Transcript Topic 1 - Fundamentals

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Topic 1 – Fundamentals
CEE 763
CEE 763
Fall 2011
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BASIC TERMS

Traffic crash – event(s) resulting in injury or
property damage
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Crash frequency – number of crashes in a certain
period (year)
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Crash severity – KABCO levels
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K – Fatal injury
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A – Incapacitating injury
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B – Non-incapacitating evident injury
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C – Possible injury
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O – Property damage only (PDO)
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BASIC TERMS (CONTINUED)
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Crash type
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Rear-end; sideswipe; angle; turning; head-on; run-off the road; fixed
object; animal; pedestrian; out of control; work zone
Collision diagrams
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COLLISION DIAGRAMS
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BASIC TERMS (CONTINUED)
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Expected crash frequency – long-term average
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Crash rate – number of crashes per unit exposure
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Safety performance function (SPF) – one of the
methods to predict the expected crash frequency
N predicted  AADT * L * 365*106 * e0.4865
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Accident modification factor – % crash reduction
due to a treatment
Nexpected,treatment 1  Nexpected,base * AMF
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EXAMPLE
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A roadway section has a length of 2.5 miles and an AADT of
20,000. What is the expected crash frequency per year for
this roadway section if the SPF is as shown:
N predicted  AADT * L * 365*106 * e0.4865
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An intersection with a permitted LT control is converted to a
protected LT control. If the AMF for protected LT is 0.90.
What is the percent reduction in crash after the control
change? Suppose the intersection has a crash frequency of
10 crashes per year with permitted LT control, what is the
expected number of crashes per year after the change of the
control?
Comment on the relationship between SPF and AMF
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REVIEW OF STATISTICS
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Traffic crash can normally be estimated according to the
Poisson Distribution.
i
P{ X  i}  e  


i!
For Poisson distribution, the variance is equal to the mean.
VAR( X )  E ( X )  
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Central Limit Theorem – Regardless of the population
distribution, the sample means follow a normal distribution.
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The standard deviation of the mean (also called standard
error) can be estimated by:
s.e. 
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EXAMPLE
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On average, a railroad crossing has about 2 crashes in three
years. What is the probability that there are more than 1
crashes in a year?
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EXAMPLE
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Ten random samples were obtained as the following:
2, 4, 6, 1, 6, 8, 10, 3, 5, 3. Calculate the standard error of the
sample. What is the implication of this calculated standard
error?
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Exercise: In Excel, generate 100 random samples from a
uniform distribution with a mean of 10 (i.e., U[0,20]). Repeat
10 times of the sampling process. Compare the estimated
standard error from the initial 100 samples and the standard
deviation of the sample means from the 10-time sampling
data.
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REVIEW OF STATISTICS
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Mean and variance for linear functions of random variables
Z  X Y
E[ Z ]  E[ X ]  E[ Y ]
VAR[ Z ]  VAR[ X ]  VAR[ Y ]
Z  X Y
E[ Z ]  E[ X ]  E[Y ]
VAR[ Z ]  VAR[ X ]  VAR[Y ]
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Coefficient of variation – normalized standard deviation

C .V . 

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REVIEW OF STATISTICS
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Confidence interval
( X  z / 2
z / 2  1.96
z / 2  2.58

n

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
n
, X  z / 2

n
)
  0.05
when   0.01
when
= the standard deviation of the sample
= the standard deviation of the population
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EXAMPLE
Two sites have the following crash data:
Road section
X
Y
Length, mi
1
0.2
Expected crash this site
5±2.2
1±1.0
Expected at similar sites
2±0.5
0.4±0.1
(mean and s.d.)
Which site has more reliable data, assuming the
performance measure is “excess of crash frequency”? If the
limiting coefficient of variation is set at 0.05, what is the
typical estimation error with respect to the mean?
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REGRESSION-TO-MEAN BIAS
RTM Bias
Perceived
Expected average
crash frequency
Actual reduction
due to treatment
Actual crash frequency
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EMPIRICAL BAYES METHODS
E{ k / K }  E( k )  ( 1   )K
s.d.  (1   )  E{k / K}
1
VAR{ k }
1 Y
E{ k }
1

1  YE(k ) / 
Crash Frequence

K - Observed # of crashes
E{k/K} is best
estimate for the
expected # of
crashes
SPF
E(k) -Modeled # of crashes
E(k) is the predicted value at similar sites,
in crash/year
Y is the analysis period in number of years
φ is over-dispersion factor
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EXAMPLE
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A road segment is 1.8 miles long, has an ADT of
4,000 and recorded 12 accidents in the last year.
The SPF for similar roads is shown in the
equation, where L is length of the segment in
miles:
E{ k }  0.0224 L  ADT 0.564
If the standard deviation of the accidents is 10
accident/year, what is the expected number of
accidents and the standard deviation for this site?
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Homework
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Now the same road segment has 3 years of
accident counts (12, 16, 8). What is the expected
number of accidents and the standard deviation
for this site?
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