Geographically weighted regression
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Transcript of 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
e3
e2
e1
e4
Stationary process
e3
e2
e1
e4
Non-stationary process
yi= i0 + i1x1i
Assumed More realistic
Simpson’s paradox
House density
Hou
se P
rice
Spatially aggregated data Spatially disaggregated data
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
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
GWR Residuals
-.76 - -.35-.34 - -.09-.08 - .09.10 - .26.27 - .56
OLS Residuals
-1.34 - -.53-.52 - -.19-.18 - .08.09 - .37.38 - .92
0 100 200 30050Kilometers
±Moran's I = 0.144 Moran's I = 0.372
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
Bandwidth
Weighting function
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
Bandwidth
Weighting function
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 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
calibration: 176 (adaptive scheme) AIC: 76317.39 (global: 81731.63)
GWR performs better than global model
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: 81731.63)
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
Soil & Imp. SfcHigh : 34357.96
Low : -220301.55
F
House AgeHigh : 929.44
Low : -1402.30
E
Fire PlaceHigh : 74706.97
Low : -6722.29
C
Air ConditionerHigh : 55860.63
Low : -7098.88
B
±
0 10 205
Kilometers
Floor SizeHigh : 119.49
Low : 17.63
A
Num. of BathrmHigh : 39931.12
Low : -2044.24
D
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 [email protected]
GWR R codes are available from Danlin Yu directly ([email protected])
Any interested groups can contact either Professor Yehua Dennis Wei ([email protected]) 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