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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?
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