1. Geographically weighted regression
Danlin Yu
Yehua Dennis Wei
Dept. of Geog., UWM

2. Outline of the presentation
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
Further information

3. 1. Stationary v.s non-stationary
yi= i0 + i1x1i
yi= 0 + 1x1i
e1
e1
e2
e2
Stationary process
Non-stationary process
e4
e3
e3
e4
Assumed
More realistic

4. Simpson’s paradox Spatially aggregated data
Spatially disaggregated data
House Price
House density
House density

5. 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

6. 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?

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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)

13. 3. 6 good reasons why using GWR
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

14. 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

15. 6 good reasons why using GWR
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

16. 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

17. 6 good reasons why using GWR
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

18. 6 good reasons why using GWR
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

19. 6 good reasons why using GWR
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

21. 6 good reasons why using GWR
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)

22. 6 good reasons why using GWR
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

23. 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

24. 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

25. Fixed weighting scheme
Weighting function
Bandwidth

26. 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)

27. Adaptive weighting schemes
Weighting function
Bandwidth

28. 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

29. 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)

30. 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

31. 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

32. Tests Are the coefficients really varying across space
F-tests based on the variance of coefficients
Monte Carlo tests: random permutation of the data

33. 5. An example Housing hedonic model in Milwaukee
Data: MPROP 2004 – 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)

34. The global model

35. 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

36. GWR run: summary
Number of nearest neighbors for calibration: 176 (adaptive scheme)
AIC: (global: )
GWR performs better than global model

37. GWR run: non-stationarity check
F statistic
Numerator DF
Denominator DF*
Pr (> F)
Floor Size
2.51
325.76
0.00
House Age
1.40
192.81 1
001.69
Fireplace
1.46
80.62
0.01
Air Conditioner
1.23
429.17
Number of Bathrooms
2.49
262.39
Soil&Imp. Surface
1.42
375.71
Tests are based on variance of coefficients, all independent variables vary significantly over space

39. 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

40. 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

41. 6. Interested Groups
GWR 3.0 software package can be obtained from Professor Stewart Fotheringham
GWR R codes are available from Danlin Yu directly
Any interested groups can contact either Professor Yehua Dennis Wei or me for further info.

42. 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

43. 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

44. Questions and comments
Thank you all
Questions and comments