7A: Calibrating Hedonic Pricing Model for Private Highrise Property with GWR Method

In this exercise, we will learn to build hedonic pricing models for private high-rise property using Geographically Weighted Regression (GWR) methods to account for non-stationary variables.

Author

Teng Kok Wai (Walter)

Published

September 27, 2024

Modified

October 14, 2024

1 Exercise 7A Reference

R for Geospatial Data Science and Analytics - 13  Calibrating Hedonic Pricing Model for Private Highrise Property with GWR Method

2 Overview

In this exercise, we will learn to build hedonic pricing models for private high-rise property using Geographically Weighted Regression (GWR) methods to account for non-stationary variables.

Geographically weighted regression (GWR) is a spatial statistical technique that takes non-stationary variables into consideration (e.g., climate; demographic factors; physical environment characteristics) and models the local relationships between these independent variables and an outcome of interest (also known as dependent variable).

In this exercise, The dependent variable is the resale prices of condominium in 2015. The independent variables are divided into either structural and locational.

We will be using the GWmodel package. It provides a collection of localised spatial statistical methods, namely: GW summary statistics, GW principal components analysis, GW discriminant analysis and various forms of GW regression; some of which are provided in basic and robust (outlier resistant) forms. Commonly, outputs or parameters of the GWmodel are mapped to provide a useful exploratory tool, which can often precede (and direct) a more traditional or sophisticated statistical analysis.

3 Learning Outcome

  • Understand the basics of geographically weighted regression (GWR).
  • Build and evaluate hedonic pricing models using GWR.
  • Convert geospatial and aspatial data into appropriate formats for analysis.
  • Perform exploratory data analysis (EDA) using statistical graphics.
  • Visualize the model outputs and spatial patterns using various R packages.
  • Assess spatial autocorrelation and other statistical properties of model residuals.
  • Compare and interpret fixed and adaptive bandwidth GWR models.

4 The Data

The following 2 datasets will be used in this study:

Data Set Description Format
MP14_SUBZONE_WEB_PL URA Master Plan 2014’s planning subzone boundaries represented as polygon features. ESRI Shapefile
condo_resale_2015.csv Data on condominium resale prices in 2015, including structural and locational attributes. CSV

5 Installing and Launching the R Packages

The following R packages will be used in this exercise:

Package Purpose Use Case in Exercise
sf Manages, processes, and manipulates vector-based geospatial data. Handling and converting geospatial data such as the URA Master Plan boundaries.
GWmodel Calibrates geographically weighted models for local spatial analysis. Building hedonic pricing models using fixed and adaptive bandwidth GWR methods.
olsrr Performs diagnostics and builds better multiple linear regression models. Testing for multicollinearity, normality, and linearity of the regression model.
corrplot Provides visual tools to explore data correlation matrices. Visualizing relationships between independent variables in the dataset.
tmap Creates static and interactive thematic maps using cartographic quality elements. Visualizing the geospatial distribution of condominium resale prices and other model outputs.
tidyverse A collection of packages for data science tasks such as data manipulation, visualization, and modeling. Importing CSV files, wrangling data, and performing data transformations and visualizations.
ggpubr Enhances ‘ggplot2’ for easier ‘publication-ready’ plots. Creating small multiple histograms and scatterplots for exploratory data analysis.

To install and load these packages, use the following code:

pacman::p_load(olsrr, corrplot, ggpubr, sf, spdep, GWmodel, tmap, tidyverse, gtsummary)

6 Geospatial Data Wrangling

6.1 Importing Geospatial Data

To import MP_SUBZONE_WEB_PL shapefile:

mpsz = st_read(dsn = "data/geospatial", layer = "MP14_SUBZONE_WEB_PL")
Reading layer `MP14_SUBZONE_WEB_PL' from data source
  `/Users/walter/code/isss626/isss626-gaa/Hands-on_Ex/Hands-on_Ex07/data/geospatial'
  using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21

The output above shows that the R object used to contain the imported MP14_SUBZONE_WEB_PL shapefile is called mpsz and it is a simple feature object. The geometry type is multipolygon. It is also important to note that mpsz simple feature object does not have EPSG information.

6.2 Updating CRS Information

The code below updates the newly imported mpsz with the correct ESPG code (i.e. 3414) 

mpsz_svy21 <- st_transform(mpsz, 3414)
st_crs(mpsz_svy21)
Coordinate Reference System:
  User input: EPSG:3414
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

Notice that the EPSG: is indicated as 3414 now.

6.3 Revealing the Extent of mpsz_svy21

To reveal the extent of mpsz_svy21:

st_bbox(mpsz_svy21) #view extent
     xmin      ymin      xmax      ymax
 2667.538 15748.721 56396.440 50256.334

7 Aspatial Data Wrangling

7.1 Importing Aspatial Data

The condo_resale_2015 is in csv file format. The codes chunk below uses read_csv() function of readr package to import condo_resale_2015 into R as a tibble data frame called condo_resale. 

condo_resale = read_csv("data/aspatial/Condo_resale_2015.csv")
glimpse(condo_resale)
Rows: 1,436
Columns: 23
$ LATITUDE             <dbl> 1.287145, 1.328698, 1.313727, 1.308563, 1.321437,…
$ LONGITUDE            <dbl> 103.7802, 103.8123, 103.7971, 103.8247, 103.9505,…
$ POSTCODE             <dbl> 118635, 288420, 267833, 258380, 467169, 466472, 3…
$ SELLING_PRICE        <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1320…
$ AREA_SQM             <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 168,…
$ AGE                  <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22, 6,…
$ PROX_CBD             <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783402…
$ PROX_CHILDCARE       <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543, 0…
$ PROX_ELDERLYCARE     <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.121…
$ PROX_URA_GROWTH_AREA <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.410632,…
$ PROX_HAWKER_MARKET   <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969, 0…
$ PROX_KINDERGARTEN    <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076, 0…
$ PROX_MRT             <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.528…
$ PROX_PARK            <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.116…
$ PROX_PRIMARY_SCH     <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.709…
$ PROX_TOP_PRIMARY_SCH <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.709…
$ PROX_SHOPPING_MALL   <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.307…
$ PROX_SUPERMARKET     <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.581…
$ PROX_BUS_STOP        <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340, 0…
$ NO_Of_UNITS          <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34, 3…
$ FAMILY_FRIENDLY      <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0…
$ FREEHOLD             <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR       <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…

To display summary statistics of condo_resale:

summary(condo_resale)
    LATITUDE       LONGITUDE        POSTCODE      SELLING_PRICE
 Min.   :1.240   Min.   :103.7   Min.   : 18965   Min.   :  540000
 1st Qu.:1.309   1st Qu.:103.8   1st Qu.:259849   1st Qu.: 1100000
 Median :1.328   Median :103.8   Median :469298   Median : 1383222
 Mean   :1.334   Mean   :103.8   Mean   :440439   Mean   : 1751211
 3rd Qu.:1.357   3rd Qu.:103.9   3rd Qu.:589486   3rd Qu.: 1950000
 Max.   :1.454   Max.   :104.0   Max.   :828833   Max.   :18000000
    AREA_SQM          AGE           PROX_CBD       PROX_CHILDCARE
 Min.   : 34.0   Min.   : 0.00   Min.   : 0.3869   Min.   :0.004927
 1st Qu.:103.0   1st Qu.: 5.00   1st Qu.: 5.5574   1st Qu.:0.174481
 Median :121.0   Median :11.00   Median : 9.3567   Median :0.258135
 Mean   :136.5   Mean   :12.14   Mean   : 9.3254   Mean   :0.326313
 3rd Qu.:156.0   3rd Qu.:18.00   3rd Qu.:12.6661   3rd Qu.:0.368293
 Max.   :619.0   Max.   :37.00   Max.   :19.1804   Max.   :3.465726
 PROX_ELDERLYCARE  PROX_URA_GROWTH_AREA PROX_HAWKER_MARKET PROX_KINDERGARTEN
 Min.   :0.05451   Min.   :0.2145       Min.   :0.05182    Min.   :0.004927
 1st Qu.:0.61254   1st Qu.:3.1643       1st Qu.:0.55245    1st Qu.:0.276345
 Median :0.94179   Median :4.6186       Median :0.90842    Median :0.413385
 Mean   :1.05351   Mean   :4.5981       Mean   :1.27987    Mean   :0.458903
 3rd Qu.:1.35122   3rd Qu.:5.7550       3rd Qu.:1.68578    3rd Qu.:0.578474
 Max.   :3.94916   Max.   :9.1554       Max.   :5.37435    Max.   :2.229045
    PROX_MRT         PROX_PARK       PROX_PRIMARY_SCH  PROX_TOP_PRIMARY_SCH
 Min.   :0.05278   Min.   :0.02906   Min.   :0.07711   Min.   :0.07711
 1st Qu.:0.34646   1st Qu.:0.26211   1st Qu.:0.44024   1st Qu.:1.34451
 Median :0.57430   Median :0.39926   Median :0.63505   Median :1.88213
 Mean   :0.67316   Mean   :0.49802   Mean   :0.75471   Mean   :2.27347
 3rd Qu.:0.84844   3rd Qu.:0.65592   3rd Qu.:0.95104   3rd Qu.:2.90954
 Max.   :3.48037   Max.   :2.16105   Max.   :3.92899   Max.   :6.74819
 PROX_SHOPPING_MALL PROX_SUPERMARKET PROX_BUS_STOP       NO_Of_UNITS
 Min.   :0.0000     Min.   :0.0000   Min.   :0.001595   Min.   :  18.0
 1st Qu.:0.5258     1st Qu.:0.3695   1st Qu.:0.098356   1st Qu.: 188.8
 Median :0.9357     Median :0.5687   Median :0.151710   Median : 360.0
 Mean   :1.0455     Mean   :0.6141   Mean   :0.193974   Mean   : 409.2
 3rd Qu.:1.3994     3rd Qu.:0.7862   3rd Qu.:0.220466   3rd Qu.: 590.0
 Max.   :3.4774     Max.   :2.2441   Max.   :2.476639   Max.   :1703.0
 FAMILY_FRIENDLY     FREEHOLD      LEASEHOLD_99YR
 Min.   :0.0000   Min.   :0.0000   Min.   :0.0000
 1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000
 Median :0.0000   Median :0.0000   Median :0.0000
 Mean   :0.4868   Mean   :0.4227   Mean   :0.4882
 3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:1.0000
 Max.   :1.0000   Max.   :1.0000   Max.   :1.0000

7.2 Converting an Aspatial Data Frame to an sf Object

In this step, we will:

  • Use the st_as_sf() function from the sf package to convert the aspatial data frame into a spatial (sf) object.
  • Apply st_transform() to reproject the coordinates from the WGS 84 coordinate system (CRS: 4326) to SVY21 (CRS: 3414), commonly used in Singapore. 
condo_resale.sf <- st_as_sf(condo_resale,
                            coords = c("LONGITUDE", "LATITUDE"),
                            crs = 4326) %>%
  st_transform(crs = 3414)

head(condo_resale.sf)
Simple feature collection with 6 features and 21 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 22085.12 ymin: 29951.54 xmax: 41042.56 ymax: 34546.2
Projected CRS: SVY21 / Singapore TM
# A tibble: 6 × 22
  POSTCODE SELLING_PRICE AREA_SQM   AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE
     <dbl>         <dbl>    <dbl> <dbl>    <dbl>          <dbl>            <dbl>
1   118635       3000000      309    30     7.94          0.166            2.52
2   288420       3880000      290    32     6.61          0.280            1.93
3   267833       3325000      248    33     6.90          0.429            0.502
4   258380       4250000      127     7     4.04          0.395            1.99
5   467169       1400000      145    28    11.8           0.119            1.12
6   466472       1320000      139    22    10.3           0.125            0.789
# ℹ 15 more variables: PROX_URA_GROWTH_AREA <dbl>, PROX_HAWKER_MARKET <dbl>,
#   PROX_KINDERGARTEN <dbl>, PROX_MRT <dbl>, PROX_PARK <dbl>,
#   PROX_PRIMARY_SCH <dbl>, PROX_TOP_PRIMARY_SCH <dbl>,
#   PROX_SHOPPING_MALL <dbl>, PROX_SUPERMARKET <dbl>, PROX_BUS_STOP <dbl>,
#   NO_Of_UNITS <dbl>, FAMILY_FRIENDLY <dbl>, FREEHOLD <dbl>,
#   LEASEHOLD_99YR <dbl>, geometry <POINT [m]>
  • The resulting output is a point-based feature data frame.

8 Exploratory Data Analysis

In the section, we will use statistical graphics functions of ggplot2 package to perform EDA.

8.1 EDA using Statistical Graphics

8.2 6.1.1 Plot Distribution

To plot the distribution of SELLING_PRICE by using histograms: 

ggplot(data=condo_resale.sf, aes(x=`SELLING_PRICE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

Note

Observations:

  • A right skewed distribution.
  • This means that more condominium units were transacted at relative lower prices.
  • Statistically, the skewed distribution can be normalised by using log transformation. 
condo_resale.sf <- condo_resale.sf %>%
  mutate(`LOG_SELLING_PRICE` = log(SELLING_PRICE))

ggplot(data=condo_resale.sf, aes(x=`LOG_SELLING_PRICE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

Notice that the distribution is relatively less skewed after the transformation.

8.3 Multiple Histogram Plots for Variable Distribution

In this section, we will create multiple histograms (also known as a trellis plot or small multiples) to visualize the distribution of several variables. We will use the ggarrange() function from the ggpubr package to organize these histograms into a 3-column by 4-row grid layout.

Show the code 
# List of variables to plot
variables <- c("AREA_SQM", "AGE", "PROX_CBD", "PROX_CHILDCARE", 
               "PROX_ELDERLYCARE", "PROX_URA_GROWTH_AREA", 
               "PROX_HAWKER_MARKET", "PROX_KINDERGARTEN", 
               "PROX_MRT", "PROX_PARK", "PROX_PRIMARY_SCH", 
               "PROX_TOP_PRIMARY_SCH")

histograms <- lapply(variables, function(var) {
  ggplot(condo_resale.sf, aes_string(x = var)) + 
    geom_histogram(bins = 20, color = "black", fill = "lightblue")
})

ggarrange(plotlist = histograms, ncol = 3, nrow = 4)

8.4 Drawing a Statistical Point Map

In this section, we will visualize the geospatial distribution of condominium resale prices in Singapore.

  • The map will be prepared by using tmap package.
    • tmap_mode("view") to use the interactive mode of tmap
  • Then, create an interactive point symbol map
    • tm_dots() is used instead of tm_bubbles()
    • set.zoom.limits argument of tm_view() sets the minimum and maximum zoom level to 11 and 14 respectively.
  • Lastly, tmap_mode("plot") to display plot mode
Show the code 
tmap_mode("view")
tmap_options(check.and.fix = TRUE)

tm_shape(mpsz_svy21)+
  tm_polygons() +
tm_shape(condo_resale.sf) +  
  tm_dots(col = "SELLING_PRICE",
          alpha = 0.6,
          style="quantile") +
  tm_view(set.zoom.limits = c(11,14))
+−
SELLING_PRICE
540,000 to 1,040,000
1,040,000 to 1,270,000
1,270,000 to 1,550,000
1,550,000 to 2,130,000
2,130,000 to 18,000,000
Leaflet | Tiles © Esri — Esri, DeLorme, NAVTEQ
Show the code 
tmap_mode("plot")

9 Hedonic Pricing Model in R

In this section, we will build a hedonic pricing model for condominium resale units using the lm() function from base R.

9.1 Simple Linear Regression

We start by modeling SELLING_PRICE as the dependent variable and AREA_SQM as the independent variable. 

condo.slr <- lm(SELLING_PRICE ~ AREA_SQM, data = condo_resale.sf)
summary(condo.slr)

Call:
lm(formula = SELLING_PRICE ~ AREA_SQM, data = condo_resale.sf)

Residuals:
     Min       1Q   Median       3Q      Max
-3695815  -391764   -87517   258900 13503875

Coefficients:
             Estimate Std. Error t value Pr(>|t|)
(Intercept) -258121.1    63517.2  -4.064 5.09e-05 ***
AREA_SQM      14719.0      428.1  34.381  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 942700 on 1434 degrees of freedom
Multiple R-squared:  0.4518,    Adjusted R-squared:  0.4515
F-statistic:  1182 on 1 and 1434 DF,  p-value: < 2.2e-16
Note

Observations:

  • The model equation is:
    SELLING_PRICE = -258121.1 + 14719 * AREA_SQM
  • R-squared: 0.4518, indicating the model explains 45% of the variation in resale prices.
  • P-value: The very small p-value (< 0.0001) indicates strong evidence to reject the null hypothesis, suggesting the model is a good fit.
  • Coefficients: Both the intercept and slope have p-values < 0.001, confirming that the parameters are significant predictors.

To visualize the fit, we can plot the data and the regression line: 

ggplot(data = condo_resale.sf, aes(x = AREA_SQM, y = SELLING_PRICE)) +
  geom_point() +
  geom_smooth(method = lm)

This scatterplot with a fitted regression line highlights some high-price outliers in the dataset.

9.2 Multiple Linear Regression

9.2.1 Checking for Multicollinearity

Before building a multiple regression model, it is essential to ensure that the independent variables are not highly correlated, as this can lead to multicollinearity, which compromises the model’s quality.

A correlation matrix is useful for visualizing relationships between variables. In this case, we use the corrplot package to plot a correlation matrix for the independent variables. 

corrplot(cor(condo_resale[, 5:23]), diag = FALSE, order = "AOE",
         tl.pos = "td", tl.cex = 0.5, method = "number", type = "upper")

Tip

Matrix reorder is very important for mining the hiden structure and patter in the matrix. There are four methods in corrplot (parameter order), named “AOE”, “FPC”, “hclust”, “alphabet”. In the code chunk above, AOE order is used. It orders the variables by using the angular order of the eigenvectors method suggested by Michael Friendly.

Note

Observations:

  • From the matrix, we observe that Freehold and LEASE_99YEAR are highly correlated, so only Freehold will be included in the model.

9.3 Building the Multiple Linear Regression Model

We now build a hedonic pricing model using multiple linear regression. The model predicts SELLING_PRICE based on several property characteristics. 

condo.mlr <- lm(SELLING_PRICE ~ AREA_SQM + AGE + 
                  PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
                  PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + 
                  PROX_KINDERGARTEN + PROX_MRT + PROX_PARK + 
                  PROX_PRIMARY_SCH + PROX_TOP_PRIMARY_SCH + 
                  PROX_SHOPPING_MALL + PROX_SUPERMARKET + 
                  PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                data = condo_resale.sf)
summary(condo.mlr)

Call:
lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE +
    PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET +
    PROX_KINDERGARTEN + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH +
    PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET +
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
    data = condo_resale.sf)

Residuals:
     Min       1Q   Median       3Q      Max
-3475964  -293923   -23069   241043 12260381

Coefficients:
                       Estimate Std. Error t value Pr(>|t|)
(Intercept)           481728.40  121441.01   3.967 7.65e-05 ***
AREA_SQM               12708.32     369.59  34.385  < 2e-16 ***
AGE                   -24440.82    2763.16  -8.845  < 2e-16 ***
PROX_CBD              -78669.78    6768.97 -11.622  < 2e-16 ***
PROX_CHILDCARE       -351617.91  109467.25  -3.212  0.00135 **
PROX_ELDERLYCARE      171029.42   42110.51   4.061 5.14e-05 ***
PROX_URA_GROWTH_AREA   38474.53   12523.57   3.072  0.00217 **
PROX_HAWKER_MARKET     23746.10   29299.76   0.810  0.41782
PROX_KINDERGARTEN     147468.99   82668.87   1.784  0.07466 .
PROX_MRT             -314599.68   57947.44  -5.429 6.66e-08 ***
PROX_PARK             563280.50   66551.68   8.464  < 2e-16 ***
PROX_PRIMARY_SCH      180186.08   65237.95   2.762  0.00582 **
PROX_TOP_PRIMARY_SCH    2280.04   20410.43   0.112  0.91107
PROX_SHOPPING_MALL   -206604.06   42840.60  -4.823 1.57e-06 ***
PROX_SUPERMARKET      -44991.80   77082.64  -0.584  0.55953
PROX_BUS_STOP         683121.35  138353.28   4.938 8.85e-07 ***
NO_Of_UNITS             -231.18      89.03  -2.597  0.00951 **
FAMILY_FRIENDLY       140340.77   47020.55   2.985  0.00289 **
FREEHOLD              359913.01   49220.22   7.312 4.38e-13 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 755800 on 1417 degrees of freedom
Multiple R-squared:  0.6518,    Adjusted R-squared:  0.6474
F-statistic: 147.4 on 18 and 1417 DF,  p-value: < 2.2e-16
Note

Observations:

  • Significant predictors:
    • AREA_SQM (positive): Larger units have higher selling prices.
    • AGE (negative): Older units tend to sell for less.
    • PROX_CBD (negative): Proximity to the central business district reduces selling prices.
    • PROX_PARK (positive): Proximity to parks increases resale prices.
    • FREEHOLD (positive): Freehold properties sell for more than leasehold properties.
  • Non-significant predictors (p > 0.05):
    • PROX_HAWKER_MARKET, PROX_TOP_PRIMARY_SCH, PROX_SUPERMARKET show weak or no significant relationship with SELLING_PRICE.

9.4 Revising the Hedonic Pricing Model

After reviewing the initial model, we will now remove non-significant variables to improve the model. The revised multiple linear regression model is calibrated as follows: 

condo.mlr1 <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + 
                   PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
                   PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + 
                   PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP +
                   NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                 data = condo_resale.sf)
ols_regress(condo.mlr1)
                                Model Summary
-----------------------------------------------------------------------------
R                            0.807       RMSE                     751998.679
R-Squared                    0.651       MSE                571471422208.592
Adj. R-Squared               0.647       Coef. Var                    43.168
Pred R-Squared               0.638       AIC                       42966.758
MAE                     414819.628       SBC                       43051.072
-----------------------------------------------------------------------------
 RMSE: Root Mean Square Error
 MSE: Mean Square Error
 MAE: Mean Absolute Error
 AIC: Akaike Information Criteria
 SBC: Schwarz Bayesian Criteria

                                     ANOVA
--------------------------------------------------------------------------------
                    Sum of
                   Squares          DF         Mean Square       F         Sig.
--------------------------------------------------------------------------------
Regression    1.512586e+15          14        1.080418e+14    189.059    0.0000
Residual      8.120609e+14        1421    571471422208.592
Total         2.324647e+15        1435
--------------------------------------------------------------------------------

                                               Parameter Estimates
-----------------------------------------------------------------------------------------------------------------
               model           Beta    Std. Error    Std. Beta       t        Sig           lower          upper
-----------------------------------------------------------------------------------------------------------------
         (Intercept)     527633.222    108183.223                   4.877    0.000     315417.244     739849.200
            AREA_SQM      12777.523       367.479        0.584     34.771    0.000      12056.663      13498.382
                 AGE     -24687.739      2754.845       -0.167     -8.962    0.000     -30091.739     -19283.740
            PROX_CBD     -77131.323      5763.125       -0.263    -13.384    0.000     -88436.469     -65826.176
      PROX_CHILDCARE    -318472.751    107959.512       -0.084     -2.950    0.003    -530249.889    -106695.613
    PROX_ELDERLYCARE     185575.623     39901.864        0.090      4.651    0.000     107302.737     263848.510
PROX_URA_GROWTH_AREA      39163.254     11754.829        0.060      3.332    0.001      16104.571      62221.936
            PROX_MRT    -294745.107     56916.367       -0.112     -5.179    0.000    -406394.234    -183095.980
           PROX_PARK     570504.807     65507.029        0.150      8.709    0.000     442003.938     699005.677
    PROX_PRIMARY_SCH     159856.136     60234.599        0.062      2.654    0.008      41697.849     278014.424
  PROX_SHOPPING_MALL    -220947.251     36561.832       -0.115     -6.043    0.000    -292668.213    -149226.288
       PROX_BUS_STOP     682482.221    134513.243        0.134      5.074    0.000     418616.359     946348.082
         NO_Of_UNITS       -245.480        87.947       -0.053     -2.791    0.005       -418.000        -72.961
     FAMILY_FRIENDLY     146307.576     46893.021        0.057      3.120    0.002      54320.593     238294.560
            FREEHOLD     350599.812     48506.485        0.136      7.228    0.000     255447.802     445751.821
-----------------------------------------------------------------------------------------------------------------

9.5 Publication-Ready Table with gtsummary

The gtsummary package provides an elegant way to create publication-ready regression tables. The following code generates a formatted regression report: 

tbl_regression(condo.mlr1, intercept = TRUE)

Characteristic

Beta

95% CI

1

p-value
(Intercept) 527,633 315,417, 739,849 <0.001
AREA_SQM 12,778 12,057, 13,498 <0.001
AGE -24,688 -30,092, -19,284 <0.001
PROX_CBD -77,131 -88,436, -65,826 <0.001
PROX_CHILDCARE -318,473 -530,250, -106,696 0.003
PROX_ELDERLYCARE 185,576 107,303, 263,849 <0.001
PROX_URA_GROWTH_AREA 39,163 16,105, 62,222 <0.001
PROX_MRT -294,745 -406,394, -183,096 <0.001
PROX_PARK 570,505 442,004, 699,006 <0.001
PROX_PRIMARY_SCH 159,856 41,698, 278,014 0.008
PROX_SHOPPING_MALL -220,947 -292,668, -149,226 <0.001
PROX_BUS_STOP 682,482 418,616, 946,348 <0.001
NO_Of_UNITS -245 -418, -73 0.005
FAMILY_FRIENDLY 146,308 54,321, 238,295 0.002
FREEHOLD 350,600 255,448, 445,752 <0.001
1

CI = Confidence Interval

You can further enhance the report by adding model statistics using add_glance_table() or add_glance_source_note(): 

tbl_regression(condo.mlr1, intercept = TRUE) %>%
  add_glance_source_note(
    label = list(sigma ~ "\U03C3"),
    include = c(r.squared, adj.r.squared, AIC, statistic, p.value, sigma))

9.5.1 Checking for Multicollinearity

We can check for multicollinearity using the olsrr package’s ols_vif_tol() function: 

ols_vif_tol(condo.mlr1)
              Variables Tolerance      VIF
1              AREA_SQM 0.8728554 1.145665
2                   AGE 0.7071275 1.414172
3              PROX_CBD 0.6356147 1.573280
4        PROX_CHILDCARE 0.3066019 3.261559
5      PROX_ELDERLYCARE 0.6598479 1.515501
6  PROX_URA_GROWTH_AREA 0.7510311 1.331503
7              PROX_MRT 0.5236090 1.909822
8             PROX_PARK 0.8279261 1.207837
9      PROX_PRIMARY_SCH 0.4524628 2.210126
10   PROX_SHOPPING_MALL 0.6738795 1.483945
11        PROX_BUS_STOP 0.3514118 2.845664
12          NO_Of_UNITS 0.6901036 1.449058
13      FAMILY_FRIENDLY 0.7244157 1.380423
14             FREEHOLD 0.6931163 1.442759
Note

Observations:

Since the VIF values are all below 10, there is no indication of multicollinearity among the independent variables.

9.5.2 Test for Non-Linearity

In multiple linear regression, it is important for us to test the assumption that linearity and additivity of the relationship between dependent and independent variables.

To test the linearity assumption, we use the ols_plot_resid_fit() function. 

ols_plot_resid_fit(condo.mlr1)

Note

Observations:

Most points are scattered around the 0 line, hence we can safely conclude that the relationships between the dependent variable and independent variables are linear.

9.5.3 Test for Normality Assumption

We can assess the normality of residuals with a histogram:

ols_plot_resid_hist(condo.mlr1)

The residuals appear normally distributed. For a formal test, we can use ols_test_normality(): 

ols_test_normality(condo.mlr1)
-----------------------------------------------
       Test             Statistic       pvalue
-----------------------------------------------
Shapiro-Wilk              0.6856         0.0000
Kolmogorov-Smirnov        0.1366         0.0000
Cramer-von Mises         121.0768        0.0000
Anderson-Darling         67.9551         0.0000
-----------------------------------------------
Note

Observations:

The summary table above reveals that the p-values of the four tests are way smaller than the alpha value of 0.05. Hence we will reject the null hypothesis and infer that there is statistical evidence that the residual are not normally distributed.

9.5.4 Testing for Spatial Autocorrelation

Since the hedonic model involves geographically referenced data, it is important to visualize the residuals and test for spatial autocorrelation.

First, we export the residuals from the hedonic pricing model into a data frame. 

mlr.output <- as.data.frame(condo.mlr1$residuals)

Next, we join the residuals with the condo_resale.sf spatial data frame: 

condo_resale.res.sf <- cbind(condo_resale.sf, condo.mlr1$residuals) %>%
  rename(MLR_RES = condo.mlr1.residuals)

We convert the condo_resale.res.sf from an sf object to a SpatialPointsDataFrame for use with the spdep package: 

condo_resale.sp <- as_Spatial(condo_resale.res.sf)
condo_resale.sp
class       : SpatialPointsDataFrame
features    : 1436
extent      : 14940.85, 43352.45, 24765.67, 48382.81  (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   : 23
names       : POSTCODE, SELLING_PRICE, AREA_SQM, AGE,    PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN,    PROX_MRT,   PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH, PROX_SHOPPING_MALL, ...
min values  :    18965,        540000,       34,   0, 0.386916393,    0.004927023,      0.054508623,          0.214539508,        0.051817113,       0.004927023, 0.052779424, 0.029064164,      0.077106132,          0.077106132,                  0, ...
max values  :   828833,       1.8e+07,      619,  37, 19.18042832,     3.46572633,      3.949157205,           9.15540001,        5.374348075,       2.229045366,  3.48037319,  2.16104919,      3.928989144,          6.748192062,        3.477433767, ...

Using tmap, we visualize the spatial distribution of the residuals: 

tmap_mode("view")

tm_shape(mpsz_svy21) +
  tmap_options(check.and.fix = TRUE) +
  tm_polygons(alpha = 0.4) +
tm_shape(condo_resale.res.sf) +  
  tm_dots(col = "MLR_RES", alpha = 0.6, style = "quantile") +
  tm_view(set.zoom.limits = c(11, 14))
+−
MLR_RES
-3,470,778 to -368,646
-368,646 to -119,192
-119,192 to 68,991
68,991 to 324,802
324,802 to 12,234,210
Leaflet | Tiles © Esri — Esri, DeLorme, NAVTEQ
tmap_mode("plot")
Note

Observations:

The figure above reveal that there is sign of spatial autocorrelation.

To formally test for spatial autocorrelation using Moran’s I Test, we first compute a distance-based weight matrix: 

nb <- dnearneigh(coordinates(condo_resale.sp), 0, 1500, longlat = FALSE)
summary(nb)
Neighbour list object:
Number of regions: 1436
Number of nonzero links: 66266
Percentage nonzero weights: 3.213526
Average number of links: 46.14624
10 disjoint connected subgraphs
Link number distribution:

  1   3   5   7   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24
  3   3   9   4   3  15  10  19  17  45  19   5  14  29  19   6  35  45  18  47
 25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44
 16  43  22  26  21  11   9  23  22  13  16  25  21  37  16  18   8  21   4  12
 45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64
  8  36  18  14  14  43  11  12   8  13  12  13   4   5   6  12  11  20  29  33
 65  66  67  68  69  70  71  72  73  74  75  76  77  78  79  80  81  82  83  84
 15  20  10  14  15  15  11  16  12  10   8  19  12  14   9   8   4  13  11   6
 85  86  87  88  89  90  91  92  93  94  95  96  97  98  99 100 101 102 103 104
  4   9   4   4   4   6   2  16   9   4   5   9   3   9   4   2   1   2   1   1
105 106 107 108 109 110 112 116 125
  1   5   9   2   1   3   1   1   1
3 least connected regions:
193 194 277 with 1 link
1 most connected region:
285 with 125 links

Convert the neighbor list into spatial weights:

nb_lw <- nb2listw(nb, style = 'W')
summary(nb_lw)
Characteristics of weights list object:
Neighbour list object:
Number of regions: 1436
Number of nonzero links: 66266
Percentage nonzero weights: 3.213526
Average number of links: 46.14624
10 disjoint connected subgraphs
Link number distribution:

  1   3   5   7   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24
  3   3   9   4   3  15  10  19  17  45  19   5  14  29  19   6  35  45  18  47
 25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44
 16  43  22  26  21  11   9  23  22  13  16  25  21  37  16  18   8  21   4  12
 45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64
  8  36  18  14  14  43  11  12   8  13  12  13   4   5   6  12  11  20  29  33
 65  66  67  68  69  70  71  72  73  74  75  76  77  78  79  80  81  82  83  84
 15  20  10  14  15  15  11  16  12  10   8  19  12  14   9   8   4  13  11   6
 85  86  87  88  89  90  91  92  93  94  95  96  97  98  99 100 101 102 103 104
  4   9   4   4   4   6   2  16   9   4   5   9   3   9   4   2   1   2   1   1
105 106 107 108 109 110 112 116 125
  1   5   9   2   1   3   1   1   1
3 least connected regions:
193 194 277 with 1 link
1 most connected region:
285 with 125 links

Weights style: W
Weights constants summary:
     n      nn   S0       S1       S2
W 1436 2062096 1436 94.81916 5798.341

Finally, perform Moran’s I test on the model residuals:

lm.morantest(condo.mlr1, nb_lw)

    Global Moran I for regression residuals

data:
model: lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_MRT +
PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP +
NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, data = condo_resale.sf)
weights: nb_lw

Moran I statistic standard deviate = 24.366, p-value < 2.2e-16
alternative hypothesis: greater
sample estimates:
Observed Moran I      Expectation         Variance
    1.438876e-01    -5.487594e-03     3.758259e-05
Note

Observations:

  • P-value: The p-value is extremely small (< 0.05), leading us to reject the null hypothesis that residuals are randomly distributed.
  • Observed Moran’s I: The value of 0.1424418 (greater than 0) indicates that residuals show a clustered pattern, confirming the presence of spatial autocorrelation.

10 Building Hedonic Pricing Models using GWmodel

In this section, we will model hedonic pricing using both fixed and adaptive bandwidth schemes.

10.1 Building Fixed Bandwidth GWR Model

10.1.1 Computing Fixed Bandwidth

We use the bw.gwr() function from the GWModel package to determine the optimal fixed bandwidth for the model.

Setting adaptive = FALSE ensures that we are calculating a fixed bandwidth. There are two possible approaches for determining the stopping rule: CV cross-validation and AICc. In this case, we use the cross-validation approach (approach = "CV"). 

bw.fixed <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
                     PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
                     PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + 
                     PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + 
                     FAMILY_FRIENDLY + FREEHOLD, 
                   data = condo_resale.sp, 
                   approach = "CV", 
                   kernel = "gaussian", 
                   adaptive = FALSE, 
                   longlat = FALSE)
Fixed bandwidth: 17660.96 CV score: 8.259118e+14
Fixed bandwidth: 10917.26 CV score: 7.970454e+14
Fixed bandwidth: 6749.419 CV score: 7.273273e+14
Fixed bandwidth: 4173.553 CV score: 6.300006e+14
Fixed bandwidth: 2581.58 CV score: 5.404958e+14
Fixed bandwidth: 1597.687 CV score: 4.857515e+14
Fixed bandwidth: 989.6077 CV score: 4.722431e+14
Fixed bandwidth: 613.7939 CV score: 1.379526e+16
Fixed bandwidth: 1221.873 CV score: 4.778717e+14
Fixed bandwidth: 846.0596 CV score: 4.791629e+14
Fixed bandwidth: 1078.325 CV score: 4.751406e+14
Fixed bandwidth: 934.7772 CV score: 4.72518e+14
Fixed bandwidth: 1023.495 CV score: 4.730305e+14
Fixed bandwidth: 968.6643 CV score: 4.721317e+14
Fixed bandwidth: 955.7206 CV score: 4.722072e+14
Fixed bandwidth: 976.6639 CV score: 4.721387e+14
Fixed bandwidth: 963.7202 CV score: 4.721484e+14
Fixed bandwidth: 971.7199 CV score: 4.721293e+14
Fixed bandwidth: 973.6083 CV score: 4.721309e+14
Fixed bandwidth: 970.5527 CV score: 4.721295e+14
Fixed bandwidth: 972.4412 CV score: 4.721296e+14
Fixed bandwidth: 971.2741 CV score: 4.721292e+14
Fixed bandwidth: 970.9985 CV score: 4.721293e+14
Fixed bandwidth: 971.4443 CV score: 4.721292e+14
Fixed bandwidth: 971.5496 CV score: 4.721293e+14
Fixed bandwidth: 971.3793 CV score: 4.721292e+14
Fixed bandwidth: 971.3391 CV score: 4.721292e+14
Fixed bandwidth: 971.3143 CV score: 4.721292e+14
Fixed bandwidth: 971.3545 CV score: 4.721292e+14
Fixed bandwidth: 971.3296 CV score: 4.721292e+14
Fixed bandwidth: 971.345 CV score: 4.721292e+14
Fixed bandwidth: 971.3355 CV score: 4.721292e+14
Fixed bandwidth: 971.3413 CV score: 4.721292e+14
Fixed bandwidth: 971.3377 CV score: 4.721292e+14
Fixed bandwidth: 971.34 CV score: 4.721292e+14
Fixed bandwidth: 971.3405 CV score: 4.721292e+14
Fixed bandwidth: 971.3396 CV score: 4.721292e+14
Fixed bandwidth: 971.3402 CV score: 4.721292e+14
Fixed bandwidth: 971.3398 CV score: 4.721292e+14
Fixed bandwidth: 971.34 CV score: 4.721292e+14
Fixed bandwidth: 971.3399 CV score: 4.721292e+14
Fixed bandwidth: 971.34 CV score: 4.721292e+14

The recommended bandwidth is 971.34 meters.

Note

Quiz Answer:
The bandwidth is in meters because the model is using projected coordinates, SVY21, where distances are measured in meters.

10.1.2 GWModel Method - Fixed Bandwidth

We can now calibrate the GWR model using a fixed bandwidth and a Gaussian kernel with the following code: 

gwr.fixed <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
                         PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
                         PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + 
                         PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + 
                         FAMILY_FRIENDLY + FREEHOLD, 
                       data = condo_resale.sp, 
                       bw = bw.fixed, 
                       kernel = 'gaussian', 
                       longlat = FALSE)
gwr.fixed
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2024-10-14 19:55:45.822271
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA +
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL +
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
    data = condo_resale.sp, bw = bw.fixed, kernel = "gaussian",
    longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max
-3470778  -298119   -23481   248917 12234210

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 **
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 **
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 **
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 **
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian
   Fixed bandwidth: 971.34
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -3.5988e+07 -5.1998e+05  7.6780e+05  1.7412e+06
   AREA_SQM              1.0003e+03  5.2758e+03  7.4740e+03  1.2301e+04
   AGE                  -1.3475e+05 -2.0813e+04 -8.6260e+03 -3.7784e+03
   PROX_CBD             -7.7047e+07 -2.3608e+05 -8.3599e+04  3.4646e+04
   PROX_CHILDCARE       -6.0097e+06 -3.3667e+05 -9.7426e+04  2.9007e+05
   PROX_ELDERLYCARE     -3.5001e+06 -1.5970e+05  3.1970e+04  1.9577e+05
   PROX_URA_GROWTH_AREA -3.0170e+06 -8.2013e+04  7.0749e+04  2.2612e+05
   PROX_MRT             -3.5282e+06 -6.5836e+05 -1.8833e+05  3.6922e+04
   PROX_PARK            -1.2062e+06 -2.1732e+05  3.5383e+04  4.1335e+05
   PROX_PRIMARY_SCH     -2.2695e+07 -1.7066e+05  4.8472e+04  5.1555e+05
   PROX_SHOPPING_MALL   -7.2585e+06 -1.6684e+05 -1.0517e+04  1.5923e+05
   PROX_BUS_STOP        -1.4676e+06 -4.5207e+04  3.7601e+05  1.1664e+06
   NO_Of_UNITS          -1.3170e+03 -2.4822e+02 -3.0846e+01  2.5496e+02
   FAMILY_FRIENDLY      -2.2749e+06 -1.1140e+05  7.6214e+03  1.6107e+05
   FREEHOLD             -9.2067e+06  3.8074e+04  1.5169e+05  3.7528e+05
                             Max.
   Intercept            112794435
   AREA_SQM                 21575
   AGE                     434203
   PROX_CBD               2704604
   PROX_CHILDCARE         1654086
   PROX_ELDERLYCARE      38867861
   PROX_URA_GROWTH_AREA  78515805
   PROX_MRT               3124325
   PROX_PARK             18122439
   PROX_PRIMARY_SCH       4637517
   PROX_SHOPPING_MALL     1529953
   PROX_BUS_STOP         11342209
   NO_Of_UNITS              12907
   FAMILY_FRIENDLY        1720745
   FREEHOLD               6073642
   ************************Diagnostic information*************************
   Number of data points: 1436
   Effective number of parameters (2trace(S) - trace(S'S)): 438.3807
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 997.6193
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 42263.61
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41632.36
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 42515.71
   Residual sum of squares: 2.534069e+14
   R-square value:  0.8909912
   Adjusted R-square value:  0.8430418

   ***********************************************************************
   Program stops at: 2024-10-14 19:55:47.015626
Note

Observations:

The AICc for this fixed bandwidth GWR model is 42263.61, which is lower than the global multiple linear regression model (AICc = 42967.1), indicating a better fit.

10.2 Building Adaptive Bandwidth GWR Model

10.2.1 Computing the Adaptive Bandwidth

To compute the adaptive bandwidth, we set adaptive = TRUE in the bw.gwr() function: 

bw.adaptive <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE + 
                        PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + 
                        PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + 
                        PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP + 
                        NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                      data = condo_resale.sp, 
                      approach = "CV", 
                      kernel = "gaussian", 
                      adaptive = TRUE, 
                      longlat = FALSE)
Adaptive bandwidth: 895 CV score: 7.952401e+14
Adaptive bandwidth: 561 CV score: 7.667364e+14
Adaptive bandwidth: 354 CV score: 6.953454e+14
Adaptive bandwidth: 226 CV score: 6.15223e+14
Adaptive bandwidth: 147 CV score: 5.674373e+14
Adaptive bandwidth: 98 CV score: 5.426745e+14
Adaptive bandwidth: 68 CV score: 5.168117e+14
Adaptive bandwidth: 49 CV score: 4.859631e+14
Adaptive bandwidth: 37 CV score: 4.646518e+14
Adaptive bandwidth: 30 CV score: 4.422088e+14
Adaptive bandwidth: 25 CV score: 4.430816e+14
Adaptive bandwidth: 32 CV score: 4.505602e+14
Adaptive bandwidth: 27 CV score: 4.462172e+14
Adaptive bandwidth: 30 CV score: 4.422088e+14
Note

Observations:

The result suggests using 30 data points for the adaptive bandwidth.

10.2.2 Constructing the Adaptive Bandwidth GWR Model

We can now calibrate the GWR model with the adaptive bandwidth: 

gwr.adaptive <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + 
                            PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + 
                            PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + 
                            PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP + 
                            NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                          data = condo_resale.sp, 
                          bw = bw.adaptive, 
                          kernel = 'gaussian', 
                          adaptive = TRUE, 
                          longlat = FALSE)
gwr.adaptive
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2024-10-14 19:55:56.103874
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA +
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL +
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
    data = condo_resale.sp, bw = bw.adaptive, kernel = "gaussian",
    adaptive = TRUE, longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max
-3470778  -298119   -23481   248917 12234210

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 **
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 **
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 **
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 **
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian
   Adaptive bandwidth: 30 (number of nearest neighbours)
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -1.3487e+08 -2.4669e+05  7.7928e+05  1.6194e+06
   AREA_SQM              3.3188e+03  5.6285e+03  7.7825e+03  1.2738e+04
   AGE                  -9.6746e+04 -2.9288e+04 -1.4043e+04 -5.6119e+03
   PROX_CBD             -2.5330e+06 -1.6256e+05 -7.7242e+04  2.6624e+03
   PROX_CHILDCARE       -1.2790e+06 -2.0175e+05  8.7158e+03  3.7778e+05
   PROX_ELDERLYCARE     -1.6212e+06 -9.2050e+04  6.1029e+04  2.8184e+05
   PROX_URA_GROWTH_AREA -7.2686e+06 -3.0350e+04  4.5869e+04  2.4613e+05
   PROX_MRT             -4.3781e+07 -6.7282e+05 -2.2115e+05 -7.4593e+04
   PROX_PARK            -2.9020e+06 -1.6782e+05  1.1601e+05  4.6572e+05
   PROX_PRIMARY_SCH     -8.6418e+05 -1.6627e+05 -7.7853e+03  4.3222e+05
   PROX_SHOPPING_MALL   -1.8272e+06 -1.3175e+05 -1.4049e+04  1.3799e+05
   PROX_BUS_STOP        -2.0579e+06 -7.1461e+04  4.1104e+05  1.2071e+06
   NO_Of_UNITS          -2.1993e+03 -2.3685e+02 -3.4699e+01  1.1657e+02
   FAMILY_FRIENDLY      -5.9879e+05 -5.0927e+04  2.6173e+04  2.2481e+05
   FREEHOLD             -1.6340e+05  4.0765e+04  1.9023e+05  3.7960e+05
                            Max.
   Intercept            18758355
   AREA_SQM                23064
   AGE                     13303
   PROX_CBD             11346650
   PROX_CHILDCARE        2892127
   PROX_ELDERLYCARE      2465671
   PROX_URA_GROWTH_AREA  7384059
   PROX_MRT              1186242
   PROX_PARK             2588497
   PROX_PRIMARY_SCH      3381462
   PROX_SHOPPING_MALL   38038564
   PROX_BUS_STOP        12081592
   NO_Of_UNITS              1010
   FAMILY_FRIENDLY       2072414
   FREEHOLD              1813995
   ************************Diagnostic information*************************
   Number of data points: 1436
   Effective number of parameters (2trace(S) - trace(S'S)): 350.3088
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 1085.691
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 41982.22
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41546.74
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 41914.08
   Residual sum of squares: 2.528227e+14
   R-square value:  0.8912425
   Adjusted R-square value:  0.8561185

   ***********************************************************************
   Program stops at: 2024-10-14 19:55:57.424832
Note

Observations:

The AICc for the adaptive bandwidth GWR model is 41982.22, which is even lower than the fixed bandwidth GWR model (AICc = 42263.61), indicating an improved model fit.

10.3 Visualizing GWR Output

The output from GWR includes fields for observed and predicted values, residuals, local R², condition numbers, and explanatory variable coefficients:

  • Condition Number: Identifies areas with strong local collinearity. Values over 30 may indicate unreliable results.
  • Local R²: Indicates how well the local model fits the observed values (ranges between 0.0 and 1.0). Mapping this can reveal where the model predicts well or poorly.
  • Predicted Values: The estimated y values (fitted) by GWR.
  • Residuals: Difference between observed and predicted values. Standardized residuals should have a mean of zero and a standard deviation of 1.
  • Coefficient Standard Error: Measures the reliability of coefficient estimates. Smaller standard errors relative to coefficients indicate greater confidence in estimates.

These fields are stored in the SDF object from the GWR output, which is a SpatialPointsDataFrame or SpatialPolygonsDataFrame.

10.4 Converting SDF to sf Data Frame

We first convert the SDF into an sf data frame for visualization: 

condo_resale.sf.adaptive <- st_as_sf(gwr.adaptive$SDF) %>%
  st_transform(crs = 3414)

condo_resale.sf.adaptive.svy21 <- st_transform(condo_resale.sf.adaptive, 3414)
condo_resale.sf.adaptive.svy21  
Simple feature collection with 1436 features and 51 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 14940.85 ymin: 24765.67 xmax: 43352.45 ymax: 48382.81
Projected CRS: SVY21 / Singapore TM
First 10 features:
    Intercept  AREA_SQM        AGE  PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE
1   2050011.7  9561.892  -9514.634 -120681.9      319266.92       -393417.79
2   1633128.2 16576.853 -58185.479 -149434.2      441102.18        325188.74
3   3433608.2 13091.861 -26707.386 -259397.8     -120116.82        535855.81
4    234358.9 20730.601 -93308.988 2426853.7      480825.28        314783.72
5   2285804.9  6722.836 -17608.018 -316835.5       90764.78       -137384.61
6  -3568877.4  6039.581 -26535.592  327306.1     -152531.19       -700392.85
7  -2874842.4 16843.575 -59166.727 -983577.2     -177810.50       -122384.02
8   2038086.0  6905.135 -17681.897 -285076.6       70259.40        -96012.78
9   1718478.4  9580.703 -14401.128  105803.4     -657698.02       -123276.00
10  3457054.0 14072.011 -31579.884 -234895.4       79961.45        548581.04
   PROX_URA_GROWTH_AREA    PROX_MRT  PROX_PARK PROX_PRIMARY_SCH
1            -159980.20  -299742.96 -172104.47        242668.03
2            -142290.39 -2510522.23  523379.72       1106830.66
3            -253621.21  -936853.28  209099.85        571462.33
4           -2679297.89 -2039479.50 -759153.26       3127477.21
5             303714.81   -44567.05  -10284.62         30413.56
6             -28051.25   733566.47 1511488.92        320878.23
7            1397676.38 -2745430.34  710114.74       1786570.95
8             269368.71   -14552.99   73533.34         53359.73
9            -361974.72  -476785.32 -132067.59        -40128.92
10           -150024.38 -1503835.53  574155.47        108996.67
   PROX_SHOPPING_MALL PROX_BUS_STOP  NO_Of_UNITS FAMILY_FRIENDLY  FREEHOLD
1          300881.390     1210615.4  104.8290640       -9075.370  303955.6
2          -87693.378     1843587.2 -288.3441183      310074.664  396221.3
3         -126732.712     1411924.9   -9.5532945        5949.746  168821.7
4          -29593.342     7225577.5 -161.3551620     1556178.531 1212515.6
5           -7490.586      677577.0   42.2659674       58986.951  328175.2
6          258583.881     1086012.6 -214.3671271      201992.641  471873.1
7         -384251.210     5094060.5   -0.9212521      359659.512  408871.9
8          -39634.902      735767.1   30.1741069       55602.506  347075.0
9          276718.757     2815772.4  675.1615559      -30453.297  503872.8
10        -454726.822     2123557.0  -21.3044311     -100935.586  213324.6
         y    yhat    residual CV_Score Stud_residual Intercept_SE AREA_SQM_SE
1  3000000 2886532   113468.16        0    0.38207013     516105.5    823.2860
2  3880000 3466801   413198.52        0    1.01433140     488083.5    825.2380
3  3325000 3616527  -291527.20        0   -0.83780678     963711.4    988.2240
4  4250000 5435482 -1185481.63        0   -2.84614670     444185.5    617.4007
5  1400000 1388166    11834.26        0    0.03404453    2119620.6   1376.2778
6  1320000 1516702  -196701.95        0   -0.72065801   28572883.7   2348.0091
7  3410000 3266881   143118.77        0    0.41291992     679546.6    893.5893
8  1420000 1431955   -11955.27        0   -0.03033109    2217773.1   1415.2604
9  2025000 1832799   192200.83        0    0.52018109     814281.8    943.8434
10 2550000 2223364   326635.53        0    1.10559735    2410252.0   1271.4073
      AGE_SE PROX_CBD_SE PROX_CHILDCARE_SE PROX_ELDERLYCARE_SE
1   5889.782    37411.22          319111.1           120633.34
2   6226.916    23615.06          299705.3            84546.69
3   6510.236    56103.77          349128.5           129687.07
4   6010.511   469337.41          304965.2           127150.69
5   8180.361   410644.47          698720.6           327371.55
6  14601.909  5272846.47         1141599.8          1653002.19
7   8970.629   346164.20          530101.1           148598.71
8   8661.309   438035.69          742532.8           399221.05
9  11791.208    89148.35          704630.7           329683.30
10  9941.980   173532.77          500976.2           281876.74
   PROX_URA_GROWTH_AREA_SE PROX_MRT_SE PROX_PARK_SE PROX_PRIMARY_SCH_SE
1                 56207.39    185181.3     205499.6            152400.7
2                 76956.50    281133.9     229358.7            165150.7
3                 95774.60    275483.7     314124.3            196662.6
4                470762.12    279877.1     227249.4            240878.9
5                474339.56    363830.0     364580.9            249087.7
6               5496627.21    730453.2    1741712.0            683265.5
7                371692.97    375511.9     297400.9            344602.8
8                517977.91    423155.4     440984.4            261251.2
9                153436.22    285325.4     304998.4            278258.5
10               239182.57    571355.7     599131.8            331284.8
   PROX_SHOPPING_MALL_SE PROX_BUS_STOP_SE NO_Of_UNITS_SE FAMILY_FRIENDLY_SE
1               109268.8         600668.6       218.1258           131474.7
2                98906.8         410222.1       208.9410           114989.1
3               119913.3         464156.7       210.9828           146607.2
4               177104.1         562810.8       361.7767           108726.6
5               301032.9         740922.4       299.5034           160663.7
6              2931208.6        1418333.3       602.5571           331727.0
7               249969.5         821236.4       532.1978           129241.2
8               351634.0         775038.4       338.6777           171895.1
9               289872.7         850095.5       439.9037           220223.4
10              265529.7         631399.2       259.0169           189125.5
   FREEHOLD_SE Intercept_TV AREA_SQM_TV     AGE_TV PROX_CBD_TV
1     115954.0    3.9720784   11.614302  -1.615447 -3.22582173
2     130110.0    3.3460017   20.087361  -9.344188 -6.32792021
3     141031.5    3.5629010   13.247868  -4.102368 -4.62353528
4     138239.1    0.5276150   33.577223 -15.524302  5.17080808
5     210641.1    1.0784029    4.884795  -2.152474 -0.77155660
6     374347.3   -0.1249043    2.572214  -1.817269  0.06207388
7     182216.9   -4.2305303   18.849348  -6.595605 -2.84136028
8     216649.4    0.9189786    4.879056  -2.041481 -0.65080678
9     220473.7    2.1104224   10.150733  -1.221345  1.18682383
10    206346.2    1.4343123   11.068059  -3.176418 -1.35360852
   PROX_CHILDCARE_TV PROX_ELDERLYCARE_TV PROX_URA_GROWTH_AREA_TV PROX_MRT_TV
1         1.00048819          -3.2612693            -2.846248368 -1.61864578
2         1.47178634           3.8462625            -1.848971738 -8.92998600
3        -0.34404755           4.1319138            -2.648105057 -3.40075727
4         1.57665606           2.4756745            -5.691404992 -7.28705261
5         0.12990138          -0.4196596             0.640289855 -0.12249416
6        -0.13361179          -0.4237096            -0.005103357  1.00426206
7        -0.33542751          -0.8235874             3.760298131 -7.31116712
8         0.09462126          -0.2405003             0.520038994 -0.03439159
9        -0.93339393          -0.3739225            -2.359121712 -1.67102293
10        0.15961128           1.9461735            -0.627237944 -2.63204802
   PROX_PARK_TV PROX_PRIMARY_SCH_TV PROX_SHOPPING_MALL_TV PROX_BUS_STOP_TV
1   -0.83749312           1.5923022            2.75358842        2.0154464
2    2.28192684           6.7019454           -0.88662640        4.4941192
3    0.66565951           2.9058009           -1.05686949        3.0419145
4   -3.34061770          12.9836105           -0.16709578       12.8383775
5   -0.02820944           0.1220998           -0.02488294        0.9145046
6    0.86781794           0.4696245            0.08821750        0.7656963
7    2.38773567           5.1844351           -1.53719231        6.2029165
8    0.16674816           0.2042469           -0.11271635        0.9493299
9   -0.43301073          -0.1442145            0.95462153        3.3123012
10   0.95831249           0.3290120           -1.71252687        3.3632555
   NO_Of_UNITS_TV FAMILY_FRIENDLY_TV FREEHOLD_TV  Local_R2
1     0.480589953        -0.06902748    2.621347 0.8846744
2    -1.380026395         2.69655779    3.045280 0.8899773
3    -0.045279967         0.04058290    1.197050 0.8947007
4    -0.446007570        14.31276425    8.771149 0.9073605
5     0.141120178         0.36714544    1.557983 0.9510057
6    -0.355762335         0.60891234    1.260522 0.9247586
7    -0.001731033         2.78285441    2.243875 0.8310458
8     0.089093858         0.32346758    1.602012 0.9463936
9     1.534793921        -0.13828365    2.285410 0.8380365
10   -0.082251138        -0.53369623    1.033819 0.9080753
                    geometry
1  POINT (22085.12 29951.54)
2   POINT (25656.84 34546.2)
3   POINT (23963.99 32890.8)
4  POINT (27044.28 32319.77)
5  POINT (41042.56 33743.64)
6   POINT (39717.04 32943.1)
7   POINT (28419.1 33513.37)
8  POINT (40763.57 33879.61)
9  POINT (23595.63 28884.78)
10 POINT (24586.56 33194.31)
gwr.adaptive.output <- as.data.frame(gwr.adaptive$SDF)
condo_resale.sf.adaptive <- cbind(condo_resale.res.sf, as.matrix(gwr.adaptive.output))

We can check the contents using glimpse and summary: 

glimpse(condo_resale.sf.adaptive)
Rows: 1,436
Columns: 77
$ POSTCODE                <dbl> 118635, 288420, 267833, 258380, 467169, 466472…
$ SELLING_PRICE           <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ AREA_SQM                <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 1…
$ AGE                     <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22,…
$ PROX_CBD                <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783…
$ PROX_CHILDCARE          <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543…
$ PROX_ELDERLYCARE        <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.…
$ PROX_URA_GROWTH_AREA    <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.4106…
$ PROX_HAWKER_MARKET      <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969…
$ PROX_KINDERGARTEN       <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076…
$ PROX_MRT                <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.…
$ PROX_PARK               <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.…
$ PROX_PRIMARY_SCH        <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.…
$ PROX_TOP_PRIMARY_SCH    <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.…
$ PROX_SHOPPING_MALL      <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.…
$ PROX_SUPERMARKET        <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.…
$ PROX_BUS_STOP           <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340…
$ NO_Of_UNITS             <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34…
$ FAMILY_FRIENDLY         <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0…
$ FREEHOLD                <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR          <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ LOG_SELLING_PRICE       <dbl> 14.91412, 15.17135, 15.01698, 15.26243, 14.151…
$ MLR_RES                 <dbl> -1489099.55, 415494.57, 194129.69, 1088992.71,…
$ Intercept               <dbl> 2050011.67, 1633128.24, 3433608.17, 234358.91,…
$ AREA_SQM.1              <dbl> 9561.892, 16576.853, 13091.861, 20730.601, 672…
$ AGE.1                   <dbl> -9514.634, -58185.479, -26707.386, -93308.988,…
$ PROX_CBD.1              <dbl> -120681.94, -149434.22, -259397.77, 2426853.66…
$ PROX_CHILDCARE.1        <dbl> 319266.925, 441102.177, -120116.816, 480825.28…
$ PROX_ELDERLYCARE.1      <dbl> -393417.795, 325188.741, 535855.806, 314783.72…
$ PROX_URA_GROWTH_AREA.1  <dbl> -159980.203, -142290.389, -253621.206, -267929…
$ PROX_MRT.1              <dbl> -299742.96, -2510522.23, -936853.28, -2039479.…
$ PROX_PARK.1             <dbl> -172104.47, 523379.72, 209099.85, -759153.26, …
$ PROX_PRIMARY_SCH.1      <dbl> 242668.03, 1106830.66, 571462.33, 3127477.21, …
$ PROX_SHOPPING_MALL.1    <dbl> 300881.390, -87693.378, -126732.712, -29593.34…
$ PROX_BUS_STOP.1         <dbl> 1210615.44, 1843587.22, 1411924.90, 7225577.51…
$ NO_Of_UNITS.1           <dbl> 104.8290640, -288.3441183, -9.5532945, -161.35…
$ FAMILY_FRIENDLY.1       <dbl> -9075.370, 310074.664, 5949.746, 1556178.531, …
$ FREEHOLD.1              <dbl> 303955.61, 396221.27, 168821.75, 1212515.58, 3…
$ y                       <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ yhat                    <dbl> 2886531.8, 3466801.5, 3616527.2, 5435481.6, 13…
$ residual                <dbl> 113468.16, 413198.52, -291527.20, -1185481.63,…
$ CV_Score                <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ Stud_residual           <dbl> 0.38207013, 1.01433140, -0.83780678, -2.846146…
$ Intercept_SE            <dbl> 516105.5, 488083.5, 963711.4, 444185.5, 211962…
$ AREA_SQM_SE             <dbl> 823.2860, 825.2380, 988.2240, 617.4007, 1376.2…
$ AGE_SE                  <dbl> 5889.782, 6226.916, 6510.236, 6010.511, 8180.3…
$ PROX_CBD_SE             <dbl> 37411.22, 23615.06, 56103.77, 469337.41, 41064…
$ PROX_CHILDCARE_SE       <dbl> 319111.1, 299705.3, 349128.5, 304965.2, 698720…
$ PROX_ELDERLYCARE_SE     <dbl> 120633.34, 84546.69, 129687.07, 127150.69, 327…
$ PROX_URA_GROWTH_AREA_SE <dbl> 56207.39, 76956.50, 95774.60, 470762.12, 47433…
$ PROX_MRT_SE             <dbl> 185181.3, 281133.9, 275483.7, 279877.1, 363830…
$ PROX_PARK_SE            <dbl> 205499.6, 229358.7, 314124.3, 227249.4, 364580…
$ PROX_PRIMARY_SCH_SE     <dbl> 152400.7, 165150.7, 196662.6, 240878.9, 249087…
$ PROX_SHOPPING_MALL_SE   <dbl> 109268.8, 98906.8, 119913.3, 177104.1, 301032.…
$ PROX_BUS_STOP_SE        <dbl> 600668.6, 410222.1, 464156.7, 562810.8, 740922…
$ NO_Of_UNITS_SE          <dbl> 218.1258, 208.9410, 210.9828, 361.7767, 299.50…
$ FAMILY_FRIENDLY_SE      <dbl> 131474.73, 114989.07, 146607.22, 108726.62, 16…
$ FREEHOLD_SE             <dbl> 115954.0, 130110.0, 141031.5, 138239.1, 210641…
$ Intercept_TV            <dbl> 3.9720784, 3.3460017, 3.5629010, 0.5276150, 1.…
$ AREA_SQM_TV             <dbl> 11.614302, 20.087361, 13.247868, 33.577223, 4.…
$ AGE_TV                  <dbl> -1.6154474, -9.3441881, -4.1023685, -15.524301…
$ PROX_CBD_TV             <dbl> -3.22582173, -6.32792021, -4.62353528, 5.17080…
$ PROX_CHILDCARE_TV       <dbl> 1.000488185, 1.471786337, -0.344047555, 1.5766…
$ PROX_ELDERLYCARE_TV     <dbl> -3.26126929, 3.84626245, 4.13191383, 2.4756745…
$ PROX_URA_GROWTH_AREA_TV <dbl> -2.846248368, -1.848971738, -2.648105057, -5.6…
$ PROX_MRT_TV             <dbl> -1.61864578, -8.92998600, -3.40075727, -7.2870…
$ PROX_PARK_TV            <dbl> -0.83749312, 2.28192684, 0.66565951, -3.340617…
$ PROX_PRIMARY_SCH_TV     <dbl> 1.59230221, 6.70194543, 2.90580089, 12.9836104…
$ PROX_SHOPPING_MALL_TV   <dbl> 2.753588422, -0.886626400, -1.056869486, -0.16…
$ PROX_BUS_STOP_TV        <dbl> 2.0154464, 4.4941192, 3.0419145, 12.8383775, 0…
$ NO_Of_UNITS_TV          <dbl> 0.480589953, -1.380026395, -0.045279967, -0.44…
$ FAMILY_FRIENDLY_TV      <dbl> -0.06902748, 2.69655779, 0.04058290, 14.312764…
$ FREEHOLD_TV             <dbl> 2.6213469, 3.0452799, 1.1970499, 8.7711485, 1.…
$ Local_R2                <dbl> 0.8846744, 0.8899773, 0.8947007, 0.9073605, 0.…
$ coords.x1               <dbl> 22085.12, 25656.84, 23963.99, 27044.28, 41042.…
$ coords.x2               <dbl> 29951.54, 34546.20, 32890.80, 32319.77, 33743.…
$ geometry                <POINT [m]> POINT (22085.12 29951.54), POINT (25656.…
summary(gwr.adaptive$SDF$yhat)
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
  171347  1102001  1385528  1751842  1982307 13887901

10.5 Visualizing Local R2

The following code creates an interactive map showing local R2 values: 

tmap_mode("view")

tm_shape(mpsz_svy21) +
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "Local_R2", border.col = "gray60", border.lwd = 1) +
  tm_view(set.zoom.limits = c(11, 14))
+−
Local_R2
0.65 to 0.70
0.70 to 0.75
0.75 to 0.80
0.80 to 0.85
0.85 to 0.90
0.90 to 0.95
0.95 to 1.00
Leaflet | Tiles © Esri — Esri, DeLorme, NAVTEQ
tmap_mode("plot")

10.6 Visualizing Coefficient Estimates

We can visualize the standard errors and t-values of the AREA_SQM coefficient using the code below: 

tmap_mode("view")

AREA_SQM_SE <- tm_shape(mpsz_svy21) +
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "AREA_SQM_SE", border.col = "gray60", border.lwd = 1) +
  tm_view(set.zoom.limits = c(11, 14))

AREA_SQM_TV <- tm_shape(mpsz_svy21) +
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "AREA_SQM_TV", border.col = "gray60", border.lwd = 1) +
  tm_view(set.zoom.limits = c(11, 14))

tmap_arrange(AREA_SQM_SE, AREA_SQM_TV, asp = 1, ncol = 2, sync = TRUE)
+−
AREA_SQM_SE
0 to 500
500 to 1,000
1,000 to 1,500
1,500 to 2,000
2,000 to 2,500
2,500 to 3,000
3,000 to 3,500
3,500 to 4,000
Leaflet | Tiles © Esri — Esri, DeLorme, NAVTEQ
+−
AREA_SQM_TV
0 to 10
10 to 20
20 to 30
30 to 40
40 to 50
Leaflet | Tiles © Esri — Esri, DeLorme, NAVTEQ
tmap_mode("plot")

10.6.1 By URA Planning Region 

tm_shape(mpsz_svy21[mpsz_svy21$REGION_N == "CENTRAL REGION", ]) +
  tm_polygons() +
tm_shape(condo_resale.sf.adaptive) + 
  tm_bubbles(col = "Local_R2", size = 0.15, border.col = "gray60", border.lwd = 1)