow_data_log,
              na.action = na.exclude)

uncSIM


Call:  glm(formula = formula, family = poisson(link = "log"), data = flow_data_log, 
    na.action = na.exclude)

Coefficients:
    (Intercept)  o_housing_count      o_biz_count   o_school_count  
       14.83292          0.12078         -0.13711          0.18786  
o_busstop_count   o_mrtlrt_count      d_biz_count   d_school_count  
        0.55197          0.08988          0.04594          0.36963  
    d_fin_count  d_busstop_count   d_mrtlrt_count             dist  
        0.31038          0.23907         -0.01979         -1.45763  

Degrees of Freedom: 65050 Total (i.e. Null);  65039 Residual
Null Deviance:      98250000 
Residual Deviance: 49120000     AIC: 49470000

Positive coefficients suggest that as the counts of the explanatory variable increase by one unit, the number of trips is also expected to increase by the value of the coefficient, holding other variables constant. Conversely, negative coefficients suggest that as the counts of the explanatory variable increase by one unit, the number of trips is expected to decrease by the value of the coefficient, holding other variables constant.

Insights

By absolute value, the most influential variable is the Distance between origin and destination. The negative coefficient means there is an inverse relationship between Distance and Number of Trips: More trips are made when the distance is nearer, while less trips are made when the distance is further.
The most propulsive origin variable is Bus Stop counts with a coefficient of 0.551: More trips are made when there is higher bus stop density at the origin site, while less trips are made when there is lower bus stop density at the origin site.
There is an inverse relationship between Business counts at the origin and number of trips: More trips are made when there are fewer businesses at the origin site, while less trips are made when there are more businesses at the origin.
Conversely, the most attractive destination variables are School and Financial Services counts with a coefficient of 0.369 and 0.310 respectively: More trips are made when there is higher school / financial services density at the destination site, while less trips are made when there is lower school / financial services density at the destination site.
Notably, there appears to be an inverse relationship between the number of trips with the Business density at the origin and the MRT / LRT station density at the destination. This means that more trips are made when there are less businesses at the origin, while less trips are made when there are more businesses at the origin. Similarly, more bus trips are made when there are less MRT / LRT stations at the destination – suggesting a complementary role between bus services and MRT / LRT services.

In statistical modelling, the goodness-of-fit of a model looks at how well the proportion of variance in the dependent variable (i.e. TOTAL_TRIPS) can be explained by the explanatory variables. This can be answered by comparing the R² statistics. However, R² is not an output of glm(). Hence, in the code chunk below, a function called CalcRSquared is written to measure how much variation of the trips can be accounted for by the model.

Code

CalcRSquared <- function(observed,estimated){
  r <- cor(observed,estimated)
  R2 <- r^2
  R2
}

Next, compute the R-squared of the unconstrained SIM by using the code chunk below.

Code

CalcRSquared(uncSIM$data$TOTAL_TRIPS, uncSIM$fitted.values)

[1] 0.237972

Insights

With reference to the R² above, it can be concluded that the model accounts for 23.79% of the variation of flows.

Origin (Production) Constrained Spatial Interaction Model

Next, fit an origin constrained SIM by using the code chunk below.

For origin constrained SIM, only explanatory variables representing the attractiveness at the destinations will be used. This is because such models emphasize the limitations or capacities of the origins rather than the demand or attractiveness of the destinations. The capacity or limitation at the origin sites determines the potential for generating interactions or flows.
Additionally, “-1” is added to the formula to remove the intercept that is inserted by glm into the model by default. Since the origin has already been constrained, the concept of an intercept would not be relevant.

Code

# Generate the formula dynamically
formula_string <- paste("TOTAL_TRIPS ~ ORIGIN_hex +", 
                        paste(destin_var, collapse = " + "), "+ dist - 1")

# Convert the string to a formula
formula <- as.formula(formula_string)

orcSIM <- glm(formula,
              family = poisson(link = "log"),
              data = flow_data_log,
              na.action = na.exclude)

summary() is used to print out the results of the model.

Code

options(max.print=9999, scipen = 999, digits = 10)

summary(orcSIM)


Call:
glm(formula = formula, family = poisson(link = "log"), data = flow_data_log, 
    na.action = na.exclude)

Deviance Residuals: 
       Min          1Q      Median          3Q         Max  
-166.23959   -12.10816    -5.36754     0.60571   618.58598  

Coefficients:
                     Estimate    Std. Error     z value               Pr(>|z|)
ORIGIN_hex118   13.9366038450  0.1270153766   109.72375 < 0.000000000000000222
ORIGIN_hex146   14.7724327742  0.1414385229   104.44420 < 0.000000000000000222
ORIGIN_hex174   14.1811015041  0.1507727138    94.05615 < 0.000000000000000222
ORIGIN_hex175   12.7981750246  0.1104490056   115.87406 < 0.000000000000000222
ORIGIN_hex200   14.0645945955  0.0803371769   175.06956 < 0.000000000000000222
ORIGIN_hex201   15.0041443507  0.0259650906   577.85835 < 0.000000000000000222
ORIGIN_hex202   16.3629847660  0.0227318405   719.82666 < 0.000000000000000222
ORIGIN_hex203   13.9838534024  0.0496043336   281.90790 < 0.000000000000000222
ORIGIN_hex227   14.5175802768  0.0643115124   225.73844 < 0.000000000000000222
ORIGIN_hex228   14.1747020359  0.0319354049   443.85540 < 0.000000000000000222
ORIGIN_hex229   14.2080246363  0.0762749124   186.27389 < 0.000000000000000222
ORIGIN_hex230   16.5682661182  0.0130600811  1268.61893 < 0.000000000000000222
ORIGIN_hex231   13.4089529693  0.0685507709   195.60616 < 0.000000000000000222
ORIGIN_hex254   16.8102537450  0.0159997761  1050.65556 < 0.000000000000000222
ORIGIN_hex255   13.9459358616  0.0672993562   207.22243 < 0.000000000000000222
ORIGIN_hex256   13.9915576944  0.0500906588   279.32469 < 0.000000000000000222
ORIGIN_hex257   15.7845295619  0.0185539684   850.73604 < 0.000000000000000222
ORIGIN_hex258   13.3447722108  0.0702141352   190.05820 < 0.000000000000000222
ORIGIN_hex259   13.5843468688  0.0493097055   275.49033 < 0.000000000000000222
ORIGIN_hex281   15.6546677854  0.0446788383   350.38216 < 0.000000000000000222
ORIGIN_hex282   14.3220124593  0.0518166927   276.39766 < 0.000000000000000222
ORIGIN_hex284   14.4397501353  0.0375854726   384.18434 < 0.000000000000000222
ORIGIN_hex285   15.0946379471  0.0225632418   668.99243 < 0.000000000000000222
ORIGIN_hex286   17.5298391153  0.0061030080  2872.32772 < 0.000000000000000222
ORIGIN_hex312   13.1751808870  0.0672985707   195.77208 < 0.000000000000000222
ORIGIN_hex313   13.6131827521  0.0496728307   274.05692 < 0.000000000000000222
ORIGIN_hex314   14.4287843204  0.0347131528   415.65756 < 0.000000000000000222
ORIGIN_hex336   13.8038852484  0.1324702304   104.20368 < 0.000000000000000222
ORIGIN_hex338   14.3249593202  0.0404754517   353.91722 < 0.000000000000000222
ORIGIN_hex339   16.7243501681  0.0078763664  2123.35859 < 0.000000000000000222
ORIGIN_hex340   12.4957978248  0.1443512868    86.56520 < 0.000000000000000222
ORIGIN_hex366   17.2541268742  0.0068662177  2512.90124 < 0.000000000000000222
ORIGIN_hex367   14.6178866306  0.0256597487   569.68160 < 0.000000000000000222
ORIGIN_hex391   14.3570155902  0.0864163279   166.13777 < 0.000000000000000222
ORIGIN_hex392   17.0803817912  0.0077756831  2196.64068 < 0.000000000000000222
ORIGIN_hex393   14.6843821525  0.0254803344   576.30257 < 0.000000000000000222
ORIGIN_hex394   12.1220097544  0.2425429317    49.97882 < 0.000000000000000222
ORIGIN_hex419   12.9360137792  0.0860895662   150.26227 < 0.000000000000000222
ORIGIN_hex420   14.7421233994  0.0311575024   473.14843 < 0.000000000000000222
ORIGIN_hex421   13.9219648479  0.1204063391   115.62485 < 0.000000000000000222
ORIGIN_hex445   15.1874988120  0.0324022221   468.71782 < 0.000000000000000222
ORIGIN_hex446   15.9409967658  0.0129167397  1234.13471 < 0.000000000000000222
ORIGIN_hex447   13.7539372110  0.0555966795   247.38775 < 0.000000000000000222
ORIGIN_hex472   13.0895312866  0.1336467077    97.94129 < 0.000000000000000222
ORIGIN_hex473   13.8823193646  0.0488365696   284.26074 < 0.000000000000000222
ORIGIN_hex474   13.6713025472  0.0497331481   274.89317 < 0.000000000000000222
ORIGIN_hex499   14.0271765285  0.0422268382   332.18629 < 0.000000000000000222
ORIGIN_hex500   16.0901119155  0.0108577453  1481.90176 < 0.000000000000000222
ORIGIN_hex526   14.7821542251  0.0296256364   498.96495 < 0.000000000000000222
ORIGIN_hex527   14.5320400510  0.0504200501   288.21947 < 0.000000000000000222
ORIGIN_hex528   14.6552449305  0.0336942210   434.94832 < 0.000000000000000222
ORIGIN_hex552   14.1076316515  0.0547551111   257.64959 < 0.000000000000000222
ORIGIN_hex553   11.8380040852  0.1291152595    91.68555 < 0.000000000000000222
ORIGIN_hex554   14.5030024103  0.0365685920   396.59723 < 0.000000000000000222
ORIGIN_hex555   14.1516010769  0.0406715593   347.94833 < 0.000000000000000222
ORIGIN_hex581   14.4698579965  0.0303441269   476.85860 < 0.000000000000000222
ORIGIN_hex582   14.0263363936  0.0310332399   451.97783 < 0.000000000000000222
ORIGIN_hex607   14.3969622526  0.0426472801   337.58219 < 0.000000000000000222
ORIGIN_hex608   13.8914937966  0.0411139037   337.87825 < 0.000000000000000222
ORIGIN_hex609   18.6593137625  0.0032343889  5769.03838 < 0.000000000000000222
ORIGIN_hex610   14.3431170006  0.0534901000   268.14526 < 0.000000000000000222
ORIGIN_hex611   13.2113692050  0.2672716131    49.43050 < 0.000000000000000222
ORIGIN_hex634   15.1746946281  0.0343392210   441.90562 < 0.000000000000000222
ORIGIN_hex635   13.0063760475  0.0619292756   210.01983 < 0.000000000000000222
ORIGIN_hex636   15.1722421188  0.0138614762  1094.56178 < 0.000000000000000222
ORIGIN_hex638   15.2277011780  0.0223668947   680.81427 < 0.000000000000000222
ORIGIN_hex661   14.2974102581  0.0578703891   247.05917 < 0.000000000000000222
ORIGIN_hex662   14.2980793531  0.0363742022   393.08297 < 0.000000000000000222
ORIGIN_hex663   14.2019040267  0.0240744165   589.91685 < 0.000000000000000222
ORIGIN_hex664   14.4493189375  0.0457374314   315.91890 < 0.000000000000000222
ORIGIN_hex665   16.0763453980  0.0128956841  1246.64541 < 0.000000000000000222
ORIGIN_hex689   13.6115409380  0.0402421232   338.24112 < 0.000000000000000222
ORIGIN_hex690   14.0765386666  0.0316916175   444.17230 < 0.000000000000000222
ORIGIN_hex692   16.3439901989  0.0125038303  1307.11869 < 0.000000000000000222
ORIGIN_hex693   15.4124118062  0.0249905803   616.72885 < 0.000000000000000222
ORIGIN_hex715   13.9297297024  0.0553212217   251.79722 < 0.000000000000000222
ORIGIN_hex716   14.4524792597  0.0362127470   399.09923 < 0.000000000000000222
ORIGIN_hex717   15.1216437349  0.0171843287   879.96709 < 0.000000000000000222
ORIGIN_hex718   18.4758103791  0.0040185204  4597.66494 < 0.000000000000000222
ORIGIN_hex719   15.9780639760  0.0179327501   890.99909 < 0.000000000000000222
ORIGIN_hex720   11.7524781736  0.2582068688    45.51575 < 0.000000000000000222
ORIGIN_hex743   15.0737405181  0.0239868106   628.41787 < 0.000000000000000222
ORIGIN_hex744   14.7538191780  0.0252595868   584.08791 < 0.000000000000000222
ORIGIN_hex745   16.7702224080  0.0061774940  2714.72905 < 0.000000000000000222
ORIGIN_hex746   17.9904158440  0.0040863943  4402.51584 < 0.000000000000000222
ORIGIN_hex747   14.9589736422  0.0242710418   616.33010 < 0.000000000000000222
ORIGIN_hex748   13.7228298831  0.0597952196   229.49711 < 0.000000000000000222
ORIGIN_hex769   13.6609312603  0.0664019117   205.73099 < 0.000000000000000222
ORIGIN_hex770   14.7189717423  0.0245448867   599.67568 < 0.000000000000000222
ORIGIN_hex771   15.6544773835  0.0112489650  1391.63713 < 0.000000000000000222
ORIGIN_hex772   17.7232960619  0.0039684593  4466.03948 < 0.000000000000000222
ORIGIN_hex773   16.7518142194  0.0095406990  1755.82672 < 0.000000000000000222
ORIGIN_hex774   14.4629329738  0.0393713837   367.34632 < 0.000000000000000222
ORIGIN_hex775   15.1762305040  0.0254527823   596.25036 < 0.000000000000000222
ORIGIN_hex776   12.7295260947  0.2041355695    62.35820 < 0.000000000000000222
ORIGIN_hex777   12.9567368541  0.2085275273    62.13442 < 0.000000000000000222
ORIGIN_hex797   14.1246149718  0.0353986721   399.01539 < 0.000000000000000222
ORIGIN_hex798   13.5625514381  0.1078508311   125.75287 < 0.000000000000000222
ORIGIN_hex799   17.6802199736  0.0042956563  4115.83670 < 0.000000000000000222
ORIGIN_hex800   18.3746434120  0.0035175631  5223.68550 < 0.000000000000000222
ORIGIN_hex802   12.7918863617  0.0857738282   149.13508 < 0.000000000000000222
ORIGIN_hex803   12.2856360847  0.1313218010    93.55367 < 0.000000000000000222
ORIGIN_hex805   13.1094664846  0.0976107535   134.30351 < 0.000000000000000222
ORIGIN_hex806   15.3756989012  0.0436191335   352.49895 < 0.000000000000000222
ORIGIN_hex823   13.8123658636  0.0558679551   247.23235 < 0.000000000000000222
ORIGIN_hex824   13.6395184928  0.0523594178   260.49790 < 0.000000000000000222
ORIGIN_hex825   15.5323502359  0.0210751493   736.99835 < 0.000000000000000222
ORIGIN_hex826   18.7202357726  0.0026927158  6952.17654 < 0.000000000000000222
ORIGIN_hex827   17.9820864572  0.0041712390  4310.97011 < 0.000000000000000222
ORIGIN_hex832   12.3904342198  1.0000023263    12.39041 < 0.000000000000000222
ORIGIN_hex833   12.6977880164  0.1280528801    99.16050 < 0.000000000000000222
ORIGIN_hex851   12.3311089432  0.0980767963   125.72912 < 0.000000000000000222
ORIGIN_hex852   15.1920348314  0.0216729532   700.96746 < 0.000000000000000222
ORIGIN_hex853   15.7731292432  0.0090816336  1736.81631 < 0.000000000000000222
ORIGIN_hex854   17.0996487663  0.0049065281  3485.08116 < 0.000000000000000222
ORIGIN_hex856   11.7187939348  0.2041342199    57.40730 < 0.000000000000000222
ORIGIN_hex861   13.1092242594  0.1091280534   120.12699 < 0.000000000000000222
ORIGIN_hex862   15.0352693364  0.1010467995   148.79511 < 0.000000000000000222
ORIGIN_hex878   14.1868546739  0.0382308458   371.08399 < 0.000000000000000222
ORIGIN_hex879   14.2859123018  0.0251636143   567.72100 < 0.000000000000000222
ORIGIN_hex880   16.3075478175  0.0065182812  2501.81716 < 0.000000000000000222
ORIGIN_hex881   15.9868102466  0.0112198283  1424.87120 < 0.000000000000000222
ORIGIN_hex888   13.7972218246  0.1857111846    74.29397 < 0.000000000000000222
ORIGIN_hex889   15.5127869366  0.0591791702   262.13255 < 0.000000000000000222
ORIGIN_hex905   13.9474062868  0.0451314294   309.03976 < 0.000000000000000222
ORIGIN_hex906   11.6887837221  0.1270173726    92.02508 < 0.000000000000000222
ORIGIN_hex907   14.1038012619  0.0251065238   561.75842 < 0.000000000000000222
ORIGIN_hex908   17.9412459325  0.0035499503  5053.94281 < 0.000000000000000222
ORIGIN_hex910   13.8093769117  0.0658333289   209.76270 < 0.000000000000000222
ORIGIN_hex932   15.2289968379  0.0173364634   878.43734 < 0.000000000000000222
ORIGIN_hex933   15.1713886772  0.0148312274  1022.93548 < 0.000000000000000222
ORIGIN_hex934   16.4995524378  0.0061093352  2700.71160 < 0.000000000000000222
ORIGIN_hex935   17.5487381623  0.0042634970  4116.04324 < 0.000000000000000222
ORIGIN_hex937   17.7054062984  0.0069394186  2551.42503 < 0.000000000000000222
ORIGIN_hex959   15.7591370993  0.0149801551  1052.00093 < 0.000000000000000222
ORIGIN_hex960   16.4028299786  0.0068670638  2388.62349 < 0.000000000000000222
ORIGIN_hex961   17.4393923777  0.0043070758  4049.00988 < 0.000000000000000222
ORIGIN_hex962   17.8625860023  0.0037148873  4808.37898 < 0.000000000000000222
ORIGIN_hex986   12.5561997709  0.0894653714   140.34704 < 0.000000000000000222
ORIGIN_hex987   17.4955979303  0.0042207052  4145.18357 < 0.000000000000000222
ORIGIN_hex988   16.5519406633  0.0067086353  2467.25898 < 0.000000000000000222
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ORIGIN_hex2444  17.1338375036  0.0038518663  4448.19110 < 0.000000000000000222
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ORIGIN_hex2448  14.3106592717  0.0312013057   458.65578 < 0.000000000000000222
ORIGIN_hex2449  17.2909034002  0.0056883533  3039.70276 < 0.000000000000000222
ORIGIN_hex2450  17.2898537241  0.0046893474  3687.04900 < 0.000000000000000222
ORIGIN_hex2451  18.4004367171  0.0030970043  5941.36628 < 0.000000000000000222
ORIGIN_hex2452  13.8559503550  0.0350627036   395.17633 < 0.000000000000000222
ORIGIN_hex2471  16.3491031740  0.0056468185  2895.27688 < 0.000000000000000222
ORIGIN_hex2472  15.6866895534  0.0151758956  1033.65824 < 0.000000000000000222
ORIGIN_hex2473  17.0205296649  0.0049154848  3462.63496 < 0.000000000000000222
ORIGIN_hex2476  14.9280920966  0.0249038668   599.42868 < 0.000000000000000222
ORIGIN_hex2478  16.9524548047  0.0056726407  2988.45913 < 0.000000000000000222
ORIGIN_hex2479  15.6374714616  0.0114871440  1361.30195 < 0.000000000000000222
ORIGIN_hex2480  14.6009489096  0.0358857610   406.87305 < 0.000000000000000222
ORIGIN_hex2497  17.2475029464  0.0041023106  4204.33868 < 0.000000000000000222
ORIGIN_hex2498  15.4996608430  0.0090754686  1707.86341 < 0.000000000000000222
ORIGIN_hex2499  17.3202011631  0.0046274433  3742.93106 < 0.000000000000000222
ORIGIN_hex2500  13.2108453141  0.0430836773   306.63226 < 0.000000000000000222
ORIGIN_hex2503  11.8434267553  0.1525124995    77.65545 < 0.000000000000000222
ORIGIN_hex2504  15.8771741110  0.0136713611  1161.34553 < 0.000000000000000222
ORIGIN_hex2505  17.1056942501  0.0056347079  3035.77301 < 0.000000000000000222
ORIGIN_hex2525  15.9121030422  0.0077150619  2062.47250 < 0.000000000000000222
ORIGIN_hex2526  16.8530902174  0.0049346014  3415.28910 < 0.000000000000000222
ORIGIN_hex2527  17.1899687015  0.0043310900  3968.97059 < 0.000000000000000222
ORIGIN_hex2531  11.2484178635  0.4472188434    25.15193 < 0.000000000000000222
ORIGIN_hex2532  17.1734508373  0.0058979408  2911.77063 < 0.000000000000000222
ORIGIN_hex2533  17.6522720128  0.0066489371  2654.90133 < 0.000000000000000222
ORIGIN_hex2551  16.2059525734  0.0072130021  2246.76943 < 0.000000000000000222
ORIGIN_hex2552  15.3813930894  0.0122670732  1253.87636 < 0.000000000000000222
ORIGIN_hex2553  16.6994498055  0.0077821203  2145.87403 < 0.000000000000000222
ORIGIN_hex2554  15.5871402334  0.0104013505  1498.56889 < 0.000000000000000222
ORIGIN_hex2557  15.0365464083  0.0255846997   587.71635 < 0.000000000000000222
ORIGIN_hex2559  16.9434660775  0.0076236606  2222.48432 < 0.000000000000000222
ORIGIN_hex2579  16.2121783792  0.0069186399  2343.26090 < 0.000000000000000222
ORIGIN_hex2580  17.0779548188  0.0043857764  3893.94111 < 0.000000000000000222
ORIGIN_hex2581  16.7671725536  0.0059726654  2807.31826 < 0.000000000000000222
ORIGIN_hex2584  13.4894108372  0.0622923682   216.54998 < 0.000000000000000222
ORIGIN_hex2605  16.2874561167  0.0093552524  1740.99591 < 0.000000000000000222
ORIGIN_hex2606  15.8243353178  0.0145586533  1086.93675 < 0.000000000000000222
ORIGIN_hex2607  17.6886169612  0.0036561097  4838.09801 < 0.000000000000000222
ORIGIN_hex2608  16.2425616619  0.0079058924  2054.48807 < 0.000000000000000222
ORIGIN_hex2609  15.6032070216  0.0142714008  1093.31993 < 0.000000000000000222
ORIGIN_hex2610  14.9348288298  0.0238811954   625.38029 < 0.000000000000000222
ORIGIN_hex2611  13.2459825346  0.0825043271   160.54894 < 0.000000000000000222
ORIGIN_hex2633  16.2667238857  0.0068099321  2388.67635 < 0.000000000000000222
ORIGIN_hex2634  17.8564475041  0.0032086243  5565.14127 < 0.000000000000000222
ORIGIN_hex2635  17.1729740700  0.0050536519  3398.13157 < 0.000000000000000222
ORIGIN_hex2636  17.3539115150  0.0045625599  3803.54710 < 0.000000000000000222
ORIGIN_hex2637  15.9735653470  0.0097736717  1634.34643 < 0.000000000000000222
ORIGIN_hex2638  15.8623298594  0.0103508540  1532.46580 < 0.000000000000000222
ORIGIN_hex2660  15.7322691797  0.0081811960  1922.97913 < 0.000000000000000222
ORIGIN_hex2661  17.1459183703  0.0041521345  4129.42270 < 0.000000000000000222
ORIGIN_hex2662  16.2526716693  0.0063709592  2551.05569 < 0.000000000000000222
ORIGIN_hex2663  17.8792847258  0.0037691651  4743.56631 < 0.000000000000000222
ORIGIN_hex2664  16.4909270659  0.0066690783  2472.74457 < 0.000000000000000222
ORIGIN_hex2665  17.2499732949  0.0050480264  3417.17179 < 0.000000000000000222
ORIGIN_hex2687  16.2857549433  0.0071065218  2291.66326 < 0.000000000000000222
ORIGIN_hex2688  17.1314855241  0.0041614477  4116.71296 < 0.000000000000000222
ORIGIN_hex2689  17.2042078323  0.0043515028  3953.62443 < 0.000000000000000222
ORIGIN_hex2690  16.5481077419  0.0056416238  2933.21716 < 0.000000000000000222
ORIGIN_hex2691  17.9236206878  0.0035396498  5063.67067 < 0.000000000000000222
ORIGIN_hex2693  17.6222241704  0.0046109143  3821.85031 < 0.000000000000000222
ORIGIN_hex2714  16.6948865125  0.0054627973  3056.10583 < 0.000000000000000222
ORIGIN_hex2715  17.4942744090  0.0043912339  3983.90858 < 0.000000000000000222
ORIGIN_hex2716  13.4240585092  0.0258614638   519.07574 < 0.000000000000000222
ORIGIN_hex2717  16.7356469362  0.0047075466  3555.06772 < 0.000000000000000222
ORIGIN_hex2718  17.0403137496  0.0056103140  3037.31906 < 0.000000000000000222
ORIGIN_hex2719  17.7470326441  0.0043093044  4118.30567 < 0.000000000000000222
ORIGIN_hex2742  17.3744207353  0.0041675865  4168.94059 < 0.000000000000000222
ORIGIN_hex2743  16.1230301639  0.0106745414  1510.41900 < 0.000000000000000222
ORIGIN_hex2744  17.2425195750  0.0037837110  4557.03923 < 0.000000000000000222
ORIGIN_hex2745  17.2725363958  0.0039722265  4348.32621 < 0.000000000000000222
ORIGIN_hex2746  16.1606560527  0.0101694797  1589.13303 < 0.000000000000000222
ORIGIN_hex2747  16.9859025078  0.0063559257  2672.45141 < 0.000000000000000222
ORIGIN_hex2768  17.0697518131  0.0052338807  3261.39490 < 0.000000000000000222
ORIGIN_hex2769  17.5557690900  0.0039884418  4401.66107 < 0.000000000000000222
ORIGIN_hex2770  15.6246563414  0.0082958405  1883.43260 < 0.000000000000000222
ORIGIN_hex2771  17.9771628872  0.0031073948  5785.28440 < 0.000000000000000222
ORIGIN_hex2772  16.4208304044  0.0085502874  1920.50041 < 0.000000000000000222
ORIGIN_hex2773  17.1044941396  0.0050453342  3390.16081 < 0.000000000000000222
ORIGIN_hex2795  13.5090544628  0.0985601494   137.06406 < 0.000000000000000222
ORIGIN_hex2796  14.7468421273  0.0212268206   694.72685 < 0.000000000000000222
ORIGIN_hex2797  15.8568425542  0.0078093432  2030.49631 < 0.000000000000000222
ORIGIN_hex2798  16.3672904552  0.0056532072  2895.22213 < 0.000000000000000222
ORIGIN_hex2799  17.0089310330  0.0044905326  3787.73131 < 0.000000000000000222
ORIGIN_hex2800  17.0427544706  0.0051841637  3287.46460 < 0.000000000000000222
ORIGIN_hex2801  14.9765461024  0.0269667488   555.37085 < 0.000000000000000222
ORIGIN_hex2822  14.7158635607  0.0177932983   827.04529 < 0.000000000000000222
ORIGIN_hex2823  15.1335300561  0.0121030819  1250.38649 < 0.000000000000000222
ORIGIN_hex2824  16.8635918156  0.0056046565  3008.85379 < 0.000000000000000222
ORIGIN_hex2825  17.4450595854  0.0037252956  4682.86581 < 0.000000000000000222
ORIGIN_hex2826  17.3823299925  0.0051742740  3359.37566 < 0.000000000000000222
ORIGIN_hex2827  17.4818189274  0.0041911253  4171.15159 < 0.000000000000000222
ORIGIN_hex2850  16.0822304293  0.0107468821  1496.45547 < 0.000000000000000222
ORIGIN_hex2851  14.5781690079  0.0159200533   915.71107 < 0.000000000000000222
ORIGIN_hex2852  17.0880700847  0.0047222466  3618.63146 < 0.000000000000000222
ORIGIN_hex2853  17.8032769294  0.0034042358  5229.74255 < 0.000000000000000222
ORIGIN_hex2854  17.7035393463  0.0040416911  4380.23063 < 0.000000000000000222
ORIGIN_hex2877  15.1296973669  0.0171480366   882.29910 < 0.000000000000000222
ORIGIN_hex2878  16.5257430178  0.0060125002  2748.56425 < 0.000000000000000222
ORIGIN_hex2879  17.2881670937  0.0057116727  3026.81334 < 0.000000000000000222
ORIGIN_hex2880  16.9707275911  0.0049876065  3402.57947 < 0.000000000000000222
ORIGIN_hex2881  17.5867596855  0.0045340413  3878.82655 < 0.000000000000000222
ORIGIN_hex2905  16.6352374382  0.0073934341  2250.00146 < 0.000000000000000222
ORIGIN_hex2906  16.2287268983  0.0095937500  1691.59367 < 0.000000000000000222
ORIGIN_hex2907  17.0231874462  0.0047597130  3576.51550 < 0.000000000000000222
ORIGIN_hex2908  17.3036661334  0.0049111967  3523.30953 < 0.000000000000000222
ORIGIN_hex2909  16.8143153474  0.0199914578   841.07500 < 0.000000000000000222
ORIGIN_hex2931  13.3611215059  0.0434828727   307.27320 < 0.000000000000000222
ORIGIN_hex2932  12.2307805882  0.0695341837   175.89594 < 0.000000000000000222
ORIGIN_hex2933  15.5426108415  0.0124840373  1244.99875 < 0.000000000000000222
ORIGIN_hex2934  16.6308266570  0.0059920957  2775.46080 < 0.000000000000000222
ORIGIN_hex2959  14.0438879995  0.0287974686   487.67786 < 0.000000000000000222
ORIGIN_hex2960  16.4357804140  0.0180023718   912.97861 < 0.000000000000000222
ORIGIN_hex2961  16.5958259456  0.0076583860  2167.01352 < 0.000000000000000222
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ORIGIN_hex118   ***
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ORIGIN_hex227   ***
ORIGIN_hex228   ***
ORIGIN_hex229   ***
ORIGIN_hex230   ***
ORIGIN_hex231   ***
ORIGIN_hex254   ***
ORIGIN_hex255   ***
ORIGIN_hex256   ***
ORIGIN_hex257   ***
ORIGIN_hex258   ***
ORIGIN_hex259   ***
ORIGIN_hex281   ***
ORIGIN_hex282   ***
ORIGIN_hex284   ***
ORIGIN_hex285   ***
ORIGIN_hex286   ***
ORIGIN_hex312   ***
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ORIGIN_hex339   ***
ORIGIN_hex340   ***
ORIGIN_hex366   ***
ORIGIN_hex367   ***
ORIGIN_hex391   ***
ORIGIN_hex392   ***
ORIGIN_hex393   ***
ORIGIN_hex394   ***
ORIGIN_hex419   ***
ORIGIN_hex420   ***
ORIGIN_hex421   ***
ORIGIN_hex445   ***
ORIGIN_hex446   ***
ORIGIN_hex447   ***
ORIGIN_hex472   ***
ORIGIN_hex473   ***
ORIGIN_hex474   ***
ORIGIN_hex499   ***
ORIGIN_hex500   ***
ORIGIN_hex526   ***
ORIGIN_hex527   ***
ORIGIN_hex528   ***
ORIGIN_hex552   ***
ORIGIN_hex553   ***
ORIGIN_hex554   ***
ORIGIN_hex555   ***
ORIGIN_hex581   ***
ORIGIN_hex582   ***
ORIGIN_hex607   ***
ORIGIN_hex608   ***
ORIGIN_hex609   ***
ORIGIN_hex610   ***
ORIGIN_hex611   ***
ORIGIN_hex634   ***
ORIGIN_hex635   ***
ORIGIN_hex636   ***
ORIGIN_hex638   ***
ORIGIN_hex661   ***
ORIGIN_hex662   ***
ORIGIN_hex663   ***
ORIGIN_hex664   ***
ORIGIN_hex665   ***
ORIGIN_hex689   ***
ORIGIN_hex690   ***
ORIGIN_hex692   ***
ORIGIN_hex693   ***
ORIGIN_hex715   ***
ORIGIN_hex716   ***
ORIGIN_hex717   ***
ORIGIN_hex718   ***
ORIGIN_hex719   ***
ORIGIN_hex720   ***
ORIGIN_hex743   ***
ORIGIN_hex744   ***
ORIGIN_hex745   ***
ORIGIN_hex746   ***
ORIGIN_hex747   ***
ORIGIN_hex748   ***
ORIGIN_hex769   ***
ORIGIN_hex770   ***
ORIGIN_hex771   ***
ORIGIN_hex772   ***
ORIGIN_hex773   ***
ORIGIN_hex774   ***
ORIGIN_hex775   ***
ORIGIN_hex776   ***
ORIGIN_hex777   ***
ORIGIN_hex797   ***
ORIGIN_hex798   ***
ORIGIN_hex799   ***
ORIGIN_hex800   ***
ORIGIN_hex802   ***
ORIGIN_hex803   ***
ORIGIN_hex805   ***
ORIGIN_hex806   ***
ORIGIN_hex823   ***
ORIGIN_hex824   ***
ORIGIN_hex825   ***
ORIGIN_hex826   ***
ORIGIN_hex827   ***
ORIGIN_hex832   ***
ORIGIN_hex833   ***
ORIGIN_hex851   ***
ORIGIN_hex852   ***
ORIGIN_hex853   ***
ORIGIN_hex854   ***
ORIGIN_hex856   ***
ORIGIN_hex861   ***
ORIGIN_hex862   ***
ORIGIN_hex878   ***
ORIGIN_hex879   ***
ORIGIN_hex880   ***
ORIGIN_hex881   ***
ORIGIN_hex888   ***
ORIGIN_hex889   ***
ORIGIN_hex905   ***
ORIGIN_hex906   ***
ORIGIN_hex907   ***
ORIGIN_hex908   ***
ORIGIN_hex910   ***
ORIGIN_hex932   ***
ORIGIN_hex933   ***
ORIGIN_hex934   ***
ORIGIN_hex935   ***
ORIGIN_hex937   ***
ORIGIN_hex959   ***
ORIGIN_hex960   ***
ORIGIN_hex961   ***
ORIGIN_hex962   ***
ORIGIN_hex986   ***
ORIGIN_hex987   ***
ORIGIN_hex988   ***
ORIGIN_hex989   ***
ORIGIN_hex991   ***
ORIGIN_hex1013  ***
ORIGIN_hex1014  ***
ORIGIN_hex1015  ***
ORIGIN_hex1016  ***
ORIGIN_hex1042  ***
ORIGIN_hex1043  ***
ORIGIN_hex1045  ***
ORIGIN_hex1067  ***
ORIGIN_hex1068  ***
ORIGIN_hex1069  ***
ORIGIN_hex1070  ***
ORIGIN_hex1078  ***
ORIGIN_hex1093  ***
ORIGIN_hex1094  ***
ORIGIN_hex1095  ***
ORIGIN_hex1096  ***
ORIGIN_hex1097  ***
ORIGIN_hex1099  ***
ORIGIN_hex1100  ***
ORIGIN_hex1122  ***
ORIGIN_hex1123  ***
ORIGIN_hex1124  ***
ORIGIN_hex1125  ***
ORIGIN_hex1126  ***
ORIGIN_hex1127  ***
ORIGIN_hex1128  ***
ORIGIN_hex1132  ***
ORIGIN_hex1147  ***
ORIGIN_hex1148  ***
ORIGIN_hex1149  ***
ORIGIN_hex1150  ***
ORIGIN_hex1151  ***
ORIGIN_hex1152  ***
ORIGIN_hex1153  ***
ORIGIN_hex1154  ***
ORIGIN_hex1155  ***
ORIGIN_hex1158  ***
ORIGIN_hex1174  ***
ORIGIN_hex1175  ***
ORIGIN_hex1176  ***
ORIGIN_hex1177  ***
ORIGIN_hex1178  ***
ORIGIN_hex1179  ***
ORIGIN_hex1180  ***
ORIGIN_hex1181  ***
ORIGIN_hex1182  ***
ORIGIN_hex1183  ***
ORIGIN_hex1185  ***
ORIGIN_hex1201  ***
ORIGIN_hex1202  ***
ORIGIN_hex1203  ***
ORIGIN_hex1205  ***
ORIGIN_hex1206  ***
ORIGIN_hex1207  ***
ORIGIN_hex1209  ***
ORIGIN_hex1210  ***
ORIGIN_hex1211  ***
ORIGIN_hex1212  ***
ORIGIN_hex1228  ***
ORIGIN_hex1229  ***
ORIGIN_hex1230  ***
ORIGIN_hex1231  ***
ORIGIN_hex1232  ***
ORIGIN_hex1233  ***
ORIGIN_hex1235  ***
ORIGIN_hex1236  ***
ORIGIN_hex1237  ***
ORIGIN_hex1238  ***
ORIGIN_hex1239  ***
ORIGIN_hex1240  ***
ORIGIN_hex1255  ***
ORIGIN_hex1256  ***
ORIGIN_hex1257  ***
ORIGIN_hex1258  ***
ORIGIN_hex1259  ***
ORIGIN_hex1261  ***
ORIGIN_hex1262  ***
ORIGIN_hex1263  ***
ORIGIN_hex1264  ***
ORIGIN_hex1265  ***
ORIGIN_hex1266  ***
ORIGIN_hex1282  ***
ORIGIN_hex1283  ***
ORIGIN_hex1284  ***
ORIGIN_hex1285  ***
ORIGIN_hex1286  ***
ORIGIN_hex1287  ***
ORIGIN_hex1288  ***
ORIGIN_hex1289  ***
ORIGIN_hex1293  ***
ORIGIN_hex1308  ***
ORIGIN_hex1309  ***
ORIGIN_hex1310  ***
ORIGIN_hex1311  ***
ORIGIN_hex1312  ***
ORIGIN_hex1314  ***
ORIGIN_hex1315  ***
ORIGIN_hex1316  ***
ORIGIN_hex1318  ***
ORIGIN_hex1319  ***
ORIGIN_hex1320  ***
ORIGIN_hex1335  ***
ORIGIN_hex1336  ***
ORIGIN_hex1337  ***
ORIGIN_hex1338  ***
ORIGIN_hex1339  ***
ORIGIN_hex1340  ***
ORIGIN_hex1341  ***
ORIGIN_hex1342  ***
ORIGIN_hex1343  ***
ORIGIN_hex1347  ***
ORIGIN_hex1348  ***
ORIGIN_hex1362  ***
ORIGIN_hex1363  ***
ORIGIN_hex1364  ***
ORIGIN_hex1366  ***
ORIGIN_hex1367  ***
ORIGIN_hex1368  ***
ORIGIN_hex1369  ***
ORIGIN_hex1370  ***
ORIGIN_hex1372  ***
ORIGIN_hex1374  ***
ORIGIN_hex1375  ***
ORIGIN_hex1389  ***
ORIGIN_hex1390  ***
ORIGIN_hex1391  ***
ORIGIN_hex1392  ***
ORIGIN_hex1394  ***
ORIGIN_hex1396  ***
ORIGIN_hex1397  ***
ORIGIN_hex1401  ***
ORIGIN_hex1402  ***
ORIGIN_hex1416  ***
ORIGIN_hex1417  ***
ORIGIN_hex1418  ***
ORIGIN_hex1419  ***
ORIGIN_hex1420  ***
ORIGIN_hex1422  ***
ORIGIN_hex1423  ***
ORIGIN_hex1426  ***
ORIGIN_hex1428  ***
ORIGIN_hex1429  ***
ORIGIN_hex1443  ***
ORIGIN_hex1444  ***
ORIGIN_hex1445  ***
ORIGIN_hex1446  ***
ORIGIN_hex1447  ***
ORIGIN_hex1455  ***
ORIGIN_hex1456  ***
ORIGIN_hex1457  ***
ORIGIN_hex1469  ***
ORIGIN_hex1470  ***
ORIGIN_hex1471  ***
ORIGIN_hex1472  ***
ORIGIN_hex1480  ***
ORIGIN_hex1482  ***
ORIGIN_hex1483  ***
ORIGIN_hex1496  ***
ORIGIN_hex1497  ***
ORIGIN_hex1498  ***
ORIGIN_hex1499  ***
ORIGIN_hex1500  ***
ORIGIN_hex1501  ***
ORIGIN_hex1507  ***
ORIGIN_hex1509  ***
ORIGIN_hex1510  ***
ORIGIN_hex1511  ***
ORIGIN_hex1523  ***
ORIGIN_hex1524  ***
ORIGIN_hex1525  ***
ORIGIN_hex1526  ***
ORIGIN_hex1527  ***
ORIGIN_hex1534  ***
ORIGIN_hex1536  ***
ORIGIN_hex1537  ***
ORIGIN_hex1550  ***
ORIGIN_hex1551  ***
ORIGIN_hex1552  ***
ORIGIN_hex1553  ***
ORIGIN_hex1554  ***
ORIGIN_hex1555  ***
ORIGIN_hex1562  ***
ORIGIN_hex1563  ***
ORIGIN_hex1564  ***
ORIGIN_hex1565  ***
ORIGIN_hex1576  ***
ORIGIN_hex1578  ***
ORIGIN_hex1579  ***
ORIGIN_hex1581  ***
ORIGIN_hex1588  ***
ORIGIN_hex1590  ***
ORIGIN_hex1591  ***
ORIGIN_hex1592  ***
ORIGIN_hex1604  ***
ORIGIN_hex1605  ***
ORIGIN_hex1606  ***
ORIGIN_hex1607  ***
ORIGIN_hex1616  ***
ORIGIN_hex1617  ***
ORIGIN_hex1618  ***
ORIGIN_hex1619  ***
ORIGIN_hex1620  ***
ORIGIN_hex1630  ***
ORIGIN_hex1631  ***
ORIGIN_hex1632  ***
ORIGIN_hex1633  ***
ORIGIN_hex1634  ***
ORIGIN_hex1635  ***
ORIGIN_hex1642  ***
ORIGIN_hex1644  ***
ORIGIN_hex1645  ***
ORIGIN_hex1646  ***
ORIGIN_hex1647  ***
ORIGIN_hex1658  ***
ORIGIN_hex1659  ***
ORIGIN_hex1660  ***
ORIGIN_hex1661  ***
ORIGIN_hex1662  ***
ORIGIN_hex1669  ***
ORIGIN_hex1670  ***
ORIGIN_hex1672  ***
ORIGIN_hex1673  ***
ORIGIN_hex1674  ***
ORIGIN_hex1684  ***
ORIGIN_hex1685  ***
ORIGIN_hex1686  ***
ORIGIN_hex1687  ***
ORIGIN_hex1688  ***
ORIGIN_hex1689  ***
ORIGIN_hex1695  ***
ORIGIN_hex1699  ***
ORIGIN_hex1700  ***
ORIGIN_hex1701  ***
ORIGIN_hex1712  ***
ORIGIN_hex1713  ***
ORIGIN_hex1714  ***
ORIGIN_hex1716  ***
ORIGIN_hex1723  ***
ORIGIN_hex1726  ***
ORIGIN_hex1727  ***
ORIGIN_hex1728  ***
ORIGIN_hex1738  ***
ORIGIN_hex1739  ***
ORIGIN_hex1740  ***
ORIGIN_hex1741  ***
ORIGIN_hex1743  ***
ORIGIN_hex1744  ***
ORIGIN_hex1748  ***
ORIGIN_hex1749  ***
ORIGIN_hex1753  ***
ORIGIN_hex1754  ***
ORIGIN_hex1765  ***
ORIGIN_hex1766  ***
ORIGIN_hex1767  ***
ORIGIN_hex1768  ***
ORIGIN_hex1770  ***
ORIGIN_hex1771  ***
ORIGIN_hex1776  ***
ORIGIN_hex1777  ***
ORIGIN_hex1778  ***
ORIGIN_hex1780  ***
ORIGIN_hex1781  ***
ORIGIN_hex1792  ***
ORIGIN_hex1793  ***
ORIGIN_hex1794  ***
ORIGIN_hex1795  ***
ORIGIN_hex1796  ***
ORIGIN_hex1798  ***
ORIGIN_hex1800  ***
ORIGIN_hex1802  ***
ORIGIN_hex1804  ***
ORIGIN_hex1805  ***
ORIGIN_hex1806  ***
ORIGIN_hex1807  ***
ORIGIN_hex1808  ***
ORIGIN_hex1820  ***
ORIGIN_hex1821  ***
ORIGIN_hex1822  ***
ORIGIN_hex1823  ***
ORIGIN_hex1824  ***
ORIGIN_hex1827  ***
ORIGIN_hex1828  ***
ORIGIN_hex1829  ***
ORIGIN_hex1831  ***
ORIGIN_hex1833  ***
ORIGIN_hex1834  ***
ORIGIN_hex1835  ***
ORIGIN_hex1846  ***
ORIGIN_hex1847  ***
ORIGIN_hex1848  ***
ORIGIN_hex1849  ***
ORIGIN_hex1850  ***
ORIGIN_hex1852  ***
ORIGIN_hex1853  ***
ORIGIN_hex1854  ***
ORIGIN_hex1855  ***
ORIGIN_hex1856  ***
ORIGIN_hex1857  ***
ORIGIN_hex1858  ***
ORIGIN_hex1859  ***
ORIGIN_hex1860  ***
ORIGIN_hex1861  ***
ORIGIN_hex1862  ***
ORIGIN_hex1874  ***
ORIGIN_hex1875  ***
ORIGIN_hex1876  ***
ORIGIN_hex1877  ***
ORIGIN_hex1878  ***
ORIGIN_hex1879  ***
ORIGIN_hex1882  ***
ORIGIN_hex1883  ***
ORIGIN_hex1884  ***
ORIGIN_hex1886  ***
ORIGIN_hex1887  ***
ORIGIN_hex1888  ***
ORIGIN_hex1889  ***
ORIGIN_hex1890  ***
ORIGIN_hex1900  ***
ORIGIN_hex1901  ***
ORIGIN_hex1902  ***
ORIGIN_hex1903  ***
ORIGIN_hex1904  ***
ORIGIN_hex1905  ***
ORIGIN_hex1906  ***
ORIGIN_hex1907  ***
ORIGIN_hex1908  ***
ORIGIN_hex1909  ***
ORIGIN_hex1910  ***
ORIGIN_hex1912  ***
ORIGIN_hex1913  ***
ORIGIN_hex1914  ***
ORIGIN_hex1916  ***
ORIGIN_hex1928  ***
ORIGIN_hex1929  ***
ORIGIN_hex1930  ***
ORIGIN_hex1931  ***
ORIGIN_hex1932  ***
ORIGIN_hex1933  ***
ORIGIN_hex1934  ***
ORIGIN_hex1935  ***
ORIGIN_hex1936  ***
ORIGIN_hex1937  ***
ORIGIN_hex1938  ***
ORIGIN_hex1940  ***
ORIGIN_hex1941  ***
ORIGIN_hex1942  ***
ORIGIN_hex1954  ***
ORIGIN_hex1955  ***
ORIGIN_hex1956  ***
ORIGIN_hex1958  ***
ORIGIN_hex1959  ***
ORIGIN_hex1960  ***
ORIGIN_hex1961  ***
ORIGIN_hex1962  ***
ORIGIN_hex1963  ***
ORIGIN_hex1964  ***
ORIGIN_hex1967  ***
ORIGIN_hex1968  ***
ORIGIN_hex1982  ***
ORIGIN_hex1983  ***
ORIGIN_hex1984  ***
ORIGIN_hex1985  ***
ORIGIN_hex1986  ***
ORIGIN_hex1987  ***
ORIGIN_hex1988  ***
ORIGIN_hex1989  ***
ORIGIN_hex1990  ***
ORIGIN_hex1991  ***
ORIGIN_hex1992  ***
ORIGIN_hex1994  ***
ORIGIN_hex1995  ***
ORIGIN_hex2009  ***
ORIGIN_hex2010  ***
ORIGIN_hex2011  ***
ORIGIN_hex2012  ***
ORIGIN_hex2013  ***
ORIGIN_hex2014  ***
ORIGIN_hex2015  ***
ORIGIN_hex2016  ***
ORIGIN_hex2017  ***
ORIGIN_hex2021  ***
ORIGIN_hex2036  ***
ORIGIN_hex2037  ***
ORIGIN_hex2038  ***
ORIGIN_hex2039  ***
ORIGIN_hex2040  ***
ORIGIN_hex2041  ***
ORIGIN_hex2042  ***
ORIGIN_hex2043  ***
ORIGIN_hex2044  ***
ORIGIN_hex2046  ***
ORIGIN_hex2049  ***
ORIGIN_hex2063  ***
ORIGIN_hex2064  ***
ORIGIN_hex2065  ***
ORIGIN_hex2066  ***
ORIGIN_hex2067  ***
ORIGIN_hex2068  ***
ORIGIN_hex2069  ***
ORIGIN_hex2070  ***
ORIGIN_hex2071  ***
ORIGIN_hex2072  ***
ORIGIN_hex2074  ***
ORIGIN_hex2089  ***
ORIGIN_hex2090  ***
ORIGIN_hex2091  ***
ORIGIN_hex2092  ***
ORIGIN_hex2093  ***
ORIGIN_hex2094  ***
ORIGIN_hex2095  ***
ORIGIN_hex2096  ***
ORIGIN_hex2097  ***
ORIGIN_hex2098  ***
ORIGIN_hex2100  ***
ORIGIN_hex2101  ***
ORIGIN_hex2102  ***
ORIGIN_hex2116  ***
ORIGIN_hex2117  ***
ORIGIN_hex2119  ***
ORIGIN_hex2120  ***
ORIGIN_hex2121  ***
ORIGIN_hex2122  ***
ORIGIN_hex2123  ***
ORIGIN_hex2124  ***
ORIGIN_hex2125  ***
ORIGIN_hex2126  ***
ORIGIN_hex2129  ***
ORIGIN_hex2146  ***
ORIGIN_hex2147  ***
ORIGIN_hex2148  ***
ORIGIN_hex2149  ***
ORIGIN_hex2150  ***
ORIGIN_hex2151  ***
ORIGIN_hex2152  ***
ORIGIN_hex2153  ***
ORIGIN_hex2154  ***
ORIGIN_hex2155  ***
ORIGIN_hex2171  ***
ORIGIN_hex2172  ***
ORIGIN_hex2174  ***
ORIGIN_hex2175  ***
ORIGIN_hex2176  ***
ORIGIN_hex2177  ***
ORIGIN_hex2178  ***
ORIGIN_hex2179  ***
ORIGIN_hex2180  ***
ORIGIN_hex2181  ***
ORIGIN_hex2182  ***
ORIGIN_hex2200  ***
ORIGIN_hex2201  ***
ORIGIN_hex2202  ***
ORIGIN_hex2203  ***
ORIGIN_hex2204  ***
ORIGIN_hex2205  ***
ORIGIN_hex2206  ***
ORIGIN_hex2207  ***
ORIGIN_hex2208  ***
ORIGIN_hex2209  ***
ORIGIN_hex2227  ***
ORIGIN_hex2228  ***
ORIGIN_hex2229  ***
ORIGIN_hex2230  ***
ORIGIN_hex2231  ***
ORIGIN_hex2233  ***
ORIGIN_hex2234  ***
ORIGIN_hex2235  ***
ORIGIN_hex2254  ***
ORIGIN_hex2255  ***
ORIGIN_hex2256  ***
ORIGIN_hex2257  ***
ORIGIN_hex2258  ***
ORIGIN_hex2259  ***
ORIGIN_hex2260  ***
ORIGIN_hex2261  ***
ORIGIN_hex2262  ***
ORIGIN_hex2281  ***
ORIGIN_hex2282  ***
ORIGIN_hex2283  ***
ORIGIN_hex2284  ***
ORIGIN_hex2285  ***
ORIGIN_hex2286  ***
ORIGIN_hex2287  ***
ORIGIN_hex2288  ***
ORIGIN_hex2308  ***
ORIGIN_hex2309  ***
ORIGIN_hex2310  ***
ORIGIN_hex2311  ***
ORIGIN_hex2312  ***
ORIGIN_hex2313  ***
ORIGIN_hex2314  ***
ORIGIN_hex2315  ***
ORIGIN_hex2316  ***
ORIGIN_hex2317  ***
ORIGIN_hex2335  ***
ORIGIN_hex2336  ***
ORIGIN_hex2337  ***
ORIGIN_hex2338  ***
ORIGIN_hex2339  ***
ORIGIN_hex2340  ***
ORIGIN_hex2341  ***
ORIGIN_hex2342  ***
ORIGIN_hex2343  ***
ORIGIN_hex2362  ***
ORIGIN_hex2363  ***
ORIGIN_hex2364  ***
ORIGIN_hex2365  ***
ORIGIN_hex2366  ***
ORIGIN_hex2367  ***
ORIGIN_hex2368  ***
ORIGIN_hex2369  ***
ORIGIN_hex2370  ***
ORIGIN_hex2371  ***
ORIGIN_hex2389  ***
ORIGIN_hex2390  ***
ORIGIN_hex2391  ***
ORIGIN_hex2392  ***
ORIGIN_hex2393  ***
ORIGIN_hex2394  ***
ORIGIN_hex2395  ***
ORIGIN_hex2396  ***
ORIGIN_hex2397  ***
ORIGIN_hex2398  ***
ORIGIN_hex2416  ***
ORIGIN_hex2417  ***
ORIGIN_hex2418  ***
ORIGIN_hex2419  ***
ORIGIN_hex2420  ***
ORIGIN_hex2422  ***
ORIGIN_hex2423  ***
ORIGIN_hex2424  ***
ORIGIN_hex2425  ***
ORIGIN_hex2426  ***
ORIGIN_hex2443  ***
ORIGIN_hex2444  ***
ORIGIN_hex2445  ***
ORIGIN_hex2448  ***
ORIGIN_hex2449  ***
ORIGIN_hex2450  ***
ORIGIN_hex2451  ***
ORIGIN_hex2452  ***
ORIGIN_hex2471  ***
ORIGIN_hex2472  ***
ORIGIN_hex2473  ***
ORIGIN_hex2476  ***
ORIGIN_hex2478  ***
ORIGIN_hex2479  ***
ORIGIN_hex2480  ***
ORIGIN_hex2497  ***
ORIGIN_hex2498  ***
ORIGIN_hex2499  ***
ORIGIN_hex2500  ***
ORIGIN_hex2503  ***
ORIGIN_hex2504  ***
ORIGIN_hex2505  ***
ORIGIN_hex2525  ***
ORIGIN_hex2526  ***
ORIGIN_hex2527  ***
ORIGIN_hex2531  ***
ORIGIN_hex2532  ***
ORIGIN_hex2533  ***
ORIGIN_hex2551  ***
ORIGIN_hex2552  ***
ORIGIN_hex2553  ***
ORIGIN_hex2554  ***
ORIGIN_hex2557  ***
ORIGIN_hex2559  ***
ORIGIN_hex2579  ***
ORIGIN_hex2580  ***
ORIGIN_hex2581  ***
ORIGIN_hex2584  ***
ORIGIN_hex2605  ***
ORIGIN_hex2606  ***
ORIGIN_hex2607  ***
ORIGIN_hex2608  ***
ORIGIN_hex2609  ***
ORIGIN_hex2610  ***
ORIGIN_hex2611  ***
ORIGIN_hex2633  ***
ORIGIN_hex2634  ***
ORIGIN_hex2635  ***
ORIGIN_hex2636  ***
ORIGIN_hex2637  ***
ORIGIN_hex2638  ***
ORIGIN_hex2660  ***
ORIGIN_hex2661  ***
ORIGIN_hex2662  ***
ORIGIN_hex2663  ***
ORIGIN_hex2664  ***
ORIGIN_hex2665  ***
ORIGIN_hex2687  ***
ORIGIN_hex2688  ***
ORIGIN_hex2689  ***
ORIGIN_hex2690  ***
ORIGIN_hex2691  ***
ORIGIN_hex2693  ***
ORIGIN_hex2714  ***
ORIGIN_hex2715  ***
ORIGIN_hex2716  ***
ORIGIN_hex2717  ***
ORIGIN_hex2718  ***
ORIGIN_hex2719  ***
ORIGIN_hex2742  ***
ORIGIN_hex2743  ***
ORIGIN_hex2744  ***
ORIGIN_hex2745  ***
ORIGIN_hex2746  ***
ORIGIN_hex2747  ***
ORIGIN_hex2768  ***
ORIGIN_hex2769  ***
ORIGIN_hex2770  ***
ORIGIN_hex2771  ***
ORIGIN_hex2772  ***
ORIGIN_hex2773  ***
ORIGIN_hex2795  ***
ORIGIN_hex2796  ***
ORIGIN_hex2797  ***
ORIGIN_hex2798  ***
ORIGIN_hex2799  ***
ORIGIN_hex2800  ***
ORIGIN_hex2801  ***
ORIGIN_hex2822  ***
ORIGIN_hex2823  ***
ORIGIN_hex2824  ***
ORIGIN_hex2825  ***
ORIGIN_hex2826  ***
ORIGIN_hex2827  ***
ORIGIN_hex2850  ***
ORIGIN_hex2851  ***
ORIGIN_hex2852  ***
ORIGIN_hex2853  ***
ORIGIN_hex2854  ***
ORIGIN_hex2877  ***
ORIGIN_hex2878  ***
ORIGIN_hex2879  ***
ORIGIN_hex2880  ***
ORIGIN_hex2881  ***
ORIGIN_hex2905  ***
ORIGIN_hex2906  ***
ORIGIN_hex2907  ***
ORIGIN_hex2908  ***
ORIGIN_hex2909  ***
ORIGIN_hex2931  ***
ORIGIN_hex2932  ***
ORIGIN_hex2933  ***
ORIGIN_hex2934  ***
ORIGIN_hex2959  ***
ORIGIN_hex2960  ***
ORIGIN_hex2961  ***
ORIGIN_hex2962  ***
ORIGIN_hex2963  ***
ORIGIN_hex2987  ***
ORIGIN_hex2988  ***
ORIGIN_hex2989  ***
ORIGIN_hex2990  ***
ORIGIN_hex3015  ***
ORIGIN_hex3016  ***
ORIGIN_hex3017  ***
ORIGIN_hex3040  ***
ORIGIN_hex3043  ***
ORIGIN_hex3044  ***
ORIGIN_hex3068  ***
ORIGIN_hex3070  ***
ORIGIN_hex3072  ***
ORIGIN_hex3092  ***
ORIGIN_hex3098  ***
ORIGIN_hex3123  ***
ORIGIN_hex3126  ***
ORIGIN_hex3151  ***
ORIGIN_hex3152  ***
ORIGIN_hex3173  ***
ORIGIN_hex3178  ***
ORIGIN_hex3179  ***
ORIGIN_hex3205  ***
ORIGIN_hex3206  ***
ORIGIN_hex3232  ***
ORIGIN_hex3233  ***
ORIGIN_hex3308  ***
d_biz_count     ***
d_school_count  ***
d_fin_count     ***
d_busstop_count ***
d_mrtlrt_count  ***
dist            ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 341321396  on 65051  degrees of freedom
Residual deviance:  37509430  on 64225  degrees of freedom
AIC: 37868106

Number of Fisher Scoring iterations: 7

Insights

By absolute value, the most influential variable is the Distance between origin and destination. The negative coefficient means there is an inverse relationship between Distance and Number of Trips: More trips are made when the distance is nearer, while less trips are made when the distance is further.
The most attractive variables appear to be Financial Services and School counts with a coefficient of 0.396 and 0.334 respectively: More trips are made when there is higher financial services / school density at the destination site, while less trips are made when there is lower financial services / school density at the destination site.

Code

CalcRSquared(orcSIM$data$TOTAL_TRIPS, orcSIM$fitted.values)

[1] 0.3269854683

Insights

With reference to the R² above, it can be concluded that the model accounts for about 32.69% of the variation of flows.

Destination Constrained Spatial Interaction Model

Next, fit a destination constrained SIM by using the code chunk below.

For destination constrained SIM, only explanatory variables which represent how propulsive the origins are will be used. This is because such models emphasize the demand or attractiveness of the destinations rather than the limitations or capacities of the origins. The demand or attractiveness of the destination sites determines the potential for generating interactions or flows.
Additionally, “-1” is added to the formula to remove the intercept that is inserted by glm into the model by default. Since the destination has already been constrained, the concept of an intercept would not be relevant.

Code

# Generate the formula dynamically
formula_string <- paste("TOTAL_TRIPS ~ DESTIN_hex +", 
                        paste(origin_var, collapse = " + "), "+ dist - 1")

# Convert the string to a formula
formula <- as.formula(formula_string)

decSIM <- glm(formula,
              family = poisson(link = "log"),
              data = flow_data_log,
              na.action = na.exclude)

summary() is used to print out the results of the model.

Code

summary(decSIM)


Call:
glm(formula = formula, family = poisson(link = "log"), data = flow_data_log, 
    na.action = na.exclude)

Deviance Residuals: 
       Min          1Q      Median          3Q         Max  
-219.74686   -12.20759    -5.68564     0.48324   626.37247  

Coefficients:
                     Estimate    Std. Error     z value               Pr(>|z|)
DESTIN_hex118   15.7875802451  0.0504969831   312.64403 < 0.000000000000000222
DESTIN_hex146   16.8330086924  0.0226830957   742.09486 < 0.000000000000000222
DESTIN_hex174   16.4524339851  0.0283408596   580.51994 < 0.000000000000000222
DESTIN_hex175   16.5289299839  0.0193330155   854.95871 < 0.000000000000000222
DESTIN_hex200   15.9654171155  0.0344714917   463.14843 < 0.000000000000000222
DESTIN_hex201   16.5453668063  0.0137470867  1203.55441 < 0.000000000000000222
DESTIN_hex202   15.4972965279  0.0237653288   652.09687 < 0.000000000000000222
DESTIN_hex203   16.3180049162  0.0200188828   815.13065 < 0.000000000000000222
DESTIN_hex227   14.7128755555  0.0404384688   363.83365 < 0.000000000000000222
DESTIN_hex228   15.8176219351  0.0169450151   933.46756 < 0.000000000000000222
DESTIN_hex229   12.8346838421  0.1005256798   127.67567 < 0.000000000000000222
DESTIN_hex230   16.9513311517  0.0111565958  1519.39995 < 0.000000000000000222
DESTIN_hex231   14.3505285039  0.0324154012   442.70711 < 0.000000000000000222
DESTIN_hex254   16.6633813658  0.0184568291   902.83013 < 0.000000000000000222
DESTIN_hex255   15.6888781519  0.0195312878   803.26901 < 0.000000000000000222
DESTIN_hex256   16.7141394313  0.0124910296  1338.09141 < 0.000000000000000222
DESTIN_hex257   15.0893631494  0.0209887771   718.92531 < 0.000000000000000222
DESTIN_hex258   15.6453136303  0.0213193723   733.85433 < 0.000000000000000222
DESTIN_hex259   15.5798036489  0.0151417717  1028.92871 < 0.000000000000000222
DESTIN_hex281   14.4873855065  0.0433583864   334.13110 < 0.000000000000000222
DESTIN_hex282   14.8049118760  0.0284373566   520.61491 < 0.000000000000000222
DESTIN_hex284   16.6487815564  0.0095343271  1746.19366 < 0.000000000000000222
DESTIN_hex285   15.7606431712  0.0119050254  1323.86472 < 0.000000000000000222
DESTIN_hex286   16.0802142973  0.0100875116  1594.07145 < 0.000000000000000222
DESTIN_hex312   16.2837191369  0.0120037112  1356.55706 < 0.000000000000000222
DESTIN_hex313   15.1203991634  0.0242482863   623.56568 < 0.000000000000000222
DESTIN_hex314   15.8032211277  0.0166619690   948.46060 < 0.000000000000000222
DESTIN_hex336   14.4556717763  0.0416147571   347.36889 < 0.000000000000000222
DESTIN_hex338   15.0038331236  0.0233940195   641.35337 < 0.000000000000000222
DESTIN_hex339   14.0894908005  0.0250953830   561.43757 < 0.000000000000000222
DESTIN_hex340   12.9987498401  0.2582091920    50.34193 < 0.000000000000000222
DESTIN_hex366   15.0478450764  0.0155410981   968.26138 < 0.000000000000000222
DESTIN_hex367   16.0233734108  0.0112919545  1419.00797 < 0.000000000000000222
DESTIN_hex391   15.5780397168  0.0290424392   536.38882 < 0.000000000000000222
DESTIN_hex392   15.2472723665  0.0177594137   858.54593 < 0.000000000000000222
DESTIN_hex393   16.4979427100  0.0087405849  1887.51016 < 0.000000000000000222
DESTIN_hex394   13.6720360748  0.0785955434   173.95434 < 0.000000000000000222
DESTIN_hex419   16.6173789989  0.0176651682   940.68615 < 0.000000000000000222
DESTIN_hex420   15.8574556919  0.0153421529  1033.58738 < 0.000000000000000222
DESTIN_hex421   16.0987615305  0.0234180562   687.45080 < 0.000000000000000222
DESTIN_hex445   16.9092537549  0.0123031993  1374.37859 < 0.000000000000000222
DESTIN_hex446   14.7410773034  0.0199467947   739.01985 < 0.000000000000000222
DESTIN_hex447   15.3349457998  0.0173829471   882.18331 < 0.000000000000000222
DESTIN_hex472   14.5045715836  0.0321581062   451.03936 < 0.000000000000000222
DESTIN_hex473   16.4015776419  0.0123860716  1324.19529 < 0.000000000000000222
DESTIN_hex474   15.4668241484  0.0161447261   958.01094 < 0.000000000000000222
DESTIN_hex499   16.3729764669  0.0125387136  1305.79395 < 0.000000000000000222
DESTIN_hex500   15.2435540250  0.0149356487  1020.61546 < 0.000000000000000222
DESTIN_hex526   17.0305828675  0.0101218757  1682.55207 < 0.000000000000000222
DESTIN_hex527   16.5275504496  0.0157777507  1047.52260 < 0.000000000000000222
DESTIN_hex528   17.2912231415  0.0081600176  2119.01787 < 0.000000000000000222
DESTIN_hex552   16.1481224170  0.0193949216   832.59540 < 0.000000000000000222
DESTIN_hex553   13.1155134162  0.0591700025   221.65815 < 0.000000000000000222
DESTIN_hex554   16.6558798231  0.0104848202  1588.57086 < 0.000000000000000222
DESTIN_hex555   16.9586839668  0.0095303127  1779.44675 < 0.000000000000000222
DESTIN_hex581   16.8593593216  0.0084309575  1999.69687 < 0.000000000000000222
DESTIN_hex582   15.5407496544  0.0120839140  1286.06920 < 0.000000000000000222
DESTIN_hex607   16.0809821600  0.0175538211   916.09582 < 0.000000000000000222
DESTIN_hex608   16.1356606822  0.0121964031  1322.98519 < 0.000000000000000222
DESTIN_hex609   16.5550066110  0.0054386383  3043.96167 < 0.000000000000000222
DESTIN_hex610   16.7543297870  0.0083231044  2012.99046 < 0.000000000000000222
DESTIN_hex611   14.8230705108  0.0272558313   543.84951 < 0.000000000000000222
DESTIN_hex634   16.7247627768  0.0141426566  1182.57575 < 0.000000000000000222
DESTIN_hex635   16.4615003621  0.0102459640  1606.63265 < 0.000000000000000222
DESTIN_hex636   15.1780384311  0.0114314843  1327.73996 < 0.000000000000000222
DESTIN_hex638   17.6573135800  0.0045499248  3880.79239 < 0.000000000000000222
DESTIN_hex661   15.6941049406  0.0213987652   733.41171 < 0.000000000000000222
DESTIN_hex662   16.8602925805  0.0115714996  1457.05338 < 0.000000000000000222
DESTIN_hex663   15.2824479107  0.0098304553  1554.60225 < 0.000000000000000222
DESTIN_hex664   16.0328953083  0.0090457358  1772.42578 < 0.000000000000000222
DESTIN_hex665   15.4115196882  0.0117201924  1314.95449 < 0.000000000000000222
DESTIN_hex689   15.3107966826  0.0168543026   908.42066 < 0.000000000000000222
DESTIN_hex690   15.7289931344  0.0098040792  1604.33150 < 0.000000000000000222
DESTIN_hex692   14.9413460042  0.0132031196  1131.65270 < 0.000000000000000222
DESTIN_hex693   14.5035365282  0.0250503715   578.97491 < 0.000000000000000222
DESTIN_hex715   15.2470186024  0.0217890486   699.75605 < 0.000000000000000222
DESTIN_hex716   15.1608504845  0.0163297728   928.41772 < 0.000000000000000222
DESTIN_hex717   14.9371944442  0.0108253269  1379.83773 < 0.000000000000000222
DESTIN_hex718   16.4939281714  0.0053350919  3091.59214 < 0.000000000000000222
DESTIN_hex719   14.8379136176  0.0191894910   773.23122 < 0.000000000000000222
DESTIN_hex720   12.9237971174  0.0518914552   249.05444 < 0.000000000000000222
DESTIN_hex743   16.3932147306  0.0088335216  1855.79607 < 0.000000000000000222
DESTIN_hex744   15.7779459910  0.0090856821  1736.57253 < 0.000000000000000222
DESTIN_hex745   14.8137587204  0.0086160873  1719.31390 < 0.000000000000000222
DESTIN_hex746   14.9063300245  0.0090653606  1644.31738 < 0.000000000000000222
DESTIN_hex747   16.5031217042  0.0072738757  2268.82097 < 0.000000000000000222
DESTIN_hex748   14.7713243641  0.0195764377   754.54608 < 0.000000000000000222
DESTIN_hex769   15.2329909686  0.0162516953   937.31704 < 0.000000000000000222
DESTIN_hex770   17.0806186638  0.0066337040  2574.82374 < 0.000000000000000222
DESTIN_hex771   15.5803146386  0.0072287008  2155.34092 < 0.000000000000000222
DESTIN_hex772   16.6860339490  0.0037570066  4441.31079 < 0.000000000000000222
DESTIN_hex773   14.3036628475  0.0136785059  1045.70360 < 0.000000000000000222
DESTIN_hex774   15.4246522874  0.0120375792  1281.37493 < 0.000000000000000222
DESTIN_hex775   15.0234943922  0.0154315127   973.55941 < 0.000000000000000222
DESTIN_hex776   11.2249185885  0.1715118023    65.44692 < 0.000000000000000222
DESTIN_hex777   13.9069128845  0.0426980358   325.70381 < 0.000000000000000222
DESTIN_hex797   14.9898016794  0.0171641279   873.32148 < 0.000000000000000222
DESTIN_hex798   14.3194681041  0.0464708087   308.13899 < 0.000000000000000222
DESTIN_hex799   16.2808633599  0.0044946504  3622.27577 < 0.000000000000000222
DESTIN_hex800   15.6151879164  0.0058729432  2658.83516 < 0.000000000000000222
DESTIN_hex802   12.1071103105  0.0731615177   165.48468 < 0.000000000000000222
DESTIN_hex803   14.1893919633  0.0328824057   431.51928 < 0.000000000000000222
DESTIN_hex805   13.9828242405  0.0529791340   263.93078 < 0.000000000000000222
DESTIN_hex806   15.9198312663  0.0159737123   996.62689 < 0.000000000000000222
DESTIN_hex823   15.7593775425  0.0178317648   883.78115 < 0.000000000000000222
DESTIN_hex824   16.3658513715  0.0113474195  1442.25314 < 0.000000000000000222
DESTIN_hex825   15.5984650582  0.0109109570  1429.61475 < 0.000000000000000222
DESTIN_hex826   17.5205062676  0.0027216418  6437.47695 < 0.000000000000000222
DESTIN_hex827   16.1312655016  0.0050738279  3179.30877 < 0.000000000000000222
DESTIN_hex832   11.4994578118  0.2000110747    57.49411 < 0.000000000000000222
DESTIN_hex833   13.9777548019  0.0503014064   277.88000 < 0.000000000000000222
DESTIN_hex851   15.0466182866  0.0206141345   729.91754 < 0.000000000000000222
DESTIN_hex852   16.5391972635  0.0093320388  1772.30267 < 0.000000000000000222
DESTIN_hex853   15.0428989487  0.0072891136  2063.74872 < 0.000000000000000222
DESTIN_hex854   15.2837894053  0.0061190594  2497.73511 < 0.000000000000000222
DESTIN_hex856   10.3452640750  0.1961290036    52.74724 < 0.000000000000000222
DESTIN_hex861   13.7175340261  0.0510122026   268.90691 < 0.000000000000000222
DESTIN_hex862   14.9453734432  0.0355242654   420.70887 < 0.000000000000000222
DESTIN_hex878   16.0104629061  0.0124197520  1289.11294 < 0.000000000000000222
DESTIN_hex879   16.0106445558  0.0071352872  2243.86828 < 0.000000000000000222
DESTIN_hex880   14.7807665792  0.0073914818  1999.70275 < 0.000000000000000222
DESTIN_hex881   15.1737564376  0.0077179192  1966.04241 < 0.000000000000000222
DESTIN_hex888   13.2316761638  0.1361074137    97.21496 < 0.000000000000000222
DESTIN_hex889   14.6109592039  0.0477369792   306.07214 < 0.000000000000000222
DESTIN_hex905   15.0876949653  0.0174181433   866.20570 < 0.000000000000000222
DESTIN_hex906   14.5696529606  0.0185853457   783.93231 < 0.000000000000000222
DESTIN_hex907   15.9644587890  0.0065569094  2434.75360 < 0.000000000000000222
DESTIN_hex908   15.7213832505  0.0053233225  2953.30281 < 0.000000000000000222
DESTIN_hex910   15.9417618479  0.0126266925  1262.54456 < 0.000000000000000222
DESTIN_hex932   15.7787395373  0.0090225250  1748.81638 < 0.000000000000000222
DESTIN_hex933   14.8534100455  0.0114701439  1294.96284 < 0.000000000000000222
DESTIN_hex934   15.4521352882  0.0058135915  2657.93275 < 0.000000000000000222
DESTIN_hex935   15.7790572079  0.0051981078  3035.53868 < 0.000000000000000222
DESTIN_hex937   15.8148949490  0.0101394520  1559.73863 < 0.000000000000000222
DESTIN_hex959   16.9352365229  0.0058237660  2907.95278 < 0.000000000000000222
DESTIN_hex960   15.7084048349  0.0063962320  2455.88417 < 0.000000000000000222
DESTIN_hex961   15.3148375563  0.0066272049  2310.90449 < 0.000000000000000222
DESTIN_hex962   17.4802985719  0.0028787425  6072.19949 < 0.000000000000000222
DESTIN_hex986   14.5544569409  0.0185608432   784.14848 < 0.000000000000000222
DESTIN_hex987   16.1486996399  0.0050024690  3228.14585 < 0.000000000000000222
DESTIN_hex988   14.3375544106  0.0114313683  1254.22907 < 0.000000000000000222
DESTIN_hex989   15.8453922518  0.0048542064  3264.26011 < 0.000000000000000222
DESTIN_hex991   15.5850491740  0.0106812897  1459.09807 < 0.000000000000000222
DESTIN_hex1013  15.2136717909  0.0197767908   769.26899 < 0.000000000000000222
DESTIN_hex1014  13.9777799953  0.0150266039   930.20220 < 0.000000000000000222
DESTIN_hex1015  13.3367178410  0.0212307862   628.17824 < 0.000000000000000222
DESTIN_hex1016  15.1660253664  0.0060002078  2527.58335 < 0.000000000000000222
DESTIN_hex1042  16.0551035905  0.0054453646  2948.39829 < 0.000000000000000222
DESTIN_hex1043  15.6128945868  0.0055221808  2827.30594 < 0.000000000000000222
DESTIN_hex1045  13.3746046212  0.0259878733   514.64791 < 0.000000000000000222
DESTIN_hex1067  16.6912893083  0.0061432200  2717.02614 < 0.000000000000000222
DESTIN_hex1068  13.3734642996  0.0322913282   414.15033 < 0.000000000000000222
DESTIN_hex1069  13.3691227755  0.0179814547   743.49506 < 0.000000000000000222
DESTIN_hex1070  15.7666099582  0.0046108584  3419.45221 < 0.000000000000000222
DESTIN_hex1078  14.3060853160  0.0378878261   377.59056 < 0.000000000000000222
DESTIN_hex1093  15.8905290620  0.0107859892  1473.25654 < 0.000000000000000222
DESTIN_hex1094  14.4612360995  0.0140930893  1026.12250 < 0.000000000000000222
DESTIN_hex1095  12.9945739805  0.0232437570   559.05652 < 0.000000000000000222
DESTIN_hex1096  15.3113914219  0.0057466166  2664.41848 < 0.000000000000000222
DESTIN_hex1097  15.3861433447  0.0063775228  2412.55794 < 0.000000000000000222
DESTIN_hex1099  15.7427328522  0.0062138246  2533.50131 < 0.000000000000000222
DESTIN_hex1100  12.7782939370  0.0461208750   277.06096 < 0.000000000000000222
DESTIN_hex1122  15.2864151082  0.0077980751  1960.28058 < 0.000000000000000222
DESTIN_hex1123  16.9648809875  0.0031799672  5334.92327 < 0.000000000000000222
DESTIN_hex1124  15.7087448411  0.0049000163  3205.85563 < 0.000000000000000222
DESTIN_hex1125  15.1167874099  0.0072396275  2088.06150 < 0.000000000000000222
DESTIN_hex1126  11.4466483818  0.0601221502   190.38987 < 0.000000000000000222
DESTIN_hex1127  15.2350981295  0.0065835293  2314.12324 < 0.000000000000000222
DESTIN_hex1128  15.8283753604  0.0075037416  2109.39772 < 0.000000000000000222
DESTIN_hex1132  15.4306421172  0.0196438670   785.51958 < 0.000000000000000222
DESTIN_hex1147  13.4950719471  0.0328792702   410.44317 < 0.000000000000000222
DESTIN_hex1148  15.3119112805  0.0080922254  1892.17559 < 0.000000000000000222
DESTIN_hex1149  14.7178438991  0.0087794126  1676.40418 < 0.000000000000000222
DESTIN_hex1150  15.7059752049  0.0051027569  3077.93915 < 0.000000000000000222
DESTIN_hex1151  15.1827764964  0.0057308517  2649.30541 < 0.000000000000000222
DESTIN_hex1152  14.9506043852  0.0072339366  2066.73145 < 0.000000000000000222
DESTIN_hex1153  15.1129147671  0.0073453708  2057.47472 < 0.000000000000000222
DESTIN_hex1154  17.1198293876  0.0030866347  5546.43846 < 0.000000000000000222
DESTIN_hex1155  12.7492810523  0.0359117359   355.01712 < 0.000000000000000222
DESTIN_hex1158  14.0254731886  0.0295885861   474.01634 < 0.000000000000000222
DESTIN_hex1174  15.7079565960  0.0114218398  1375.25626 < 0.000000000000000222
DESTIN_hex1175  16.9749978205  0.0077793652  2182.05438 < 0.000000000000000222
DESTIN_hex1176  15.8227951312  0.0052732058  3000.60263 < 0.000000000000000222
DESTIN_hex1177  15.7687029395  0.0053313395  2957.73756 < 0.000000000000000222
DESTIN_hex1178  15.2015101090  0.0058831654  2583.89983 < 0.000000000000000222
DESTIN_hex1179  14.3801072071  0.0089860766  1600.26536 < 0.000000000000000222
DESTIN_hex1180  13.0150834867  0.0178463397   729.28587 < 0.000000000000000222
DESTIN_hex1181  15.8493837535  0.0045327965  3496.60169 < 0.000000000000000222
DESTIN_hex1182  15.5181584561  0.0069495551  2232.97151 < 0.000000000000000222
DESTIN_hex1183  15.5938181659  0.0097643428  1597.01666 < 0.000000000000000222
DESTIN_hex1185  15.1622432967  0.0154788296   979.54714 < 0.000000000000000222
DESTIN_hex1201  15.0151968023  0.0136749331  1098.00880 < 0.000000000000000222
DESTIN_hex1202  15.3841574498  0.0077301871  1990.14037 < 0.000000000000000222
DESTIN_hex1203  16.4245740709  0.0049732344  3302.59396 < 0.000000000000000222
DESTIN_hex1205  16.6937626861  0.0032315609  5165.85125 < 0.000000000000000222
DESTIN_hex1206  15.6750402494  0.0053665717  2920.86665 < 0.000000000000000222
DESTIN_hex1207  15.5511865143  0.0050127203  3102.34477 < 0.000000000000000222
DESTIN_hex1209  17.0012171758  0.0041988133  4049.05290 < 0.000000000000000222
DESTIN_hex1210  15.2130295552  0.0142760067  1065.63620 < 0.000000000000000222
DESTIN_hex1211  14.9627263586  0.0166910068   896.45439 < 0.000000000000000222
DESTIN_hex1212  15.0678429430  0.0169159850   890.74582 < 0.000000000000000222
DESTIN_hex1228  15.2720672244  0.0136104048  1122.08766 < 0.000000000000000222
DESTIN_hex1229  15.6699341128  0.0067569735  2319.07585 < 0.000000000000000222
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dist            -1.4609624634  0.0002608931 -5599.85044 < 0.000000000000000222
                   
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o_housing_count ***
o_biz_count     ***
o_school_count  ***
o_busstop_count ***
o_mrtlrt_count  ***
dist            ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 341321396  on 65051  degrees of freedom
Residual deviance:  36294465  on 64224  degrees of freedom
AIC: 36653143

Number of Fisher Scoring iterations: 8

Insights

By absolute value, the most influential variable is the Distance between origin and destination. The negative coefficient means there is an inverse relationship between Distance and Number of Trips: More trips are made when the distance is nearer, while less trips are made when the distance is further.
The most propulsive variable appears to be Bus Stop counts at the origin site with a coefficient of 0.553: More trips are made when there is higher bus stop density at the origin site, while less trips are made when there is lower bus stop density at the origin site.
There is an inverse relationship between the number of businesses at the origin and Number of Trips: More bus trips are made when there are less businesses at the origin, while less bus trips are made when there are more businesses at the origin.
Interestingly, unlike with business density, there is a direct relationship between school density at origin site and bus trips: More bus trips are made when there are more schools at the origin, while less bus trips are made when there are less schools at the origin. This could be due to the relationship between schools and bus stop as a result of urban planning – most schools have at least one bus stop immediately within the vicinity of the school.

Code

CalcRSquared(decSIM$data$TOTAL_TRIPS, decSIM$fitted.values)

[1] 0.4144590931

Insights

With reference to the R² above, it can be concluded that the model accounts for almost 41.44% of the variation of flows.

Doubly Constrained Spatial Interaction Model

Next, fit a doubly constrained SIM by using the code chunk below.

Code

dbcSIM <- glm(formula = TOTAL_TRIPS ~ 
                ORIGIN_hex + 
                DESTIN_hex + 
                dist,
              family = poisson(link = "log"),
              data = flow_data_log,
              na.action = na.exclude)

Code

summary(dbcSIM)


Call:
glm(formula = TOTAL_TRIPS ~ ORIGIN_hex + DESTIN_hex + dist, family = poisson(link = "log"), 
    data = flow_data_log, na.action = na.exclude)

Deviance Residuals: 
       Min          1Q      Median          3Q         Max  
-189.23777    -9.59267    -3.37029     2.29728   372.68284  

Coefficients:
                    Estimate    Std. Error     z value               Pr(>|z|)
(Intercept)    14.3800438474  0.1374345620   104.63193 < 0.000000000000000222
ORIGIN_hex146   2.1835203793  0.1912090937    11.41954 < 0.000000000000000222
ORIGIN_hex174   2.1384201895  0.1978768354    10.80682 < 0.000000000000000222
ORIGIN_hex175   0.1038432960  0.1693595948     0.61315             0.53977536
ORIGIN_hex200  -0.4966627544  0.1501239095    -3.30835             0.00093847
ORIGIN_hex201   1.7389270413  0.1300958785    13.36650 < 0.000000000000000222
ORIGIN_hex202   3.3861740906  0.1299735913    26.05279 < 0.000000000000000222
ORIGIN_hex203   0.7830595914  0.1373789614     5.70000 0.00000001198099271729
ORIGIN_hex227   2.3025870011  0.1435896399    16.03589 < 0.000000000000000222
ORIGIN_hex228   0.9826573869  0.1315819019     7.46803 0.00000000000008140525
ORIGIN_hex229   2.3885492037  0.1495380994    15.97285 < 0.000000000000000222
ORIGIN_hex230   4.2239518256  0.1287714476    32.80193 < 0.000000000000000222
ORIGIN_hex231   0.8030467017  0.1451949786     5.53082 0.00000003187446490472
ORIGIN_hex254   4.5153366053  0.1292211758    34.94270 < 0.000000000000000222
ORIGIN_hex255   1.0923572017  0.1447844683     7.54471 0.00000000000004532860
ORIGIN_hex256   1.6197444529  0.1375821561    11.77293 < 0.000000000000000222
ORIGIN_hex257   2.5860306846  0.1293532026    19.99201 < 0.000000000000000222
ORIGIN_hex258   0.4537789331  0.1457984916     3.11237             0.00185591
ORIGIN_hex259   0.9984929285  0.1369997050     7.28829 0.00000000000031392479
ORIGIN_hex281   2.5308770666  0.1356518268    18.65715 < 0.000000000000000222
ORIGIN_hex282   1.9952119647  0.1383730855    14.41908 < 0.000000000000000222
ORIGIN_hex284   1.9893878335  0.1330659238    14.95039 < 0.000000000000000222
ORIGIN_hex285   2.9964803672  0.1299802758    23.05335 < 0.000000000000000222
ORIGIN_hex286   5.4024286875  0.1279344002    42.22812 < 0.000000000000000222
ORIGIN_hex312   0.9634289130  0.1446105219     6.66223 0.00000000002696989882
ORIGIN_hex313   1.1345973471  0.1371344092     8.27362 < 0.000000000000000222
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ORIGIN_hex610   2.0871312399  0.1388114781    15.03573 < 0.000000000000000222
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ORIGIN_hex634   2.6149026263  0.1328263584    19.68662 < 0.000000000000000222
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ORIGIN_hex770   2.1604964494  0.1304145844    16.56637 < 0.000000000000000222
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ORIGIN_hex775   3.6652120004  0.1305865258    28.06731 < 0.000000000000000222
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ORIGIN_hex777   1.3471612575  0.2447334287     5.50461 0.00000003699934393892
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ORIGIN_hex798   0.2842201538  0.1675023593     1.69681             0.08973205
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ORIGIN_hex800   6.3341958744  0.1280516717    49.46594 < 0.000000000000000222
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ORIGIN_hex803   1.2227584447  0.1836232295     6.65906 0.00000000002755809303
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ORIGIN_hex806   4.2201004146  0.1354357575    31.15943 < 0.000000000000000222
ORIGIN_hex823   1.0310643875  0.1398457249     7.37287 0.00000000000016699298
ORIGIN_hex824   0.4661112920  0.1383760340     3.36844             0.00075595
ORIGIN_hex825   3.4658439272  0.1297321380    26.71538 < 0.000000000000000222
ORIGIN_hex826   7.0227782170  0.1280309535    54.85219 < 0.000000000000000222
ORIGIN_hex827   6.2125831295  0.1280784164    48.50609 < 0.000000000000000222
ORIGIN_hex832   1.0318505074  1.0092941839     1.02235             0.30661593
ORIGIN_hex833   0.6530078899  0.1815461262     3.59693             0.00032200
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ORIGIN_hex854   5.2618821499  0.1281013634    41.07593 < 0.000000000000000222
ORIGIN_hex856  -0.0188771466  0.2409847424    -0.07833             0.93756287
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ORIGIN_hex880   4.2701028279  0.1281743554    33.31480 < 0.000000000000000222
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ORIGIN_hex906  -0.2815118999  0.1803470870    -1.56095             0.11853671
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ORIGIN_hex962   6.3482019582  0.1280678120    49.56907 < 0.000000000000000222
ORIGIN_hex986   0.1193060381  0.1562154009     0.76373             0.44502949
ORIGIN_hex987   5.9071990274  0.1280852093    46.11929 < 0.000000000000000222
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ORIGIN_hex1128  6.2189896785  0.1281595757    48.52536 < 0.000000000000000222
ORIGIN_hex1132  0.5027881736  0.2747078396     1.83027             0.06721031
ORIGIN_hex1147  0.3914722631  0.1507807079     2.59630             0.00942332
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ORIGIN_hex1374  4.5539372961  0.1282304513    35.51370 < 0.000000000000000222
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ORIGIN_hex1402  5.9199477919  0.1280963422    46.21481 < 0.000000000000000222
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ORIGIN_hex1646  3.7654048447  0.1285997680    29.28003 < 0.000000000000000222
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ORIGIN_hex1662  4.6128317513  0.1281693293    35.99014 < 0.000000000000000222
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ORIGIN_hex1670  1.9242799612  0.1574855520    12.21877 < 0.000000000000000222
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ORIGIN_hex1699  4.8143981451  0.1281847038    37.55829 < 0.000000000000000222
ORIGIN_hex1700  0.9638404381  0.1409257591     6.83935 0.00000000000795539085
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ORIGIN_hex1744  0.8541006167  0.1868582792     4.57085 0.00000485756705328517
ORIGIN_hex1748 -0.4403221127  0.2182903687    -2.01714             0.04368096
ORIGIN_hex1749  5.7536669837  0.1281751408    44.88910 < 0.000000000000000222
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ORIGIN_hex1776 -0.0182809001  0.1782876696    -0.10254             0.91833126
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ORIGIN_hex1800  2.3229349892  0.1328113446    17.49049 < 0.000000000000000222
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ORIGIN_hex2365  3.0876682526  0.1284232955    24.04290 < 0.000000000000000222
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DESTIN_hex3232  0.0709654671  0.0565663077     1.25455             0.20964084
DESTIN_hex3233 -0.7682907737  0.0554352681   -13.85924 < 0.000000000000000222
DESTIN_hex3308  0.2419880563  0.0547464312     4.42016 0.00000986272592849379
dist           -1.6001837283  0.0002897849 -5521.97139 < 0.000000000000000222
                  
(Intercept)    ***
ORIGIN_hex146  ***
ORIGIN_hex174  ***
ORIGIN_hex175     
ORIGIN_hex200  ***
ORIGIN_hex201  ***
ORIGIN_hex202  ***
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ORIGIN_hex227  ***
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ORIGIN_hex254  ***
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ORIGIN_hex257  ***
ORIGIN_hex258  ** 
ORIGIN_hex259  ***
ORIGIN_hex281  ***
ORIGIN_hex282  ***
ORIGIN_hex284  ***
ORIGIN_hex285  ***
ORIGIN_hex286  ***
ORIGIN_hex312  ***
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ORIGIN_hex314  ***
ORIGIN_hex336     
ORIGIN_hex338  ***
ORIGIN_hex339  ***
ORIGIN_hex340  *  
ORIGIN_hex366  ***
ORIGIN_hex367  ***
ORIGIN_hex391  ***
ORIGIN_hex392  ***
ORIGIN_hex393  ***
ORIGIN_hex394     
ORIGIN_hex419  ***
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ORIGIN_hex445  ***
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ORIGIN_hex447  ***
ORIGIN_hex472     
ORIGIN_hex473  ***
ORIGIN_hex474  ***
ORIGIN_hex499  ***
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ORIGIN_hex526  ***
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ORIGIN_hex528  ***
ORIGIN_hex552  ** 
ORIGIN_hex553  ***
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ORIGIN_hex555  ***
ORIGIN_hex581  ***
ORIGIN_hex582  ***
ORIGIN_hex607  ***
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ORIGIN_hex610  ***
ORIGIN_hex611     
ORIGIN_hex634  ***
ORIGIN_hex635     
ORIGIN_hex636  ***
ORIGIN_hex638  ***
ORIGIN_hex661  ***
ORIGIN_hex662  ***
ORIGIN_hex663  ***
ORIGIN_hex664  ***
ORIGIN_hex665  ***
ORIGIN_hex689  ***
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ORIGIN_hex692  ***
ORIGIN_hex693  ***
ORIGIN_hex715  ***
ORIGIN_hex716  ***
ORIGIN_hex717  ***
ORIGIN_hex718  ***
ORIGIN_hex719  ***
ORIGIN_hex720  *  
ORIGIN_hex743  ***
ORIGIN_hex744  ***
ORIGIN_hex745  ***
ORIGIN_hex746  ***
ORIGIN_hex747  ***
ORIGIN_hex748  ***
ORIGIN_hex769  ***
ORIGIN_hex770  ***
ORIGIN_hex771  ***
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ORIGIN_hex773  ***
ORIGIN_hex774  ***
ORIGIN_hex775  ***
ORIGIN_hex776  ***
ORIGIN_hex777  ***
ORIGIN_hex797  ***
ORIGIN_hex798  .  
ORIGIN_hex799  ***
ORIGIN_hex800  ***
ORIGIN_hex802  ***
ORIGIN_hex803  ***
ORIGIN_hex805  ***
ORIGIN_hex806  ***
ORIGIN_hex823  ***
ORIGIN_hex824  ***
ORIGIN_hex825  ***
ORIGIN_hex826  ***
ORIGIN_hex827  ***
ORIGIN_hex832     
ORIGIN_hex833  ***
ORIGIN_hex851  ***
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ORIGIN_hex856     
ORIGIN_hex861  ***
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ORIGIN_hex878  ***
ORIGIN_hex879  ***
ORIGIN_hex880  ***
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ORIGIN_hex888  ***
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ORIGIN_hex905  ***
ORIGIN_hex906     
ORIGIN_hex907  ***
ORIGIN_hex908  ***
ORIGIN_hex910  ***
ORIGIN_hex932  ***
ORIGIN_hex933  ***
ORIGIN_hex934  ***
ORIGIN_hex935  ***
ORIGIN_hex937  ***
ORIGIN_hex959  ***
ORIGIN_hex960  ***
ORIGIN_hex961  ***
ORIGIN_hex962  ***
ORIGIN_hex986     
ORIGIN_hex987  ***
ORIGIN_hex988  ***
ORIGIN_hex989  ***
ORIGIN_hex991  ***
ORIGIN_hex1013    
ORIGIN_hex1014 ***
ORIGIN_hex1015 ***
ORIGIN_hex1016 ***
ORIGIN_hex1042 ***
ORIGIN_hex1043 ***
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ORIGIN_hex1067 ***
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ORIGIN_hex1078 ***
ORIGIN_hex1093 ***
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ORIGIN_hex1100 ***
ORIGIN_hex1122 ***
ORIGIN_hex1123 ***
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ORIGIN_hex1125 ***
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ORIGIN_hex1127 ***
ORIGIN_hex1128 ***
ORIGIN_hex1132 .  
ORIGIN_hex1147 ** 
ORIGIN_hex1148 ***
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ORIGIN_hex1150 ***
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ORIGIN_hex1185 ***
ORIGIN_hex1201 ***
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ORIGIN_hex1228 ***
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ORIGIN_hex1318 ***
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ORIGIN_hex1335 .  
ORIGIN_hex1336 ***
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ORIGIN_hex1372    
ORIGIN_hex1374 ***
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ORIGIN_hex1389 ***
ORIGIN_hex1390 ***
ORIGIN_hex1391 ***
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ORIGIN_hex1397 ***
ORIGIN_hex1401 ***
ORIGIN_hex1402 ***
ORIGIN_hex1416 ** 
ORIGIN_hex1417 ***
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ORIGIN_hex1419 ***
ORIGIN_hex1420 ***
ORIGIN_hex1422 ***
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ORIGIN_hex1426 ***
ORIGIN_hex1428 ***
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ORIGIN_hex1496 ***
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ORIGIN_hex1500 ***
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ORIGIN_hex1536 ***
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ORIGIN_hex1669    
ORIGIN_hex1670 ***
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ORIGIN_hex1699 ***
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ORIGIN_hex1748 *  
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ORIGIN_hex1800 ***
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ORIGIN_hex1820 ***
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ORIGIN_hex2089 ***
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ORIGIN_hex2091 ***
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DESTIN_hex3016 ***
DESTIN_hex3017 ***
DESTIN_hex3040 ***
DESTIN_hex3043 ***
DESTIN_hex3044 ***
DESTIN_hex3068 ***
DESTIN_hex3070 ***
DESTIN_hex3072 ***
DESTIN_hex3092 ***
DESTIN_hex3098 ***
DESTIN_hex3123 ***
DESTIN_hex3126 ***
DESTIN_hex3151 ***
DESTIN_hex3152 ***
DESTIN_hex3173 ***
DESTIN_hex3178 ***
DESTIN_hex3179 ***
DESTIN_hex3205 ***
DESTIN_hex3206 ***
DESTIN_hex3232    
DESTIN_hex3233 ***
DESTIN_hex3308 ***
dist           ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 98250529  on 65050  degrees of freedom
Residual deviance: 25093776  on 63410  degrees of freedom
AIC: 25454082

Number of Fisher Scoring iterations: 7

Insights

There is an inverse relationship between Distance and Number of Trips: More trips are made when the distance is nearer, while less trips are made when the distance is further.

Code

CalcRSquared(dbcSIM$data$TOTAL_TRIPS, dbcSIM$fitted.values)

[1] 0.5736803132

Insights

With reference to the R² above, it can be concluded that the model accounts for 57.36% of the variation of flows.

Poisson Regression Models Comparison

Lastly, use compare_performance() of the performance package to identify the better model. First, create a list called model_list using the code chunk below.

Code

model_list <- list(unconstrained = uncSIM,
                   originConstrained=orcSIM,
                   destinConstrained=decSIM,
                   doublyConstrained=dbcSIM)

Next, compute the RMSE of all the models in model_list using the code chunk below.

Code

compare_performance(model_list, metrics = "RMSE")

# Comparison of Model Performance Indices

Name              | Model |     RMSE
------------------------------------
unconstrained     |   glm | 1569.416
originConstrained |   glm | 1474.109
destinConstrained |   glm | 1375.017
doublyConstrained |   glm | 1174.174

Insights

The print above reveals that doubly constrained SIM is the best model as it has the smallest RMSE value of 1174.174. This is supported by the R² value of doubly constrained SIM being the highest at 0.5736. However, given that doubly constrained SIM considers only the distance decay effect and not propulsive and/or attractiveness variables, practically, it may not be the best SIM in terms of uncovering the factors driving urban commuting flows. In contrast, the second best model of destination constrained SIM identifies the key propulsive variables driving commuters to destination sites.

Visualise Fitted Values

The observed and fitted values of the destination constrained SIM will be visualised on a scatterplot.

In the code chunk below, the fitted values from the destination constrained SIM is extracted then appended to the flow_data data frame. rename() is used to rename the newly added column.

Code

df <- as.data.frame(decSIM$fitted.values) %>%
  round(digits = 0)

inter_zonal_flow <- flow_data %>%
  cbind(df) %>%
  rename(decTRIPS = "decSIM$fitted.values")

The scatterplot will then be created using geom_point() and other appropriate functions of ggplot2 package.

Code

ggplot(data = inter_zonal_flow,
                aes(x = decTRIPS,
                    y = TOTAL_TRIPS)) +
  geom_point() +
  geom_smooth(method = lm) +
  coord_cartesian(xlim=c(0,100000),
                  ylim=c(0,100000)) + 
  labs(title = "Observed vs. Fitted Values for Destination constrained SIM",
       x = "Fitted Values", y = "Observed Values")

Insights

Most of the values fall far from the linear regression line, suggesting that the model can be further refined in terms of accuracy.

Calibrate Spatial Econometric Interaction Model usng Maximum Likelihood Estimation

Spatial Econometric Interaction Models (SEIM) could also be used as they extend the traditional SIM by incorporating additional econometric considerations and addressing specific complexities that standard SIM might not adequately capture. For example, a limitation of SIM is that it assumes independence among observations. However, SEIM explicitly address spatial dependencies among observations, acknowledging that nearby locations might influence each other’s behavior or outcomes. SEIM can also handle spatial autocorrelation more effectively and address issues of multicollinearity.

To calibrate SEIM using spflow package, three data sets are required:

Spatial Weights
Distance Matrix
Explanatory Variables

Prepare Spatial Weights

There are three different matrices that can be used to describe the connectivity between TAZ: contiguity, fixed distance and adaptive distance.

Before fixed distance weights can be derived, there is a need to determine the upper limit for distance band by using the steps below:

st_centroid() is used to convert the hexagon grids into a point geometry.
knearneigh() of spdep is then used to return a matrix with the indices of points belonging to the set of the k nearest neighbours of each other by using. The returned knn object is then converted into a neighbours list of class nb with a list of integer vectors containing neighbour region number ids by using knn2nb().
Lastly, return the length of neighbour relationship edges by using nbdists() of spdep. The function returns in the units of the coordinates if the coordinates are projected (longlat = FALSE), and in km if otherwise (longlat = TRUE). Remove the list structure of the returned object by using unlist().

Code

coords <- hex_grid_bounded2 %>%
  filter(busstop_count > 0) %>%
  select(geometry) %>%
  st_centroid()
k1 <- knn2nb(knearneigh(coords))
k1dists <- unlist(nbdists(k1, coords, longlat = FALSE))

# Print summary report
summary(k1dists)

     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
 750.0000  750.0000  750.0000  753.6959  750.0000 1984.3135

The summary report shows that the largest first nearest neighbour distance is 1299.03m, so using this as the upper threshold (rounded up to the next integer) gives certainty that all units will have at least one neighbour.

Next , the code chunk below will be used to compute the three spatial weights at one go as follows:

poly2nb() of spdep package is used to build a neighbours list based on regions with contiguous boundaries.
dnearneigh() of **spdep* package is used to identifies neighbours of region centroids by Euclidean distance in the metric of the points between lower and and upper (less than or equal to) bounds.
knn2nb() and knearneigh() is used to to build the adaptive spatial weights.

Lastly, list() is used to keep these three spatial weights in a single list class called hexgrid_nb.

Code

centroids <- suppressWarnings({
    st_point_on_surface(st_geometry(hex_grid_bounded2))})

hexgrid_nb <- list(
  "by_contiguity" = poly2nb(hex_grid_bounded2),
  "by_distance" = dnearneigh(centroids, 
                             d1 = 0, d2 = 1300),
  "by_knn" = knn2nb(knearneigh(centroids, 6))
)

The code chunk below prints a summary of the spatial weights, which reveals that by using contiguity weights, there is at least one TAZ that does not have any neighbours.

Code

hexgrid_nb

$by_contiguity
Neighbour list object:
Number of regions: 1945 
Number of nonzero links: 10790 
Percentage nonzero weights: 0.2852214828 
Average number of links: 5.547557841 
1 region with no links:
1935

$by_distance
Neighbour list object:
Number of regions: 1945 
Number of nonzero links: 21126 
Percentage nonzero weights: 0.5584419876 
Average number of links: 10.86169666 

$by_knn
Neighbour list object:
Number of regions: 1945 
Number of nonzero links: 11670 
Percentage nonzero weights: 0.3084832905 
Average number of links: 6 
Non-symmetric neighbours list

Prepare Flow Data

From the existing flow_data data frame that was assembled earlier, only the required variables are selected using the select() function.

Code

flow_data1 <- flow_data %>%
  select(ORIGIN_hex, DESTIN_hex, TOTAL_TRIPS, dist)

Prepare Explanatory Variables

As both origin propulsive and destination attractiveness variables are required, this data set will be prepared by dropping the geometry (using st_drop_geometry()) from the existing hex_grid_bounded3 sf data frame that was assembled earlier.

Code

explanatory <- hex_grid_bounded3 %>%
  st_drop_geometry()

Prepare spflow objects

The development version (0.1.0.9010) of spflow will be used instead of the released version (0.1.0). The code chunk below will be used to install the development version of spflow package.

Code

devtools::install_github("LukeCe/spflow")

Next, load spflow package into the R environment.

Code

library(spflow)

Three spflow objects are required:

spflow_network-class is an S4 class that contains all information on a spatial network which is composed by a set of nodes that are linked by some neighborhood relation.
spflow_network_pair-class is an S4 class that holds information on O-D pairs. Each O-D pair is composed of two nodes, each belonging to one network. All origin nodes must belong to the same origin network should be contained in one spflow_network-class, and likewise for the destinations.
spflow_network_multi-class is an S4 class that gathers information on multiple objects of types spflow_network-class and spflow_network_pair-class. Its purpose is to ensure that the identification between the nodes that serve as origins or destinations, and the O-D pairs is consistent (similar to relational databases).

Creating spflow_network-class objects

spflow_network-class can be created using spflow_network() function of spflow package. For this model, the fixed distance based neighbourhood structure will be chosen.

Code

hex_net <- spflow_network(
  id_net = "sg",  # assign an id name, can give it any input
  node_neighborhood = nb2mat(hexgrid_nb$by_distance),
  node_data = explanatory,
  node_key_column = "index"
)

hex_net

Spatial network nodes with id: sg
--------------------------------------------------
Number of nodes: 1945
Average number of links per node: 10.862
Density of the neighborhood matrix: 0.56% (non-zero connections)

Data on nodes:
     index busstop_count housing_count biz_count school_count fin_count
1       33             0             0         0            0         0
2       34             0             0         0            0         0
3       35             0             0         0            0         0
4       36             0             0         0            0         0
5       37             0             0         0            0         0
6       60             0             0         0            0         0
---    ---           ---           ---       ---          ---       ---
1940  3884             0             0         0            0         0
1941  3904             0             0         0            0         0
1942  3908             0             0         0            0         0
1943  3909             0             0         0            0         0
1944  3910             0             0         0            0         0
1945  3937             0             0         0            0         0
     mrtlrt_count
1               0
2               0
3               0
4               0
5               0
6               0
---           ---
1940            0
1941            0
1942            0
1943            0
1944            0
1945            0

spflow_network_pair-class can be created using spflow_network_pair() function of spflow package.

Code

hex_net_pairs <- spflow_network_pair(
  id_orig_net = "sg",
  id_dest_net = "sg",
  pair_data = flow_data1,
  orig_key_column = "ORIGIN_hex",
  dest_key_column = "DESTIN_hex"
)

hex_net_pairs

Spatial network pair with id: sg_sg
--------------------------------------------------
Origin network id: sg (with 820 nodes)
Destination network id: sg (with 821 nodes)
Number of pairs: 65051
Completeness of pairs: 9.66% (65051/673220)

Data on node-pairs:
    DESTIN_hex ORIGIN_hex TOTAL_TRIPS    dist
1          201        118           1    2250
2          228        118           2    3000
3          254        118           2 1984.31
4          281        118           1 2598.08
5          285        118          56 6873.86
6          228        146          13    2250
---        ---        ---         ---     ---
11        2769       3308         535 7611.67
21        2742       3308           8 7901.74
31        2877       3308          37 6139.01
41        2931       3308           3 5408.33
51        3173       3308           7 1984.31
61        3232       3308           1 5952.94

spflow_network_multi-class can be created using spflow_network_multi() function of spflow package and only works on spflow_network-class and spflow_network_pair-class.

Code

hex_multi_net <- spflow_network_multi(hex_net, hex_net_pairs)

hex_multi_net

Collection of spatial network nodes and pairs
--------------------------------------------------
Contains 1 spatial network nodes  
    With id :  sg
Contains 1 spatial network pairs  
    With id :  sg_sg

Availability of origin-destination pair information:

 ID_ORIG_NET ID_DEST_NET ID_NET_PAIR COMPLETENESS       C_PAIRS    C_ORIG
          sg          sg       sg_sg        1.72% 65051/3783025 1945/1945
    C_DEST
 1945/1945

Correlation Analysis

Multicollinearity refers to a situation in which more than two explanatory variables in a multiple regression model are highly linearly related. In this situation, the coefficient estimates of the multiple regression may change erratically in response to small changes in the data or the procedure used to fit the model. To avoid including explanatory variables that are highly correlated, spflow provides two functions:

pair_cor() to create a correlation matrix, and
cor_image() to plot the correlation matrix as a correlogram.

Code

# Generate explanatory variables names
var_name <- explanatory %>%
  select(-(index)) %>%
  names()

# Generate the formula dynamically
cor_formula <- log(1 + TOTAL_TRIPS) ~ 
  busstop_count + 
  housing_count + 
  biz_count + 
  school_count + 
  fin_count + 
  mrtlrt_count + 
  P_(log(dist + 1))

cor_mat <- pair_cor(
  hex_multi_net, 
  spflow_formula = cor_formula, 
  add_lags_x = FALSE)

colnames(cor_mat) <- paste0(
  substr(
    colnames(cor_mat),1,3),"...")

cor_image(cor_mat)

Given that there are no variable pairs that are very highly correlated with one another, all variables will be used to calibrate the SEIM.

Model Calibration

There are currently three estimators of spatial econometric interaction models supported by spflow package:

Maximum likelihood estimation (MLE) – default estimation procedure.
Spatial two-stage least squares (S2SLS) – activate the S2SLS estimation via the estimation_control argument using the input spflow_control(estimation_method = "s2sls").
Bayesian Markov Chain Monte Carlo (MCMC) – activate the MCMC estimation via the estimation_control argument using the input spflow_control(estimation_method = "mcmc").

The code chunk below will be used to calibrate a base model based on the defaults (model 9 and MLE estimator). The spflow function offers a formula interface adapted to spatial interaction models, which has the following structure:

Y ~ O_(X1) + D_(X2) + I_(X3) + P_(X4)

O_(…) and D_(…) indicate which variables are used as characteristics of the origins and destinations respectively.
I_(…) indicates variables that should be used for the intra-regional parameters.
P_(…) declares which variables describe origin-destination pairs, which usually will include a measure of distance (distance decay).

Code

base_model <- spflow(
  spflow_formula = log(1 + TOTAL_TRIPS) ~ 
    O_(busstop_count +
         housing_count + 
         biz_count + 
         school_count + 
         mrtlrt_count) +
    D_(busstop_count +
         biz_count + 
         school_count + 
         fin_count + 
         mrtlrt_count) +
    P_(log(dist + 1)),
  spflow_networks = hex_multi_net)

base_model

--------------------------------------------------
Spatial interaction model estimated by: MLE  
Spatial correlation structure: SDM (model_9)
Dependent variable: log(1 + TOTAL_TRIPS)

--------------------------------------------------
Coefficients:
                         est     sd   t.stat  p.val
rho_d                  0.812  0.006  134.701  0.000
rho_o                  0.820  0.006  135.153  0.000
rho_w                 -0.999  0.013  -76.331  0.000
(Intercept)            5.991  0.099   60.529  0.000
(Intra)                   NA     NA       NA     NA
D_busstop_count        0.017  0.002   10.336  0.000
D_busstop_count.lag1  -0.030  0.004   -7.046  0.000
D_biz_count            0.005  0.000   10.637  0.000
D_biz_count.lag1      -0.001  0.001   -0.845  0.398
D_school_count         0.061  0.006    9.599  0.000
D_school_count.lag1    0.018  0.026    0.700  0.484
D_fin_count            0.004  0.000   10.558  0.000
D_fin_count.lag1      -0.005  0.001   -3.399  0.001
D_mrtlrt_count         0.047  0.008    5.614  0.000
D_mrtlrt_count.lag1   -0.079  0.027   -2.941  0.003
O_busstop_count        0.021  0.002   11.996  0.000
O_busstop_count.lag1  -0.038  0.005   -7.943  0.000
O_housing_count        0.000  0.000   30.082  0.000
O_housing_count.lag1   0.000  0.000    5.384  0.000
O_biz_count            0.000  0.000    0.993  0.321
O_biz_count.lag1      -0.005  0.001   -5.389  0.000
O_school_count         0.017  0.006    2.589  0.010
O_school_count.lag1    0.148  0.027    5.427  0.000
O_mrtlrt_count         0.058  0.008    7.496  0.000
O_mrtlrt_count.lag1   -0.194  0.019  -10.081  0.000
P_log(dist + 1)       -0.510  0.012  -43.006  0.000

--------------------------------------------------
R2_corr: 0.6213425511  
Observations: 65051  
Model coherence: Unknown

t.stat refers to the coefficients: a positive number means there is a direct relationship between the explanatory variable and the dependent variable, while a negative number implies an inverse relationship. p.val shows whether the coefficients are statistically significant, and hence, a good explanatory variable, or not statistically significant, and hence, not a good explanatory variable.

Insights

The R² value is 0.6213, which means the model accounts for 62.13% of the variation of flows. While not perfect, this value is much higher than the R² values of the SIM seen earlier.

Among the destination explanatory variables:

Bus stop counts, Financial services counts and MRT/LRT station counts and their lags all have a t.stat that is statistically significant. This means that such counts within the TAZ and in neighbouring TAZ will affect the attractiveness of the specific destination TAZ.
However, for all three variables, their coefficients are positive while the coefficients of their lags are negative. This means that the more of such counts there are in the destination TAZ, the more trips are made to that TAZ. And the more of such counts there are in neighbouring TAZ, the less trips are made to that TAZ.
School and Business counts have a t.stat that is statistically significant, but their lags are not. This means that such counts within the TAZ will affect the attractiveness of the TAZ but counts in neighbouring zones do not affect the attractiveness of the specific TAZ.

Among the origin explanatory variables:

Bus stop counts and MRT/LRT station counts and their lags both have a t.stat that is statistically significant. This means that they are good explanatory variable as such counts within the TAZ and in neighbouring TAZ will affect the propulsiveness of the specific origin TAZ.
However, for both variables, their coefficients are positive while the coefficients of their lags are negative. This means that the more of such counts there are in the origin TAZ, the more trips are made from that origin TAZ. And the more of such counts there are in neighbouring TAZ, the less trips are made from that origin TAZ.
School and Housing counts and their lags also have a t.stat that is statistically significant, making them good explanatory variables. For these two variables, their coefficients and the coefficients of their lags are positive, which means the more of such counts there are in the origin TAZ and its neighbouring TAZ, the more trips are made from that origin TAZ.
Business counts has a t.stat that is not statistically significant, but their lags are statistically significant. This means that while the business counts of the origin TAZ is not a good explanatory variable, counts in the neighbouring TAZ are: the more of such counts there are in the neighbouring TAZ, the less trips are made from that origin TAZ.

Residual diagnostics

In building explanatory models, it is important to check if the model calibrate conform to the statistical assumption of the statistical methods used. *spflow package provides several functions to support residual diagnostics needs. In the code chunk below, spflow_moran_plots() is used.

Code

old_par <- par(mfrow = c(1, 3), 
               mar = c(2,2,2,2))

spflow_moran_plots(base_model)

Visualise Fitted Values

Lastly, the observed and fitted values of the SEIM will be visualised on a scatterplot.

In the code chunk below, the fitted values from the destination constrained SIM is extracted then appended to the flow_data data frame. rename() is used to rename the newly added column.

Code

model.df <- as_tibble(base_model@spflow_indicators) %>%
  mutate(FITTED_Y = round(exp(FITTED),0))

inter_zonal_flow2 <- flow_data %>%
  left_join(model.df) %>%
  mutate(diff = (FITTED_Y-TOTAL_TRIPS))

The scatterplot will then be created using geom_point() and other appropriate functions of ggplot2 package.

Code

ggplot(data = inter_zonal_flow2,
                aes(x = FITTED,
                    y = ACTUAL)) +
  geom_point() +
  geom_smooth(method = lm) +
  coord_cartesian(xlim=c(0,14),
                  ylim=c(0,14)) + 
  labs(title = "Observed vs. Fitted Values for SEIM",
       x = "Fitted Values", y = "Observed Values")

Insights

Most of the values fall closer to the linear regression line as compared to the destination constrained SIM.

Conclusion

During weekday morning peak periods, most travelling is happening to and from residential areas (e.g. Tampines, Jurong East, Punggol). Notably, there are several long-distance bus routes linking the East to the North, and the Central region to the North-Western part of Singapore. However, bus stop density does not appear to directly correlate with the number of desire lines or the line thickness. As such, there is a need to further understand what are the other factors that drive such trends in commuting flows apart.

Using the SEIM (model 9, MLE estimator) which accounts for 62.13% of the variation of flows, the explanatory variables for the flow are:

Destination Explanatory Variables

Higher public transportation node density and financial services density in the destination TAZ drives the number of trips to the destination TAZ. However, higher similar density in the neighbouring TAZ results in less trips to the destination TAZ.
Higher school density and employment opportunities in the destination TAZ also drives up the number of trips to the destination TAZ.

Origin Explanatory Variables

Higher public transportation node density in the origin TAZ increase the number of trips from the origin TAZ. However, a higher similar density in the neighbouring TAZ results in less trips from the origin TAZ.
Higher school and population density also increases the number of trips from the origin TAZ. A higher similar density in the neighbouring TAZ also results in more trips from the origin TAZ.
However, more employment opportunities in the neighbouring TAZ results in less trips from the origin TAZ.