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Logit Gravity Trip Distribution Model for a Personal Travel External Trip Table

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Logit Gravity Trip Distribution Model for a Personal Travel External Trip Table

Authors:

Dr. John Douglas HUNT

Zoran CARKIC

Presentation prepared by:

Zoran CARKIC

Nina GANCHEV

September 2005

RTM Study Area & Zone System

Current RTM was updated in 2001 and expanded to include the surrounding region

1187 Zones in City

235 Zones in Region

25 External Entry/Exit Points

Total of 1447 Zones

Regional Transportation Model

- Nested Logit Structure
- Personal travel model with choice behaviour for trip generation, mode choice, time-of-day and distribution
- 25 travel segments (person/purpose)
- 9 possible modes choices
- 3 times of day choices
- Crown/Shoulder choice for auto 1, 2, 3+ persons

378 practical combinations of person/purpose/time of day/mode

Full Trip Matrix for Total Trips

D e s t i n a t i o n T r a n s p . Z o n e s

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Data Sources

Traffic Volume Counts

2001 Household Activity Survey

Population and Employment Data

1981 External Cordon Survey

Traffic Volume Counts

Traffic volume counts were conducted on roads and highways bordering with the Region

Provided information about the total number of journeys Teto/from External Zones

Completed a total of 9769 surveys

City of Calgary 8537

Regional Households 950

Banff/Canmore 282

Travel Activity Questions Included:

Start and end times

How did you travel?

Where was the activity? (address)

Where you picked up or dropped off by someone?

Was a vehicle available for the trip?

2001 Household Activity Survey

2001 Household Activity Survey

Trip Purpose

TZ O-H H-Sc H-W H-O NH Sc-H W-H

101 0 31 345 1535 1696 0 0

102 1357 0 2040 2149 2300 362 719

103 195 0 1520 483 1092 0 291

104 1713 137 4380 1734 4700 0 1187

105 1614 252 4230 641 3737 0 686

106 946 40 400 3427 3275 0 370

107 59 0 2415 918 2753 62 351

108 280 0 6487 1678 6619 0 205

109 74 133 6774 1216 7698 0 151

110 0 0 5997 1525 4846 0 0

111 0 0 604 1115 2749 0 0

112 0 0 2011 855 3478 0 0

113 158 0 3617 1345 4051 0 0

114 0 0 1639 44 1908 0 0

115 0 69 1749 2917 6157 0 0

. .

. .

Summary statistics from the 1981 External Cordon Survey provided ratios for all external trip types

In 1981 it was found that 5.9% of external trips were through trips with no stops in Calgary or the region

The impact of through traffic on regional traffic is relatively small

1981 External Cordon SurveyModel Selection

A simple growth factor trip distribution model was considered but rejected due to its disadvantages.

Advantages:

- Simple

- Direct use of observed values

Disadvantages:

- Requires extensive data (therefore expensive)

- Accuracy of results are heavily influenced by accuracy of input trip matrix

- Components of input trip matrix with zero in cells continue to have zero in solution

- Preserving patterns in observed behaviour - only applicable for short term planning horizon

- Changes in transport cost ignored - therefore of limited use when analysing policies involving new modes, new links, pricing changes and/or new regulations

A gravity model is appropriate if the assumptions made are acceptable for the situation modelled

The general expression for the Gravity Model comes from:

Analogy approach isbased on Newton\'s gravitational law – estimates trips without using observed trip pattern directly (synthetic). Entropy-maximization and Intervening Opportunities –two other approaches to form a gravity model

The generalized cost function must reflect the influences of relevant factors.

Model SelectionG * m1* m2

F = -----------------------

r2

F

- Calibration of a gravity model involves:
- -Selecting a functional form for the function
- -Selecting the variables to include in the function
- -Estimating values for the associated coefficients

Transportation Zone Variables

Distance

Population

Employment

Decisions regarding which variables to combine for the utility functions were made independently of the t-statistic analysis.

Distance between internal TZ and external entry/exit points was calculated directly from the model network

Population and Employment values for internal zones were used to develop general attraction ratios in the Logit Gravity Model calculations.

Utility Function

A linear function that assigns utility values to the trip maker alternatives is called a utility function:

U i = V ( a, i ) + E ( a, i )

or :

U i = F1 *X1i + F2 *X2i +….. Fn*Xni +…

where:

V ( a, i ) – the measurable conditioning component of the utility individual i associates with alternative a,

E ( a, i ) – the error component of the utility individual i associates with alternative a

Utility Function

General formula for the utility function chosen is:

U i = a * Ln(Attr i ) + b * Distance i

Where:

a, b - parameters to be estimated

Attr i - weighted attraction factor

Distance i - distance between TZ

Using different mathematical combinations for selected variables, alternative utility functions were examined.

General Logit Expression

The logit expression could be presented in the following form:

Regardless of the magnitude of the coefficients of the variable values, this model will always produce values in the range 0 to 1.

exp( l V(a*,i ))

Pi* =

S exp( l V(a ,i ))

a A

Pi* is a PROBABILITY that trip maker alternative a * is selected out of full set of alternatives A being considered

By combining the logit expression with a utility function that assigns utility values to the trip maker alternatives, we created a logit gravity model formula:

Logit Gravity Trip Distribution Model

exp (a * Ln(Attr i ) + b * Distance i)

Tei = Te * ---------------------------------------------------

S i exp(a * Ln(Attr i ) + b * Distance i)

Constants Determination

Maximum likelihood technique was used for constant determination

The A-logit computer program is used for the calibration of utility functions

Some of the outputs are:

1) Initial likelihood

2) Final value of likelihood

3) r2(0) - "Rho-Squared" with respect to zero

4) r2(C) - "Rho -Squared" with respect to constants

5) Standard error

6) T - Ratio

Model Quality of Fit

The quality of fit of the logit model could be considered by using a goodness-of-fit index labeled as r2. The following is the mathematical definition of r2 :

r2(0)= 1 -L (K*) - N

L (0)

Where: L (K*) – log-likelihood for model with the full vector of parameters K*

L (0) - log-likelihood for model with no parameters, where all parameters are set to 0

N – number of coefficients in estimated models

The r2(0) index is used in the same way that the R2 is used with linear regression models.

Larger values indicate a better fit.

The mathematical definition for the t-ratio is:

If the absolute value of the t-ratio is greater than 1.96 than there is a less than 5% chance that the associated difference is due to random effects only

The Zero Hypotheses Testt - ratio [K(1)] = _ ___K(1)_______

{VAR [K (1)]} 0.5

The t-ratio is used to test the zero hypotheses that there is no difference between a given parameter K (1) and 0.

If a given parameter is not significantly different from 0 it is not going to be used in the utility function

because it has no influence on the choice behavior of a trip maker.

Results

In absence of the field data it was hard to find attributes and/or variables that influence trip maker decisions.

Even though decisions regarding choice of variables combined for the utility functions were made independently of the t-statistic analysis,

the logic behind the gravity model gave us a lot of comfort in choosing the distance and the attraction between transportation zones as variables.

Results obtained from statistical analyses in this assignment suggest that the model was reasonable.

Journeys originating outside the region and destined to a TZ within the region

Final logit formula for the logit gravity model:

External - Internal TripsUi = 0.6589 * Ln(Attr i) – 0.07737 * Distancei [1]

(2.1) (-5.7)

exp (0.6589 * Ln(Attr i) – 0.07737 * Distancei)

Tei = Te * ------------------------------------------------------------- [2]

S i exp (0.6589 * Ln(Attr i) – 0.07737 * Distancei)

The coefficient estimates signs were consistent with what would be expected in case corresponding variable values were changed.

For example, if distance is increased then that alternative would be less attractive, so the coefficient for distance is negative.

Journeys originating from a TZ within the region destined outside the region

Final logit formula for the logit gravity model:

Internal - External TripsUi = 0.4744 * Ln(Attr i) – 0.06328 * Distancei [3]

(1.3) (-3.7)

exp (0.4744 * Ln(Attr i) – 0.06328 * Distancei)

Tie = Te * ------------------------------------------------------------- [4]

S i exp (0.4744 * Ln(Attr i) – 0.06328 * Distancei)

Represents through traffic in the model

1981 External Cordon Survey was used to obtain the percentage of the regional through traffic

External - External TripsTee = S Te* 0.059

It was found that 5.9% of the trips were through trips with no stop in Calgary or the Region.

A simple adjustment of the Logit Gravity Model is appropriate to reflect the overall influence of the through traffic.

Authors:

Dr. John Douglas HUNT

Zoran CARKIC

Presentation prepared by:

Nina GANCHEV

Zoran CARKIC

With limited resources to do an external travel survey, the logit gravity model was found appropriate for the situation modelled.

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