TRB Planning Applications Conference May 2009, Houston,TX A Caveat on O-D Matrix Estimation/Adjustment: Deviations from a seed matrix and Simultaneous

TRB Planning Applications ConferenceMay 2009, Houston,TX

A Caveat on O-D Matrix Estimation/Adjustment:

Deviations from a seed matrix and Simultaneous multi-class adjustments

Michael Florian and Yolanda NoriegaCIRRELT and INRO


Contents of presentation

1. Motivation 2. The gradient method for adjusting O-D matrices 3. Deviations from the matrix to be adjusted 4. Using deviations from the reference O-D matrix 5. Extension of the method to multi-class adjustments

of O-D matrices 6. Conclusions


Motivation

– The use of counts to adjust out of date origin-destination matrices is a commonly used method;

– Practically all the transportation planning software packages offer a way to adjust matrices;

– The benefits are obvious: the counts are reliable and far less expensive than an O-D survey;

.


Motivation

– The analysis of sub-areas that are then used for dynamic traffic assignment or micro-simulation applications benefit from an adjusted O-D matrix that better replicates counts;

– Other traffic operations applications benefit from more reliable turning movements that can be obtained as a by-product of the O-D adjustment.


More an Art than a Science

– The adjustment of an O-D matrix by using counts requires good judgment in addition to the method used for the adjustment;

– The quality and consistency of the counts must be analyzed and verified;

– The network coding and the volume/delay functions used should be free of errors, as much as possible;


More an Art than a Science

– Finally, the adjustment process should not distort the structure of the O-D matrix that is being adjusted;

– A balance must be achieved between the fit to the counts and the changes of the O-D matrix that result from the adjustment.


The Gradient Method for O-D Adjustment

– Let’s recall what is being done in this O-D adjustment method:

– The inputs are the link counts and the existing O-D matrix;

– The aim is to obtain an assignment that fits the counts as best as possible;

– The objective then is to obtain the smallest value of the difference between the assigned flows and the counts:

.



Find new O-D matrix g such that the objective function

SUM (assigned flows(g) – counts)^2is as small as possible. SUM is over all links that have counts– The adjusted matrix g is computed with a

method that is based on computing the rate of decrease of the objective function.

– Constraints can include screen lines, cordon counts, productions, attractions…



Step 0. Initialization; Step 1. Equilibrium assignment to obtain the link volumes;Step 2. Computation of the link derivatives and the objective function. Step 3 Computation the gradient matrix;Step 4. Computation of the link derivatives.;Step 5. Compute the maximal gradient; Compute the optimal step length; Update the demand matrix;

Step 6. Update iteration counter; If the maximum number of iterations is reached STOP.

otherwise go to Step 1;


But considering only the link counts may not be enough

– An example of a matrix adjustment for the City of Montreal illustrates the issue:

– The heavy links indicate the location of counts


Montreal flow comparison for SOV’s without adjustment

Link R^2=.89


Montreal flow and O-D matrixcomparison for SOV’s after adjustment

Link R^2=.99O-D R^2=.89


How can one control the deviations of the O-D matrix?

– It is possible to introduce a demand term in the O-D matrix adjustment objective:

Find new O-D matrix g such that the a weighted sum

(weight) [SUM (Adj.O-D – Initial O-D)^2]+(1-weight) [SUM (assigned flows(g) – counts)^2]

where weight is less than 1, is as small as possible.

SUM is over all links that have counts

– The adjusted matrix g is computed with a similar method that is based on computing the rate of decrease of the objective function

by using the gradient method.


Results obtained by varying the weight

O-D R^2=.91 Link R^2=.96


Results obtained by varying the weight

O-D R^2=.99 Link R^2=.93


Original link fit Considering the demand term


Best link fit Considering the demand term


Which is the right adjustment?

The analyst is the judge!

The ability to inspect the structure of the adjusted O-D matrix is important.

(It is also possible to give weights for particular link counts and elements of the O-D matrix to

be adjusted).


Considering only the link counts may not be enough:

another example


O-D matrix comparisons

weight=0 weight=0.05


Link flow comparisons

weight=0 weight=0.05


O-D matrix comparisons

weight =0.1 weight=0.2


Multi-Class OD Matrix adjustment

• Over the past 15 years, the use of multi-class equlibrium assignments has become quite common;

• So there is an interest to extend the gradient method for simultaneous multi-class OD matrix adjustment;

• The gradient method was extended for multi-class OD matrix adjustment

• It is implemented and sample results using data from the Montreal region network will be shown.


Extension of the method to multi-class adjustments of O-D matrices

Find new O-D matrices g(c) for each class such that the a weighted sum

(weight) [SUM (Adj.O-D – Initial O-D)2]+(1-weight) [SUM (assigned flows(g) – counts)2]

where weight is less than 1, is as small as possible.

SUM is over classes and links that have counts


The Multi-Class Method

Step 0. Initialization; Step 1. Multi-class assignment to obtain the link volumes;Step 2. Computation of the link derivatives and the objective function; Step 3. Compute the gradient matrices for each class;Step 4. Obtain the link flow derivatives; Step 5. For each class

Compute the maximal gradient;Compute the optimal step length; Update of the demand matrices;

Step 6. Update the iteration counter If the maximum number of iterations is reached STOP;

Otherwise go to Step 1.


We tried three approaches :

•The first approach is the multi-class adjustment where the demand for every class is adjusted iteration by iteration (MC Adjustment);

•The second approach consists on adjusting the demand for one class at the time, leaving however the flows of all classes variable during the assignments (SEQ Adjustment);

•Similarly, in the third approach the demand of one class is adjusted at the time, but here the volumes of the other classes are considered as fixed (FIX Adjustment).

Three Approaches for Multi-Class Adjustments


Computational results

Objective function. AM

-

100 000

200 000

300 000

400 000

500 000

600 000

700 000

0 2 4 6 8 10 12 14 16 18

Iteration

Valu

e

MC Adj

SEQ Adj

FIX Adj



Objective function. OD

-

200 000

400 000

600 000

800 000

1 000 000

1 200 000

1 400 000

1 600 000

1 800 000

2 000 000

0 2 4 6 8 10 12 14 16 18

Iteration

Valu

e MC Adj

SEQ Adj

FIX Adj



Objective function. PM

-

100 000

200 000

300 000

400 000

500 000

600 000

700 000

800 000

900 000

1 000 000

0 2 4 6 8 10 12 14 16 18

Iteration

Valu

e MC Adj

SEQ Adj

FIX Adj


Auto Demand to be Adjusted


MC Adjustment of Auto Demand


Regular Trucks to be Adjusted


MC Adjustment of Regular Trucks Demand


Demand to be Adjusted Heavy Trucks


MC Adjustment of Heavy Trucks Demand



100% on flows, cars 99.99% on flows, cars



99.97% on flows, cars 100% on flows, regular trucks



99.97% on flows, regular trucks99.99% on flows, regular trucks


Conclusion

• The adjustment of O-D matrices is a process that should be done carefully with inspection of the adjusted matrix; comparison with the matrix to be adjusted is important;

• One can carry out the simultaneous adjustment of the O-D matrices for several classes.


THE END

Documents

TRB Planning Applications Conference May 2009, Houston,TX A Caveat on O-D Matrix Estimation/Adjustment: Deviations from a seed matrix and Simultaneous