Geographically
weighted regression
Danlin Yu
Yehua Dennis Wei
Dept. of Geog., UWM
Outline of the
presentation
1.
2.
3.
4.
5.
Spatial non-stationarity: an example
GWR – some definitions
6 good reasons using GWR
Calibration and tests of GWR
An example: housing hedonic model
in Milwaukee
6. Further information
1. Stationary v.s nonstationary
yi= 0 + 1x1i
e1
e2
Stationary process
e3
e4
Assumed
yi= i0 + i1x1i
e1
e2
Non-stationary process
e3
e4
More realistic
Simpson’s paradox
Spatially disaggregated data
House Price
Spatially aggregated data
House density
House density
Stationary v.s. nonstationary

If non-stationarity is modeled by
stationary models
– Possible wrong conclusions might be
drawn
– Residuals of the model might be highly
spatial autocorrelated
Why do relationships
vary spatially?

Sampling variation
– Nuisance variation, not real spatial nonstationarity

Relationships intrinsically different across
space
– Real spatial non-stationarity

Model misspecification
– Can significant local variations be removed?
2. Some definitions
Spatial non-stationarity: the same
stimulus provokes a different response
in different parts of the study region
 Global models: statements about
processes which are assumed to be
stationary and as such are location
independent

Some definitions

Local models: spatial decompositions
of global models, the results of local
models are location dependent – a
characteristic we usually anticipate
from geographic (spatial) data
Regression




Regression establishes relationship among
a dependent variable and a set of
independent variable(s)
A typical linear regression model looks like:
yi=0 + 1x1i+ 2x2i+……+ nxni+i
With yi the dependent variable, xji (j from 1
to n) the set of independent variables, and i
the residual, all at location i
Regression

When applied to spatial data, as can
be seen, it assumes a stationary
spatial process
– The same stimulus provokes the same
response in all parts of the study region
– Highly untenable for spatial process
Geographically
weighted regression
Local statistical technique to analyze
spatial variations in relationships
 Spatial non-stationarity is assumed
and will be tested
 Based on the “First Law of
Geography”: everything is related with
everything else, but closer things are
more related

GWR

Addresses the non-stationarity directly
– Allows the relationships to vary over space, i.e.,
s do not need to be everywhere the same
– This is the essence of GWR, in the linear form:
– yi=i0 + i1x1i+ i2x2i+……+ inxni+i
– Instead of remaining the same everywhere, s
now vary in terms of locations (i)
3. 6 good reasons why
using GWR
1. GWR is part of a growing trend in
GIS towards local analysis
•
•
Local statistics are spatial
disaggregations of global ones
Local analysis intends to understand the
spatial data in more detail
Global v.s. local
statistics

Global statistics

– Similarity across
space
– Single-valued statistics
– Not mappable
– GIS “unfriendly”
– Search for regularities
– aspatial
Local statistics
– Difference across
space
– Multi-valued statistics
– Mappable
– GIS “friendly”
– Search for exceptions
– spatial
6 good reasons why
using GWR
2. Provides useful link to GIS
•
•
•
•
GISs are very useful for the storage,
manipulation and display of spatial data
Analytical functions are not fully developed
In some cases the link between GIS and
spatial analysis has been a step backwards
Better spatial analytical tools are called for to
take advantage of GIS’s functions
GWR and GIS
An important catalyst for the better
integration of GIS and spatial analysis
has been the development of local
spatial statistical techniques
 GWR is among the recently new
developments of local spatial analytical
techniques

6 good reasons why
using GWR
3. GWR is widely applicable to almost
any form of spatial data
•
•
•
•
•
Spatial link between “health” and
“wealth”
Presence/absence of a disease
Determinants of house values
Regional development mechanisms
Remote sensing
6 good reasons why
using GWR
4. GWR is truly a spatial technique
•
•
•
It uses geographic information as well
as attribute information
It employs a spatial weighting function
with the assumption that near places
are more similar than distant ones
(geography matters)
The outputs are location specific hence
mappable for further analysis
6 good reasons why
using GWR
5. Residuals from GWR are generally
much lower and usually much less
spatially dependent
•
•
GWR models give much better fits to
data, EVEN accounting for added model
complexity and number of parameters
(decrease in degrees of freedom)
GWR residuals are usually much less
spatially dependent
Moran's I = 0.144
Moran's I = 0.372
±
GWR Residuals
-.76 - -.35
-.34 - -.09
-.08 - .09
.10 - .26
.27 - .56
0
50 100
OLS Residuals
-1.34 - -.53
-.52 - -.19
-.18 - .08
.09 - .37
.38 - .92
200
300
Kilometers
6 good reasons why
using GWR
6. GWR as a “spatial microscope”
•
•
Instead of determining an optimal
bandwidth (nearest neighbors), they can
be input a priori
A series of bandwidths can be selected
and the resulting parameter surface
examined at different levels of
smoothing (adjusting amplifying factor in
a microscope)
6 good reasons why
using GWR
6. GWR as a “spatial microscope”
•
Different details will exhibit different
spatial varying patterns, which enables
the researchers to be more flexible in
discovering interesting spatial patterns,
examining theories, and determining
further steps
4. Calibration of GWR

Local weighted least squares
– Weights are attached with locations
– Based on the “First Law of Geography”:
everything is related with everything else,
but closer things are more related than
remote ones
Weighting schemes

Determines weights
– Most schemes tend to be Gaussian or
Gaussian-like reflecting the type of
dependency found in most spatial
processes
– It can be either Fixed or Adaptive
– Both schemes based on Gaussian or
Gaussian-like functions are implemented
in GWR3.0 and R
Fixed weighting
scheme
Weighting function
Bandwidth
Problems of fixed
schemes


Might produce large estimate variances
where data are sparse, while mask subtle
local variations where data are dense
In extreme condition, fixed schemes might
not be able to calibrate in local areas where
data are too sparse to satisfy the calibration
requirements (observations must be more
than parameters)
Adaptive weighting
schemes
Weighting function
Bandwidth
Adaptive weighting
schemes

Adaptive schemes adjust itself
according to the density of data
– Shorter bandwidths where data are dense
and longer where sparse
– Finding nearest neighbors are one of the
often used approaches
Calibration


Surprisingly, the results of GWR appear to
be relatively insensitive to the choice of
weighting functions as long as it is a
continuous distance-based function
(Gaussian or Gaussian-like functions)
Whichever weighting function is used,
however the result will be sensitive to the
bandwidth(s)
Calibration

An optimal bandwidth (or nearest
neighbors) satisfies either
– Least cross-validation (CV) score
 CV
score: the difference between observed
value and the GWR calibrated value using the
bandwidth or nearest neighbors
– Least Akaike Information Criterion (AIC)
 An
information criterion, considers the added
complexity of GWR models
Tests

Are GWR really better than OLS
models?
– An ANOVA table test (done in GWR 3.0,
R)
– The Akaike Information Criterion (AIC)
 Less
the AIC, better the model
 Rule of thumbs: a decrease of AIC of 3 is
regarded as successful improvement
Tests

Are the coefficients really varying
across space
– F-tests based on the variance of
coefficients
– Monte Carlo tests: random permutation of
the data
5. An example

Housing hedonic model in Milwaukee
– Data: MPROP 2004 – 3430+ samples
used
– Dependent variable: the assessed value
(price)
– Independent variables: air conditioner,
floor size, fire place, house age, number
of bathrooms, soil and Impervious surface
(remote sensing acquired)
The global model
Estimate
Std. Error
t value
(Intercept)
18944.05
4112.79
4.61
Floor Size
78.88
2.00
39.42
House Age
-508.56
33.45
-15.20
Fireplace
14688.13
1609.53
9.13
Air Conditioner
13412.99
1296.51
10.35
Number of Bathrooms 19697.65
1725.64
11.42
Soil&Imp. Surface
-27926.77
5179.42
-5.39
Residual standard error: 35230 on 3430 degrees of freedom
Multiple R-Squared: 0.6252, Adjusted R-squared: 0.6246
F-statistic: 953.7 on 6 and 3430 DF, p-value: < 2.2e-16
Akaike Information Criterion: 81731.63
Pr(>|t|)
4.25e-06
<2e-16
<2e-16
<2e-16
<2e-16
<2e-16
7.44e-08
The global model




62% of the dependent variable’s variation is
explained
All determinants are statistically significant
Floor size is the largest positive
determinant; house age is the largest
negative determinant
Deteriorated environment condition (large
portion of soil&impervious surface) has
significant negative impact
GWR run: summary
Number of nearest neighbors for
calibration: 176 (adaptive scheme)
 AIC: 76317.39 (global: 81731.63)

ANOVA Test
Source
SS
DF
MS
OLS Residuals
4257667878068.3 7.00
GWR Improvement
3544862425088.0 327.83
10813043388.63
GWR Residuals
712805558309.1 3102.17
229776586.89
GWR Akaike Information Criterion: 76317.39 (OLS: 81731.63)

GWR performs better than global
model
F
47.06
GWR run: nonstationarity check
F statistic
Floor Size
2.51
House Age
1.40
Fireplace
1.46
Air Conditioner
1.23
Number of Bathrooms 2.49
Soil&Imp. Surface
1.42
Numerator DF
325.76
192.81
80.62
429.17
262.39
375.71
Denominator DF*
1001.69
1001.69
1001.69
1001.69
1001.69
1001.69
Tests are based on variance of coefficients, all
independent variables vary significantly over space
Pr (> F)
0.00
0.00
0.01
0.00
0.00
0.00
Floor Size
Air Conditioner
High : 119.49
High : 55860.63
Low : 17.63
Low : -7098.88
B
A
Fire Place
High : 74706.97
Low : -6722.29
C
±
Num. of Bathrm
House Age
Soil & Imp. Sfc
High : 39931.12
High : 929.44
High : 34357.96
Low : -2044.24
Low : -1402.30
Low : -220301.55
E
D
0
5
10
Kilometers
20
F
General conclusions


Except for floor size, the established
relationship between house values and the
predictors are not necessarily significant
everywhere in the City
Same amount of change in these attributes
(ceteris paribus) will bring larger amount of
change in house values for houses locate
near the Lake than those farther away
General conclusions

In the northwest and central eastern
part of the City, house ages and house
values hold opposite relationship as
the global model suggests
– This is where the original immigrants built
their house, and historical values weight
more than house age’s negative impact
on house values
6. Interested Groups



GWR 3.0 software package can be obtained
from Professor Stewart Fotheringham
stewart.fotheringham@MAY.IE
GWR R codes are available from Danlin Yu
directly (danlinyu@uwm.edu)
Any interested groups can contact either
Professor Yehua Dennis Wei
(weiy@uwm.edu) or me for further info.
Interested Groups

The book: Geographically Weighted
Regression: the analysis of spatially
varying relationships is HIGHLY
recommended for anyone who are
interested in applying GWR in their
own problems
Acknowledgement
Parts of the contents in this workshop
are from CSISS 2004 summer
workshop Geographically Weighted
Regression & Associated Statistics
 Specific thanks go to Professors
Stewart Fotheringham, Chris
Brunsdon, Roger Bivand and Martin
Charlton

Thank you all
Questions and comments