Assessing spatial heterogeneity

Assessing spatial heterogeneity in
crime prediction
Using geographically weighted regression to explore local patterns
in crime prediction in Belgian municipalities
November 2016
Johan Blomme
Leenstraat 11
8340 Damme-Sijsele
URL : www.johanblomme.com
Email : j.blomme@telenet.be

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Assessingspatialheterogeneityincrimeprediction
ASSESSING SPATIAL HETEROGENEITY IN CRIME PREDICTION
Contents
1. Analytical framework 4
2. Exploratory spatial data analysis 7
3. Global non-spatial regression model 11
4. Global spatial regression model 13
5. Local spatial regression model 15
5.1. Visualising GWR results 18
5.2. Cluster analysis 28
6. Conclusions 36

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Assessing spatial heterogeneity in crime
prediction
Traditional regression analysis describes a modelled relationship between a dependent variable
and a set of independent variables. When applied to spatial data, the regression analysis often
assumes that the modelled relationship is stationary over space and produces a global model
which is supposed to describe the relationship at every location in the study area. This would be
misleading, however, if relationships being modelled are intrinsically different across space. One
of the spatial statistical methods that attempts to solve this problem and explain local variation
in complex relationships is Geographically Weighted Regression (GWR).
In a global regression model, the dependent variable is often modelled as a linear combination
of independent variables that is stationary over the whole area (i.e. the model returns one value
for each parameter). GWR extends this framework by dropping the stationarity assumption : the
parameters are assumed to be continuous functions of location. The result of the GWR analysis
is a set of continuous localised parameter estimate surfaces, which describe the geography of
the parameter space. These estimates are usually mapped or analysed statistically to examine
the plausibility of the stationarity assumption of the traditional regression and different possible
causes of nonstationarity.
The use of linear regression is common in many areas of science. Ordinary linear regression
implicitly assumes spatial stationarity of the regression-model that is, the relationships between
the variables remain constant over geographical space. We refer to a model in which the
parameter estimates for every observation in the sample are identical as a global model.

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Spatial nonstationarity occurs when a relationship (or pattern) that applies in one region does
not apply in another. Global models are statements about processes or patterns which are
assumed to be stationary and as such are local independent, i.e. are assumed to apply to all
locations. In contrast local models are spatial disaggregations of global models, the results of
which are location-specific. The template of the model is the same : the model is a linear
regression model with certain variables, but the coefficients alter geographically. If the
parameter estimates are allowed to vary across the study area such that every observation has
its own separate set of parameter estimates we have a local model.
GWR does not assume the relationships between independent and dependent variables are
constant across space. Instead, GWR explores whether the relationships between a set of
predictors and an outcome vary by geographical location. GWR is suggested to be a powerful
tool for investigating spatial nonstationarity in the relationship between predictors and the
outcome variable.
Theoretically, spatial nonstationarity is based on the concept of the social construction of
space. The interaction between individuals with each other and their physical environment
produces space. Human beings are just as much spatial as temporaral beings. By temporal, we
mean that we are most influenced by what is immediate in space. What happens near us
matters more than non-proximal events. Human’s spatiality and temporality are essential and
equal powerful in explaining human behavior. Consequently, everything that is social is
inherently spatial, just as everything spatial is inherently socialized.
From this perspective, we analyse how the macro-level relationship between crime and various
socio-economic and demographic variables unfolds over geographical space.

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1. Analytical framework
Our analysis strategy entails estimating regression models that summarize the “global”, or
average, effects of the predictor variables on crime rates across our sample of Belgian
municipalities. Given the well-known spatial autocorrelation evident in crime data, we generate
the global models using Ordinary Least Squares (OLS) and Spatial AutoRegression (SAR)
estimators. OLS and the spatial autoregression model are “global” models in the sense that
they both assume that a single set of parameters sufficiently describe the relationships between
predictor variables and crime rates.
The classical ordinary least Squares (OLS) model is widely used to model the global relationship
between a response variable and one or more explanatory variables. OLS assumes, among
other things that residuals are spatially independent. Residual autocorrelation captures
unexplained similarities between neighboring municipalities, which can be the result of omitted
variables or a misspecification of the regression model. Assuming a global model does exist, an
exploration of spatial patterns in the data can help determine whether a global model is
misspecified – whether the model is missing important predictor variables (spatial error model)
or if a spatial term should be included in the model (spacial lag model) – which would improve
the accuracy of the global model in explaining crime levels across the study area.
Global models that account for spatial effects are spatial autoregressive models (SAR). The
spatial error model addresses the presence of spatial autocorrelation by defining a spatial
autoregressive process for the error term and, by doing so, captures unexplained similarities.
The spacial lag model extends the standard OLS regression model by including a spatially lagged
dependent variable, which can be mostly interpreted as spill-over effects.
Global regression models assume a homogeneous behavior of the estimated parameters across
space. We expect spatial homogeneity to be rare and assume that most social phenomena are
not geographically stationary. A way to deal with spacial heterogeneity is the application of
geographically weighted regression (GWR) to investigate spatially varying relationships.
GWR models spatial autocorrelation and spatial heterogeneity for subsets of the entire data set.
Each subset is established around a regression point with near data points exhibiting a higher
influence than more distant data points. This weighting is often based on a bi-square kernel
function. Of crucial importance is the specification of an appropriate bandwidth length. The

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most common is the adaptive bandwidth, where is length is allowed to vary across space,
depending on the density of the data points. In densely populated areas the kernel possesses a
shorter bandwith in contrast to regions with larger inter-point distances, where the bandwidth
is longer.
While it is often argued that GWR is more suitable for exploratory analysis, it is a technique to
test whether local models yield a significant improvement in fit over the global models.
The following analysis models both spatial autocorrelation and nonstationarity by means of
global and local spatial statistical models. An exploratory spatial data analysis, a global non-
spatial regression model, a global spatial regression model and finally a local spatial regression
model were applied to explore the association between various predictors and crime in Belgian
municipalities. We rely on crime data in municipalities, the main political and administrative
unit of the Belgian territory.
The dependent variable in this study is the crime rate/1000 residents (calculated as a mean
over the period 2008-2012) in Belgian municipalities (N= 589, source data : statistics Belgian
Federal Police).
To test the impact of social deprivation on crime, we collected data at municipality level about
various indicators of inequality . Besides mean family income and the percentage unemployed,
we use the Gini coefficient as a measure of income variation, indicating the distribution of
income in each municipality (between extremes of 0 (absolute equality) and 1 (maximum
inequality). As control variables we include various socio-demographic indicators : population
density, the share of males in the age group 15 to 64, the percentage of young people (15-24) in
the population, the percentage of residents that are foreign born, the percentage of non-Euro
foreign born residents and the degree of female labour force participation (source data :
statistics Federal Government Belgium, 2011).

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Since the original data for the dependent variable and five of the independent variables are not normally distributed (skewness marked in red in the above table) and
normality of data is a basic assumption for both ordinary least squares regression and spatial regression, natural log values (ln) were used for these variables.

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2. Exploratory spatial data analysis
The first step in an exploratory spatial data analysis (ESDA) is to verify if spatial data are
randomly distributed. To do this, it is necessary to use global autocorrelation statistics. The
global indicators of spatial autocorrelation are not capable of identifying local patterns of spatial
association, such as local spatial clusters or local outliers in data that are statistically significant.
To overcome this obstacle, it is necessary to implement a spatial clustering analysis (we made
use of GeoDa open-source spatial regression software of the GeoDa Center for Geospatial
Analysis and Computation, http://geodacenter.asu.edu).
A significant Moran’s I statistic is a first clue that parameter estimates in an OLS regression can
be affected by spatial residual autocorrelation. For this reason, the Moran’s I statistic was
calculated for the dependent variable and the nine independent variables included in this study.
The neighborhood relationships for calculating the Moran’s I statistic are defined as first order
queen contiguity, which is commonly used (a municipality’s spatial lag is a weighted average of
its neighboring localities ; neighbors are typically defined in terms of their physical proximity to
the local geographic unit).
Results indicate that both the dependent and all independent variables exhibit significant
positive spatial autocorrelation. The hypothesis of spatial randomness is clearly rejected. A
positive and significant spatial dependence in the dependent variable (crime rate) indicates that
the crime rate in a particular municipality is associated with (not independent of) crime rates in
surrounding counties. The value of the spatial autocorrelation coefficient (0,297) indicates that
a 10 percentage point increase in the crime rate in a municipality results in an increase of nearly
3% in the crime rate in a neighboring municipality. This, together with the results of the LISA
cluster analysis, is evidence of the existence of significant spillover effects between
municipalities with respect to crime, and implies that there is a need of a coordination of the
municipal efforts to fight criminal activities that spill over the municipal borders.

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Prevalence of crime in Belgian municipalities (N = 589)
(crimerate/1000 inhabitants ; crimerate percentiles)
Global Moran’s I statistic for variables included in this study

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Cartograms of the geographical distribution of independent variables

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LISA cluster map for criminality in Belgian municipalities, N= 589

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3. Global non-spatial regression model
Exploring the relationship between the independent variables and crime rates starts with a
multivariate OLS regression model. None of the correlations between the predictors is
excessively high enough to yield a major concern about multicollinearity. Nevertheless, we
evaluated the diagnostics to assess the issue of multicollinearity more formally. In particular,
Variance Inflation Factors (VIFs) were investigated. Since all VIF scores are below the critical
value of 5, multicollinearity is rejected1.
Results show that the nine predictors explain about 54,2% of the variance in crime rates. Of
those, the variables representing the percentage of males in the age group 15-64, the
percentage of the age group 15-24 in the population and the percentage of foreign born
residents do not contribute significantly to the explanation of the variability in crime rates
between municipalities2.
A more detailed analysis of the error residuals reveals that they are not normally distributed
(Jarque Bera test = 410.059 ; p < 0.001) but not heteroscedastic (Koenker-Bassett test = 14.115 ;
p=0.118). Finally, residual independence is tested by the Moran I-statistic. This test shows
significant spatial residual autocorrelation (Moran’s I = 0.155 ; p < 0.001), violating the model’s
independence assumption. This residual pattern in the OLS model can be the result of existing
spatial effects and can be accounted for by means of a spatial regression model.
______________________________________________________________________________
1
Collinearity diagnostics were estimated using SPSS Base Statistics and no problems of multicollinearity were
found among the independent variables. The collinearity diagnostics used were the variance inflation factors (VIF)
and tolerances for individual variables. Multicollinearity is said to exist if the VIF is 5 or higher (or equivalently,
tolerances of 0,20 or less). The highest VIF-value in this analysis was 4,852 and the lowest tolerance was 0,206,
both for mean income.
2
Initially, two dummy variables representing the regions in Belgium were added to the regression equation.
However, VIF scores indicated the presence of multicollinearity. Therefore, these dummy variables were no
longer withheld in the OLS regression.

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4. Global spatial regression model
The clustering of crime rates indicates that the data are not randomly distributed, but
instead follow a systematic pattern. The spatial clustering of variables, and the
possibility of omitted variables that relate to the connectivity of neighboring localities,
raise model specification issues. Evidence for the latter also comes from the residual
autocorrelations present in the OLS model.
We employ two alternative specifications to correct for spatial dependence. One is the
spatial lag model. This specification is relevant when the spatial dependence works
through a spatial lag of the dependent variable. The other specification is the spatial
error model. This specification is relevant when the spacial dependence works through
the disturbance term (spatial regression models ware developed by making use of
GeoDa, regression software of the GeoDa Center for Geospatial Analysis and
Computation, http://geodacenter.asu.edu).
The value of the LMLAG
-test is only weakly significant (LMLAG
= 3.598 ; p < 0.1) but the results of
the LMERROR
-test (56.900 ; p < 0.001) suggest that a spatial error must be considered in the
global spatial regression model.
The results from the spatial lag model shown in the table on the page, suggest that this model
does not perform as well as the spatial error model. The effect of the spatial lag term is
statistically weak (rho = 0.084 ; p= 0.101). The robust Lagrange Multiplier (LM) test also
recommends the use of the spatial error model and the lower AIC value combined with the
higher R
2
value for the spatial error model signals that this model outperforms the spatial lag
model. In the spatial error model, all predictor variables except one (the percentage of foreign
born residents) yield a statistically significant effect.

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Global OLS versus global spatial regression models
Based on the results of the global spatial regression model it is difficult to defend similarities in
municipality-level crime as arising from imitation of one’s neighbors, that is, a spatial lag
process. Criminality results from a complex mix of social, economic and cultural factors, only a
small number of which can be brought into a statistical model of the process. Much of it
remains unaccounted for and is summarized in the model’s error term.
Although we observe a very small Moran’s I value (-0.022) associated with the spatial error
model, the residuals are not in compliance with the assumption of being spatially independent
of each other (Breusch-Pagan test for heteroscedasticity = 54.060 ; p< 0.001).

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5. Local spatial regression model
As a global model, local regression modeling carries the assumption that the processes being
modeled are uniform throughout the study area : the relationships between the dependent and
the independent variables remain stationary (constant) across the entire study area of Belgium.
Local spatial regression models take nonstationarity into account. We use GWR4 to perform
geographically weighted regression analysis (GWR4 is release of a Microsoft Windows based
application for calibrating geographically weighted regression models, which can be used to
explore geographically varying relationships between dependent/response variables and
independent/explanatory variables ; see Nakaya, 2012).
The results of fitting the dataset to different GWR descriptive models are shown below. Four
alternatives of GWR modeling were applied considering the four possible combinations
between two different types of kernels (fixed or adaptive) and two different bandwidth
methods (AICC
and CV). GWR models 3 and 4 (both models use an adaptive kernel) offered
lower residual squares, meaning that these models provided a better fit to the data. The R
2
value of both GWR-models is nearly the same. We chose GWR model 3 with the lowest AICC
value to provide an exploratory analysis of the data.
GWR model 1 GWR model 2 GWR model 3 GWR model 4
Kernel fixed fixed adaptive adaptive
bandwidth method AICc CV AICc CV
adjusted R2 0,628 0,570 0,633 0,639
residual squares 342571,621 444329,261 337812,559 323290,833
AICc 5584,352 5630,776 5578,562 5581,763
Anova test residuals OLS/GWR p < 0,01 p < 0,01 p < 0,01 p < 0,01
GWR models applied to dataset of Belgian municipalities

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Results reveal that the GWR model exhibits a significant improvement in explained variance as
compared to the OLS regression model (63,3% vs. 54,2%). The AIC score for the GWR model
(5578.562) is substantially lower than the AIC score for the global OLS model (5657.654), which
reflects a better goodness of fit (AIC is a measure of spatial collinearity. The lower its value, the
better the fit of the model to the observed data).
Another method to evaluate the GWR model is the ANOVA test which verifies the null
hypothesis that the GWR model represents no improvement over the global model. The
computed F-value of 2.753 is in excess of the critical value of F (2.41 ; α = 0.01) with 10 and 496
degress of freedom. The ANOVA test thus suggests that the GWR model is a significant
improvement on the global model for the data of Belgian municipalities.
The results obtained by the GWR method provide information about locally differing estimation
coefficients. Therefore, the GWR results do not report a global estimate for each explanatory
variable but rather they provide insights into local ranges of the estimates (minimum, 25%
quantile, median, 75% quantile and maximum). The 5-number summary (see page 16) is
helpful to get a feel of the degree of spatial nonstationarity in a relationship by comparing the
range of the local parameter estimates with a confidence interval around the global estimate of
the parameter. This is accomplished by dividing the interquartile range of the GWR coefficient
by twice the standard error of the same variable from the global regression (OLS). Ratio values
> 1 suggest nonstationarity in the relationship between an independent variable and the
dependent variable.
The results of the Monte Carlo test indicate that the parameter estimates do vary significantly
across space. As shown on the map on the next slide, the total variance explained by the local
model ranges from 47,8% to 83,4%. In general, there is a north-south divide with higher R
2
values in the northern part of the country. Explained variance is lowest in the southern part of
the province of East Flanders and its surrounding municipalities in Wallonia.

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minimum lower quartile median upper quartile maximum status significance
Intercept -513,965 -28,093 210,299 361,918 512,299 non-stationary p < 0.001
ln(Gini inequality) -0,706 0,285 1,062 1,439 2,529 non-stationary p < 0.001
mean income -0,010 -0,006 -0,004 -0,002 0,001 non-stationary p < 0.001
ln(unemployment) -0,012 0,216 0,325 0,444 0,720 non-stationary p < 0.001
ln(population density) -0,057 0,006 0,056 0,132 0,227 non-stationary p < 0.001
% male in age group 15-64 -0,100 0,019 0,044 0,065 0,129 non-stationary p < 0.001
% 15-24 in population -0,119 -0,058 -0,011 0,022 0,087 non-stationary p < 0.001
ln(% foreign born) -0,119 -0,009 0,036 0,083 0,167 non-stationary p < 0.001
ln(% non-Euro foreign) -0,015 0,076 0,106 0,147 0,319 no spatial variability p < 0.001
female labour force participation 0,008 0,038 0,048 0,064 0,096 non-stationary p < 0.001
5-number parameter summary Monte Carlo test
Geographically weighted regression 5-number parameter summary results and
Monte Carlo significance test for spatial variability of parameters
(Belgian municipalities, N = 589)
Local R
2
values of the GWR model (Belgian municipalities, N = 589)

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5.1. Visualising GWR results
To better understand and interpret nonstationarity in individual parameters it is necessary to
visualize the local parameter estimates and their associated diagnostics. The output of a GWR
analysis includes data that can be used to generate surfaces for each model parameter that can
be mapped, where each surface depicts the spatial variation of the relationship between a
predictor and the outcome variable.
A challenge in GWR analysis is to visually represent the large number of results through the use
of cartographic design. Mapping only the parameter estimates is misleading, as the map reader
has no way of knowing whether the local parameter estimates are significant. As Mennis (2006
: 172) notes, a main issue is that “the spatial distribution of the parameter estimates must be
presented in concert with the distribution of significance, as indicated by the t-value, in order to
yield meaningful interpretation of results”.
There are several possibilities. The most popular and easiest way to visualise the results of
GWR is to make use of choropleth maps and colour the regions according to the values of
parameter estimates or the associated t-values in order to interpret the significance of the
parameters. Because the patterns of t-values for the parameter estimates are important to
reveal which areas have statistically significant estimates, we initially mapped the t-values for all
variables (see appendix).
Another possibility to map the GWR results is to create raster surfaces for both the parameter
estimates and the t-values. Geostatistical methods, e.g. inverse distance weighting (IDW) and
ordinary kriging (OK), are applied in spatial interpolation from point measurement to
continuous surfaces. Both IDW and OK estimate values at unmeasured points by the weighted
average of observed data at surrounding points. The weight of each measured value is a
function of its distance from the point we are trying to predict. The difference between both
methods is that in IDW the weights are arbitrarily specified while in OK the weights are
estimated from the data itself 1.
____________________________________________________________________________
1
See Dorman (2014, chapter 8) for an extensive explanation of spatial interpolation.

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To create raster surfaces of estimated coefficients and local t-values for each of the
parameters, we use R’s gstat package and the gstat function that lets us create a spatial
prediction model. The latter is then applied to a grid that represents the area we are working
with, to yield a new raster with predicted values (this new raster is obtained through the use of
the interpolation function in R’s raster package). For IDW, we created prediction models with
IDP-parameters set to 1, 2 and 3. A low IDP-parameter (1) results in a smoother surface while
higher values result in sharper boundaries. For OK, a model is automatically created through
the use of the autofitVariogram function of the automap package in R.
To evaluate the predictive ability of the interpolation models the process of cross-validation is
used to compare the predicted values to the observed ones. The raster surfaces with the
lowest root mean square error (RMSE) are finally chosen for the visual representation of
parameter estimates and t-values1.
_____________________________________________________________________________________________
1
R code for the various steps to construct a raster surface :
# GWR parameter estimates for a variable (e.g. income)
gwr_income <- read.csv("parameter_estimates_income.csv",header=T,sep=";")
# extract centroids of municipalities
mun.centroids <- data.frame(coordinates(belgie),belgie@data$ID_4)
names(mun.centroids) <- c("lon","lat","id")
# add lat lon to data
gwr_income <- merge(gwr_income,mun.centroids,by="id")
names(gwr_income) [3] <- "x"
names(gwr_income) [4] <- "y"
# make datafile a SpatialPointsDataFrame
coordinates(gwr_income) <- ~ x + y
# create grid (grid and datafile must have the same projection)
r <- raster(nrow=500,ncol=500,
xmn=bbox(belgie)["x","min"],xmx=bbox(belgie)["x","max"],
ymn=bbox(belgie)["y","min"],ymx=bbox(belgie)["y","max"],
crs=proj4string(gwr_income))
# model creation with gstat
model <- gstat(formula = inc ~ 1, data = gwr_income,set=list(idp=3))
print(model)
z <- interpolate(r,model)
z <- mask(z,belgie)
# cross-validation
cv <- gstat.cv(model)
rmse <- function(x) sqrt(sum((-x$residual)^2)/nrow(x))
rmse(cv)

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For a selected parameter, the surface created for the estimated coefficients and the local t-
values can be mapped together. In the map below, the t-values for the income parameter are
added as contour lines on top of the parameter estimate surface. While it is possible for the
reader to distinguish significant parameter estimates from those that are not significant, the
contour lines may not allways be easy to interpret.
Overlay of t-values as contour lines on parameter estimate map for income
In order to identify directly zones with significant parameter estimates, it is possible to set up a
mask. Insignificant values (between -1.96 and 1.96) in the raster surface layer of t-values are
set to NA, and subsequently using the mask function removes all values from the parameter
raster layer that are NA in the t-surface layer. This allows the visualisation of only the
significant parameter estimates. We used R’s plotGoogleMaps package to map significant
parameter estimates with a Google maps background (the resulting html-files also allow to
interactively explore the GWR results). The maps provide strong evidence of significant spatial
heterogeneity in the effect of predictor variables on crime across municipalities (significant
positive parameter erstimates are coloured yellow to red while significant negative estimates
are shades of blue).

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Significant GWR-estimates for Gini inequality (ordinary kriging)
Significant GWR-estimates for income (IDW, ß = 3)

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Significant GWR-estimates for unemployment (ordinary kriging)
Significant GWR-estimates for population density (ordinary kriging)

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Significant GWR-estimates for female labour force participation (IDW, ß= 3)
Significant GWR-estimates for % males in age group 15-64 (ordinary kriging)

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Significant GWR-estimates for % age group 15-24 in population (ordinary kriging)
Significant GWR-estimates for % foreign born in population (IDW, ß= 3)

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Significant GWR-estimates for % non-Euro in population (IDW, ß= 3)
The results of the geographically weighted regression analysis indicate that spatially varying
processes operate in Belgian municipalities with respect to the relationships between socio-
economic and socio-demographic variables and crime rates.
Several local results are of particular note. First, when we examine the incidence of significant
parameter estimates at the local level, 61 % of all parameter estimates are insignificant (see
graphs on pages 25-26). With the exception of unemployment and female labour force
participation, the majority of parameter estimates for all other independent variables and the
intercept are insignificant. Positively of negatively signed global effects of covariates do not
hold across all municipalities. This proves it is important to analyze beyond the global level
(OLS) and to examine variation at the local level (GWR).
Secondly, the global parameter estimates mask a great deal of variation at the local level. For
example, while the global parameter estimate for unemployment is 0,217, the parameter
estimates at the local level range from -0,012 to 0,720. Where the global estimate for the

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percentage of non-Euro foreign born inhabitants is 0,114, the local parameter estimates range
from -0,015 to 0,319.
Finally, insignificant global results mask countervailing positive and negative effects of
covariates at the local level. The negatively signed but insignificant global effect of the
percentages of 15-24 aged youngsters in the population (age) reaches negative significance in
23,2 % of the municipalities while the effect of this covariate reverses to positive significance in
a minority (2,9 %) of all municipalities. In a similar way, the positively signed but insignificant
global effect of the percentage of males aged 15-64 in the local population (gender) reaches
positive significance in 39,2 % of the municipalities while the effect of this variable is negative
significant in 2 % of the municipalities.
GWR model significant estimates

27

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5.2. Cluster analysis
We can further explore the results of the GWR analysis by clustering locations with similar
parameter values for the variables considered. This synthesizes the output that is generated by
the GWR model and can help to interpret the results .
A two-step cluster analysis based on the nine parameter estimates and the intercept was
applied. We experimented with a range of clusters between 4 and 8. The optimal choice in
terms of the number of clusters was 6 (municipalities were divided in evenly sized clusters).

29
Although latitude and longitude were not included in clustering municipalities’ parameter
estimates, the six clusters are geographically coherent. A discriminant analysis with cluster
membership as the dependent variable and both lat/lon-coordinates as predictors confirms that
70,6% of the cluster members are correctly classified based on their location which means that
70,6% of the municipalities were geographically near other members of the same cluster. By
cluster, the percentage of correctly classified members varies from 57,8% to 83,8%.

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Distribution of t-values within clusters

31
Distribution of t-values within clusters (con’d)

32

33

34
Although the parameter estimate of non-Euro inhabitants does not vary spatially (see 5-number
parameter summary), it is by far the most important predictor of criminality in cluster 1. In
comparison with the other clusters, the effect of the percentage of males in the age group 15-
64 is significant in a large majority of municipalities covered by cluster 1.
Like cluster 1, cluster 2 represents a contiguous area of municipalities but the percentage of
correctly classified municipalities is lowest (57,8%) of all clusters. Within this cluster the
percentage of explained variance strongly differs when moving from west to east (R
2
between
47,7% and 80,7).
Cluster 3 covers large parts of Wallonia, where local R
2
values are relative low. In cluster 3 as
well as in cluster 4 and cluster 6, the parameter estimates for socio-economic variables (Gini
inequality, mean income and unemployment) are significant in resp. 80,6 %, 65,9 % and 80,6 %
of the municipalities. In the other clusters, the effect of these variables is significant in less than
one third of the municipalities.

35
Apart from the effect of socio-economic variables, the effect of non-Euro inhabitants on crime is
also significant in a majority of municipalities in cluster 4.
In cluster 5 the local R
2
values are also relative low and the estimate of the intercept factor is
significant. Criminality in the east cantons of cluster 5 also correlates significant and
independent of other predictors with population density and the presence of young people in
the population inhibits criminality in this area of cluster 5.
As stated, in the area that represents cluster 6 (the largest cluster in terms of the number of
municipalities), the measures of inequality are the most significant determinants of crime.
Criminality also varies in an independent way with population density.

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6. Conclusions
The objectives of this study were to examine the extent of geographic variation in the
relationship between socio-economic and demographic variables on the one hand and crime
rates on the other. The goals of our study were (i) to compare the performance of global and
local spatial regression with OLS regression (ii) examine spatial nonstationarity throught the
use of GWR (iii) map the parameter coefficients of GWR for further interpretation and (iv)
examine whether there are spatial groupings of parameter estimates.
The analysis revealed that there is evidence of overall clustering in crime rates in Belgium. Local
spatial analysis uncovered that places with the highest crime rates are often proximate.
The finding of the existence of local spatial autocorrelation in crime rates suggests that failing to
utilize spatially-oriented methodologies may result in biased parameter values in explanatory
models. As far as global models are concerned, this analysis demonstrated that a spatial error
model adds significantly to the understanding and interpretation of spatially varying crime rates.
The use of a GWR model allowed for an assessment of spatial heterogeneity when exploring the
relationships between predictor variables and crime rates by local area. Geographically
weighted estimations provided the best fit to the data. Predictor variables as well as crime rates
showing strong local variation point to problems that policy makers best address at the local
level and the situation in particular areas.
Significant local parameter estimates were found for the predictor variables, confirming spatial
heterogeneity in the effects of these variables on crime and providing insights into the spatial
scale at which processes may be operating. Furthermore, a two-steps cluster analysis revealed
distinct zones of spatial effects.

37
APPENDIX
Spatial mapping of the coefficients from GWR modeling

38
Spatial mapping of the coefficients from GWR modeling (con’d)

39

40

41

42
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