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CHAPTER 4. LOCALLY LINEAR SPATIAL MODELS

170

rameter smoothing prior. We will provide an example of this in the next

section.

The function bgwr returns the entire sequence of draws made for the

parameters in the problem, allowing one to: check for convergence of the

Gibbs sampler, plot posterior density functions using a function pltdens

from the Econometrics Toolbox or compute statistics from the posterior

distributions of the parameters. Most users will rely on the prt function

that calls a related function to produce printed output very similar to the

resultsprintedfor the gwr function. If youdo want to access the  estimates,

note that they are stored ina MATLAB 3-dimensional matrix structure. We

illustrate how to access these in Example 4.5 of the next section.

4.5 An applied exercise

The program in example 4.5 shows how to use the bgwr function to produce

estimates for the Anselin neighborhood crime data set. We begin by using

an improper prior for  = 1000000 and setting r = 30 to demonstrate that

the BGWR model can replicate the estimates from the GWR model. The

program plots these estimates for comparison. We produce estimates for

all three parameter smoothing priors, but given the large  = 1000000,

these relationships should not eectively enter the model. This will result in

all three sets of estimates identical to those from the GWR. We did this to

illustrate that the BGWR can replicatethe GWR estimates withappropriate

settings for the hyperparameters.

% ----- example 4.5 Using the bgwr() function

load anselin.data; % load the Anselin data set

y = anselin(:,1); nobs = length(y); x = [ones(nobs,1) anselin(:,2:3)];

east = anselin(:,4); north = anselin(:,5); tt=1:nobs;

ndraw = 250; nomit = 50;

prior.ptype = 'contiguity'; prior.rval = 30; prior.dval = 1000000;

tic; r1 = bgwr(y,x,east,north,ndraw,nomit,prior); toc;

prior.ptype = 'concentric'; prior.ctr = 20;

tic; r2 = bgwr(y,x,east,north,ndraw,nomit,prior); toc;

dist = res2.dist; [dists di] = sort(dist); % recover distance vector

prior.ptype = 'distance';

tic; r3 = bgwr(y,x,east,north,ndraw,nomit,prior); toc;

vnames = strvcat('crime','constant','income','hvalue');

% compare gwr estimates with posterior means

info2.dtype = 'exponential';

result = gwr(y,x,east,north,info2);

bgwr = result.beta(di,:);

b1 = r1.bdraw(:,di,1); b2 = r1.bdraw(:,di,2); b3 = r1.bdraw(:,di,3);

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CHAPTER 4. LOCALLY LINEAR SPATIAL MODELS

171

b1m = mean(b1); b2m = mean(b2); b3m = mean(b3);

c1 = r2.bdraw(:,:,1); c2 = r2.bdraw(:,:,2); c3 = r2.bdraw(:,:,3);

c1m = mean(c1); c2m = mean(c2); c3m = mean(c3);

d1 = r3.bdraw(:,di,1); d2 = r3.bdraw(:,di,2); d3 = r3.bdraw(:,di,3);

d1m = mean(d1); d2m = mean(d2); d3m = mean(d3);

% plot mean of vi draws (sorted by distance from #20)

plot(tt,r1.vmean(1,di),'-b',tt,r2.vmean,'--r',tt,r3.vmean(1,di),'-.k');

title('vi means'); legend('contiguity','concentric','distance');

pause;

% plot beta estimates (sorted by distance from #20)

subplot(3,1,1),

plot(tt,bgwr(:,1),'-k',tt,b1m,'--k',tt,c1m,'-.k',tt,d1m,':k');

legend('gwr','contiguity','concentric','distance');

xlabel('b1 parameter');

subplot(3,1,2),

plot(tt,bgwr(:,2),'-k',tt,b2m,'--k',tt,c2m,'-.k',tt,d2m,':k');

xlabel('b2 parameter');

subplot(3,1,3),

plot(tt,bgwr(:,3),'-k',tt,b3m,'--k',tt,c3m,'-.k',tt,d3m,':k');

xlabel('b3 parameter');

As we can see from the graph of the GWR and BGWR estimates shown

in Figure4.6, the three sets of BGWR estimates are nearly identical to the

GWR. Keep in mind that we don't recommend using the BGWR model

to replicate GWR estimates, as this problem involving 250 draws took 193

seconds for the contiguity prior, 185 for the concentric city prior and 190

seconds for the distance prior.

Example 4.6 produces estimates based on the contiguity smoothing re-

lationship with a value of r = 4 for this hyperparameter to indicate a prior

belief in heteroscedasticity or outliers. We keep the parameter smoothing

relationship from entering the model by setting an improper prior based

on  = 1000000, so there is no need to run more than a single parameter

smoothingmodel. Estimates basedonany of the three parameter smoothing

relationships would produce the same results because the large value for 

keeps this relationfrom entering the model. The focus here is on the impact

of outliers in the sample data and how BGWR robust estimates compare

to the GWR estimates based on the assumption of homoscedasticity. We

specify 250 draws with the rst 50 to be discarded for \burn-in" of the Gibbs

sampler. Figure4.7 shows a graph of the mean of the 200 draws which rep-

resent the posterior parameter estimates, compared to the non-parametric

estimates.

% ----- example 4.6 Producing robust BGWR estimates

% load the Anselin data set

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CHAPTER 4. LOCALLY LINEAR SPATIAL MODELS

172

load anselin.data;

y = anselin(:,1); nobs = length(y);

x = [ones(nobs,1) anselin(:,2:3)];

east = anselin(:,4); north = anselin(:,5);

ndraw = 250; nomit = 50;

prior.ptype = 'contiguity'; prior.rval = 4; prior.dval = 1000000;

% use diffuse prior for distance decay smoothing of parameters

result = bgwr(y,x,east,north,ndraw,nomit,prior);

vnames = strvcat('crime','constant','income','hvalue');

info.dtype = 'exponential';

result2 = gwr(y,x,east,north,info);

% compare gwr and bgwr estimates

b1 = result.bdraw(:,:,1); b1mean = mean(b1);

b2 = result.bdraw(:,:,2); b2mean = mean(b2);

b3 = result.bdraw(:,:,3); b3mean = mean(b3);

betagwr = result2.beta;

tt=1:nobs;

subplot(3,1,1),

plot(tt,betagwr(:,1),'-k',tt,b1mean,'--k');

100

b1 parameter

gwr

contiguity

concentric

distance

-4

-2

b2 parameter

-2

-1

b3 parameter

Figure 4.6: GWR and BGWR diuse prior estimates

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CHAPTER 4. LOCALLY LINEAR SPATIAL MODELS

173

legend('gwr','bgwr');

xlabel('b1 parameter');

subplot(3,1,2),

plot(tt,betagwr(:,2),'-k',tt,b2mean,'--k');

xlabel('b2 parameter');

subplot(3,1,3),

plot(tt,betagwr(:,3),'-k',tt,b3mean,'--k');

xlabel('b3 parameter');

pause;

plot(result.vmean);

xlabel('Observations');

ylabel('V_{i} estimates');

100

b1 parameter

gwr

bgwr

-4

-2

b2 parameter

-2

-1

b3 parameter

Figure 4.7: GWR and Robust BGWR estimates

We see a departure of the two sets of estimates around the rst 10

observations and around observations 30 to 45. To understand why, we

need to consider the V

terms that represent the only dierence between the

BGWR and GWR models given that we used a large  value. Figure4.9

shows the mean of the draws for the V

parameters. Large estimates for the

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CHAPTER 4. LOCALLY LINEAR SPATIAL MODELS

174

terms at observations 2, 4 and 34 indicate aberrant observations. This

accounts for the dierence in trajectory taken by the non-parametric GWR

estimates and the Bayesian estimates that robustify against these aberrant

observations.

Toillustratehow robusticationtakes place, Figure4.8shows the weight-

ing terms W

from the GWR model plotted alongside the weights adjusted

by the V

terms, W

1=2

−1

1=2

from the BGWR model. A sequence of six

observations from 30 to 35 are plotted, with a symbol `o' placed at observa-

tion #34 on the BGWR weights to help distinguish this observation in the

gure.

Beginning with observation #30, the aberrant observation #34 is down-

weighted when estimates are produced for observations #30 to #35. This

downweightingof the distance-based weight for observation #34 occurs dur-

ing estimation of 

for observations #30 through #35, all of which are near

#34 in terms of the GWR distance measure. This alternative weighting

produces the divergence between the GWR and BGWR estimates that we

observe in Figure4.7 starting around observation #30.

0.5

Observation 30

BGWR

GWR

0.5

1.5

Observation 31

0.5

1.5

Observation 32

0.5

1.5

Observation 33

0.5

Observation 34

0.5

1.5

Observation 35

Figure 4.8: GWR and BGWR Distance-based weights adjusted by V

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CHAPTER 4. LOCALLY LINEAR SPATIAL MODELS

175

Ultimately, the role of the parameters V

in the model and the prior

assigned to theseparameters reﬂects our prior knowledge that distance alone

may not be reliable as the basis for spatial relationships between variables.

If distance-based weights are used in the presence of aberrant observations,

inferences will be contaminated for whole neighborhoods and regions in our

analysis. Incorporatingthis prior knowledge turns out to be relativelysimple

in the Bayesian framework, and it appears to eectively robustify estimates

against the presence of spatial outliers.

0.8

1.2

1.4

1.6

1.8

2.2

2.4

2.6

neighborhoods

-Vi

Figure 4.9: Average V

estimates over all draws and observations

The function bgwr has associated prt and plt methods to produced

printed and graphical presentationofthe results. Some of the printedoutput

is shown below. Note that the time needed to carry out 550 draws was 289

seconds, making this estimation approach quite competitive to DARP or

GWR. The plt function produces the same output as for the GWR model.

Gibbs sampling geographically weighted regression model

Dependent Variable =

crime

R-squared

0.5650

sigma^2

0.7296

Nobs, Nvars

49,

ndraws,nomit

550,

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CHAPTER 4. LOCALLY LINEAR SPATIAL MODELS

176

r-value

4.0000

delta-value

25.0776

gam(m,k) d-prior =

50,

time in secs

= 289.2755

prior type

distance

***************************************************************

Obs =

1, x-coordinate= 35.6200, y-coordinate= 42.3800

Variable

Coefficient

t-statistic

t-probability

constant

74.130145

15.143596

0.000000

income

-2.208382

-9.378908

0.000000

hvalue

-0.197050

-5.166565

0.000004

Obs =

2, x-coordinate= 36.5000, y-coordinate= 40.5200

Variable

Coefficient

t-statistic

t-probability

constant

82.308344

20.600005

0.000000

income

-2.559334

-11.983369

0.000000

hvalue

-0.208478

-4.682454

0.000023

The next task was to implement the BGWR model with a diuse prior

on the  parameter. The results indicated that the mean of the draws for 

was: around 32 for the contiguity prior, 43 for the concentric prior and 33

for the distance prior. The BGWR estimates were almost identical to those

from the improper prior  = 1000000,so wedo not present these graphically.

Given these estimates for  with a diuse prior, we can impose the pa-

rameter restrictions by setting smaller values for , say in the range of 1

to 10. This should produce diering estimates that rely on the alternative

parameter smoothing relationships. We used  = 1 to impose the parameter

smoothing relationships fairly tightly. The resulting parameter estimates

are shown in Figure4.10.

Here we see some departure between the estimates based on alterna-

tive smoothing relationships. This raises the question of which smoothing

relationship is most consistent with the data.

We can compute posterior probabilities for each of the three models

based on alternative parameter smoothing relationships using an approxi-

mationfromLeamer (1983). Since this is agenerallyuseful wayofcomparing

alternative Bayesian models, the bgwr() function returns a vector of the

approximate log posterior for each observation. The nature of this approx-

imation as well as the computation is beyond the scope of our discussion

here. The program in example 4.7 demonstrates how to use the vector of

log posterior magnitudes to compute posterior probabilities for each model.

We set the parameter  = 0:5, to impose the three alternative parameter

smoothingrelationships even tighter than the hyperparameter value of  = 1

used to generate the estimates in Figure4.10. A successive tightening of the

CHAPTER 4. LOCALLY LINEAR SPATIAL MODELS

177

parameter smoothing relationships will show whichrelationship is most con-

sistent withthe sampledata andwhich relationship is rejectedby the sample

data. A fourth model based on  = 1000 is also estimated to test whether

the sample data rejects all three parameter smoothing relationships.

% ----- example 4.7 Posterior probabilities for models

% load the Anselin data set

load anselin.data; y = anselin(:,1); nobs = length(y);

x = [ones(nobs,1) anselin(:,2:3)]; [junk nvar] = size(x);

east = anselin(:,4); north = anselin(:,5);

ndraw = 550; nomit = 50; % estimate all three models

prior.ptype = 'contiguity';

prior.rval = 4; prior.dval = 0.5;

res1 = bgwr(y,x,east,north,ndraw,nomit,prior);

prior2.ptype = 'concentric';

prior2.ctr = 20; prior2.rval = 4; prior2.dval = 0.5;

res2 = bgwr(y,x,east,north,ndraw,nomit,prior2);

prior3.ptype = 'distance';

prior3.rval = 4; prior3.dval = 0.5;

100

b1 parameter

contiguity

concentric

distance

-4

-3

-2

-1

b2 parameter

-0.5

0.5

b3 parameter

Figure 4.10: Alternative smoothing BGWR estimates

CHAPTER 4. LOCALLY LINEAR SPATIAL MODELS

178

res3 = bgwr(y,x,east,north,ndraw,nomit,prior3);

prior4.ptype = 'distance';

prior4.rval = 4; prior4.dval = 1000;

res4 = bgwr(y,x,east,north,ndraw,nomit,prior4);

% compute posterior model probabilities

nmodels = 4;

pp = zeros(nobs,nmodels); lpost = zeros(nobs,nmodels);

lpost(:,1) = res1.logpost;

lpost(:,2) = res2.logpost;

lpost(:,3) = res3.logpost;

lpost(:,4) = res4.logpost;

psum = sum(lpost');

for j=1:nmodels

pp(:,j) = lpost(:,j)./psum';

end;

% compute posterior means for beta

bb = zeros(nobs,nvar*nmodels);

b1 = res1.bdraw(:,:,1); bb(:,1) = mean(b1)';

b2 = res1.bdraw(:,:,2); bb(:,2) = mean(b2)';

b3 = res1.bdraw(:,:,3); bb(:,3) = mean(b3)';

c1 = res2.bdraw(:,:,1); bb(:,4) = mean(c1)';

c2 = res2.bdraw(:,:,2); bb(:,5) = mean(c2)';

c3 = res2.bdraw(:,:,3); bb(:,6) = mean(c3)';

d1 = res3.bdraw(:,:,1); bb(:,7) = mean(d1)';

d2 = res3.bdraw(:,:,2); bb(:,8) = mean(d2)';

d3 = res3.bdraw(:,:,3); bb(:,9) = mean(d3)';

e1 = res4.bdraw(:,:,1); bb(:,10) = mean(e1)';

e2 = res4.bdraw(:,:,2); bb(:,11) = mean(e2)';

e3 = res4.bdraw(:,:,3); bb(:,12) = mean(e3)';

tt=1:nobs;

plot(tt,pp(:,1),'ok',tt,pp(:,2),'*k',tt,pp(:,3),'+k', tt,pp(:,4),'-');

legend('contiguity','concentric','distance','diffuse');

xlabel('observations'); ylabel('probabilities');

pause;

subplot(3,1,1),

plot(tt,bb(:,1),'-k',tt,bb(:,4),'--k',tt,bb(:,7),'-.',tt,bb(:,10),'+');

legend('contiguity','concentric','distance','diffuse');

xlabel('b1 parameter');

subplot(3,1,2),

plot(tt,bb(:,2),'-k',tt,bb(:,5),'--k',tt,bb(:,8),'-.',tt,bb(:,11),'+');

xlabel('b2 parameter');

subplot(3,1,3),

plot(tt,bb(:,3),'-k',tt,bb(:,6),'--k',tt,bb(:,9),'-.',tt,bb(:,12),'+');

xlabel('b3 parameter');

% produce a Bayesian model averaging set of estimates

bavg = zeros(nobs,nvar); cnt = 1;

for j=1:nmodels

bavg = bavg + matmul(pp(:,j),bb(:,cnt:cnt+nvar-1)); cnt = cnt+nvar;

end;

ttp = tt';

b1out = [ttp bavg(:,1) bb(:,1) bb(:,4) bb(:,7) bb(:,10)];

CHAPTER 4. LOCALLY LINEAR SPATIAL MODELS

179

in.fmt = strvcat('%4d','%8.2f','%8.2f','%8.2f','%8.2f','%8.2f');

in.cnames = strvcat('Obs','avg','contiguity','concentric',...

'distance','diffuse');

fprintf(1,'constant term parameter \n');

mprint(b1out,in);

b2out = [ttp bavg(:,2) bb(:,2) bb(:,5) bb(:,8) bb(:,11)];

fprintf(1,'household income parameter \n');

mprint(b2out,in);

b3out = [ttp bavg(:,3) bb(:,3) bb(:,6) bb(:,9) bb(:,12)];

in.fmt = strvcat('%4d','%8.3f','%8.3f','%8.3f','%8.3f','%8.3f');

fprintf(1,'house value parameter \n');

mprint(b3out,in);

The graphical display of the posterior probabilities produced by the pro-

gram in example 4.7 are shown in Figure4.11 and the parameter estimates

are shown in Figure4.12. We see some divergence between the parameter es-

timates produced by the alternative spatial smoothing priors and the model

with no smoothing prior whose estimates are graphed using \+" symbols,

but not a dramatic amount. Given the similarity of the parameter esti-

mates, we would expect relatively uniform posterior probabilities for the

four models.

In Figure4.11 the posterior probabilities for the model with no parame-

ter smoothing is graphed as a line tomake comparisonwiththe three models

that impose parameter smoothing easy. We see certain sample observations

where the parameter smoothing relationships produce lower model probabil-

ities than the model without parameter smoothing. For most observations

however, the parameter smoothing relationships are relatively consistent

with the sample data, producing posterior probabilities above the model

with no smoothing relationship. As we would expect, none of the models

dominates.

ABayesian solution to the problem of model specication and choice is

to produce a \mixed" or averaged model that relies on the posterior prob-

abilities as weights. We would simply multiply the four sets of coecient

estimates by the four probability vectors to produce a Bayesian model av-

eraging solution to the problem of which estimates are best.

Atabular presentation of this type of result is shown below. The aver-

aged results have the virtue that a single set of estimates are available from

which to draw inferences and one can feel comfortable that the inferences

are valid for a wide variety of model specications.

constant term parameter

Obs

average contiguity concentric

distance

diffuse

65.92

62.72

63.53

77.64

57.16

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