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Transportation leadership you can trust.
presented topresented to
12th TRB National Planning Applications Conference12th TRB National Planning Applications ConferenceHouston, TX Houston, TX
presented bypresented by
Dan BeaganDan BeaganCambridge Systematics, Inc.Cambridge Systematics, Inc.
May 18, 2009May 18, 2009
Trip Table Estimation from CountsScience or Magic?
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Science vs. Magic
Any sufficiently advanced technology is indistinguishable from magic
− Arthur C. Clarke, “Profiles of The Future,” (Clarke’s third law)
When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong
− Arthur C. Clarke, “Profiles of The Future”, (Clarke’s first law)
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Science vs. Magic
Magic
If results are unexpected, the conditions for the spell weren’t “right”
Not expected to duplicate results
Works only for believers
Science
If results are unexpected, the expectations were wrong
Will always duplicate results
Works for believers and non-believers
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Transportation Planning
Expected to be based on science
Most methods accepted as scientific
Trip Table Estimation from counts not always accepted
• Method not always understood
–”If you can believe results”
• Method is widely available− Included in standard software packages
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Scientific JustificationTrip Table Estimation from Counts
Statistical Principle behind Maximum Entropy
Maximum Entropy Techniques in Transportation
Applications of Matrix Estimation from Counts
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Maximum Entropy
Most probable state is the one with the Maximum Entropy
Statistically, for a given macrostate, the most probable mesostate is the one with the maximum number of microstates
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Snake Snake EyesEyes
Acey Acey DeuceDeuce
Easy Easy FourFour
Fever Fever FiveFive Easy SixEasy Six
Natural Natural or Seven or Seven
OutOut
Acey Acey DeuceDeuce
Hard Hard FourFour
Fever Fever FiveFive Easy SixEasy Six
Natural Natural or Seven or Seven
OutOut
Easy Easy EightEight
Easy Easy FourFour
Fever Fever FiveFive Hard SixHard Six
Natural Natural or Seven or Seven
OutOut
Easy Easy EightEight
Nine Nine (Nina)(Nina)
Fever Fever FiveFive Easy SixEasy Six
Natural Natural or Seven or Seven
OutOut
Hard Hard EightEight
Nine Nine (Nina)(Nina)
Easy Easy TenTen
Easy SixEasy SixNatural Natural
or Seven or Seven OutOut
Easy Easy EightEight
Nine Nine (Nina)(Nina) Hard TenHard Ten
Yo Yo (Yo-(Yo-
leven)leven)
Natural Natural or Seven or Seven
OutOut
Easy Easy EightEight
Nine Nine (Nina)(Nina)
Easy Easy TenTen
Yo Yo (Yo-(Yo-
leven)leven)BoxcarsBoxcars
In “craps” (macrostate)
the most probable roll (mesostate) is a seven, a natural,
because there are more ways (microstates)to make a seven than any other roll
Game of Dice
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The economic impact of three individuals traveling from one home to three geographically different jobs (microstates) may not be the same, but the traffic impact of the trip table (mesostates) is identical
Trip Tables
CurlyCurly
LarryLarry
MoeMoe
CurlyCurly
MoeMoe
LarryLarry
LarryLarry
MoeMoe
CurlyCurly
MoeMoe
CurlyCurly
LarryLarry
MoeMoe
LarryLarry
CurlyCurly
LarryLarry
CurlyCurly
MoeMoe
HOMEHOME HOMEHOME
Job 1Job 1
Job 2Job 2
Job 3Job 3
Job 1Job 1
Job 2Job 2
Job 3Job 3
HOMEHOME
Job 1Job 1
Job 2Job 2
Job 3Job 3
HOMEHOME
Job 1Job 1
Job 2Job 2
Job 3Job 3
HOMEHOME
Job 1Job 1
Job 2Job 2
Job 3Job 3
HOMEHOME
Job 1Job 1
Job 2Job 2
Job 3Job 3
MICROSTATE 1MICROSTATE 1 MICROSTATE 2MICROSTATE 2 MICROSTATE 3MICROSTATE 3
MICROSTATE 4MICROSTATE 4 MICROSTATE 5MICROSTATE 5 MICROSTATE 6MICROSTATE 6
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A solution trip table, t ij, given an existing trip table, T ij , will be a maximum entropy trip table, if the following equation is solved
The solution will depend on the constraints imposed
Trip Tables Maximum Entropy
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Trip Tables Maximum Entropy
Solving for the trip table relies on the following mathematical principles
• The maximum of any monotonically increasing function of tij will have the same solution trip table, tij
• Sterling’s approximation of X !, X ln X – X, is a monotonically increasing function
• LaGrangian multipliers can be used to combine the target and constraint equations
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Fratar Growth Factor
For an existing table, Tij,
find a new table, tij,
given growth targetsoi for the origins and
dj for the destinations
Also known as Furness or IPF, Iterative Proportional Fitting
Choose values for K’i; solve
for K’’j, resolve for K’j;iterate
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A. G. Wilson’s Gravity Model
Traditionally there is no initial table, Tij, so Tij =1
Total cost, C, does not need to be known
Choose values for K’i,
solve for K’’j, then K’i
and iterate
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Logit Mode Split
Traditionally there is no initial Table, Tm, so Tm =1
Indices are modes m for each ij pair
Total utility, U, does not need to be known
Stating the solution as a percentage eliminates the constants
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Matrix Estimation from Counts
A “seed” table, Tij, may be available; otherwise Tij = 1
Constraints exist for those links a which have counts, Va
The probability of traveling between pair ij on link a, pija can be found from assignment scripts
• E.g., for AON, pija = 1 when link a is on the path between i and j
A set of simultaneous equations, which can be solved iteratively, can be developed by substituting the solution into the constraints
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(OD)ME Trip Table
What should you use for the initial trip table?
• Invariant to Uniform Scaling
How many counts and where should they be located?
• Network Sensor Location Problem
How good is the solution?
• Maximum Possible Relative Error
How well does the solution table validate to counts?
• Maximum Entropy
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(OD)ME Trip Table Applications
Subareas
• TAZs are small
• Many traffic counts / turning movements available
• The seed trip table might be disaggregated from a regional travel demand model
• Examples− Traffic Microsimulation OD tables
− Traffic Impact Reports
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(OD)ME Trip Table Applications
Truck tables in TDF Models
• Behavioral based trip table for autos or freight OD table
• Highway network for assignment
• Sufficient link counts for trucks
• Examples− Indiana DOT
− Virginia DOT
− Nashville MPO
− New York City MPO
− Binghamton MPO
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(OD)ME Trip Table Applications
State and multistate models
• No behavioral based trip tables for autos or trucks
• Highway network for assignment
• Sufficient link counts
• Examples− Georgia DOT
− Tennessee DOT
− I-95 Corridor Coalition
− Appalachian Regional Commission
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Transportation leadership you can trust.
presented topresented to
12th TRB National Planning Applications Conference12th TRB National Planning Applications ConferenceHouston, TX Houston, TX
presented bypresented by
Dan BeaganDan BeaganCambridge Systematics, Inc.Cambridge Systematics, Inc.
May 18, 2009May 18, 2009
Trip Table Estimation from CountsScience or Magic?
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