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CEE 763 Fall 2011

Topic 1 – Fundamentals

CEE 763

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CEE 763 Fall 2011

BASIC TERMS

Traffic crash – event(s) resulting in injury or property damage

Crash frequency – number of crashes in a certain period (year)

Crash severity – KABCO levels K – Fatal injury

A – Incapacitating injury

B – Non-incapacitating evident injury

C – Possible injury

O – Property damage only (PDO)

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CEE 763 Fall 2011

BASIC TERMS (CONTINUED)

Crash type 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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CEE 763 Fall 2011

COLLISION DIAGRAMS

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CEE 763 Fall 2011

BASIC TERMS (CONTINUED)

Expected crash frequency – long-term average

Crash rate – number of crashes per unit exposure

Safety performance function (SPF) – one of the methods to predict the expected crash frequency

Accident modification factor – % crash reduction due to a treatment

4865.06 *10*365** eLAADTN predicted

AMF*NN base,ectedexp1 treatmentexpected,

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CEE 763 Fall 2011

EXAMPLE

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:

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?

4865.06 *10*365** eLAADTN predicted

Comment on the relationship between SPF and AMF

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CEE 763 Fall 2011

REVIEW OF STATISTICS

Traffic crash can normally be estimated according to the Poisson Distribution.

For Poisson distribution, the variance is equal to the mean.

Central Limit Theorem – Regardless of the population distribution, the sample means follow a normal distribution.

The standard deviation of the mean (also called standard error) can be estimated by:

!}{

ieiXP

i

)()( XEXVAR

n

ses ..

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CEE 763 Fall 2011

EXAMPLE

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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CEE 763 Fall 2011

EXAMPLE

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?

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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CEE 763 Fall 2011

REVIEW OF STATISTICS

Mean and variance for linear functions of random variables

Coefficient of variation – normalized standard deviation

]Y[VAR]X[VAR]Z[VAR

]Y[E]X[E]Z[E

YXZ

.V.C

][][][

][][][

YVARXVARZVAR

YEXEZE

YXZ

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CEE 763 Fall 2011

REVIEW OF STATISTICS

Confidence interval

),( 2/2/n

zXn

zX

01.058.2

05.096.1

2/

2/

whenz

whenz

n

= the standard deviation of the sample

= the standard deviation of the population

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CEE 763 Fall 2011

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 (mean and s.d.)

Expected at similar sites 2±0.5 0.4±0.1

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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CEE 763 Fall 2011

REGRESSION-TO-MEAN BIAS

Expected average crash frequency

Perceived

Actual reduction due to treatment

RTM Bias

Actual crash frequency

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CEE 763 Fall 201114

EMPIRICAL BAYES METHODS

Volume

Cra

sh F

requ

ence

E() -Modeled # of crashes

SPF

K - Observed # of crashes

is best estimate for the expected # of

crashes

K)()k(E}K/k{E 1

}k{E}k{VAR

Y

1

1

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

}/{)1(.. KkEds

/)(1

1

kYE

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CEE 763 Fall 2011

EXAMPLE

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:

If the standard deviation of the accidents is accident/year, what is the expected number of accidents and the standard deviation for this site?

564002240 .ADTL.}k{E

10

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CEE 763 Fall 2011

Homework

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?