1.0 Overview
Before we start, we need to know: just what is geographically weighted regression? Well, 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).
Our goal for this hands-on exercise is to build hedonic pricing models by using GWR methods, with the dependent variable being the resale prices of condominium in 2015 and the independent variables divided into either structural and locational.
2.0 Setup
2.1 Packages Used
The R package we’ll be introducing today is the GWmodel package for geospatial statistical modelling!
A short explanation of our GWmodel package: it provides a collection of localised spatial statistical methods:
- GW summary statistics
- GW principal components analysis
- GW discriminant analysis
- various forms of GW regression (basic and robust forms, the latter of which is outlier-resistant)
It’s used to calibrate a geographically weighted family of modes. 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.
In addition, we’ll be using the packages from our previous exercises:
- sf: used for importing, managing, and processing geospatial data
- tmap: used for plotting thematic maps, such as choropleth and bubble maps
- tidyverse: used for importing, wrangling and visualising data (and other data science tasks!)
- spdep: used to create spatial weights matrix objects, global and local spatial autocorrelation statistics and related calculations (e.g. spatially lag attributes)
- coorplot + ggpubr for multivariate data visualisation & analysis
- olsrr for building least squares regression models
In addition, the following tidyverse packages will be used (for attribute data handling):
- readr for importing delimited files (.csv)
- dplyr for wrangling and transforming data
- ggplot2 for visualising data
Show code
packages = c('olsrr', 'corrplot', 'ggpubr', 'sf', 'tmap', 'tidyverse', 'spdep', 'GWmodel')
for (p in packages){
if(!require(p, character.only = T)){
install.packages(p)
}
library(p,character.only = T)
}
2.2 Data Used
The datasets used for this exercise are:
MP14_SUBZONE_WEB_PL, a polygon feature data in ESRI shapefile format from data.gov.sg, providing information of URA 2014 Master Plan Planning Subzone boundary data, in SVY21 projected coordinates systemcondo_resale_2015: a .csv of condo resales in 2015
2.3 Geospatial Data Importing + Wrangling
Show code
# output: simple features object
mpsz = st_read(dsn = "data/geospatial", layer = "MP14_SUBZONE_WEB_PL")
Reading layer `MP14_SUBZONE_WEB_PL' from data source
`C:\IS415\IS415_msty\_posts\2021-10-17-hands-on-exercise-9\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: SVY21As we can see, our mpsz does not have any EPSG information. Let’s address that, and use ESPG 3414 which is the SVY21 projected coordinates system specific to Singapore:
Show code
mpsz_svy21 <- st_transform(mpsz, 3414)
# check the newly-transformed sf for the correct ESPG code 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]]We can also reveal the extent of mpsz_svy21 with st_bbox():
Show code
st_bbox(mpsz_svy21) #view extent
xmin ymin xmax ymax
2667.538 15748.721 56396.440 50256.334 2.4 Aspatial Data Importing + Wrangling
Importing the aspatial data
Show code
# output: tibble dataframe
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,~
$ LONGITUDE <dbl> 103.7802, 103.8123, 103.7971, 103.8247,~
$ POSTCODE <dbl> 118635, 288420, 267833, 258380, 467169,~
$ SELLING_PRICE <dbl> 3000000, 3880000, 3325000, 4250000, 140~
$ AREA_SQM <dbl> 309, 290, 248, 127, 145, 139, 218, 141,~
$ AGE <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, ~
$ PROX_CBD <dbl> 7.941259, 6.609797, 6.898000, 4.038861,~
$ PROX_CHILDCARE <dbl> 0.16597932, 0.28027246, 0.42922669, 0.3~
$ PROX_ELDERLYCARE <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910~
$ PROX_URA_GROWTH_AREA <dbl> 6.618741, 7.505109, 6.463887, 4.906512,~
$ PROX_HAWKER_MARKET <dbl> 1.76542207, 0.54507614, 0.37789301, 1.6~
$ PROX_KINDERGARTEN <dbl> 0.05835552, 0.61592412, 0.14120309, 0.3~
$ PROX_MRT <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910~
$ PROX_PARK <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832~
$ PROX_PRIMARY_SCH <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546~
$ PROX_TOP_PRIMARY_SCH <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006~
$ PROX_SHOPPING_MALL <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525~
$ PROX_SUPERMARKET <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162~
$ PROX_BUS_STOP <dbl> 0.10336166, 0.28673408, 0.28504777, 0.2~
$ NO_Of_UNITS <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32,~
$ FAMILY_FRIENDLY <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, ~
$ FREEHOLD <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, ~
$ LEASEHOLD_99YR <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ~Show code
head(condo_resale$LONGITUDE) #see the data in XCOORD column
[1] 103.7802 103.8123 103.7971 103.8247 103.9505 103.9386Show code
head(condo_resale$LATITUDE) #see the data in YCOORD column
[1] 1.287145 1.328698 1.313727 1.308563 1.321437 1.314198Show code
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
Min. :0.05451 Min. :0.2145 Min. :0.05182
1st Qu.:0.61254 1st Qu.:3.1643 1st Qu.:0.55245
Median :0.94179 Median :4.6186 Median :0.90842
Mean :1.05351 Mean :4.5981 Mean :1.27987
3rd Qu.:1.35122 3rd Qu.:5.7550 3rd Qu.:1.68578
Max. :3.94916 Max. :9.1554 Max. :5.37435
PROX_KINDERGARTEN PROX_MRT PROX_PARK
Min. :0.004927 Min. :0.05278 Min. :0.02906
1st Qu.:0.276345 1st Qu.:0.34646 1st Qu.:0.26211
Median :0.413385 Median :0.57430 Median :0.39926
Mean :0.458903 Mean :0.67316 Mean :0.49802
3rd Qu.:0.578474 3rd Qu.:0.84844 3rd Qu.:0.65592
Max. :2.229045 Max. :3.48037 Max. :2.16105
PROX_PRIMARY_SCH PROX_TOP_PRIMARY_SCH PROX_SHOPPING_MALL
Min. :0.07711 Min. :0.07711 Min. :0.0000
1st Qu.:0.44024 1st Qu.:1.34451 1st Qu.:0.5258
Median :0.63505 Median :1.88213 Median :0.9357
Mean :0.75471 Mean :2.27347 Mean :1.0455
3rd Qu.:0.95104 3rd Qu.:2.90954 3rd Qu.:1.3994
Max. :3.92899 Max. :6.74819 Max. :3.4774
PROX_SUPERMARKET PROX_BUS_STOP NO_Of_UNITS
Min. :0.0000 Min. :0.001595 Min. : 18.0
1st Qu.:0.3695 1st Qu.:0.098356 1st Qu.: 188.8
Median :0.5687 Median :0.151710 Median : 360.0
Mean :0.6141 Mean :0.193974 Mean : 409.2
3rd Qu.:0.7862 3rd Qu.:0.220466 3rd Qu.: 590.0
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 Converting aspatial data frame into a sf object
Show code
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 x 22
POSTCODE SELLING_PRICE AREA_SQM AGE PROX_CBD PROX_CHILDCARE
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 118635 3000000 309 30 7.94 0.166
2 288420 3880000 290 32 6.61 0.280
3 267833 3325000 248 33 6.90 0.429
4 258380 4250000 127 7 4.04 0.395
5 467169 1400000 145 28 11.8 0.119
6 466472 1320000 139 22 10.3 0.125
# ... with 16 more variables: PROX_ELDERLYCARE <dbl>,
# 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]>3.0 Exploratory Data Analysis
3.1 EDA using statistical graphics
Now, let’s move on to EDA! We can start by plotting the distribution of selling price:
Show code
ggplot(data=condo_resale.sf, aes(x=`SELLING_PRICE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
The figure above reveals 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 - which we’ll derive with the mutate() function of the dplyr package and store in a new variable, like so:
Show code
condo_resale.sf <- condo_resale.sf %>%
mutate(`LOG_SELLING_PRICE` = log(SELLING_PRICE))
Now, let’s plot the LOG_SELLING_PRICE:
Show code
ggplot(data=condo_resale.sf, aes(x=`LOG_SELLING_PRICE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
Did you notice? Comapre the pre-transformation and post-transformation distribution: the post-transformation one is relatively less skewed!
3.2 Multiple Histogram Plots distribution of variables
Now, we’ll draw multiple small histograms (also known as trellis plot) using the ggarrange() function of the ggpubr package.
We’ll create 12 histograms, one for each variable of interest - and then organise them into 3 col x 4 rows with ggarrange():
Show code
AREA_SQM <- ggplot(data=condo_resale.sf, aes(x= `AREA_SQM`)) +
geom_histogram(bins=20, color="black", fill="light blue")
AGE <- ggplot(data=condo_resale.sf, aes(x= `AGE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_CBD <- ggplot(data=condo_resale.sf, aes(x= `PROX_CBD`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_CHILDCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_CHILDCARE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_ELDERLYCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_ELDERLYCARE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_URA_GROWTH_AREA <- ggplot(data=condo_resale.sf, aes(x= `PROX_URA_GROWTH_AREA`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_HAWKER_MARKET <- ggplot(data=condo_resale.sf, aes(x= `PROX_HAWKER_MARKET`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_KINDERGARTEN <- ggplot(data=condo_resale.sf, aes(x= `PROX_KINDERGARTEN`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_MRT <- ggplot(data=condo_resale.sf, aes(x= `PROX_MRT`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_PARK <- ggplot(data=condo_resale.sf, aes(x= `PROX_PARK`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_PRIMARY_SCH <- ggplot(data=condo_resale.sf, aes(x= `PROX_PRIMARY_SCH`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_TOP_PRIMARY_SCH <- ggplot(data=condo_resale.sf, aes(x= `PROX_TOP_PRIMARY_SCH`)) +
geom_histogram(bins=20, color="black", fill="light blue")
ggarrange(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, ncol = 3, nrow = 4)
3.3 Drawing Statistical Point Map
Let’s create an interactive map that reveals the geospatial distribution condominium resale prices in Singapore:
Show 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") +
# sets minimum zoom level to 11, sets maximum zoom level to 14
tm_view(set.zoom.limits = c(11,14))
Before moving on to the next section, we’ll revert the R display into plot mode for future visualisations.
Show code
# return tmap mode to plot for future visualisations
tmap_mode("plot")
4.0 Hedonic Pricing Modelling in R
Now to the meat of today’s exercise: buildling the hedonic pricing models for condominium resale units using lm() function which is present in base R.
4.1 Simple Linear Regression Method
Firstly, let’s build a simple linear regression model with SELLING_PRICE as our dependent variable and AREA_SQM as our independent variable.
Show code
# output: class 'lm' object or class c('mlm', 'lm') multi-response
condo.slr <- lm(formula=SELLING_PRICE ~ AREA_SQM, data = condo_resale.sf)
We can use the summary() and anova() (short for analysis of variance) functions to obtain and print their related results, like so:
Show code
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-16The output report reveals that the SELLING_PRICE can be explained by using the formula:
\[ y = -258121.1 + 14719x1 \]
From our summary output above, we see that our multiple R-squared value is 0.4518. What does this signify? It means that the simple regression model built is able to explain about 45% of the resale prices.
Since our p-value is much smaller than 0.0001, we will reject the null hypothesis that the mean is a good estimator of SELLING_PRICE. This allows us to infer that the simple linear regression model above is a good estimator of SELLING_PRICE.
The Coefficients: section of the report reveals that the p-values of both the estimates of the Intercept and ARA_SQM are smaller than 0.001. Within this context, the null hypothesis of the B0 and B1 are equal to 0 will be rejected. As such, we can infer that B0 and B1 are good parameter estimates.
To visualise the best fit curve on a scatterplot, we can incorporate lm() as a method function in ggplot’s geometry, like so:
Show code
ggplot(data=condo_resale.sf,
aes(x=`AREA_SQM`, y=`SELLING_PRICE`)) +
geom_point() +
geom_smooth(method = lm)
Hmm…. as we can see, there are few statistical outliers with relatively high selling prices.
4.2 Multiple Linear Regression Method
4.2.1 Visualising the relationships of the independent variables
Just like in the previous Hands-on Exercise 08, before building a multiple regression model, we need to perform correlation analysis so as to ensure that the cluster variables are not highly correlated. “Why?” Well - if you have two variables that are highly correlated (aka collinear) - the concept they represent is effectively similar, which compromises the quality of our model as it becomes skewed towards those collinear variables.
A correlation matrix is commonly used to visualise the relationships between the independent variables: in this section, the corrplot package will be used. Now, let’s try plotting a scatterplot matrix of the relationship between the independent variables in condo_resale:
Show code
corrplot(cor(condo_resale[, 5:23]), diag = FALSE, order = "AOE",
tl.pos = "td", tl.cex = 0.5, method = "number", type = "upper")
As a matter of fact, the order of the matrix is critical for mining the hidden structures and patterns, which is why we sometimes need to reorder the matrix. There are four methods in corrplot: “AOE”, “FPC”, “hclust”, “alphabet”. You can read more about it in this article. We used the AOE order above, which orders the variables by the angular order of eigenvectors.
From our scatterplot matrix, we can clearly tell that Freehold is highly correlated to LEASE_99YEAR. As such, it is wiser to only include either of them in the subsequent model building. Here, we’ll exclude LEASE_99YEAR.
4.2.2 Building a hedonic pricing model using multiple linear regression method
Show code
condo.mlr <- 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)
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-16As we can clearly see, not all the independent variables are statistically significant, and said variables should be removed, like so:
Show code
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 755957.289
R-Squared 0.651 Coef. Var 43.168
Adj. R-Squared 0.647 MSE 571471422208.591
Pred R-Squared 0.638 MAE 414819.628
------------------------------------------------------------------------
RMSE: Root Mean Square Error
MSE: Mean Square Error
MAE: Mean Absolute Error
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 571471422208.591
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
-----------------------------------------------------------------------------------------------------------------4.2.3 Checking for multicolinearity
New package introduction! Let’s look at olsrr, a fantastic R package specially programmed for performing OLS regression that provides a collection of useful methods for building better multiple linear regression models, such as:
- comprehensive regression output
- residual diagnostics
- measures of influence
- heteroskedasticity tests
- collinearity diagnostics
- model fit assessment
- variable contribution assessment
- variable selection procedures
We’ll test for signs of multicollinearity with the ols_vif_tol() function of our olsrr package. In general, if the VIF value is less than 5, then there is usually no sign/possibility of correlations.
Show code
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 1.442759Since the VIF of the independent variables is less than 10, we can safely conclude that there are no signs of multicollinearity amongst the independent variables. 😄
4.2.4 Test for Non-Linearity
In addition to testing for multicollinearity, we also need to test the assumption that linearity and additivity of the relationship between dependent and independent variables when performing multiple linear regression.
We’ll perform the linearity assumption test with the ols_plot_resid_fit() function:
Show code
ols_plot_resid_fit(condo.mlr1)
As we can observe, most of the data points are scattered around the 0 line, hence we can safely conclude that the relationships between the dependent variable and independent variables are linear.
4.2.5 Test for Normality Assumption
Lastly, we’ll perform the normality assumption test with the ols_plot_resid_hist() function:
Show code
ols_plot_resid_hist(condo.mlr1)
As we can see, the residual of the multiple linear regression model (i.e. condo.mlr1) resembles a normal distribution.
This isn’t the only way to test for it: there are more formal statistical test methods such as ols_test_normality():
Show code
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
-----------------------------------------------The summary table above reveals that the p-values of the four tests are significantly smaller than the alpha value of 0.05. Hence, we will reject the null hypothesis that the residual does NOT resemble normal distribution.
4.2.6 Testing for Spatial Autocorrelation
Since hedonic model we aim to build is using geographically referenced attributes, it is important for us to visualize the residual of the hedonic pricing model. In order to the perform spatial autocorrelation test, we’l need to convert condo_resale.sf simple into a SpatialPointsDataFrame.
Show code
mlr.output <- as.data.frame(condo.mlr1$residuals)
condo_resale.res.sf <- cbind(condo_resale.sf,
condo.mlr1$residuals) %>%
rename(`MLR_RES` = `condo.mlr1.residuals`)
# convert from a simple features object to a SpatialPointsDataFrame
# due to requirements of using 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, ... Now, let’s display the distribution of the residuals on an interactive map:
Show code
tmap_mode("view")
tm_shape(mpsz_svy21)+
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))
Show code
#switch back to 'plot' before continuing
tmap_mode("plot")
There ARE indeed signs of spatial autocorrelation. To proof that our observation is indeed true, the Moran’s I test will be performed.
4.2.7 Moran I’s Test
Firstly, let’s compute the distance-based weight matrix by using the dnearneigh() function of the spdep package.
Show code
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
Link number distribution:
1 3 5 7 9 10 11 12 13 14 15 16 17 18 19 20 21
3 3 9 4 3 15 10 19 17 45 19 5 14 29 19 6 35
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
45 18 47 16 43 22 26 21 11 9 23 22 13 16 25 21 37
39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
16 18 8 21 4 12 8 36 18 14 14 43 11 12 8 13 12
56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
13 4 5 6 12 11 20 29 33 15 20 10 14 15 15 11 16
73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89
12 10 8 19 12 14 9 8 4 13 11 6 4 9 4 4 4
90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106
6 2 16 9 4 5 9 3 9 4 2 1 2 1 1 1 5
107 108 109 110 112 116 125
9 2 1 3 1 1 1
3 least connected regions:
193 194 277 with 1 link
1 most connected region:
285 with 125 linksWe’ll convert the output neighbours lists (i.e. nb) into a spatial weights with the nb2listw() function:
Show code
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
Link number distribution:
1 3 5 7 9 10 11 12 13 14 15 16 17 18 19 20 21
3 3 9 4 3 15 10 19 17 45 19 5 14 29 19 6 35
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
45 18 47 16 43 22 26 21 11 9 23 22 13 16 25 21 37
39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
16 18 8 21 4 12 8 36 18 14 14 43 11 12 8 13 12
56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
13 4 5 6 12 11 20 29 33 15 20 10 14 15 15 11 16
73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89
12 10 8 19 12 14 9 8 4 13 11 6 4 9 4 4 4
90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106
6 2 16 9 4 5 9 3 9 4 2 1 2 1 1 1 5
107 108 109 110 112 116 125
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 5798.341Lastly, to perform Moran’s I test for residual spatial autocorrelation, we’ll use the lm.morantest() function:
Show code
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 The Global Moran’s I test for residual spatial autocorrelation shows that its p-value is ~0.00000000000000022 which is less than the alpha value of 0.05. Hence, we will reject the null hypothesis that the residuals are randomly distributed.
Since the Observed Global Moran I = 0.1424418 which is greater than 0, we can infer that the residuals resemble cluster distribution.
5.0 Building Hedonic Pricing Models using GWmodel
5.1 Building Fixed Bandwidth GWR Model
Computing fixed bandwith
Now, time to determine the optimal fixed bandwidth to use in the model using the bw.gwr() function of the GWModel package. To indicate that we want to compute the fixed bandwidth, we’ll set the adapative argument to FALSE.
There are two possible approaches can be used to determine the stopping rule, the (a) CV cross-validation approach and the (b) AIC corrected (AICc) approach, which we’ll define in the approach argument.
Show code
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: 1.378294e+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: 971.3408 CV score: 4.721292e+14
Fixed bandwidth: 971.3403 CV score: 4.721292e+14
Fixed bandwidth: 971.3406 CV score: 4.721292e+14
Fixed bandwidth: 971.3404 CV score: 4.721292e+14
Fixed bandwidth: 971.3405 CV score: 4.721292e+14
Fixed bandwidth: 971.3405 CV score: 4.721292e+14 The result shows that the recommended bandwidth is 971.4443 metres.
Quiz: Do you know why it is in metres?
Answer: Remember our CRS? The projection coordinates system is SVY21 which is in metres - thus our results also being in metres!
GWModel method - fixed bandwith
Let’s calibrate the gwr model using fixed bandwidth and gaussian kernel:
Show 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)
We should display the model output too:
Show code
gwr.fixed
***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2021-12-03 20:10:58
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: 971.3405
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
Intercept -3.5988e+07 -5.1998e+05 7.6780e+05
AREA_SQM 1.0003e+03 5.2758e+03 7.4740e+03
AGE -1.3475e+05 -2.0813e+04 -8.6260e+03
PROX_CBD -7.7047e+07 -2.3608e+05 -8.3600e+04
PROX_CHILDCARE -6.0097e+06 -3.3667e+05 -9.7425e+04
PROX_ELDERLYCARE -3.5000e+06 -1.5970e+05 3.1971e+04
PROX_URA_GROWTH_AREA -3.0170e+06 -8.2013e+04 7.0749e+04
PROX_MRT -3.5282e+06 -6.5836e+05 -1.8833e+05
PROX_PARK -1.2062e+06 -2.1732e+05 3.5383e+04
PROX_PRIMARY_SCH -2.2695e+07 -1.7066e+05 4.8472e+04
PROX_SHOPPING_MALL -7.2585e+06 -1.6684e+05 -1.0517e+04
PROX_BUS_STOP -1.4676e+06 -4.5207e+04 3.7601e+05
NO_Of_UNITS -1.3170e+03 -2.4822e+02 -3.0846e+01
FAMILY_FRIENDLY -2.2749e+06 -1.1140e+05 7.6214e+03
FREEHOLD -9.2067e+06 3.8073e+04 1.5169e+05
3rd Qu. Max.
Intercept 1.7412e+06 112793548
AREA_SQM 1.2301e+04 21575
AGE -3.7784e+03 434201
PROX_CBD 3.4646e+04 2704596
PROX_CHILDCARE 2.9007e+05 1654087
PROX_ELDERLYCARE 1.9577e+05 38867814
PROX_URA_GROWTH_AREA 2.2612e+05 78515730
PROX_MRT 3.6922e+04 3124316
PROX_PARK 4.1335e+05 18122425
PROX_PRIMARY_SCH 5.1555e+05 4637503
PROX_SHOPPING_MALL 1.5923e+05 1529952
PROX_BUS_STOP 1.1664e+06 11342182
NO_Of_UNITS 2.5496e+02 12907
FAMILY_FRIENDLY 1.6107e+05 1720744
FREEHOLD 3.7528e+05 6073636
************************Diagnostic information*************************
Number of data points: 1436
Effective number of parameters (2trace(S) - trace(S'S)): 438.3804
Effective degrees of freedom (n-2trace(S) + trace(S'S)): 997.6196
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: 2.53407e+14
R-square value: 0.8909912
Adjusted R-square value: 0.8430417
***********************************************************************
Program stops at: 2021-12-03 20:10:59 Hooray! From the report, we can see that the adjusted r-square of the gwr is 0.8430418 which is significantly better than the global multiple linear regression model of 0.6472.
5.2 Building Adaptive Bandwidth GWR Model
Computing the adaptive bandwidth
Similar to the earlier section, we will first use bw.ger() to determine the recommended data point to use. Remember that now we’re using adapative bandswidth, the adaptive argument should change to TRUE.
Show code
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 The result shows that 30 is the recommended data points to be used.
Constructing the adaptive bandwidth gwr model
Now, we can calibrate the gwr-based hedonic pricing model by using adaptive bandwidth and gaussian kernel:
Show code
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)
Let’s display the model output:
Show code
gwr.adaptive
***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2021-12-03 20:11:08
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
Intercept -1.3487e+08 -2.4669e+05 7.7928e+05
AREA_SQM 3.3188e+03 5.6285e+03 7.7825e+03
AGE -9.6746e+04 -2.9288e+04 -1.4043e+04
PROX_CBD -2.5330e+06 -1.6256e+05 -7.7242e+04
PROX_CHILDCARE -1.2790e+06 -2.0175e+05 8.7158e+03
PROX_ELDERLYCARE -1.6212e+06 -9.2050e+04 6.1029e+04
PROX_URA_GROWTH_AREA -7.2686e+06 -3.0350e+04 4.5869e+04
PROX_MRT -4.3781e+07 -6.7282e+05 -2.2115e+05
PROX_PARK -2.9020e+06 -1.6782e+05 1.1601e+05
PROX_PRIMARY_SCH -8.6418e+05 -1.6627e+05 -7.7853e+03
PROX_SHOPPING_MALL -1.8272e+06 -1.3175e+05 -1.4049e+04
PROX_BUS_STOP -2.0579e+06 -7.1461e+04 4.1104e+05
NO_Of_UNITS -2.1993e+03 -2.3685e+02 -3.4699e+01
FAMILY_FRIENDLY -5.9879e+05 -5.0927e+04 2.6173e+04
FREEHOLD -1.6340e+05 4.0765e+04 1.9023e+05
3rd Qu. Max.
Intercept 1.6194e+06 18758355
AREA_SQM 1.2738e+04 23064
AGE -5.6119e+03 13303
PROX_CBD 2.6624e+03 11346650
PROX_CHILDCARE 3.7778e+05 2892127
PROX_ELDERLYCARE 2.8184e+05 2465671
PROX_URA_GROWTH_AREA 2.4613e+05 7384059
PROX_MRT -7.4593e+04 1186242
PROX_PARK 4.6572e+05 2588497
PROX_PRIMARY_SCH 4.3222e+05 3381462
PROX_SHOPPING_MALL 1.3799e+05 38038564
PROX_BUS_STOP 1.2071e+06 12081592
NO_Of_UNITS 1.1657e+02 1010
FAMILY_FRIENDLY 2.2481e+05 2072414
FREEHOLD 3.7960e+05 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: 2021-12-03 20:11:11 From this, we can see that the adjusted r-square of the gwr is 0.8561185 which is significantly better than the global multiple linear regression model of 0.6472!
6.0 Visualising GWR Output
In addition to regression residuals, the output feature class table includes fields for observed and predicted y values, condition number (cond), Local R2, residuals, and explanatory variable coefficients and standard errors. Let’s go through them one-by-one:
Condition Number: this diagnostic evaluates local collinearity. In the presence of strong local collinearity, results become unstable. Results associated with condition numbers larger than 30, may be unreliable.
Local R2: these values range between 0.0 and 1.0 and indicate how well the local regression model fits observed y values.
- Very low values indicate the local model is performing poorly.
- Mapping the Local R2 values to see where GWR predicts well and where it predicts poorly may provide clues about important variables that may be missing from the regression model.
- how to interpret the local R2: for example, the Central Region might see a lower value of 75% compared to the Eastern Region of 95% - this just means that the model for the Central Region can only explain 75% of variation compared to the model for the Eastern Region which can explain 95%.
Predicted: these are the estimated (or fitted) y values 3. computed by GWR.
Residuals: to obtain the residual values, the fitted y values are subtracted from the observed y values.
- Standardized residuals have a mean of zero and a standard deviation of 1.
- A cold-to-hot rendered map of standardized residuals can be produce by using these values.
Coefficient Standard Error: these values measure the reliability of each coefficient estimate.
- Confidence in those estimates are higher when standard errors are small in relation to the actual coefficient values.
- Large standard errors may indicate problems with local collinearity.
- Confidence in those estimates are higher when standard errors are small in relation to the actual coefficient values.
They are all stored in a SpatialPointsDataFrame or SpatialPolygonsDataFrame object integrated with fit.points, GWR coefficient estimates, y value, predicted values, coefficient standard errors and t-values in its “data” slot in an object called SDF of the output list.
6.1 Converting SDF into sf data.frame
To visualise the fields in SDF, we need to first covert it into sf data.frame, like so:
Show code
condo_resale.sf.adaptive <- st_as_sf(gwr.adaptive$SDF) %>%
st_transform(crs=3414)
Show code
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
1 2050011.7 9561.892 -9514.634 -120681.9 319266.92
2 1633128.2 16576.853 -58185.479 -149434.2 441102.18
3 3433608.2 13091.861 -26707.386 -259397.8 -120116.82
4 234358.9 20730.601 -93308.988 2426853.7 480825.28
5 2285804.9 6722.836 -17608.018 -316835.5 90764.78
6 -3568877.4 6039.581 -26535.592 327306.1 -152531.19
7 -2874842.4 16843.575 -59166.727 -983577.2 -177810.50
8 2038086.0 6905.135 -17681.897 -285076.6 70259.40
9 1718478.4 9580.703 -14401.128 105803.4 -657698.02
10 3457054.0 14072.011 -31579.884 -234895.4 79961.45
PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK
1 -393417.79 -159980.20 -299742.96 -172104.47
2 325188.74 -142290.39 -2510522.23 523379.72
3 535855.81 -253621.21 -936853.28 209099.85
4 314783.72 -2679297.89 -2039479.50 -759153.26
5 -137384.61 303714.81 -44567.05 -10284.62
6 -700392.85 -28051.25 733566.47 1511488.92
7 -122384.02 1397676.38 -2745430.34 710114.74
8 -96012.78 269368.71 -14552.99 73533.34
9 -123276.00 -361974.72 -476785.32 -132067.59
10 548581.04 -150024.38 -1503835.53 574155.47
PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS
1 242668.03 300881.390 1210615.4 104.8290640
2 1106830.66 -87693.378 1843587.2 -288.3441183
3 571462.33 -126732.712 1411924.9 -9.5532945
4 3127477.21 -29593.342 7225577.5 -161.3551620
5 30413.56 -7490.586 677577.0 42.2659674
6 320878.23 258583.881 1086012.6 -214.3671271
7 1786570.95 -384251.210 5094060.5 -0.9212521
8 53359.73 -39634.902 735767.1 30.1741069
9 -40128.92 276718.757 2815772.4 675.1615559
10 108996.67 -454726.822 2123557.0 -21.3044311
FAMILY_FRIENDLY FREEHOLD y yhat residual CV_Score
1 -9075.370 303955.6 3000000 2886532 113468.16 0
2 310074.664 396221.3 3880000 3466801 413198.52 0
3 5949.746 168821.7 3325000 3616527 -291527.20 0
4 1556178.531 1212515.6 4250000 5435482 -1185481.63 0
5 58986.951 328175.2 1400000 1388166 11834.26 0
6 201992.641 471873.1 1320000 1516702 -196701.94 0
7 359659.512 408871.9 3410000 3266881 143118.77 0
8 55602.506 347075.0 1420000 1431955 -11955.27 0
9 -30453.297 503872.8 2025000 1832799 192200.83 0
10 -100935.586 213324.6 2550000 2223364 326635.53 0
Stud_residual Intercept_SE AREA_SQM_SE AGE_SE PROX_CBD_SE
1 0.38207013 516105.5 823.2860 5889.782 37411.22
2 1.01433140 488083.5 825.2380 6226.916 23615.06
3 -0.83780678 963711.4 988.2240 6510.236 56103.77
4 -2.84614670 444185.5 617.4007 6010.511 469337.41
5 0.03404453 2119620.6 1376.2778 8180.361 410644.47
6 -0.72065800 28572883.7 2348.0091 14601.909 5272846.47
7 0.41291992 679546.6 893.5893 8970.629 346164.20
8 -0.03033109 2217773.1 1415.2604 8661.309 438035.69
9 0.52018109 814281.8 943.8434 11791.208 89148.35
10 1.10559735 2410252.0 1271.4073 9941.980 173532.77
PROX_CHILDCARE_SE PROX_ELDERLYCARE_SE PROX_URA_GROWTH_AREA_SE
1 319111.1 120633.34 56207.39
2 299705.3 84546.69 76956.50
3 349128.5 129687.07 95774.60
4 304965.2 127150.69 470762.12
5 698720.6 327371.55 474339.56
6 1141599.8 1653002.19 5496627.21
7 530101.1 148598.71 371692.97
8 742532.8 399221.05 517977.91
9 704630.7 329683.30 153436.22
10 500976.2 281876.74 239182.57
PROX_MRT_SE PROX_PARK_SE PROX_PRIMARY_SCH_SE PROX_SHOPPING_MALL_SE
1 185181.3 205499.6 152400.7 109268.8
2 281133.9 229358.7 165150.7 98906.8
3 275483.7 314124.3 196662.6 119913.3
4 279877.1 227249.4 240878.9 177104.1
5 363830.0 364580.9 249087.7 301032.9
6 730453.2 1741712.0 683265.5 2931208.6
7 375511.9 297400.9 344602.8 249969.5
8 423155.4 440984.4 261251.2 351634.0
9 285325.4 304998.4 278258.5 289872.7
10 571355.7 599131.8 331284.8 265529.7
PROX_BUS_STOP_SE NO_Of_UNITS_SE FAMILY_FRIENDLY_SE FREEHOLD_SE
1 600668.6 218.1258 131474.7 115954.0
2 410222.1 208.9410 114989.1 130110.0
3 464156.7 210.9828 146607.2 141031.5
4 562810.8 361.7767 108726.6 138239.1
5 740922.4 299.5034 160663.7 210641.1
6 1418333.3 602.5571 331727.0 374347.3
7 821236.4 532.1978 129241.2 182216.9
8 775038.4 338.6777 171895.1 216649.4
9 850095.5 439.9037 220223.4 220473.7
10 631399.2 259.0169 189125.5 206346.2
Intercept_TV AREA_SQM_TV AGE_TV PROX_CBD_TV PROX_CHILDCARE_TV
1 3.9720784 11.614302 -1.615447 -3.22582173 1.00048819
2 3.3460017 20.087361 -9.344188 -6.32792021 1.47178634
3 3.5629010 13.247868 -4.102368 -4.62353528 -0.34404755
4 0.5276150 33.577223 -15.524302 5.17080808 1.57665606
5 1.0784029 4.884795 -2.152474 -0.77155660 0.12990138
6 -0.1249043 2.572214 -1.817269 0.06207388 -0.13361179
7 -4.2305303 18.849348 -6.595605 -2.84136028 -0.33542751
8 0.9189786 4.879056 -2.041481 -0.65080678 0.09462126
9 2.1104224 10.150733 -1.221345 1.18682383 -0.93339393
10 1.4343123 11.068059 -3.176418 -1.35360852 0.15961128
PROX_ELDERLYCARE_TV PROX_URA_GROWTH_AREA_TV PROX_MRT_TV
1 -3.2612693 -2.846248368 -1.61864578
2 3.8462625 -1.848971738 -8.92998600
3 4.1319138 -2.648105057 -3.40075727
4 2.4756745 -5.691404992 -7.28705261
5 -0.4196596 0.640289855 -0.12249416
6 -0.4237096 -0.005103357 1.00426206
7 -0.8235874 3.760298131 -7.31116712
8 -0.2405003 0.520038994 -0.03439159
9 -0.3739225 -2.359121712 -1.67102293
10 1.9461735 -0.627237944 -2.63204802
PROX_PARK_TV PROX_PRIMARY_SCH_TV PROX_SHOPPING_MALL_TV
1 -0.83749312 1.5923022 2.75358842
2 2.28192684 6.7019454 -0.88662640
3 0.66565951 2.9058009 -1.05686949
4 -3.34061770 12.9836105 -0.16709578
5 -0.02820944 0.1220998 -0.02488294
6 0.86781794 0.4696245 0.08821750
7 2.38773567 5.1844351 -1.53719231
8 0.16674816 0.2042469 -0.11271635
9 -0.43301073 -0.1442145 0.95462153
10 0.95831249 0.3290120 -1.71252687
PROX_BUS_STOP_TV NO_Of_UNITS_TV FAMILY_FRIENDLY_TV FREEHOLD_TV
1 2.0154464 0.480589953 -0.06902748 2.621347
2 4.4941192 -1.380026395 2.69655779 3.045280
3 3.0419145 -0.045279967 0.04058290 1.197050
4 12.8383775 -0.446007570 14.31276425 8.771149
5 0.9145046 0.141120178 0.36714544 1.557983
6 0.7656963 -0.355762335 0.60891234 1.260522
7 6.2029165 -0.001731033 2.78285441 2.243875
8 0.9493299 0.089093858 0.32346758 1.602012
9 3.3123012 1.534793921 -0.13828365 2.285410
10 3.3632555 -0.082251138 -0.53369623 1.033819
Local_R2 geometry
1 0.8846744 POINT (22085.12 29951.54)
2 0.8899773 POINT (25656.84 34546.2)
3 0.8947007 POINT (23963.99 32890.8)
4 0.9073605 POINT (27044.28 32319.77)
5 0.9510057 POINT (41042.56 33743.64)
6 0.9247586 POINT (39717.04 32943.1)
7 0.8310458 POINT (28419.1 33513.37)
8 0.9463936 POINT (40763.57 33879.61)
9 0.8380365 POINT (23595.63 28884.78)
10 0.9080753 POINT (24586.56 33194.31)Show code
gwr.adaptive.output <- as.data.frame(gwr.adaptive$SDF)
condo_resale.sf.adaptive <- cbind(condo_resale.res.sf, as.matrix(gwr.adaptive.output))
Show code
glimpse(condo_resale.sf.adaptive)
Rows: 1,436
Columns: 77
$ POSTCODE <dbl> 118635, 288420, 267833, 258380, 4671~
$ SELLING_PRICE <dbl> 3000000, 3880000, 3325000, 4250000, ~
$ AREA_SQM <dbl> 309, 290, 248, 127, 145, 139, 218, 1~
$ AGE <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 3~
$ PROX_CBD <dbl> 7.941259, 6.609797, 6.898000, 4.0388~
$ PROX_CHILDCARE <dbl> 0.16597932, 0.28027246, 0.42922669, ~
$ PROX_ELDERLYCARE <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9~
$ PROX_URA_GROWTH_AREA <dbl> 6.618741, 7.505109, 6.463887, 4.9065~
$ PROX_HAWKER_MARKET <dbl> 1.76542207, 0.54507614, 0.37789301, ~
$ PROX_KINDERGARTEN <dbl> 0.05835552, 0.61592412, 0.14120309, ~
$ PROX_MRT <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6~
$ PROX_PARK <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9~
$ PROX_PRIMARY_SCH <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4~
$ PROX_TOP_PRIMARY_SCH <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3~
$ PROX_SHOPPING_MALL <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3~
$ PROX_SUPERMARKET <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4~
$ PROX_BUS_STOP <dbl> 0.10336166, 0.28673408, 0.28504777, ~
$ NO_Of_UNITS <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, ~
$ FAMILY_FRIENDLY <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, ~
$ FREEHOLD <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, ~
$ LEASEHOLD_99YR <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ~
$ LOG_SELLING_PRICE <dbl> 14.91412, 15.17135, 15.01698, 15.262~
$ MLR_RES <dbl> -1489099.55, 415494.57, 194129.69, 1~
$ Intercept <dbl> 2050011.67, 1633128.24, 3433608.17, ~
$ AREA_SQM.1 <dbl> 9561.892, 16576.853, 13091.861, 2073~
$ AGE.1 <dbl> -9514.634, -58185.479, -26707.386, -~
$ PROX_CBD.1 <dbl> -120681.94, -149434.22, -259397.77, ~
$ PROX_CHILDCARE.1 <dbl> 319266.925, 441102.177, -120116.816,~
$ PROX_ELDERLYCARE.1 <dbl> -393417.79, 325188.74, 535855.81, 31~
$ PROX_URA_GROWTH_AREA.1 <dbl> -159980.203, -142290.389, -253621.20~
$ PROX_MRT.1 <dbl> -299742.96, -2510522.23, -936853.28,~
$ PROX_PARK.1 <dbl> -172104.47, 523379.72, 209099.85, -7~
$ PROX_PRIMARY_SCH.1 <dbl> 242668.03, 1106830.66, 571462.33, 31~
$ PROX_SHOPPING_MALL.1 <dbl> 300881.390, -87693.378, -126732.712,~
$ PROX_BUS_STOP.1 <dbl> 1210615.44, 1843587.22, 1411924.90, ~
$ NO_Of_UNITS.1 <dbl> 104.8290640, -288.3441183, -9.553294~
$ FAMILY_FRIENDLY.1 <dbl> -9075.370, 310074.664, 5949.746, 155~
$ FREEHOLD.1 <dbl> 303955.61, 396221.27, 168821.75, 121~
$ y <dbl> 3000000, 3880000, 3325000, 4250000, ~
$ yhat <dbl> 2886531.8, 3466801.5, 3616527.2, 543~
$ residual <dbl> 113468.16, 413198.52, -291527.20, -1~
$ CV_Score <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ~
$ Stud_residual <dbl> 0.38207013, 1.01433140, -0.83780678,~
$ Intercept_SE <dbl> 516105.5, 488083.5, 963711.4, 444185~
$ AREA_SQM_SE <dbl> 823.2860, 825.2380, 988.2240, 617.40~
$ AGE_SE <dbl> 5889.782, 6226.916, 6510.236, 6010.5~
$ PROX_CBD_SE <dbl> 37411.22, 23615.06, 56103.77, 469337~
$ PROX_CHILDCARE_SE <dbl> 319111.1, 299705.3, 349128.5, 304965~
$ PROX_ELDERLYCARE_SE <dbl> 120633.34, 84546.69, 129687.07, 1271~
$ PROX_URA_GROWTH_AREA_SE <dbl> 56207.39, 76956.50, 95774.60, 470762~
$ PROX_MRT_SE <dbl> 185181.3, 281133.9, 275483.7, 279877~
$ PROX_PARK_SE <dbl> 205499.6, 229358.7, 314124.3, 227249~
$ PROX_PRIMARY_SCH_SE <dbl> 152400.7, 165150.7, 196662.6, 240878~
$ PROX_SHOPPING_MALL_SE <dbl> 109268.8, 98906.8, 119913.3, 177104.~
$ PROX_BUS_STOP_SE <dbl> 600668.6, 410222.1, 464156.7, 562810~
$ NO_Of_UNITS_SE <dbl> 218.1258, 208.9410, 210.9828, 361.77~
$ FAMILY_FRIENDLY_SE <dbl> 131474.7, 114989.1, 146607.2, 108726~
$ FREEHOLD_SE <dbl> 115954.0, 130110.0, 141031.5, 138239~
$ Intercept_TV <dbl> 3.9720784, 3.3460017, 3.5629010, 0.5~
$ AREA_SQM_TV <dbl> 11.614302, 20.087361, 13.247868, 33.~
$ AGE_TV <dbl> -1.6154474, -9.3441881, -4.1023685, ~
$ PROX_CBD_TV <dbl> -3.22582173, -6.32792021, -4.6235352~
$ PROX_CHILDCARE_TV <dbl> 1.000488185, 1.471786337, -0.3440475~
$ PROX_ELDERLYCARE_TV <dbl> -3.2612693, 3.8462625, 4.1319138, 2.~
$ PROX_URA_GROWTH_AREA_TV <dbl> -2.846248368, -1.848971738, -2.64810~
$ PROX_MRT_TV <dbl> -1.61864578, -8.92998600, -3.4007572~
$ PROX_PARK_TV <dbl> -0.83749312, 2.28192684, 0.66565951,~
$ PROX_PRIMARY_SCH_TV <dbl> 1.59230221, 6.70194543, 2.90580089, ~
$ PROX_SHOPPING_MALL_TV <dbl> 2.75358842, -0.88662640, -1.05686949~
$ PROX_BUS_STOP_TV <dbl> 2.0154464, 4.4941192, 3.0419145, 12.~
$ NO_Of_UNITS_TV <dbl> 0.480589953, -1.380026395, -0.045279~
$ FAMILY_FRIENDLY_TV <dbl> -0.06902748, 2.69655779, 0.04058290,~
$ FREEHOLD_TV <dbl> 2.6213469, 3.0452799, 1.1970499, 8.7~
$ Local_R2 <dbl> 0.8846744, 0.8899773, 0.8947007, 0.9~
$ coords.x1 <dbl> 22085.12, 25656.84, 23963.99, 27044.~
$ coords.x2 <dbl> 29951.54, 34546.20, 32890.80, 32319.~
$ geometry <POINT [m]> POINT (22085.12 29951.54), POI~Show code
summary(gwr.adaptive$SDF$yhat)
Min. 1st Qu. Median Mean 3rd Qu. Max.
171347 1102001 1385528 1751842 1982307 13887901 6.2 Visualising local R2
Now, we’ll create an interactive point symbol map:
Show code
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))
Show code
tmap_mode("plot")
6.3 By URA Planning Region
Show code
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)
7.0 Ending Notes
With that, we’ve learned how to calibrate geographically weighted regression models, from choosnig the regression model to building the model to visualisation. Tune in next week for more geospatial analytics tips 💪