Ofir Cohen | PB | 414-243-4645 | coheno@pbworld.com

Sample Of Alternatives11th National Transportation Planning Applications

Conference

May 6-10, 2007, Daytona Beach, Florida

Ofir Cohen, PB, San-FranciscoChristi Willison, PB, Albuquerque

Andrew Stryker, PB, Portland

Session 14: Hot and cool topics

Agenda

Motivation – Why Sampling? Sampling Algorithm

Random Sample Smart sampling Concept- S.A.L.T Correction Factor

Optimal sample size Results – Disaggregate Commercial movement -

Ohio Statewide model Run Time improvement

Motivation

Multinomial logit function can have a great number of alternatives – Destination Choice

Utilities can be cumbersome and include many parameters.

A Micro-Simulation model evaluates the utility of each alternative every time it applies the model.

This involves in a very intensive computing time.

OHIO Statewide Disaggregate Commercial Model

Ohio State Wide model has 4248 Internal zones.

The model has 4.6M trips -> utility is evaluated ~20G times. Java based software- EXP(), LOG() are rather

“expensive functions” Some parameter are calculated on the fly and

therefore utilities can't be re-used Run time is around 80 minutes.

A faster yet unbiased approach is needed.

Destination Choice Model

Utility

Simple concepts Random selection Apply model among selected alternatives only

Random Sample-100K times R2 = 0.4243

0

50

100

150

200

250

300

350

400

450

500

0 50 100 150 200 250 300 350 400 450 500

Non Sampled Frequencies

Samp

led Fr

eque

ncies

A better Algorithm is needed

Random Sample- 100K times R2 = 0.9233

0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Non Sampled Frequencies

Samp

led Fr

eque

ncies

sample size=20 sample size=200

Y = X

Y = X

Zone 873

SALT – Sample of ALTernatives

Add Correction Factor

λ = 1/avg(dist) Define a simplified utility

Uij = ln(Total_HH+Total_Jobs) + λ*dist(i, j)

Compute a pre-defined static probability matrix (N^2).

Draw a sample of alternatives (with replacement) based on the

probability matrix

Apply the full utility for each sampled alternative and draw the

chosen alternativeOn the

fly

Pre-Calculated

λ distance dispersion effect Probabiliy based on Simple Utility, Size term=1,000

0%

10%

20%

30%

40%

50%

60%

70%

1000

Dist (m

iles) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Distance (miles)

Pro

b(i

)

Prob (λ=-1)

Prob (λ=-1/8)

Prob (λ=-1/50)

Correction Factor P(alt)= P(In sample)*P(Full Utility| being

sampled) Fix the Monte-Carlo randomness error in the

sample

Cf(ij)= -ln(freq. of j in sample set /(sample size * Pre Defined probability))

Sample Size Frequency of Chosen TAZ

Pre-Calculated Probability

Correction Factor

10 1 10% 0.00

10 1 20% 0.69

10 1 3% -1.20

Sample Size=5y = 0.9765x + 0.4695

R2 = 0.9436

0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Salt Frequency

Non

Salt

Freq

uenc

y

Salt

Linear (Salt)

Sample Size=2 y = 1.0332x - 0.6638

R2 = 0.9056

-500

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Non Salt Frequency

Sal

t Fre

quen

cy

Sample Size=10 y = 1.0079x - 0.1577

R2 = 0.9936

-500

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Non Salt Frequency

Salt

Freq

uenc

y

Sample Size=20 y = 1.006x - 0.12

R2 = 0.998

-500

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Non Salt Frequency

Salt

Freq

uenc

y

Y = X

Y = X

Y = X

Y = X

RunTime Improvement and R-Square for different Sample Sizes

0.9

0.92

0.94

0.96

0.98

1

1.02

0 20 40 60 80 100 120

Samle Size

R-S

qu

are

0

20

40

60

80

100

120

Ru

nT

ime

Ra

tio

( r

ela

tiv

e t

o

no

n s

am

ple

d m

eth

od

)

r^2

Time Ratio

Optimal Sample Size

Destination Choice DistributionSALT NON SALT

Destination Choice – Cleveland, OHSALT

NON SALT

Run Time Improvments relatives to model's alternative size

5.43

41.74

19.43

0.1

1

10

100

20 50 100

137

150

200

300

395

500

1000

2000

4248

1000

0

Number of alternatives

Ra

tio

Imp

rov

me

nts

Ratio Salt/ No Salt

SFCTA- Workplace Location using SALT

Distance Function for Sampling

0%

5%

10%

15%

20%

25%

30%

0 5 10 15 20 25 30 35 40 45 50

Distance (mi)

Uti

lity

Simplified utility under-samples trips to Santa-Clara county (~25 miles)

U(j)=jobs + Exp(λ*Max(dist(j),20))

SFCTA Workplace Location

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

14.0%

16.0%

0 5 10 15 20 25 30 35 40 45 50

Distance (mi)

Per

cen

t o

f W

ork

ers

Modeled

Observed

Conclusion

Simple – minor code modifications Statistically unbiased Reduce runtime drastically Robust - various sampling method.

Acknowledge

Peter Vovsha, PB, New-York Greg Erhardt, PB, San-Francisco

Questions?