Areal Bayesian Wombling for Geographic Boundary Analysis
Haolan Lu, Haijun Ma, and Bradley P. Carlin
[e-mail], [e-mail], and [e-mail]
Division of Biostatistics School of Public Health University of Minnesota
Bayesian Areal Wombling for Geographic Boundary Analysis p. 1/3
Background
Spatial data typically classified two ways: point-referenced (geostatistical): spatial locations are points with known coordinates areal (lattice): locations are geographic regions (e.g., counties) with adjacency information
Bayesian Areal Wombling for Geographic Boundary Analysis p. 2/3
Background
Spatial data typically classified two ways: point-referenced (geostatistical): spatial locations are points with known coordinates areal (lattice): locations are geographic regions (e.g., counties) with adjacency information Important topic in spatial statistics: boundary analysis, wherein we seek to identify regions of abrupt change. Indicated by: steep gradients in a continuous surface regional boundaries separating regions with drastically different measurements in a lattice surface
Bayesian Areal Wombling for Geographic Boundary Analysis p. 2/3
Background
Spatial data typically classified two ways: point-referenced (geostatistical): spatial locations are points with known coordinates areal (lattice): locations are geographic regions (e.g., counties) with adjacency information Important topic in spatial statistics: boundary analysis, wherein we seek to identify regions of abrupt change. Indicated by: steep gradients in a continuous surface regional boundaries separating regions with drastically different measurements in a lattice surface Focus in this talk: Boundary analysis for areal data
Bayesian Areal Wombling for Geographic Boundary Analysis p. 2/3
Background (cont'd)
Spatial boundary analysis techniques often called wombling, after a foundational paper by Womble (1951).
Bayesian Areal Wombling for Geographic Boundary Analysis p. 3/3
Background (cont'd)
Spatial boundary analysis techniques often called wombling, after a foundational paper by Womble (1951). In areal (or polygonal) wombling, a dissimilarity metric measures the difference between adjacent regions. Methods for choosing boundary elements: absolute (dissimilarity metrics greater than C ) relative (dissimilarity metrics in the top k %)
Bayesian Areal Wombling for Geographic Boundary Analysis p. 3/3
Background (cont'd)
Spatial boundary analysis techniques often called wombling, after a foundational paper by Womble (1951). In areal (or polygonal) wombling, a dissimilarity metric measures the difference between adjacent regions. Methods for choosing boundary elements: absolute (dissimilarity metrics greater than C ) relative (dissimilarity metrics in the top k %) Problems: Relative (top k %) thresholding method always finds a fixed number of boundary elements Approach is algorithmic, rather than stochastic: no model or likelihood, so statements about the "significance" of a detected boundary only relative to predetermined, often unrealistic null distributions.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 3/3
Crisp vs. Fuzzy Areal Wombling
Suppose we have responses Zi for regions i = 1, . . . , n For neighboring regions i and j and some distance metric || ||, assign the boundary likelihood value (BLV)
Dij = ||Zi - Zj || .
Bayesian Areal Wombling for Geographic Boundary Analysis p. 4/3
Crisp vs. Fuzzy Areal Wombling
Suppose we have responses Zi for regions i = 1, . . . , n For neighboring regions i and j and some distance metric || ||, assign the boundary likelihood value (BLV)
Dij = ||Zi - Zj || .
Crisp wombling: Boundary is those edges having BLV's above specified thresholds, i.e., for some c > 0,
{(i, j) : Dij > c, i adjacent to j} .
Bayesian Areal Wombling for Geographic Boundary Analysis p. 4/3
Crisp vs. Fuzzy Areal Wombling
Suppose we have responses Zi for regions i = 1, . . . , n For neighboring regions i and j and some distance metric || ||, assign the boundary likelihood value (BLV)
Dij = ||Zi - Zj || .
Crisp wombling: Boundary is those edges having BLV's above specified thresholds, i.e., for some c > 0,
{(i, j) : Dij > c, i adjacent to j} .
Fuzzy wombling: Partial membership in the boundary is permitted, say by defining the boundary membership values (BMVs) Dij / max{Dij }
ij
Bayesian Areal Wombling for Geographic Boundary Analysis p. 4/3
Example: MN colorectal cancer data
Here ni is the total number of colorectal cancers occurring in county i, and Yi is the number of these that were detected late. Let
Yi Zi = SLDRi = , i = 1, . . . , N , Ei
the standardized late detection ratio (SLDR), where the expected counts are computed via internal standardization as Ei = ni r, and r = i Yi / i ni , the statewide late detection rate.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 5/3
Example: MN colorectal cancer data
Here ni is the total number of colorectal cancers occurring in county i, and Yi is the number of these that were detected late. Let
Yi Zi = SLDRi = , i = 1, . . . , N , Ei
the standardized late detection ratio (SLDR), where the expected counts are computed via internal standardization as Ei = ni r, and r = i Yi / i ni , the statewide late detection rate. Left panel of next slide shows BoundarySEER plots of traditional wombled boundaries...
Bayesian Areal Wombling for Geographic Boundary Analysis p. 5/3
Example: MN colorectal cancer data
Left: BoundarySeer choropleth map of colorectal cancer late detection SLDRs, and crisp wombled boundaries arising from the top 20% (red) and 50% (yellow) of the Dij .
Bayesian Areal Wombling for Geographic Boundary Analysis p. 6/3
Example: MN colorectal cancer data
Left: BoundarySeer choropleth map of colorectal cancer late detection SLDRs, and crisp wombled boundaries arising from the top 20% (red) and 50% (yellow) of the Dij . Right: "Tricked" BoundarySeer map of fitted Bayesian colorectal cancer detection SLDRs, and Bayesian wombled boundaries based on the top 20% of the E(ij |y) values using G = 1500 Gibbs samples.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 6/3
Hierarchical modeling approach
We employ the Poisson log-linear form
Yi P oisson(i ) where log i = log Ei + x + i . i
where xi are region-specific covariates, and
Bayesian Areal Wombling for Geographic Boundary Analysis p. 7/3
Hierarchical modeling approach
We employ the Poisson log-linear form
Yi P oisson(i ) where log i = log Ei + x + i . i
where xi are region-specific covariates, and the random effects = (1 , . . . , N ) are given a conditionally autoregressive CAR( ) specification,
i | j=i N (i , 1/( mi )) , where N denotes the normal distribution, i is the average of the j=i that are adjacent to i , and mi is the number of these adjacencies.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 7/3
Hierarchical modeling approach
Define the BLV for boundary (i, j) as
ij = |i - j | for all i adjacent to j ,
with i = i /Ei .
Bayesian Areal Wombling for Geographic Boundary Analysis p. 8/3
Hierarchical modeling approach
Define the BLV for boundary (i, j) as
ij = |i - j | for all i adjacent to j ,
with i = i /Ei . BLVs' posterior distribution = wombled boundaries!
Bayesian Areal Wombling for Geographic Boundary Analysis p. 8/3
Hierarchical modeling approach
Define the BLV for boundary (i, j) as
ij = |i - j | for all i adjacent to j ,
with i = i /Ei . BLVs' posterior distribution = wombled boundaries! Crisp: Define ij to be part of the boundary if and only if E(ij |y) > c for some constant c > 0, or if and only if P (ij c | y) > c for some constant 0 < c < 1.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 8/3
Hierarchical modeling approach
Define the BLV for boundary (i, j) as
ij = |i - j | for all i adjacent to j ,
with i = i /Ei . BLVs' posterior distribution = wombled boundaries! Crisp: Define ij to be part of the boundary if and only if E(ij |y) > c for some constant c > 0, or if and only if P (ij c | y) > c for some constant 0 < c < 1. Fuzzy: P (ij c | y) is itself the fuzzy BMV, with MC estimate and associated sd
pij = ^
(g) #ij
>c
G
and
pij (1 - pij ) ^ ^ , G
retaining only every M th sample, for a total of G.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 8/3
MN colorectal cancer data revisited
means pij : ^
associated sd's: Posterior probability areal wombling maps for the Minnesota colorectal cancer detection data using three illustrative values of c (5, 15, and 30%), M = 5, and G = 2000.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 9/3
MN colorectal cancer data revisited
The probability of each segment being a member of the boundary decreases in c (the threshold for being a BE)
Bayesian Areal Wombling for Geographic Boundary Analysis p. 10/3
MN colorectal cancer data revisited
The probability of each segment being a member of the boundary decreases in c (the threshold for being a BE) The p maps suggest little evidence of strong boundaries ^ between counties; only a few county boundaries estimated to separate regions with true SLDRs that differ by more than 15%.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 10/3
MN colorectal cancer data revisited
The probability of each segment being a member of the boundary decreases in c (the threshold for being a BE) The p maps suggest little evidence of strong boundaries ^ between counties; only a few county boundaries estimated to separate regions with true SLDRs that differ by more than 15%. The standard deviation plots reveal that the overall uncertainty associated with each segment tends to decrease for the more extreme c (5 and 30), as we become more certain that most segments either are or are not part of the boundary.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 10/3
MN colorectal cancer data revisited
The probability of each segment being a member of the boundary decreases in c (the threshold for being a BE) The p maps suggest little evidence of strong boundaries ^ between counties; only a few county boundaries estimated to separate regions with true SLDRs that differ by more than 15%. The standard deviation plots reveal that the overall uncertainty associated with each segment tends to decrease for the more extreme c (5 and 30), as we become more certain that most segments either are or are not part of the boundary. Animated sequences of crisp boundaries (as in www.biostat.umn.edu/haolanl/movie.gif) may be more enlightening than maps of posterior standard deviations...
Bayesian Areal Wombling for Geographic Boundary Analysis p. 10/3
MN colorectal cancer data revisited
To assess whether County 63 (Red Lake, a T-shaped county in the NW) is truly "isolated" from its only two neighbors (Counties 57 and 60), by evaluating
p63 P (63,57 > c 63,60 > c | y) . ~
Bayesian Areal Wombling for Geographic Boundary Analysis p. 11/3
MN colorectal cancer data revisited
To assess whether County 63 (Red Lake, a T-shaped county in the NW) is truly "isolated" from its only two neighbors (Counties 57 and 60), by evaluating
p63 P (63,57 > c 63,60 > c | y) . ~
Since the {ij } come from the joint posterior of {ij }, Monte Carlo estimates are available = simultaneous inference without a multiple comparisons problem (i.e. no need for Bonferroni correction).
(g)
Bayesian Areal Wombling for Geographic Boundary Analysis p. 11/3
MN colorectal cancer data revisited
To assess whether County 63 (Red Lake, a T-shaped county in the NW) is truly "isolated" from its only two neighbors (Counties 57 and 60), by evaluating
p63 P (63,57 > c 63,60 > c | y) . ~
Since the {ij } come from the joint posterior of {ij }, Monte Carlo estimates are available = simultaneous inference without a multiple comparisons problem (i.e. no need for Bonferroni correction). If we womble the spatial residuals i (instead of the i ),
(g)
Bayesian Areal Wombling for Geographic Boundary Analysis p. 11/3
MN colorectal cancer data revisited
To assess whether County 63 (Red Lake, a T-shaped county in the NW) is truly "isolated" from its only two neighbors (Counties 57 and 60), by evaluating
p63 P (63,57 > c 63,60 > c | y) . ~
Since the {ij } come from the joint posterior of {ij }, Monte Carlo estimates are available = simultaneous inference without a multiple comparisons problem (i.e. no need for Bonferroni correction). If we womble the spatial residuals i (instead of the i ), Boundaries now separate regions with differing unmodeled heterogeneity = help identify missing covariates!
(g)
Bayesian Areal Wombling for Geographic Boundary Analysis p. 11/3
MN colorectal cancer data revisited
To assess whether County 63 (Red Lake, a T-shaped county in the NW) is truly "isolated" from its only two neighbors (Counties 57 and 60), by evaluating
p63 P (63,57 > c 63,60 > c | y) . ~
Since the {ij } come from the joint posterior of {ij }, Monte Carlo estimates are available = simultaneous inference without a multiple comparisons problem (i.e. no need for Bonferroni correction). If we womble the spatial residuals i (instead of the i ), Boundaries now separate regions with differing unmodeled heterogeneity = help identify missing covariates! No significant boundaries = (covariate-adjusted) mapwide equity, a.k.a. "environmental justice"
(g)
Bayesian Areal Wombling for Geographic Boundary Analysis p. 11/3
Alternative: Modeling adjacency
Note that the univariate CAR model can be written as
i | (-i) N
j
wij j
j
wij
,
1
j
wij
,
where the weights wij are equal to 1 if i = j and regions i and j are adjacent, and 0 otherwise.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 12/3
Alternative: Modeling adjacency
Note that the univariate CAR model can be written as
i | (-i) N
j
wij j
j
wij
,
1
j
wij
,
where the weights wij are equal to 1 if i = j and regions i and j are adjacent, and 0 otherwise. The CAR remains a valid distributional specification provided 0 wij 1 = more possibilities for spatial smoothing:
Bayesian Areal Wombling for Geographic Boundary Analysis p. 12/3
Alternative: Modeling adjacency
Note that the univariate CAR model can be written as
i | (-i) N
j
wij j
j
wij
,
1
j
wij
,
where the weights wij are equal to 1 if i = j and regions i and j are adjacent, and 0 otherwise. The CAR remains a valid distributional specification provided 0 wij 1 = more possibilities for spatial smoothing: Choose the wij inversely proportional to the distance separating the centroids of regions i and j , or
Bayesian Areal Wombling for Geographic Boundary Analysis p. 12/3
Alternative: Modeling adjacency
Note that the univariate CAR model can be written as
i | (-i) N
j
wij j
j
wij
,
1
j
wij
,
where the weights wij are equal to 1 if i = j and regions i and j are adjacent, and 0 otherwise. The CAR remains a valid distributional specification provided 0 wij 1 = more possibilities for spatial smoothing: Choose the wij inversely proportional to the distance separating the centroids of regions i and j , or Think of the wij as additional unknown parameters to be estimated, allowing the data to help determine the degree and nature of spatial smoothing.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 12/3
Alternative: Modeling adjacency
Mimicking an idea from statistical social network analysis (Wang and Wong, 1987; Hoff et al., 2002), model the wij as
wij |pij Bernoulli(pij ) , where log pij 1 - pij = z . ij
Bayesian Areal Wombling for Geographic Boundary Analysis p. 13/3
Alternative: Modeling adjacency
Mimicking an idea from statistical social network analysis (Wang and Wong, 1987; Hoff et al., 2002), model the wij as
wij |pij Bernoulli(pij ) , where log pij 1 - pij = z . ij
Example: Let z1ij = 1 (so that 1 is an intercept parameter), z2ij = dij , the distance between the centroids of regions i and j , z3ij = (areai + areaj )/2, the average area of the two regions, and z4ij = |xi - xj |, the absolute difference of some regional covariate (say, percent urban, or percent of residents who are smokers).
Bayesian Areal Wombling for Geographic Boundary Analysis p. 13/3
Alternative: Modeling adjacency
Crisp boundaries based on wij : those segments ij having P (wij = 0 | y) > c .
Bayesian Areal Wombling for Geographic Boundary Analysis p. 14/3
Alternative: Modeling adjacency
Crisp boundaries based on wij : those segments ij having P (wij = 0 | y) > c . Fuzzy boundaries based on wij : use the P (wij = 0 | y) values themselves as the BMVs.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 14/3
Alternative: Modeling adjacency
Crisp boundaries based on wij : those segments ij having P (wij = 0 | y) > c . Fuzzy boundaries based on wij : use the P (wij = 0 | y) values themselves as the BMVs. Reilly (2001) showed is estimable even under a noninformative prior.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 14/3
Alternative: Modeling adjacency
Crisp boundaries based on wij : those segments ij having P (wij = 0 | y) > c . Fuzzy boundaries based on wij : use the P (wij = 0 | y) values themselves as the BMVs. Reilly (2001) showed is estimable even under a noninformative prior. MCMC sampling of , , , , and W can proceed by a tuned mixture of Gibbs and Metropolis steps.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 14/3
Alternative: Modeling adjacency
Crisp boundaries based on wij : those segments ij having P (wij = 0 | y) > c . Fuzzy boundaries based on wij : use the P (wij = 0 | y) values themselves as the BMVs. Reilly (2001) showed is estimable even under a noninformative prior. MCMC sampling of , , , , and W can proceed by a tuned mixture of Gibbs and Metropolis steps.
Try it with simulated data, arising from 0 = 1, 1 = -2, 0 = -1, and = 30.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 14/3
Alternative: Modeling adjacency
Crisp boundaries based on wij : those segments ij having P (wij = 0 | y) > c . Fuzzy boundaries based on wij : use the P (wij = 0 | y) values themselves as the BMVs. Reilly (2001) showed is estimable even under a noninformative prior. MCMC sampling of , , , , and W can proceed by a tuned mixture of Gibbs and Metropolis steps.
Try it with simulated data, arising from 0 = 1, 1 = -2, 0 = -1, and = 30. The simulation of is facilitated by WinBUGS, and log Ei 's are borrowed from a Minnesota breast cancer late detection data set.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 14/3
Example: Simulated data
Left: Map of simulated "true" boundaries (1 - wij ), where thick dark lines indicate boundaries and thin blue lines indicate adjacency between two regions.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 15/3
Example: Simulated data
Left: Map of simulated "true" boundaries (1 - wij ), where thick dark lines indicate boundaries and thin blue lines indicate adjacency between two regions. Right: Map of simulated "true" i on a blue color scale.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 15/3
Example: Simulated data
Left: Map of simulated "true" boundaries (1 - wij ), where thick dark lines indicate boundaries and thin blue lines indicate adjacency between two regions. Right: Map of simulated "true" i on a blue color scale. Note: Most of the boundaries in the left panel are preserved in the right panel!
Bayesian Areal Wombling for Geographic Boundary Analysis p. 15/3
Example: Simulated data
Left: Map of posterior mean of 1 - pij with corresponding boundaries shaded in color. Note: The majority of these lines match with those in L panel of previous figure!
Bayesian Areal Wombling for Geographic Boundary Analysis p. 16/3
Example: Simulated data
Left: Map of posterior mean of 1 - pij with corresponding boundaries shaded in color. Note: The majority of these lines match with those in L panel of previous figure! Right: Map of posterior means of i on blue color scale. Note: Overall spatial pattern remains, but with smoothing of the i among neighbors.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 16/3
Extension: Boundary Adjacency
The previous approaches (especially those based on ij ) tend to create networks of disconnected boundary segments. We would prefer an approach that drew continuous boundaries across the map.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 17/3
Extension: Boundary Adjacency
The previous approaches (especially those based on ij ) tend to create networks of disconnected boundary segments. We would prefer an approach that drew continuous boundaries across the map. That is, given that a segment is part of the boundary, we'd like our model to favor including neighboring segments in the boundary as well.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 17/3
Extension: Boundary Adjacency
The previous approaches (especially those based on ij ) tend to create networks of disconnected boundary segments. We would prefer an approach drew that continuous boundaries across the map. That is, given that a segment is part of the boundary, we'd like our model to favor including neighboring segments in the boundary as well. But the CAR model is ideal for this type of modeling! We simply need to define a second CAR model on the boundary segment space, in addition to the one we already have on the areal unit space.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 17/3
Extension: Boundary Adjacency
The previous approaches (especially those based on ij ) tend to create networks of disconnected boundary segments. We would prefer an approach that drew continuous boundaries across the map. That is, given that a segment is part of the boundary, we'd like our model to favor including neighboring segments in the boundary as well. But the CAR model is ideal for this type of modeling! We simply need to define a second CAR model on the boundary segment space, in addition to the one we already have on the areal unit space. That is, if W is the N N (random) adjacency matrix for the counties, now we also have W , an Nadj Nadj (fixed) adjacency matrix for the boundary segments, where Nadj is the total number of unique county adjacencies.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 17/3
New Approach: 2-Level CAR
That is, we need to expand our logit model to
log pij 1 - pij = z + ij , ij
where
CAR(, W )
and W is a 0-1 adjacency matrix on the boundary segment space, with adjacency defined by the map (fixed; no covariates this time).
Bayesian Areal Wombling for Geographic Boundary Analysis p. 18/3
New Approach: 2-Level CAR
That is, we need to expand our logit model to
log pij 1 - pij = z + ij , ij
where
CAR(, W )
and W is a 0-1 adjacency matrix on the boundary segment space, with adjacency defined by the map (fixed; no covariates this time). Boundaries may again be obtained from posterior summarization of the ij , the pij , or the wij .
Bayesian Areal Wombling for Geographic Boundary Analysis p. 18/3
New Approach: 2-Level CAR
That is, we need to expand our logit model to
log pij 1 - pij = z + ij , ij
where
CAR(, W )
and W is a 0-1 adjacency matrix on the boundary segment space, with adjacency defined by the map (fixed; no covariates this time). Boundaries may again be obtained from posterior summarization of the ij , the pij , or the wij . To check, we generate data from a 4 4 "template" having "true boundary" separating 6 regions (the 4 in the bottom row and the middle two in the 3rd row) from the rest...
Bayesian Areal Wombling for Geographic Boundary Analysis p. 18/3
New Approach: 2-Level CAR
Raw Data
6 6
L&C, Pr(delta_ij>c|data)
[-2.17,-1.51) [-1.51,-0.85) [-0.85,-0.19) [-0.19,0.47) [0.47,1.13) [1.13,1.79]
[0.06,0.75] (0.75,0.94]
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Left: Map of a simulated data set from our model.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 19/3
New Approach: 2-Level CAR
Raw Data
6 6
L&C, Pr(delta_ij>c|data)
[-2.17,-1.51) [-1.51,-0.85) [-0.85,-0.19) [-0.19,0.47) [0.47,1.13) [1.13,1.79]
[0.06,0.75] (0.75,0.94]
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Left: Map of a simulated data set from our model. Right: Posterior probability that ij exceeds the cutoff c, LC method (nonrandom adjacency matrix W ).
Bayesian Areal Wombling for Geographic Boundary Analysis p. 19/3
New Approach: 2-Level CAR
Raw Data
6 6
L&C, Pr(delta_ij>c|data)
[-2.17,-1.51) [-1.51,-0.85) [-0.85,-0.19) [-0.19,0.47) [0.47,1.13) [1.13,1.79]
[0.06,0.75] (0.75,0.94]
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Left: Map of a simulated data set from our model. Right: Posterior probability that ij exceeds the cutoff c, LC method (nonrandom adjacency matrix W ).
Note that there are six "true" boundary segments; LC finds only 3 of them. It is badly fooled by the surprisingly low values in the northeast corner and the surprisingly high value in the 3rd row, last column.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 19/3
Results: 2-Level CAR
CAR2, Pr(delta_ij>c|data)
6 6
CAR2, post.mean of (1-pij)
[0.07,0.82] (0.82,0.97]
[0.12,0.17) [0.17,0.21) [0.21,0.25) [0.25,0.29) [0.29,0.34) [0.34,0.38]
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Left: Posterior probability that ij exceeds the cutoff c (fuzzy boundaries), 2-Level CAR (CAR2).
Bayesian Areal Wombling for Geographic Boundary Analysis p. 20/3
Results: 2-Level CAR
CAR2, Pr(delta_ij>c|data)
6 6
CAR2, post.mean of (1-pij)
[0.07,0.82] (0.82,0.97]
[0.12,0.17) [0.17,0.21) [0.21,0.25) [0.25,0.29) [0.29,0.34) [0.34,0.38]
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Left: Posterior probability that ij exceeds the cutoff c (fuzzy boundaries), 2-Level CAR (CAR2). Right: Posterior mean of 1 - pij , 2-Level CAR.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 20/3
Results: 2-Level CAR
CAR2, Pr(delta_ij>c|data)
6 6
CAR2, post.mean of (1-pij)
[0.07,0.82] (0.82,0.97]
[0.12,0.17) [0.17,0.21) [0.21,0.25) [0.25,0.29) [0.29,0.34) [0.34,0.38]
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Left: Posterior probability that ij exceeds the cutoff c (fuzzy boundaries), 2-Level CAR (CAR2). Right: Posterior mean of 1 - pij , 2-Level CAR.
When applied to the ij 's, the CAR2 method performs no better than LC. But when applied to the pij 's, the method now correctly finds all six boundary segments.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 20/3
Current Work
Compare methods based on ij and wij by simulating the average probability that they correctly classify each potential boundary segment
Bayesian Areal Wombling for Geographic Boundary Analysis p. 21/3
Current Work
Compare methods based on ij and wij by simulating the average probability that they correctly classify each potential boundary segment Various remedies for handling "islands" (regions with no neighbors) when modeling adjacency
Bayesian Areal Wombling for Geographic Boundary Analysis p. 21/3
Current Work
Compare methods based on ij and wij by simulating the average probability that they correctly classify each potential boundary segment Various remedies for handling "islands" (regions with no neighbors) when modeling adjacency Investigate L1 (absolute difference) CARs, and sensitivity to the choice of priors p(i ) that control the effect of the covariate data Zij
Bayesian Areal Wombling for Geographic Boundary Analysis p. 21/3
Current Work
Compare methods based on ij and wij by simulating the average probability that they correctly classify each potential boundary segment Various remedies for handling "islands" (regions with no neighbors) when modeling adjacency Investigate L1 (absolute difference) CARs, and sensitivity to the choice of priors p(i ) that control the effect of the covariate data Zij Try to increase the impact of response data Yi on the wombled boundaries (relative to the covariates Zij ) while retaining boundary connectedness
Bayesian Areal Wombling for Geographic Boundary Analysis p. 21/3
Current Work
Compare methods based on ij and wij by simulating the average probability that they correctly classify each potential boundary segment Various remedies for handling "islands" (regions with no neighbors) when modeling adjacency Investigate L1 (absolute difference) CARs, and sensitivity to the choice of priors p(i ) that control the effect of the covariate data Zij Try to increase the impact of response data Yi on the wombled boundaries (relative to the covariates Zij ) while retaining boundary connectedness by eliminating the CAR(, W ) model and the i 's altogether (but retaining the CAR(, W ) and the ij )
Bayesian Areal Wombling for Geographic Boundary Analysis p. 21/3
Current Work
Compare methods based on ij and wij by simulating the average probability that they correctly classify each potential boundary segment Various remedies for handling "islands" (regions with no neighbors) when modeling adjacency Investigate L1 (absolute difference) CARs, and sensitivity to the choice of priors p(i ) that control the effect of the covariate data Zij Try to increase the impact of response data Yi on the wombled boundaries (relative to the covariates Zij ) while retaining boundary connectedness by eliminating the CAR(, W ) model and the i 's altogether (but retaining the CAR(, W ) and the ij ) by retaining both CAR models, but fitting them simultaneously, rather than hierarchically
Bayesian Areal Wombling for Geographic Boundary Analysis p. 21/3
Connected, Yi-dominated boundaries
Eliminating the CAR(, W ) model and the i 's: Suppose we think of the original data Yi as normal. Then the differences Dij = Yi - Yj are also normal (note differences, not absolute differences here).
Bayesian Areal Wombling for Geographic Boundary Analysis p. 22/3
Connected, Yi-dominated boundaries
Eliminating the CAR(, W ) model and the i 's: Suppose we think of the original data Yi as normal. Then the differences Dij = Yi - Yj are also normal (note differences, not absolute differences here). Suppose Dij N (ij , 1/e ), where ij = 0 + zij 1 + ij . Let zij = xi - xj (again, difference not absolute difference here), and we again let CAR(, W ).
Bayesian Areal Wombling for Geographic Boundary Analysis p. 22/3
Connected, Yi-dominated boundaries
Eliminating the CAR(, W ) model and the i 's: Suppose we think of the original data Yi as normal. Then the differences Dij = Yi - Yj are also normal (note differences, not absolute differences here). Suppose Dij N (ij , 1/e ), where ij = 0 + zij 1 + ij . Let zij = xi - xj (again, difference not absolute difference here), and we again let CAR(, W ). This model eliminates the hard-to-estimate Wij (Bernoulli) parameters, and retains only the "second" CAR model, which captures the similarity of neighboring boundary segments on the map.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 22/3
Connected, Yi-dominated boundaries
Eliminating the CAR(, W ) model and the i 's: Suppose we think of the original data Yi as normal. Then the differences Dij = Yi - Yj are also normal (note differences, not absolute differences here). Suppose Dij N (ij , 1/e ), where ij = 0 + zij 1 + ij . Let zij = xi - xj (again, difference not absolute difference here), and we again let CAR(, W ). This model eliminates the hard-to-estimate Wij (Bernoulli) parameters, and retains only the "second" CAR model, which captures the similarity of neighboring boundary segments on the map. This model should deliver ij -based wombled boundaries that are more connected than before
Bayesian Areal Wombling for Geographic Boundary Analysis p. 22/3
Connected, Yi-dominated boundaries
Simultaneous fitting of the two CAR models: Suppose we replace the Poisson mean structure with
log i = log Ei + x + i + i , i
where CAR(, W ) and CAR(, W ) as before.
Bayesian Areal Wombling for Geographic Boundary Analysis p. 23/3
Connected, Yi-dominated boundaries
Simultaneous fitting of the two CAR models: Suppose we replace the Poisson mean structure with
log i = log Ei + x + i + i , i
where CAR(, W ) and CAR(, W ) as before. Now i , the average of the edge effects for those edges that comprise region i, contributes directly to the mean structure (instead of to the variance structure via W ). The ij capture signed agreement since:
Bayesian Areal Wombling for Geographic Boundary Analysis p. 23/3
Connected, Yi-dominated boundaries
Simultaneous fitting of the two CAR models: Suppose we replace the Poisson mean structure with
log i = log Ei + x + i + i , i
where CAR(, W ) and CAR(, W ) as before. Now i , the average of the edge effects for those edges that comprise region i, contributes directly to the mean structure (instead of to the variance structure via W ). The ij capture signed agreement since: ij = 0 when Yi and Yj are both larger or both smaller than expected
Bayesian Areal Wombling for Geographic Boundary Analysis p. 23/3
Connected, Yi-dominated boundaries
Simultaneous fitting of the two CAR models: Suppose we replace the Poisson mean structure with
log i = log Ei + x + i + i , i
where CAR(, W ) and CAR(, W ) as before. Now i , the average of the edge effects for those edges that comprise region i, contributes directly to the mean structure (instead of to the variance structure via W ). The ij capture signed agreement since: ij = 0 when Yi and Yj are both larger or both smaller than expected ij 0 when one o...