What is HGWR model?

Hierarchical and Geographically Weighted Regression, shorted for HGWR, is a spatial modelling method designed for data of spatial hierarchical structures. Just as its name implies, this is a combination of Hierarchical Linear Model (HLM, also known as Multilevel Model, Raudenbush 1993) and Geographically Weighted Regression (GWR, Brunsdon, Fotheringham, and Charlton 1996). In this model, spatial effects are divided into three types: global fixed, local fixed and random. Formally, it is expressed as \[ y = G\gamma + X\beta + Z\mu + \epsilon \] with \(y\) the dependent variable, \(G\) the group level independent variables, \(\gamma\) the local fixed effects, \(X\) also the group level independent variables, \(\beta\) the global fixed effects, \(Z\) the individual level independent variables, \(\mu\) the random effects, \(\epsilon\) the individual errors.

Why HGWR model?

As we know, hierarchical structure is commonly existing in spatial data. For example, cities can be grouped by provinces or other higher-level administrative district they belong to; house prices may share some factors from the block; and students in one school have different access to education resources with those in another school. When dealing with this type of data, we usually choose HLM to address the within-group homogeneity and the between-group heterogeneity. And there are usually two types of variables: group-level variables and sample-level variables. The formal ones are used to describe the properties of groups (such as the provinces, blocks and schools); the latter ones are observations of individual samples (such as the cities, houses and students). The effect of some sample-level variables are similar in all groups, thus they are modelled with fixed coefficients (effects). For others, they are modelled individually, i.e., with random effects.

However, for group-level variables, they can only be modelled with fixed effects. For spatial data, we would encounter some problems. According to the Tobler’s first law of Geography “Everything is related to everything else, but near things are more related than distant things” (Tobler 1970). If the model is calibrated with equally weighted samples, spatial heterogeneity would be overlooked (Fotheringham, Brunsdon, and Charlton 2002). Thus, it requires us to distinguish “local fixed effects” from “global fixed effects” to discover spatial heterogeneity in group-level variables.

But why not GWR or Multiscale GWR (Fotheringham, Yang, and Kang 2017, LuBrunsdon–2017) Because when dealing with data of hierarchical structures, GWR is problematic (Hu et al. 2022). We know that GWR calibrate a model with unique coefficients on each sample by borrowing data from its neighbours. And it uses a parameter “bandwidth” to control how many neighbours are included. If samples are not hierarchically structured, everything works well. However, just imagine a situation like Figure 1. For the two samples of red color and blue color, we take the same number of their neighbours, but actually the spatial extents are not the same. In extreme cases, spatial extends of some samples could be too small to hold more than one or two location, but some are large enough. This would lead to the failure of bandwidth optimization and reduce the reliability of the optimized bandwidth.

As shown in this video, bandwidths have inequal spatial scale for two samples (represented by cubes). Both the samples represented by large red cubes and large blue cubes take 41 neighbour samples to calibrate GWR models. For the red one, neighbours on 8 nearest locations are taken. But the figure for the blue one is only 6. This situation means estimated coefficients are more smoothed for the red samples. In other words, estimations for the blue samples are much local.

To solve the problems mentioned above, we need to use HGWR model. It is able to modelling spatial hierarchical structure and spatial heterogeneity simultaneously. Examples below can show that it works well for spatial hierarchical data.

Installation

Package hgwrr is available on CRAN. Simply type the following codes to install it.

install.packages("hgwrr")

Or download latest released source package and run the following command to install this package.

R CMD INSTALL hgwrr_0.2-0.tar.gz

Note that RTools is required on Windows.

Usage

We are going to show the usage of hgwrr package with a simulated data.

First, we need to load this package in an R session.

library(hgwrr)

Then we can calibrate an HGWR model via hgwr() function.

hgwr(
  formula, data, local.fixed, coords, bw,
  alpha = 0.01, eps_iter = 1e-06, eps_gradient = 1e-06, max_iters = 1e+06,
  max_retries = 10, ml_type = HGWR_ML_TYPE_D_ONLY, verbose = 0
)

The first five arguments are mandatory.

• formula accepts a formula object in R. Its format follows lme4 package. As there are two types of effects: fixed effects and random effects, we use the following format to specify both of them:

  dependent ~ fixed1 + fixed2 + (random1 + random2 | group)

• data accepts a DataFrame object in R. All variables specified in formula are extracted from data. In this stage, Spatial*DataFrame is not supported.

• local.fixed accepts a list of character specifying which fixed effects are local. For example, if fixed1 needs to be local fixed, then set local.fixed to c("fixed1").

• coords accepts a matrix of 2 columns. Each row is the longitude and latitude of each group.

• bw accepts a integer or numeric number to specify the bandwidth used in geographically weighted process. Currently it can only be adaptive bandwidth.

Other arguments are optional which is used to control the backfitting maximum likelihood algorithm. On most occassions the default values are fine. If the default values cause some problems and you want change some of them, please check the documentation of function hgwr() for more infomation.

Example: A Small Simulated Data Set

This example is used to show the usage of this package and to test whether it works. We don’t care about how good the fitness of this model is with this data set.

A data set “multisampling” is provided with this package,

data(multisampling)
head(multisampling$data)

head(multisampling$coords)

where y is the dependent variable, g1 and g2 are two group-level variables, z1 and x1 are two sample-level variables, group are the labels of the groups they belong to, and U, V are longitude and latitude coordinate values of all groups.

We regards g1 and g2 have local fixed effects, x1 have global fixed effects and z1 have random effects. Then we can calibrate an HGWR model with like this

ms_hgwr <- hgwr(
  formula = y ~ g1 + g2 + x1 + (z1 | group),
  data = multisampling$data,
  local.fixed = c("g1", "g2"),
  coords = multisampling$coords,
  bw = 10
)
ms_hgwr

## Hierarchical and geographically weighted regression model 
## ========================================================= 
## Formula: y ~ g1 + g2 + x1 + (z1 | group) 
##  Method: Back-fitting and Maximum likelihood 
##    Data: multisampling$data 
## 
## Global Fixed Effects 
## ------------------- 
##     Intercept        x1 
##  -4538.623106  0.964530 
## 
## Local Fixed Effects 
## ------------------- 
##  Coefficient          Min  1st Quartile       Median  3rd Quartile          Max 
##    Intercept  4540.169533   4540.258191  4540.536449   4540.825440  4541.111718 
##           g1     5.919438      6.366335     7.282849      7.877878     9.261846 
##           g2    -0.935852      0.052836     0.978779      1.494663     2.178014 
## 
## Random Effects 
## -------------- 
##    Groups       Name    Std.Dev.      Corr 
##     group  Intercept  778.316317           
##                   z1   49.552648  0.063573 
##  Residual               1.900196           
## 
## Other Information 
## ----------------- 
## Number of Obs: 484 
##        Groups: group , 16

The output of the model shows estimations of global fixed effects, summary of those of local fixed effects. Also there are the standard deviations of random effects and correlation coefficients between them.

Then we can have a look on the coefficient estimations.

coef(ms_hgwr)

With ggplot2 or other packages, we can create some figures.

library(ggplot2)
ms_hgwr_coef <- as.data.frame(cbind(multisampling$coords, coef(ms_hgwr)))
ggplot(ms_hgwr_coef, aes(x = U, y = V)) +
  geom_point(aes(color = g1)) +
  coord_fixed() + theme_bw()

We can also convert it to spatial data and use tmap to visualize.

library(sf)
library(tmap)
ms_hgwr_coef_sf <- st_as_sf(ms_hgwr_coef,
                            coords = names(multisampling$coords),
                            crs = 27700)
tm_shape(ms_hgwr_coef_sf) + tm_dots(col = c("g1", "g2"), size = 0.5)

And we can also fetch the fitted and residuals.

head(data.frame(
  real = multisampling$data$y,
  fitted = fitted(ms_hgwr),
  residuals = residuals(ms_hgwr)
))

The summary() function will give some statistical information about this model.

summary(ms_hgwr)

## Hierarchical and geographically weighted regression model 
## ========================================================= 
## Formula: y ~ g1 + g2 + x1 + (z1 | group) 
##  Method: Back-fitting and Maximum likelihood 
##    Data: multisampling$data 
## 
## Diagnostics 
## ----------- 
##  Rsquared 
##  0.647278 
## 
## Scaled residuals 
## ---------------- 
##        Min         1Q     Median        3Q       Max 
##  -5.457712  -1.200908  -0.066369  1.264277  5.293872 
## 
## Other Information 
## ----------------- 
## Number of Obs: 484 
##        Groups: group , 16

On the current stage, only pesudo \(R^2\) is available. In the future, more diagnostic information will be provided in this package.

Example: Large Scale Simulated Data

In the former example, there are only 484 observations and 16 groups. They are not adequate enough to get precises estimations. Here, we are going to use a large scale simulated data set to show the performance of HGWR model. As the true value of coefficients are already known (stored in variable msl_beta), closeness between estimated and true values is an appliable performance metric.

The data set is provided here. Its structure is similar to data multisampling.

data(multisampling.large)
head(multisampling.large$data)

head(multisampling.large$coords)

##            V1       V2
## [1,] 2940.897 2851.943
## [2,] 3002.659 3157.717
## [3,] 2848.345 2904.326
## [4,] 2863.735 2907.999
## [5,] 3117.849 2800.236
## [6,] 2906.585 2972.770

msl_beta <- multisampling.large$beta
head(msl_beta)

In this data, we also regards g1 and g2 as two group-level variables, z1 and x1 as two sample-level variables, group as the labels of the groups they belong to, and U, V as longitude and latitude coordinate values of all groups. Then we calibrate an HGWR model.

As the data is large (13862 observations), it may take some time to get results.

msl_hgwr <- hgwr(
  formula = y ~ g1 + g2 + x1 + (z1 | group),
  data = multisampling.large$data,
  local.fixed = c("g1", "g2"),
  coords = multisampling.large$coords,
  bw = 32, kernel = "bisquared"
)
msl_hgwr

## Hierarchical and geographically weighted regression model 
## ========================================================= 
## Formula: y ~ g1 + g2 + x1 + (z1 | group) 
##  Method: Back-fitting and Maximum likelihood 
##    Data: multisampling.large$data 
## 
## Global Fixed Effects 
## ------------------- 
##  Intercept        x1 
##   3.142118  1.010699 
## 
## Local Fixed Effects 
## ------------------- 
##  Coefficient         Min  1st Quartile     Median  3rd Quartile        Max 
##    Intercept   -2.099433     -1.501617  -1.330840     -1.160251  -0.637086 
##           g1   -3.495954      0.264949   3.452187      6.340162  11.766479 
##           g2  -11.003983     -2.416328  -0.251891      1.704997   8.757358 
## 
## Random Effects 
## -------------- 
##    Groups       Name  Std.Dev.       Corr 
##     group  Intercept  1.743385            
##                   z1  1.841777  -0.178200 
##  Residual             1.991062            
## 
## Other Information 
## ----------------- 
## Number of Obs: 13862 
##        Groups: group , 200

library(dplyr)
library(purrr)
coef(msl_hgwr) %>%
  select(Intercept, g1, g2, z1) %>% 
  list(label = names(.), Truth = msl_beta[names(.)], Estimated = .) %>%
  pmap_df(data.frame) %>%
  mutate(label = factor(label, levels = c("Intercept", "g1", "g2", "z1"))) %>%
  ggplot(aes(x = Truth, y = Estimated)) + geom_point() +
    coord_fixed() + scale_y_continuous(limits = c(-25, 25)) +
    facet_grid(cols = vars(label)) +
    theme_bw()

msl_hgwr_err <- coef(msl_hgwr) %>%
  select(Intercept, g1, g2, z1, x1) %>% {
    as.data.frame(rbind(
      MAE = map2_dbl(msl_beta, ., ~ mean(abs(.x - .y))),
      RMSE = map2_dbl(msl_beta, ., ~ sqrt(mean((.x - .y)^2)))
    ))
  }
msl_hgwr_err

### GWR model
library(GWmodel)
msl_gwr_data <- multisampling.large$data
coordinates(msl_gwr_data) <- with(multisampling.large, coords[data$group, ])
##### Get optimized bandwidth via golden minimization algorithm.
msl_gwr_bw <- bw.gwr(
  formula = y ~ g1 + g2 + x1 + z1, data = msl_gwr_data, approach = "AIC",
  adaptive = TRUE, parallel.method = "omp"
)

## Take a cup of tea and have a break, it will take a few minutes.
##           -----A kind suggestion from GWmodel development group
## Adaptive bandwidth (number of nearest neighbours): 8574 AICc value: 64882.39 
## Adaptive bandwidth (number of nearest neighbours): 5307 AICc value: 64454.28 
## Adaptive bandwidth (number of nearest neighbours): 3286 AICc value: 64047.9 
## Adaptive bandwidth (number of nearest neighbours): 2039 AICc value: 63620.4 
## Adaptive bandwidth (number of nearest neighbours): 1266 AICc value: 62999.43 
## Adaptive bandwidth (number of nearest neighbours): 790 AICc value: 62173.75 
## Adaptive bandwidth (number of nearest neighbours): 494 AICc value: 61359.34 
## Adaptive bandwidth (number of nearest neighbours): 313 AICc value: 60434.33 
## Adaptive bandwidth (number of nearest neighbours): 199 AICc value: 1071693 
## Adaptive bandwidth (number of nearest neighbours): 381 AICc value: 60780.34 
## Adaptive bandwidth (number of nearest neighbours): 268 AICc value: 1013623 
## Adaptive bandwidth (number of nearest neighbours): 338 AICc value: 60513.49 
## Adaptive bandwidth (number of nearest neighbours): 295 AICc value: 60292.35 
## Adaptive bandwidth (number of nearest neighbours): 286 AICc value: 60250.93 
## Adaptive bandwidth (number of nearest neighbours): 278 AICc value: 1013623 
## Adaptive bandwidth (number of nearest neighbours): 288 AICc value: 60255.59 
## Adaptive bandwidth (number of nearest neighbours): 281 AICc value: 60239.58 
## Adaptive bandwidth (number of nearest neighbours): 282 AICc value: 60239.58

##### Calibrate GWR model with optimized bandwidth.
msl_gwr <- gwr.basic(
  formula = y ~ g1 + g2 + x1 + z1, data = msl_gwr_data, bw = msl_gwr_bw,
  adaptive = TRUE, parallel.method = "omp"
)
#### Get coefficient estimations.
#### As samples in one group have equal estimations,
#### we use mean value of each group to represent them.
msl_gwr_coef <- cbind(msl_gwr$SDF@data, group = msl_gwr_data$group) %>%
  select(Intercept, g1, g2, x1, z1, group) %>%
  group_by(group) %>%
  summarise_all(mean)
##### Calculate RMSE and MAE of estimations.
msl_gwr_err <- msl_gwr_coef %>%
  select(Intercept, g1, g2, z1, x1) %>% {
    as.data.frame(rbind(
      MAE = map2_dbl(msl_beta, ., ~ mean(abs(.x - .y))),
      RMSE = map2_dbl(msl_beta, ., ~ sqrt(mean((.x - .y)^2)))
    ))
  }
msl_gwr_err

library(lme4)
msl_hlm <- lmer(
  formula = y ~ g1 + g2 + x1 + (z1 | group),
  data = multisampling.large$data
)
msl_hlm_coef <- coef(msl_hlm)$group
colnames(msl_hlm_coef)[which(colnames(msl_hlm_coef) == "(Intercept)")] <- "Intercept"
msl_hlm_err <- msl_hlm_coef %>%
  select(Intercept, g1, g2, z1, x1) %>% {
    as.data.frame(rbind(
      MAE = map2_dbl(msl_beta, ., ~ mean(abs(.x - .y))),
      RMSE = map2_dbl(msl_beta, ., ~ sqrt(mean((.x - .y)^2)))
    ))
  }
msl_hlm_err

list(HGWR = msl_hgwr_err, GWR = msl_gwr_err, HLM = msl_hlm_err) %>%
  map2_dfr(., names(.), function(x, nx) {
    map2_dfr(x, colnames(x), function(i, ni) {
      data.frame(Coefficient = ni, Label = rownames(x), Value = i)
    }) %>% cbind(Algorithm = nx, .)
  }) %>%
  ggplot(aes(x = Coefficient, y = Value, group = Algorithm, fill = Algorithm)) +
    geom_col(position = "dodge") +
    geom_text(aes(label = round(Value, 2), y = pmin(Value, 10), vjust = -0.2),
              position = position_dodge(0.9)) +
    facet_grid(rows = vars(Label)) +
    scale_y_continuous(limits = c(0, 10), oob = scales::squish,
                       expand = expansion(add = c(0.5, 1))) +
    theme_bw()

Case Study: Impact Factors of House Price in Wuhan

As the fitness of HGWR has been demonstrated with simulation data, we are thing about applying it in collected data sets. Here we are going to use a case study to show the usage of HGWR on this type of data. The data set is provided in this file.

data(wuhan.hp)
head(wuhan.hp$data)

head(wuhan.hp$coords)

##             X       Y
## [1,] 511387.8 3381299
## [2,] 512928.1 3387722
## [3,] 513303.3 3392168
## [4,] 513462.6 3391487
## [5,] 513777.9 3388737
## [6,] 513813.1 3392252

This data set, collected in 2018, has 19599 properties located in 776 communities. In each observation, there are fields of house properties, locational and neighbourhood variables. In details, they are distances to nearby educational resources (kindergartens, primary schools, middle schools, high schools, universities), commercial areas (business districts, shopping malls and supermarkets), transportation facilities (metro stations and bus stations), rivers, lakes, and green lands. The following figure shows the distribution of these properties, and the table gives information about all available variables.

Map of second-hand properties

A variable selection process (Hu et al. 2022) suggests us to use the following variables to build a HGWR model:

• Fee
• d.Water
• d.Commercial
• d.PrimarySchool
• d.Kindergarten
• BuildingArea
• Floor.High

But we are going to append a variable d.ShoppingMall which is usually concerned by customers and researchers. And regards the locational and Fee as local fixed effects, the BuildingArea as global fixed effects and Floor.High as random effects. Then we can calibrate a HGWR model with the following codes.

whhp_hgwr <- hgwr(
  formula = Price ~ d.Water + d.Commercial + d.PrimarySchool + d.ShoppingMall +
            d.Kindergarten + Fee + BuildingArea + (Floor.High | group),
  data = wuhan.hp$data,
  local.fixed = c("d.Water", "d.Commercial", "d.PrimarySchool", 
                  "d.ShoppingMall", "d.Kindergarten", "Fee"),
  coords = wuhan.hp$coords,
  bw = 50,
  kernel = "bisquared",
  eps_gradient = 1e-16,
  eps_iter = 1e-16
)
whhp_hgwr

## Hierarchical and geographically weighted regression model 
## ========================================================= 
## Formula: Price ~ d.Water + d.Commercial + d.PrimarySchool + d.ShoppingMall +      d.Kindergarten + Fee + BuildingArea + (Floor.High | group) 
##  Method: Back-fitting and Maximum likelihood 
##    Data: wuhan.hp$data 
## 
## Global Fixed Effects 
## ------------------- 
##  Intercept  BuildingArea 
##  10.271682     -0.019036 
## 
## Local Fixed Effects 
## ------------------- 
##      Coefficient         Min  1st Quartile     Median  3rd Quartile        Max 
##        Intercept   -5.655311     -0.753470  -0.335763     -0.007905   2.481812 
##          d.Water  -29.363638     -7.449347  -1.706057      2.879884  16.308855 
##     d.Commercial  -56.248990     -7.554371  -2.721221      2.859166  90.103118 
##  d.PrimarySchool   -0.215276     -0.048748  -0.025535      0.004450   0.203624 
##   d.ShoppingMall   -0.204345     -0.039703  -0.006851      0.026344   0.092649 
##   d.Kindergarten   -1.094823     -0.032577   0.003839      0.017043   0.669662 
##              Fee    0.000741      0.099061   0.142559      0.187511   0.381820 
## 
## Random Effects 
## -------------- 
##    Groups        Name  Std.Dev.      Corr 
##     group   Intercept  0.081694           
##            Floor.High  0.081694  0.000000 
##  Residual              0.081694           
## 
## Other Information 
## ----------------- 
## Number of Obs: 19599 
##        Groups: group , 776

wuhan <- rgdal::readOGR("data/wuhan.geojson")

## OGR data source with driver: GeoJSON 
## Source: "C:\Users\rd21411\Develop\hlmgwr-backfitting-ml-doc\data\wuhan.geojson", layer: "Wuhan"
## with 13 features
## It has 6 fields

wuhp_coef_sf <- cbind(coef(whhp_hgwr), as.data.frame(wuhan.hp$coords)) %>%
  st_as_sf(coords = c("X", "Y"), crs = 4547)
wh_basemap <- tm_shape(wuhan) + tm_polygons(col = "white")

with(whhp_hgwr$effects, c(local.fixed, random, global.fixed, "Intercept")) %>%
  map(function(var) {
    wh_basemap +
      tm_shape(wuhp_coef_sf, is.master = T) +
      tm_dots(col = var, size = 0.1, midpoint = 0,
              palette = "-RdBu", legend.col.reverse = T)
  }) %>%
    tmap_arrange(ncol = 3)

whhp_hgwr_res <- data.frame(
  residuals = residuals(whhp_hgwr),
  group = wuhan.hp$data$group
) %>%
  group_by(group) %>%
  summarise(res.abs.mean = mean(abs(residuals)),
            res.sd = sd(residuals)) %>%
  cbind(wuhan.hp$coords) %>%
  st_as_sf(coords = c("X", "Y"), crs = 4547)
wh_basemap +
  tm_shape(whhp_hgwr_res, is.master = T) +
  tm_dots(col = "res.abs.mean", size = "res.sd", midpoint = 0,
          palette = "-RdBu", legend.col.reverse = T)

library(spdep)
whhp_hgwr_res_listw <- st_coordinates(whhp_hgwr_res) %>%
  knearneigh(k = 20) %>% knn2nb() %>% nb2listw() %>% listw2U()
moran.test(whhp_hgwr_res$res.abs.mean, whhp_hgwr_res_listw)

## 
##  Moran I test under randomisation
## 
## data:  whhp_hgwr_res$res.abs.mean  
## weights: whhp_hgwr_res_listw    
## 
## Moran I statistic standard deviate = 6.4658, p-value = 5.038e-11
## alternative hypothesis: greater
## sample estimates:
## Moran I statistic       Expectation          Variance 
##       0.066815400      -0.001290323       0.000110949

whhp_hgwr_res_localmoran <- localmoran(whhp_hgwr_res$res.abs.mean, whhp_hgwr_res_listw)
wh_basemap +
  tm_shape(cbind(whhp_hgwr_res, whhp_hgwr_res_localmoran), is.master = T) +
  tm_dots(col = "Ii", size = 0.2, palette = "-RdBu", style = "quantile",
          midpoint = 0, title = "Local Moran")
whhp_hgwr_res$res_c <- scale(whhp_hgwr_res$res.abs.mean, scale = F)
whhp_hgwr_res$lmoran_c <- scale(whhp_hgwr_res_localmoran[,1], scale = F)
whhp_hgwr_res$quadrant <- with(whhp_hgwr_res, {
  quadrant <- integer(length(res_c))
  quadrant[res_c < 0 & lmoran_c < 0] <- 1
  quadrant[res_c < 0 & lmoran_c > 0] <- 2
  quadrant[res_c > 0 & lmoran_c < 0] <- 3
  quadrant[res_c > 0 & lmoran_c > 0] <- 4
  quadrant[whhp_hgwr_res_localmoran[,5] > 0.1] <- 0
  as.factor(quadrant)
})
LISA_palette <- list(
  "Insignificant" = "green", "Low-Low" = "darkblue", "Low-High" = "blue",
  "High-Low" = "red", "High-High" = "darkred"
)
tm_layout(aes.palette = "cat") +
  wh_basemap +
  tm_shape(whhp_hgwr_res, is.master = T) +
  tm_dots(col = "quadrant", size = 0.2,
          palette = as.vector(unlist(LISA_palette)),
          labels = names(LISA_palette))