Geographically weighted regression

```					Geographically
weighted regression

Danlin Yu
Yehua Dennis Wei
Dept. of Geog., UWM
Outline of the
presentation
1. Spatial non-stationarity: an example
2. GWR – some definitions
3. 6 good reasons using GWR
4. Calibration and tests of GWR
5. An example: housing hedonic model
in Milwaukee
6. Further information
1. Stationary v.s non-
stationary
yi= 0 + 1x1i       yi= i0 + i1x1i

e1                      e1
e2                      e2

Stationary process     Non-stationary process

e3          e4          e3             e4

Assumed            More realistic
Spatially aggregated data   Spatially disaggregated data
House Price

House density               House density
Stationary v.s. non-
stationary
   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 non-
stationarity
   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
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
– 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           Local statistics
– Similarity across          – Difference across
space                        space
– Single-valued statistics   – Multi-valued statistics
– Not mappable               – Mappable
– GIS “unfriendly”           – GIS “friendly”
– Search for regularities    – Search for exceptions
– aspatial                   – 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
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                                 OLS Residuals
-.76 - -.35                                    -1.34 - -.53
-.34 - -.09                                    -.52 - -.19
-.08 - .09                                     -.18 - .08
.10 - .26                                      .09 - .37
.27 - .56                                      .38 - .92

0   50 100      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
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)
schemes
Weighting function

Bandwidth
schemes
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)
 Aninformation 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   Pr(>|t|)
(Intercept)                18944.05        4112.79       4.61      4.25e-06
Floor Size                 78.88           2.00          39.42     <2e-16
House Age                  -508.56         33.45         -15.20    <2e-16
Fireplace                  14688.13        1609.53       9.13      <2e-16
Air Conditioner            13412.99        1296.51       10.35     <2e-16
Number of Bathrooms 19697.65               1725.64       11.42     <2e-16
Soil&Imp. Surface          -27926.77       5179.42       -5.39     7.44e-08
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
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
 AIC: 76317.39 (global: 81731.63)
ANOVA Test
Source                SS                   DF        MS               F
OLS Residuals         4257667878068.3 7.00
GWR Improvement       3544862425088.0 327.83         10813043388.63
GWR Residuals         712805558309.1 3102.17         229776586.89     47.06
GWR Akaike Information Criterion: 76317.39 (OLS: 8173 1.63)

   GWR performs better than global
model
GWR run: non-
stationarity check
F statistic   Numerator DF   Denominator DF*   Pr (> F)
Floor Size          2.51          325.76         1001.69           0.00
House Age           1.40          192.81         1001.69           0.00
Fireplace           1.46          80.62          1001.69           0.01
Air Conditioner     1.23          429.17         1001.69           0.00
Number of Bathrooms 2.49          262.39         1001.69           0.00
Soil&Imp. Surface   1.42          375.71         1001.69           0.00

Tests are based on variance of coefficients, all
independent variables vary significantly over space
Fire Place
Floor Size                           Air Conditioner
High : 74706.97
High : 119.49                         High : 55860.63

Low : 17.63                           Low : -7098.88         Low : -6722.29

A                                         B                  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                        F
D

0   5      10             20
Kilometers
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