The document compares three methods for hotspot identification: crash rate, empirical Bayes, and Bayesian hierarchical models. It evaluates the performance of the methods using four tests across different time periods on crash data from intersections in Germany. The Bayesian hierarchical model outperformed the empirical Bayes approach on most tests and time periods according to the evaluation scores. Crash rate was the least recommended method. The study concludes the Bayesian hierarchical model is superior but notes limitations from a limited number of variables and comparing crash predictions rather than observed crashes.

1. A Comparative Analysis of Empirical Bayes and
Bayesian Hierarchical Models in Hotspot Identification
Xiaoyu “Sky” Guo
Lingtao Wu, Ph.D.
Yajie Zou, Ph.D.
Lee Fawcett, Ph.D.
TRB 98th Annual Meeting
January 15, 2019
Paper No.: 19-03519

2. Background
• Importance of Hotspot Identification (HSID)
– Inefficient use of limited resources
– Additional loss of lives
• Various Methods for HSID
– Crash frequency / Crash rate
– Equivalent property damage only
– Potential for improvement
– Empirical Bayes (EB)
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3. Background
• EB Limitation: Temporary Instability in Crash Data
• EB Limitation: Parameter Estimation
• Bayesian Hierarchical Model
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1
2
3
4
5
6
7
8
2010 2011 2012 2013 2014 2015
Crash
Year
Observed Crashes
EB
"Real"
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4. Research Objective
• Perform HSID Methods
– Crash Rate
– Empirical Bayes
– Bayesian Hierarchical Model
• Assess the Performance using Four Evaluation Tests
– Site Consistency Test
– Method Consistency Test
– Total Rank Differences Test
– Poisson Mean Differences Test
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5. Methodology
• Crash Rate (CR)
𝐶𝑟𝑎𝑠ℎ 𝑅𝑎𝑡𝑒 =
𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑐𝑟𝑎𝑠ℎ
𝐸𝑥𝑝𝑜𝑠𝑢𝑟𝑒
• Empirical Bayes (EB)
𝑦 = 𝑤 ∗ 𝐸 𝑦 + 1 − 𝑤 ∗ 𝑦
where,
𝑦 = EB estimated crash
𝑤 = weight
𝐸 𝑦 = estimated crashes by crash prediction model
𝑦 = observed crash
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6. Methodology
• Bayesian Hierarchical Model (BH; see Fawcett, Thorpe et al., AAP 2017)
𝑦 𝑡 |𝜆 𝑡 ~
𝑃𝑜𝑖 𝜆 𝑡 , 𝑡 ≥ 0;
𝑁𝐵 𝑟 =
𝜆 𝑡
𝑐 𝑡 − 1
, 𝑝 =
1
𝑐 𝑡
, 𝑡 < 0,
where,
𝑦 = BH estimated crash rate
𝑡 = time period
𝜆 = adjusted mean crash rate
𝑐 = time-varying parameter to inflate the variance
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Mean = 𝜆 𝑡 ; Variance = 𝑐 𝑡 ∗ 𝜆 𝑡

7. Methodology
• Adjusted Mean Crash Rate, 𝜆 𝑡 :
𝜆 𝑡 = 𝜇 ∗ 𝑎 ∗ 𝑒 𝑏𝑡
where,
𝜇 = mean crash rate estimate
= f(predictor variables, t )
𝑎 = parameter of discrepancy
𝑏 = parameter of site-specific trend
Year 1
Year 2
0
1
2
3
4
5
Site 1 Site 2 Site 3
Crash
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8. Methodology
• Priors
– Variance inflation factor 𝒄 𝒕 : We use a prior that reflects our belief about
how influential historical counts should be for making predictions
– Discrepancy parameter 𝒂: In the absence of any knowledge relating to
site-specific crash modification factors, we use a non-informative prior
– Site-specific trend parameter 𝒃: We use a prior that enables site-specific
deviations from the global trend only if there is significant evidence for this
• Implementation
– Use R-JAGS package to make inferences
– App: https://discover.ptvgroup.com/road-safety-evaluation-prediction
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9. Period 2
Site
ID
Observed
Crash
Method A Method B
Rank
Estimated
Crash
Rank
Estimated
Crash
#1 2.4 2 2.9 2 3.2
#2 4.4 1 3.8 1 4.2
#3 1.8 3 1.2 3 2.3
Methodology
• Site Consistency Test
Period 1
Site
ID
Observed
Crash
Method A Method B
Rank
Estimated
Crash
Rank
Estimated
Crash
#1 4.3 1 3.9 2 2.5
#2 2.5 2 2.6 1 2.9
#3 3.5 3 3.2 3 2.3
Method Period 1 Period 2 Score Judgement
High Risk Site Estimated Crash Sum of Period 2
Method B is more
consistent than A.
Method A Site #1 2.9 2.9
Method B Site #2 4.2 4.2
Score B > Score A
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10. Period 2
Site
ID
Observed
Crash
Method A Method B
Rank
Estimated
Crash
Rank
Estimated
Crash
#1 2.4 2 2.9 2 3.2
#2 4.4 1 3.8 1 4.2
#3 1.8 3 1.2 3 2.3
Methodology
• Method Consistency Test
Period 1
Site
ID
Observed
Crash
Method A Method B
Rank
Estimated
Crash
Rank
Estimated
Crash
#1 4.3 1 3.9 2 2.5
#2 2.5 2 2.6 1 2.9
#3 3.5 3 3.2 3 2.3
Method Period 1 Period 2 Score Judgement
High Risk Site High Risk Site # of Similar Sites
Method B is more
consistent than A.
Method A Site #1 Site #2 0
Method B Site #2 Site #2 1
Score B > Score A
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11. Period 2
Site
ID
Observed
Crash
Method A Method B
Rank
Estimated
Crash
Rank
Estimated
Crash
#1 2.4 2 2.9 2 3.2
#2 4.4 1 3.8 1 4.2
#3 1.8 3 1.2 3 2.3
Methodology
• Total Rank Differences Test
Period 1
Site
ID
Observed
Crash
Method A Method B
Rank
Estimated
Crash
Rank
Estimated
Crash
#1 4.3 1 3.9 2 2.5
#2 2.5 2 2.6 1 2.9
#3 3.5 3 3.2 3 2.3
Method Period 1 Period 2 Score Judgement
Method A Site #1: Rank 1 Site #1: Rank 2 |1-2|=1
Method B is more
consistent than A.
Method B Site #2: Rank 1 Site #2: Rank 1 |1-1|=0
Score B < Score A
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12. • Poisson Mean Differences Test
Period 1 Period 2 All Periods
Site
ID
Observed
Crash
Method A Method B Observed
Crash
Method A Method B Rank Crash
MeanRank Estimated
Crash
Rank Estimated
Crash
Rank Estimated
Crash
Rank Estimated
Crash
#1 4.3 1 3.9 2 2.5 2.4 2 2.9 2 3.2 2 3.35
#2 2.5 2 2.6 1 2.9 4.4 1 3.8 1 4.2 1 3.45
#3 3.5 3 3.2 3 2.3 1.8 3 1.2 3 2.3 3 2.65
Methodology
Method Period 1 Period 2 Score Judgement
Method A |3.35-3.45| |3.45-3.45| 0.10+0.00=0.10
Method B is more
consistent than A.
Method B |3.45-3.45| |3.45-3.45| 0.00+0.00=0.00
Score B < Score A
Critical Crash Mean
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13. Data Description
• Data sites in the city of Halle, Germany
• Unsignalized intersections in urban area of 186 sites
• Variables
– Annual Crashes : 2004 ~ 2011
– Volume: AADT, Major Road, Minor Road
– Speed Limit: 30 kmph ~ 70 kmph, with 10 kmph increments
– Type of Intersection: Four Legs, Three Legs
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14. Result
• Evaluation Results (Top 5% High Risk Sites)
Score
α = 0.050
2004 & 2005
vs.
2006 & 2007
2006 & 2007
vs.
2008 & 2009
2008 & 2009
vs.
2010 & 2011
CR EB BH CR EB BH CR EB BH
Test 1 92 270 296 76 258 268 93 257 272
Test 2 6 6 8 3 6 9 5 6 9
Test 3 71 67 25 77 92 3 67 69 3
Test 4 389 24 22 391 22 8 363 18 24
Bold and underline indicate the highest performance;
CR = Crash Rate; EB = Empirical Bayes; BH = Bayesian Hierarchical;
Test 1: Site consistency; Test 2: Method Consistency; Test 3: Total Rank Differences; Test 4: Poisson Mean Differences
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15. Result
• Evaluation Results (Top 2.5% and 7.5% High Risk Sites)
Score
2004 & 2005
vs.
2006 & 2007
2006 & 2007
vs.
2008 & 2009
2008 & 2009
vs.
2010 & 2011
CR EB BH CR EB BH CR EB BH
𝛼 = 0.025
Test 1 32 144 129 27 137 152 37 138 138
Test 2 1 1 3 0 3 4 1 3 4
Test 3 36 27 20 44 23 3 47 5 4
Test 4 208 49 8 210 1 0 212 3 0
𝛼 = 0.075
Test 1 158 293 303 149 258 290 163 276 291
Test 2 7 5 11 8 6 13 8 8 11
Test 3 95 162 29 115 271 9 112 171 15
Test 4 300 81 32 321 40 15 315 30 27
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Bold and underline indicate the highest performance;
CR = Crash Rate; EB = Empirical Bayes; BH = Bayesian Hierarchical;
Test 1: Site consistency; Test 2: Method Consistency; Test 3: Total Rank Differences; Test 4: Poisson Mean Differences

16. Summary
• Conclusion:
– Bayesian hierarchical model outperformed the empirical Bayes
approach in almost all the tests and periods
– Crash rate is not recommended
• Limitations
– Limited numbers of variables included in models
– HSID comparison on crash predictions
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17. Questions?
Xiaoyu “Sky” Guo
Graduate Research Assistant
Texas A&M University
Texas A&M Transportation Institute
Email: xiaoyuguo@tamu.edu
TRB Paper No.: 19 - 03519
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Auckland, New Zealand. Source: http://ourauckland.aucklandcouncil.govt.nz