ITE Journal – December 2019 - 31

at intersections are independent. The data z = z(S) are observed at
150 points or observations (intersections), S = {s1, s2, ..., s150}. The
data model then becomes [z | λ(S)]. For the data model, a passion
distribution was assumed:
z | λ ~ Poisson (λ)

L (β, θ) = ∫ Poisson(z | λ)N(λ | μ, C) dλ

(1)

The spatial dependence in crashes was modeled in the second
level, the process model. The process model () conditional on the
process parameters was assumed as Gaussian stochastic process
(with log-link) as:
λ ( . ) | β, θc ~ G P(μ, C)

where log (μ) = x(s) β, x(s) and β are vectors of covariates and
coefficients. The spillover effect was examined by inferencing the
presence of RLC as a covariate. θc is the parameter vector with
respect to the covariance function. C is covariance function such
that C(s1, s2) = Cov(λ(s1), λ(s2)). The spatially correlated random effect
is captured by covariance function. We defined the process with a
spatially isotropic random field. This assumption implies that the
mean is constant over space (μ(s) = μ (s + h) -- μ), and the covariance
function is a function of distance: . Hence, the covariance function
takes the form:
C = σ2ρ(h)

(3)

where σ is the variance and ρ(h) is the correlation function as a
function of h. Two correlation functions, Matérn and Exponential,
are examined in this study. The Matérn and Exponential correlation
function are specified as:28

() ()

21-υ h υ
h
Matérn: ρ(h)= -- - Kυ - , α > 0, υ > 0
r(υ) α
α

(4)

h
Exponential: ρ(h)= exp (- -), α > 0
α

(5)

where α and υ are parameters of the covariance function and
estimate along with σ which construct the parameter vector θc. These
parameters control the strength and scale of the spatial autocorrelation. The Kυ is a modified Bessel function in second kind of order υ.
For parameter inference of hierarchical spatial model, we
integrated the likelihood of data:
L (β, θ) = [z | β, θ] = ∫ [z, λ | β, θ] dλ = ∫ [z | λ][λ | β, θ] dλ

The maximum likelihood estimate (MLE) of β and θ are defined
as the values that maximize the likelihood function L (β, θ), or the
log-likelihood ℓ (β, θ) = log L(β, θ). Based on Bayesian thinking, the
parameter distribution consists of the prior distribution for β. So,
the prior distribution is [β]. In this case, the posterior is given by:
[z | β, θ][β]
[β, θ | z] = ---------
∫θ [z | β, θ][β]dθ

(2)

(6)

Given the Poisson distribution of the count data model and the
Gaussian process of the process model, the likelihood function
turns into the equation (7):

(7)

[z | β, θ][β] = L(β, θ)[β]

(8)

For this problem, the MLE was not available in a closed form
and so is the posterior. In this case, Markov Chain Monte Carlo
(MCMC) simulation was used for determining the posterior.
Three goodness-of-fit measures, Mean Absolute Deviance
(MAD), Mean-squared Predictive Error (MSPE), and Deviance
Information Criteria (DIC) were used for evluating the model's
performance (equations 9 to 11):
1
MAD = -
n

Σ|z

1
MSPE = -
n

Σ (z

n

i=1

estimate

- zobserved |

(9)

estimate

- zobserved)2

(10)

n

i=1

-
DIC = D + ρD = D(θ) + 2ρD

(11)

where n is number of samples, D is the average deviance, D(θ)
is the deviance of the posterior, and ρD is the effective number of
parameters of the hierarchical model.

Data
The model was implemented using data from the Chicago RLC
program. Chicago has one of the longest-running and largest RLC
enforcement systems in the United States and has been evaluated in
a few studies.5,30 The RLCs were installed between 2008 and 2010.
In this study, four sets of data were used including; 1) crash data (all
injuries: fatal, incapacitated, non-incapacitated, and possible injury),
2) Chicago land-use, 3) RLC locations and spatial indicators, and
4) the intersections function, control, and geometry characteristics.
The studied intersections locations are shown in Figure 1.

Crash Data
Data from the periods 2005-2007 (before RLC installation) and
2010-2012 (after RLC installation) were employed for developing
the model. Crash data were available in the form of injury crash
frequency data from 150 intersections, where 90 of them were
equipped with cameras in 2010-2012. The distribution of RLCs
across the intersection in 2010 is shown in Figure 1. Data for the
period of 2008-2009 were not included in this study assuming as an
w w w .i t e.or g

D e cember 2019

31


http://www.ite.org

ITE Journal – December 2019

Table of Contents for the Digital Edition of ITE Journal – December 2019

President’s Message
Director’s Message
People in the Profession
ITE News
10th Annual ITE Collegiate Traffic Bowl Grand Championship Tournament Recap
Board Committee: Women of ITE: Allies in Design and in the Workplace
Member to Member: Ariel Farnsworth (M)
Calendar
Where in the World?
Industry News
ITE 2019 Year in Review
Impacts of Red-Light Cameras on Intersection Safety: A Bayesian Hierarchical Spatial Model
Dynamic Flashing Yellow Arrow Operations
Advisory Bike Lanes and Shoulders: Current Status and Future Possibilities
Professional Services Directory
ITE Journal – December 2019 - 1
ITE Journal – December 2019 - 2
ITE Journal – December 2019 - 3
ITE Journal – December 2019 - President’s Message
ITE Journal – December 2019 - 5
ITE Journal – December 2019 - Director’s Message
ITE Journal – December 2019 - 7
ITE Journal – December 2019 - People in the Profession
ITE Journal – December 2019 - ITE News
ITE Journal – December 2019 - 10
ITE Journal – December 2019 - 11
ITE Journal – December 2019 - 12
ITE Journal – December 2019 - 13
ITE Journal – December 2019 - 10th Annual ITE Collegiate Traffic Bowl Grand Championship Tournament Recap
ITE Journal – December 2019 - 15
ITE Journal – December 2019 - 16
ITE Journal – December 2019 - Board Committee: Women of ITE: Allies in Design and in the Workplace
ITE Journal – December 2019 - 18
ITE Journal – December 2019 - 19
ITE Journal – December 2019 - Member to Member: Ariel Farnsworth (M)
ITE Journal – December 2019 - Where in the World?
ITE Journal – December 2019 - Industry News
ITE Journal – December 2019 - ITE 2019 Year in Review
ITE Journal – December 2019 - 24
ITE Journal – December 2019 - 25
ITE Journal – December 2019 - 26
ITE Journal – December 2019 - 27
ITE Journal – December 2019 - 28
ITE Journal – December 2019 - Impacts of Red-Light Cameras on Intersection Safety: A Bayesian Hierarchical Spatial Model
ITE Journal – December 2019 - 30
ITE Journal – December 2019 - 31
ITE Journal – December 2019 - 32
ITE Journal – December 2019 - 33
ITE Journal – December 2019 - 34
ITE Journal – December 2019 - 35
ITE Journal – December 2019 - 36
ITE Journal – December 2019 - Dynamic Flashing Yellow Arrow Operations
ITE Journal – December 2019 - 38
ITE Journal – December 2019 - 39
ITE Journal – December 2019 - 40
ITE Journal – December 2019 - 41
ITE Journal – December 2019 - 42
ITE Journal – December 2019 - 43
ITE Journal – December 2019 - Advisory Bike Lanes and Shoulders: Current Status and Future Possibilities
ITE Journal – December 2019 - 45
ITE Journal – December 2019 - 46
ITE Journal – December 2019 - 47
ITE Journal – December 2019 - 48
ITE Journal – December 2019 - 49
ITE Journal – December 2019 - Professional Services Directory
ITE Journal – December 2019 - 51
ITE Journal – December 2019 - 52
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