Geographically Weighted Regression

Transcription

1 Geographically Weighted Regression CSDE Statistics Workshop Christopher S. Fowler PhD. February 1 st 2011 Significant portions of this workshop were culled from presentations prepared by Fotheringham, Charleton and Brunsdon and presented at the 2010 Advanced Workshop on Spatial Analysis at the University of Santa Barbara. University of Washington Center for Studies in Demography and Ecology

2 Outline for the Session The motivation for GWR Examples from YOUR discipline Mapping OLS Residuals A good baseline for why we need GWR GWR Definitions, basic concepts Running GWR A straightforward implementation in ArcGIS GWR and some extensions

3 Basics of OLS y X Assumes a stationary process Same stimulus provokes the same response anywhere in the study area

4 Why might relationships vary spatially? Sampling variation Relationships intrinsically different across space (attitudes, preferences, contextual effects) Model misspecification

5 Applications: Ecology GWR works on trees Could have been differentiated sampling pattern creates predictable and changing levels of interaction among observations

6 Applications: Public Health Relationships vary systematically The relationship between mortality and occupational segregation and between mortality and unemployment varies across Tokyo

7 Applications: Sociology/Public Policy Missing variables (and they may very well be unknowable) The link between multifamily housing and residential burglaries varies widely even when controlling for numerous socioeconomic and neighborhood factors

8 Back up How do we know if we have nonstationarity in our model? Map residuals and test them for spatial autocorrelation if our model errs systematically with a spatial pattern then we may be on to something.

9 Poverty in the Southern U.S.

10 Our example Model Poverty Fem aleh eadedh ousehold U nem ployed Black 65andolder M etro AtLeastH ighschooleducation Based on the work of Paul Voss and Katherine Curtis These are all understood to be good predictors of poverty What kinds of spatial structures influence this data set?

11 Lab Part 1 Run our OLS model in ArcGIS Examine model output Map residuals Calculate Moran s I and Local Moran s I

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13 Our best aspatial model

14 So what now? Add more missing variables and try again Repeat the steps from the lab Accept that there is something about certain places that makes them different (spatial heterogeneity) Try GWR Test variables meant to explore interactions taking place at short distances (spatial dependence) Try Spatial Regression (Likely a spatial lag model) Assume that the correlation is a nuisance and control for it in the error term Try Spatial Regression (Likely a spatial error model)

15 Outline for Part II What is GWR Weighting in GWR

16 Geographically Weighted Regression Local statistical technique to analyze spatial variations in relationships We are not content with global averages of spatial data (climate for example) Why should we be satisfied with global averages in a statistical analysis?

17 Put another way.simpson s Paradox If we think of these points as our data grouped into colors by region we can see that the global and local models differ significantly Source: Rücker and Schumacher BMC Medical Research Methodology :34 doi: /

18 Basic definitions Spatial nonstationarity exists when the same stimulus provokes a different response in different parts of the study region Global models are statements about processes that are assumed to be stationary and, as such, are location GWR independent greater detail Local models are spatial disaggregations of global models, the results of which are location specific Spatial heterogeneity refers to spatial patterns resulting from broad similarities usually over time Spatial dependence refers to spatial patterns that result from interactions among observations

19 Spatial Heterogeneity and Spatial Dependence

20 GWR and Spatial Processes GWR is excellent at picking up broad scale regional differences spatial heterogeneity Not as effective at dealing with small scale interaction processes Too much bias in each local model That doesn t mean it wont try (and give you misleading results)

21 GWR in a nutshell Global model y X becomes y X i i i i Where i indicates that there is a set of coefficients estimated for every observation in our data set

22 The Key Difference We estimate a set of regression coefficients for each observation To do so we weight near observations more heavily than more distant ones. We may also estimate coefficients based on some local subset of observations

23 Some advantages of GWR Excellent tool for testing model specification Where does model fit look good, where are you missing something? Residuals generally lower and not spatially autocorrelated

24 Real values for β

25 Estimated Values of β in global model

26 Residuals from global model

27 Reasons to use GWR Identify model misspecification Identify nonstationarity in relationships Improved model fit (R 2, AIC, etc) Reduced spatial autocorrelation Represent context Address spatial heterogeneity when precise variables may not exist

28 You ve convinced me, what next? Run your aspatial model (as we did in 1 st lab) We will want the results and diagnostics to compare with what comes next. Decide how you are going to weight your nearby locations Fixed bandwidth Variable bandwidth User-defined bandwidth

29 It all comes down to how you weight the observations We can use a fixed bandwidth h h Wij = exp[-((dij/h) 2 )/2] Number of observations will vary, but area they represent will remain constant

30 Weighting option 2 Or we can employ an adaptive bandwidth Wij = [1-(d ij2 / h 2 )] 2 if j is one of i s N nearest neighbors Number of observations will remain fixed, but area will not be the same

31 Kernels and Weights Bandwidth specifies shape of weights curve Kernel type tells us whether we will define our bandwidth based on distance (fixed) or number of neighbors (adaptive) So how do we know what bandwidth to use?

32 Judging the appropriate bandwidth A tradeoff between Bias: we include observations that are not part of the same spatial group and Variance: we don t have enough points in our model to say anything with conviction AIC Variance Optimum Bias AICc or CV measure model fit Optimize fit to obtain best bandwidth. Bandwidth

33 To sum Weighting assumptions are very important to outcomes in GWR Fixed distance kernel is more appropriate when the distribution of your observations is relatively stable across space (e.g. size, number of neighbors). Adaptive kernel is appropriate when distribution varies across space (e.g. events are clustered or polygons are heterogeneous) Once a kernel type is selected optimization takes some of the guesswork out of it, but robustness checks are still needed

34 Residuals from the OLS model from last lesson Looks reasonably good Moran s I is still.22 and highly significant

35 Lab Run GWR model Check Residuals Check variation in coefficients

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37 Further topics/issues in GWR Where to go for next steps General troubleshooting Significance testing Outlier problems Poisson and Logistic model implementations Mixed form models

38 Other software implementations of GWR GWR 3.x (4.0 should be out soon) R (spgwr package) Stata Matlab Perhaps others I haven t heard of

39 General Troubleshooting Regional dummies BAD Eliminate them from model we are trying to show regional variation, not control for it Binary and low probability count variables Use caution, lack of variation may cause model to crash or have trouble finding a workable bandwidth

40 Significance Testing How do I know if the variation I see in my coefficients is meaningful? Could do t-test, but you will run into problems with multiple (1,387) tests Results in lots of false positives Standard correction (Bonferroni) will make any significance finding nearly impossible

41 Best Method: Monte Carlo simulation Randomly reassign all observation values (dependent and independent variables travel together) to different observation locations Each county s data gets assigned randomly to a different county Re-run GWR and record coefficients Repeat lots of times (at least 100) Define a distribution for coefficient values and compare your coefficients to this distribution

42 Other method: Fotheringham Significance Test Fotheringham 1 p e p e np p e is effective number of parameters p is the number of parameters

43 Fotheringham Significance Test F otheringham Fotheringham 1 p e p e np Type equation here (37.97) In Excel we can find the significant T-statistic using: TINV( ,1379) In R we use: qt(1-( /2),1379) Either way we get a value of ~

44 Results: Significant Nonstationarity for Percent Hispanic

45 Outlier problems Outliers cause problems for everybody, but their impact is greater for local regressions, particularly when bandwidth keeps number of observations low. In standard OLS Run model and identify observations with high or low residuals (~ +/- 4) Weight these observations less than 1 Re-run until none of the observations have extreme residuals Now do your GWR with weights assigned

46 Poisson and Logistic model forms Implementations exist in both R and GWR 3.x software Both require much greater care with respect to colinearity and lack of variation

47 Mixed-form models What if some of your variables are stationary and others have variation? Mixed-form models allow you to hold some coefficients constant while allowing others to vary Not yet implemented in any statistical package, but not that difficult from a technical standpoint

48 Concluding comments What comes next? Spatial regression Multilevel models